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We have presented this elementary mathematics

material from a variety of perspectives so that you...

We have presented this elementary mathematics

material from a variety of perspectives so that you will be

better equipped to address that broad range of learning

styles that you will encounter in your future students. This

book also encourages prospective teachers to gain the

ability to do the mathematics of elementary school and to

understand the underlying concepts so they will be able

to assist their students, in turn, to gain a deep understanding

of mathematics.

- BMIndex.indd 29

7/31/2013 7:29:35 AM - National Council of Teachers of Mathematics Principles

and Standards for School Mathematics

Principles for School Mathematics

r EQUITY. &YDFMMFODF JO NBUIFNBUJDT FEVDBUJPO SFRVJSFT

r LEARNING. 4UVEFOUT NVTU MFBSO NBUIFNBUJDT XJUI VOEFS-

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TUBOEJOH BDUJWFMZ CVJMEJOH OFX LOPXMFEHF GSPN FYQFSJFODF BOE

r CURRICULUM. " DVSSJDVMVN JT NPSF UIBO B DPMMFDUJPO PG

QSJPS LOPXMFEHF

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r ASSESSMENT. "TTFTTNFOU TIPVME TVQQPSU UIF MFBSOJOH PG

JDT BOE XFMM BSUJDVMBUFE BDSPTT UIF HSBEFT

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r TEACHING. &GGFDUJWF NBUIFNBUJDT UFBDIJOH SFRVJSFT VOEFS-

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BOE FOIBODFT TUVEFOUT MFBSOJOH

Standards for School Mathematics

NUMBER AND OPERATIONS

DATA ANALYSIS AND PROBABILITY

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GEOMETRY

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FMEndpaper.indd 15

7/31/2013 10:58:25 AM - CONNECTIONS

REPRESENTATION

*OTUSVDUJPOBM QSPHSBNT GSPN QSFLJOEFSHBSUFO UISPVHI HSBEF

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BOE NBUIFNBUJDBM QIFOPNFOB

Curriculum Focal Points for Prekindergarten through

Grade 8 Mathematics

PREKINDERGARTEN

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG XIPMF

EFDJNBMT JODMVEJOH UIF DPOOFDUJPOT CFUXFFO GSBDUJPOT BOE

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Geometry: *EFOUJGZJOH TIBQFT BOE EFTDSJCJOH TQBUJBM SFMBUJPO-

NJOJOH UIF BSFBT PG UXP EJNFOTJPOBM TIBQFT

TIJQT

GRADE 5

Measurement: *EFOUJGZJOH NFBTVSBCMF BUUSJCVUFT BOE DPNQBSJOH

PCKFDUT CZ VTJOH UIFTF BUUSJCVUFT

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TUBOEJOH PG BOE áVFODZ XJUI EJWJTJPO PG XIPMF OVNCFST

KINDERGARTEN

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG BOE

Number and Operations: 3FQSFTFOUJOH DPNQBSJOH BOE PSEFSJOH

áVFODZ XJUI BEEJUJPO BOE TVCUSBDUJPO PG GSBDUJPOT BOE EFDJNBMT

XIPMF OVNCFST BOE KPJOJOH BOE TFQBSBUJOH TFUT

Geometry BOE Measurement BOE Algebra: %FTDSJCJOH UISFF

Geometry: %FTDSJCJOH TIBQFT BOE TQBDF

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Measurement: 0SEFSJOH PCKFDUT CZ NFBTVSBCMF BUUSJCVUFT

WPMVNF BOE TVSGBDF BSFB

GRADE 1

GRADE 6

Number and Operations BOE Algebra: %FWFMPQJOH VOEFSTUBOE-

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG BOE áV-

JOHT PG BEEJUJPO BOE TVCUSBDUJPO BOE TUSBUFHJFT GPS CBTJD BEEJ-

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Number and Operations: $POOFDUJOH SBUJP BOE SBUF UP NVMUJQMJ-

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG XIPMF

DBUJPO BOE EJWJTJPO

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Algebra: 8SJUJOH JOUFSQSFUJOH BOE VTJOH NBUIFNBUJDBM FYQSFT-

Geometry: $PNQPTJOH BOE EFDPNQPTJOH HFPNFUSJD TIBQFT

TJPOT BOE FRVBUJPOT

GRADE 2

GRADE 7

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG UIF

Number and Operations BOE Algebra BOE Geometry: %FWFMPQ-

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JOH BO VOEFSTUBOEJOH PG BOE BQQMZJOH QSPQPSUJPOBMJUZ JODMVEJOH

Number and Operations BOE Algebra: %FWFMPQJOH RVJDL SFDBMM

TJNJMBSJUZ

PG BEEJUJPO GBDUT BOE SFMBUFE TVCUSBDUJPO GBDUT BOE áVFODZ XJUI

Measurement BOE Geometry BOE Algebra: %FWFMPQJOH BO VOEFS-

NVMUJEJHJU BEEJUJPO BOE TVCUSBDUJPO

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GRADE 3

MJOFBS FRVBUJPOT

Number and Operations BOE Algebra: %FWFMPQJOH VOEFSTUBOE-

JOHT PG NVMUJQMJDBUJPO BOE EJWJTJPO BOE TUSBUFHJFT GPS CBTJD NVM-

GRADE 8

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Algebra: "OBMZ[JOH BOE SFQSFTFOUJOH MJOFBS GVODUJPOT BOE TPMW-

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PG GSBD-

JOH MJOFBS FRVBUJPOT BOE TZTUFNT PG MJOFBS FRVBUJPOT

UJPOT BOE GSBDUJPO FRVJWBMFODF

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TJPOBM TIBQFT

Data Analysis BOE Number and Operations BOE Algebra: "OBMZ[-

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GRADE 4

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PG NVMUJQMJDBUJPO GBDUT BOE SFMBUFE EJWJTJPO GBDUT BOE áVFODZ

XJUI XIPMF OVNCFS NVMUJQMJDBUJPO

FMEndpaper.indd 16

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For Elementary Teachers

TENTH EDITION

A C O N T E M P O R A R Y A P P R O A C H

Gary L. Musser t Blake E. Peterson t William F. Burger

Oregon State University

Brigham Young University

FMWileyPlus.indd 3

7/31/2013 2:14:09 PM - To:

Irene, my wonderful wife of 52 years who is the best mother our son could have; Greg, our son, for his inquiring mind;

Maranda, our granddaughter, for her willingness to listen; my parents who have passed away, but always with me; and

Mary Burger, my initial coauthor's daughter.

G.L.M.

Shauna, my beautiful eternal companion and best friend, for her continual support of all my endeavors; my four children:

Quinn for his creative enthusiasm for life, Joelle for her quiet yet strong confidence, Taren for her unintimidated ap-

proach to life, and Riley for his good choices and his dry wit.

B.E.P.

VICE PRESIDENT & EXECUTIVE PUBLISHER

Laurie Rosatone

PROJECT EDITOR

Jennifer Brady

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Karoline Luciano

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COVER & TEXT DESIGN

Madelyn Lesure

This book was set by Laserwords and printed and bound by Courier Kendallville. The cover was printed by

Courier Kendallville.

Copyright © 2014, 2011, 2008, 2005, John Wiley & Sons, Inc. All rights reserved. No part of this publication

may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, me-

chanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the

1976 United States Copyright Act, without either the prior written permission of the Publisher, or authoriza-

tion through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 Rosewood

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addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774,

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Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in

their courses during the next academic year. These copies are licensed and may not be sold or transferred to a

third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instruc-

tions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the

United States, please contact your local representative.

Library of Congress Cataloging-in-Publication Data

Musser, Gary L.

Mathematics for elementary teachers : a contemporary approach / Gary L. Musser, Oregon State University,

William F. Burger, Blake E. Peterson, Brigham Young University. -- 10th edition.

pages cm

Includes index.

ISBN 978-1-118-45744-3 (hardback)

1. Mathematics. 2. Mathematics–Study and teaching (Elementary) I. Title.

QA39.3.M87 2014

510.2’4372–dc23

2013019907

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

FMWileyPlus.indd 4

7/31/2013 2:14:09 PM - ABOUT THE AUTHORS

Gary L. Musser is Professor Emeritus from Oregon State University. He earned both

his B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at the

University of Michigan and his Ph.D. in Mathematics (Radical Theory) in 1970 at the

University of Miami in Florida. He taught at the junior and senior high, junior college

college, and university levels for more than 30 years. He spent his final 24 years teaching

prospective teachers in the Department of Mathematics at Oregon State University.

While at OSU, Dr. Musser developed the mathematics component of the elementary

teacher program. Soon after Profesor William F. Burger joined the OSU Department

of Mathematics in a similar capacity, the two of them began to write the first edtion of

this book. Professor Burger passed away during the preparation of the second edition,

and Professor Blake E. Peterson was hired at OSU as his replacement. Professor Peter-

son joined Professor Musser as a coauthor beginning with the fifth edition.

Professor Musser has published 40 papers in many journals, including the Pacific

Journal of Mathematics, Canadian Journal of Mathematics, The Mathematics Association of America Monthly, the

NCTM’s The Mathematics Teacher, the NCTM’s The Arithmetic Teacher, School Science and Mathematics, The

Oregon Mathematics Teacher, and The Computing Teacher. In addition, he is a coauthor of two other college mathematics

books: College Geometry—A Problem-Solving Approach with Applications (2008) and A Mathematical View of Our

World (2007). He also coauthored the K-8 series Mathematics in Action. He has given more than 65 invited lectures/

workshops at a variety of conferences, including NCTM and MAA conferences, and was awarded 15 federal, state, and

local grants to improve the teaching of mathematics.

While Professor Musser was at OSU, he was awarded the university’s prestigious College of Science Carter Award

for Teaching. He is currently living in sunny Las Vegas, were he continues to write, ponder the mysteries of the stock

market, enjoy living with his wife and his faithful yellow lab, Zoey.

Blake E. Peterson is currently a Professor in the Department of Mathematics Educa-

tion at Brigham Young University. He was born and raised in Logan, Utah, where he

graduated from Logan High School. Before completing his BA in secondary mathe-

matics education at Utah State University, he spent two years in Japan as a missionary

for The Church of Jesus Christ of Latter Day Saints. After graduation, he took his

new wife, Shauna, to southern California, where he taught and coached at Chino High

School for two years. In 1988, he began graduate school at Washington State Univer-

sity, where he later completed a M.S. and Ph.D. in pure mathematics.

After completing his Ph.D., Dr. Peterson was hired as a mathematics educator in the

Department of Mathematics at Oregon State University in Corvallis, Oregon, where

he taught for three years. It was at OSU where he met Gary Musser. He has since

moved his wife and four children to Provo, Utah, to assume his position at Brigham

Young University where he is currently a full professor.

Dr. Peterson has published papers in Rocky Mountain Mathematics Journal, The American Mathematical Monthly,

The Mathematical Gazette, Mathematics Magazine, The New England Mathematics Journal, School Science and

Mathematics, The Journal of Mathematics Teacher Education, and The Journal for Research in Mathematics as well

as chapters in several books. He has also published in NCTM’s Mathematics Teacher, and Mathematics Teaching in

the Middle School. His research interests are teacher education in Japan and productive use of student mathematical

thinking during instruction, which is the basis of an NSF grant that he and 3 of his colleagues were recently awarded.

In addition to teaching, research, and writing, Dr. Peterson has done consulting for the College Board, founded

the Utah Association of Mathematics Teacher Educators, and has been the chair of the editorial panel for the

Mathematics Teacher.

Aside from his academic interests, Dr. Peterson enjoys spending time with his family, fulfilling his church responsi-

bilities, playing basketball, mountain biking, water skiing, and working in the yard.

v

FMWileyPlus.indd 5

7/31/2013 2:14:11 PM - ABOUT THE COVER

Are you puzzled by the numbers on the cover? They are 25 different randomly selected

counting numbers from 1 to 100. In that set of numbers, two different arithmetic pro-

gressions are highlighted. (An arithmetic progression is a sequence of numbers with a

common difference between consecutive pairs.) For example, the sequence highlighted

in green, namely 7, 15, 23, 31, is an arithmetic progression because the difference

between 7 and 15 is 8, between 15 and 23 is 8, and between 23 and 31 is 8. Thus, the

sequence 7, 15, 23, 31 forms an arithmetic progression of length 4 (there are 4 numbers

in the sequence) with a common difference of 8. Similarly, the numbers highlighted

in red, namely 45, 69, 93, form another arithmetic progression. This progression is of

length 3 which has a common difference of 24.

You may be wondering why these arithmetic progressions are on the cover. It is to

acknowledge the work of the mathematician Endre Szemerédi. On May 22, 2012, he

was awarded the $1,000,000 Abel prize from the Norwegian Academy of Science and

Letters for his analysis of such progressions. This award recognizes mathematicians

for their contributions to mathematics that have a far reaching impact. One of Pro-

fessor Szemerédi’s significant proofs is found in a paper he wrote in 1975. This paper

proved a famous conjecture that had been posed by Paul Erdös and Paul Turán in

1936. Szemerédi’s 1975 paper and the Erdös/Turán conjecture are about finding arith-

metic progressions in random sets of counting numbers (or integers). Namely, if one

randomly selects half of the counting numbers from 1 and 100, what lengths of arith-

metic progressions can one expect to find? What if one picks one-tenth of the numbers

from 1 to 100 or if one picks half of the numbers between 1 and 1000, what lengths

of arithmetic progressions is one assured to find in each of those situations? While the

result of Szemerédi’s paper was interesting, his greater contribution was that the tech-

nique used in the proof has been subsequently used by many other mathematicians.

Now let’s go back to the cover. Two progressions that were discussed above, one

of length 4 and one of length 3, are shown in color. Are there others of length 3?

Of length 4? Are there longer ones? It turns out that there are a total of 28 different

arithmetic progressions of length three, 3 arithmetic progressions of length four and

1 progression of length five. See how many different progressions you can find on the

cover. Perhaps you and your classmates can find all of them.

vi

FMWileyPlus.indd 6

7/31/2013 2:14:12 PM - BRIEF CONTENTS

1 Introduction to Problem Solving 2

2 Sets, Whole Numbers, and Numeration 42

3 Whole Numbers: Operations and Properties 84

4 Whole Number Computation—Mental, Electronic,

and Written 128

5 Number Theory 174

6 Fractions 206

7 Decimals, Ratio, Proportion, and Percent 250

8 Integers 302

9 Rational Numbers, Real Numbers, and Algebra 338

10 Statistics 412

11 Probability 484

12 Geometric Shapes 546

13 Measurement 644

14 Geometry Using Triangle Congruence and Similarity 716

15 Geometry Using Coordinates 780

16 Geometry Using Transformations 820

Epilogue: An Eclectic Approach to Geometry 877

Topic 1 Elementary Logic 881

Topic 2 Clock Arithmetic: A Mathematical System 891

Answers to Exercise/Problem Sets A and B, Chapter Reviews, Chapter

Tests, and Topics Section A1

Index I1

Contents of Book Companion Web Site

Resources for Technology Problems

Technology Tutorials

Webmodules

Additional Resources

Videos

vii

FMBriefContents.indd 7

7/31/2013 12:29:55 PM - CONTENTS

Preface xi

1 Introduction to Problem Solving 2

1.1 The Problem-Solving Process and Strategies 5

1.2 Three Additional Strategies 21

2 Sets, Whole Numbers, and Numeration 42

2.1 Sets as a Basis for Whole Numbers 45

2.2 Whole Numbers and Numeration 57

2.3 The Hindu–Arabic System 67

3 Whole Numbers: Operations and Properties 84

3.1 Addition and Subtraction 87

3.2 Multiplication and Division 101

3.3 Ordering and Exponents 116

4 Whole Number Computation—Mental, Electronic,

and Written 128

4.1 Mental Math, Estimation, and Calculators 131

4.2 Written Algorithms for Whole-Number Operations 145

4.3 Algorithms in Other Bases 162

5 Number Theory 174

5.1 Primes, Composites, and Tests for Divisibility 177

5.2 Counting Factors, Greatest Common Factor, and Least

Common Multiple 190

6 Fractions 206

6.1 The Set of Fractions 209

6.2 Fractions: Addition and Subtraction 223

6.3 Fractions: Multiplication and Division 233

7 Decimals, Ratio, Proportion, and Percent 250

7.1 Decimals 253

7.2 Operations with Decimals 262

7.3 Ratio and Proportion 274

7.4 Percent 283

8 Integers 302

8.1 Addition and Subtraction 305

8.2 Multiplication, Division, and Order 318

viii

FMBriefContents.indd 8

7/31/2013 12:29:55 PM - 9 Rational Numbers, Real Numbers, and Algebra 338

9.1 The Rational Numbers 341

9.2 The Real Numbers 358

9.3 Relations and Functions 375

9.4 Functions and Their Graphs 391

10 Statistics 412

10.1 Statistical Problem Solving 415

10.2 Analyze and Interpret Data 440

10.3 Misleading Graphs and Statistics 460

11 Probability 484

11.1 Probability and Simple Experiments 487

11.2 Probability and Complex Experiments 502

11.3 Additional Counting Techniques 518

11.4 Simulation, Expected Value, Odds, and Conditional

Probability 528

12 Geometric Shapes 546

12.1 Recognizing Geometric Shapes—Level 0 549

12.2 Analyzing Geometric Shapes—Level 1 564

12.3 Relationships Between Geometric Shapes—Level 2 579

12.4 An Introduction to a Formal Approach to Geometry 589

12.5 Regular Polygons, Tessellations, and Circles 605

12.6 Describing Three-Dimensional Shapes 620

13 Measurement 644

13.1 Measurement with Nonstandard and Standard Units 647

13.2 Length and Area 665

13.3 Surface Area 686

13.4 Volume 696

14 Geometry Using Triangle Congruence and

Similarity 716

14.1 Congruence of Triangles 719

14.2 Similarity of Triangles 729

14.3 Basic Euclidean Constructions 742

14.4 Additional Euclidean Constructions 755

14.5 Geometric Problem Solving Using Triangle Congruence

and Similarity 765

15 Geometry Using Coordinates 780

15.1 Distance and Slope in the Coordinate Plane 783

15.2 Equations and Coordinates 795

15.3 Geometric Problem Solving Using Coordinates 807

ix

FMBriefContents.indd 9

8/2/2013 3:24:49 PM - 16 Geometry Using Transformations 820

16.1 Transformations 823

16.2 Congruence and Similarity Using Transformations 846

16.3 Geometric Problem Solving Using Transformations 863

Epilogue: An Eclectic Approach to Geometry 877

Topic 1. Elementary Logic 881

Topic 2. Clock Arithmetic: A Mathematical System 891

Answers to Exercise/Problem Sets A and B, Chapter

Reviews, Chapter Tests, and Topics Section A1

Index I1

Contents of Book Companion Web Site

Resources for Technology Problems

eManipulatives

Spreadsheet Activities

Geometer’s Sketchpad Activities

Technology Tutorials

Spreadsheets

Geometer’s Sketchpad

Programming in Logo

Graphing Calculators

Webmodules

Algebraic Reasoning

Children’s Literature

Introduction to Graph Theory

Additional Resources

Guide to Problem Solving

Problems for Writing/Discussion

Research Articles

Web Links

Videos

Book Overview

Author Walk-Through Videos

Children’s Videos

x

FMBriefContents.indd 10

7/31/2013 12:29:55 PM - PREFACE

Welcome to the study of the foundations of ele- approach is clearly superior to the seemingly more effi-

mentary school mathematics. We hope you will

cient sequence of whole numbers-integers-rationals-reals

find your studies enlightening, useful, and fun.

that is more appropriate to use when teaching high school

We salute you for choosing teaching as a profession and

mathematics.

hope that your experiences with this book will help prepare

you to be the best possible teacher of mathematics that you

Approach to Geometry Geometry is organized

can be. We have presented this elementary mathematics

from the point of view of the five-level van Hiele model

material from a variety of perspectives so that you will be

of a child’s development in geometry. After studying

better equipped to address that broad range of learning

shapes and measurement, geometry is approached more

styles that you will encounter in your future students. This

formally through Euclidean congruence and similarity,

book also encourages prospective teachers to gain the

coordinates, and transformations. The Epilogue provides

ability to do the mathematics of elementary school and to

an eclectic approach by solving geometry problems using

understand the underlying concepts so they will be able

a variety of techniques.

to assist their students, in turn, to gain a deep understand-

ing of mathematics.

Additional Topics

We have also sought to present this material in a man-

r Topic 1, “Elementary Logic,” may be used anywhere

ner consistent with the recommendations in (1) The

in a course.

Mathematical Education of Teachers prepared by the

r Topic 2, “Clock Arithmetic: A Mathematical System,”

Conference Board of the Mathematical Sciences, (2) the

uses the concepts of opposite and reciprocal and hence

National Council of Teachers of Mathematics’ Standards

may be most instructive after Chapter 6, “Fractions,” and

Documents, and (3) The Common Core State Standards

Chapter 8, “Integers,” have been completed. This section

for Mathematics. In addition, we have received valuable

also contains an introduction to modular arithmetic.

advice from many of our colleagues around the United

States through questionnaires, reviews, focus groups, and

personal communications. We have taken great care to

respect this advice and to ensure that the content of the

Underlying Themes

book has mathematical integrity and is accessible and

Problem Solving An extensive collection of problem-

helpful to the variety of students who will use it. As al-

solving strategies is developed throughout the book;

ways, we look forward to hearing from you about your

these strategies can be applied to a generous supply of

experiences with our text.

problems in the exercise/problem sets. The depth of

GARY L. MUSSER, glmusser@cox.net

problem-solving coverage can be varied by the number

BLAKE E. PETERSON, peterson@mathed.byu.edu

of strategies selected throughout the book and by the

problems assigned.

Unique Content Features

Deductive Reasoning The use of deduction is pro-

moted throughout the book The approach is gradual,

Number Systems The order in which we present the

with later chapters having more multistep problems. In

number systems in this book is unique and most relevant

particular, the last sections of Chapters 14, 15, and 16

to elementary school teachers. The topics are covered to

and the Epilogue offer a rich source of interesting theo-

parallel their evolution historically and their development

rems and problems in geometry.

in the elementary/middle school curriculum. Fractions

and integers are treated separately as an extension of the

Technology Various forms of technology are an inte-

whole numbers. Then rational numbers can be treated at

gral part of society and can enrich the mathematical

a brisk pace as extensions of both fractions (by adjoining

understanding of students when used appropriately.

their opposites) and integers (by adjoining their appro-

Thus, calculators and their capabilities (long division

priate quotients) since students have a mastery of the

with remainders, fraction calculations, and more) are

concepts of reciprocals from fractions (and quotients)

introduced throughout the book within the body of

and opposites from integers from preceding chapters.

the text.

Longtime users of this book have commented to us

In addition, the book companion Web site has eMa-

that this whole numbers-fractions-integers-rationals-reals

nipulatives, spreadsheets, and sketches from Geometer’s

xi

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8/1/2013 12:05:27 PM - xii Preface

Sketchpad®. The eManipulatives are electronic versions

their students will view geometry at various van Hiele

of the manipulatives commonly used in the elementary

levels.

classroom, such as the geoboard, base ten blocks, black

r Chapter 13 contains several new examples to give stu-

and red chips, and pattern blocks. The spreadsheets

dents the opportunity to see how the various equations

contain dynamic representations of functions, statistics,

for area and volume are applied in different contexts.

and probability simulations. The sketches in Geometer’s

r Children’s Videos are videos of children solving math-

Sketchpad® are dynamic representations of geomet-

ematical problems linked to QR codes placed in the

ric relationships that allow exploration. Exercises and

margin of the book in locations where the content

problems that involve eManipulatives, spreadsheets, and

being discussed is related to the content of the prob-

Geometer’s Sketchpad® sketches have been integrated

lems being solved by the children. These videos will

into the problem sets throughout the text.

bring the mathematical content being studied to life.

r Author Walk-Throughs are videos linked to the QR

Course Options

code on the third page of each chapter. These brief

videos are of an author, Blake Peterson, describing

We recognize that the structure of the mathematics for

and showing points of major emphasis in each chapter

elementary teachers course will vary depending upon the

so students’ study can be more focused.

college or university. Thus, we have organized this text so

r Children’s Literature and Reflections from Research

that it may be adapted to accommodate these differences.

margin notes have been revised/refreshed.

Basic course: Chapters 1-7

r Common Core margin notes have been added through-

Basic course with logic: Topic 1, Chapters 1–7

out the text to highlight the correlation between the

Basic course with informal geometry: Chapters 1–7,

content of this text and the Common Core standards.

12

r Professional recommendation statements from the

Basic course with introduction to geometry and mea-

Common Core State Standards for Mathematics,

surement: Chapters 1–7, 12, 13

the National Council of Teachers of Mathematics’

Principles and Standards for School Mathematics, and

the Curriculum Focal Points, have been compiled on

Summary of Changes to the

the third page of each chapter.

Tenth Edition

r Mathematical Tasks have been added to sections

Pedagogy

throughout the book to allow instructors more flex-

The general organization of the book was motivated by

ibility in how they choose to organize their classroom

the following mathematics learning cube:

instruction. These tasks are designed to be investigated

by the students in class. As the solutions to these tasks

are discussed by students and the instructor, the big

ideas of the section emerge and can be solidified

through a classroom discussion.

r Chapter 6 contains a new discussion of fractions on a

number line to be consistent with the Common Core

standards.

r Chapter 10 has been revised to include a discus-

sion of recommendations by the GAISE document

and the NCTM Principles and Standards for School

Mathematics. These revisions include a discussion

of steps to statistical problem solving. Namely,

(1) formulate questions, (2) collect data, (3) organize

and display data, (4) analyze and interpret data.

These steps are then applied in several of the examples

The three dimensions of the cube—cognitive levels,

through the chapter.

representational levels, and mathematical content—are

r Chapter 12 has been substantially revised. Sections

integrated throughout the textual material as well as

12.1, 12.2, and 12.3 have been organized to parallel the

in the problem sets and chapter tests. Problem sets are

first three van Hiele levels. In this way, students will be

organized into exercises (to support knowledge, skill, and

able to pass through the levels in a more meaningful

understanding) and problems (to support problem solv-

fashion so that they will get a strong feeling about how

ing and applications).

FMPreface.indd 12

8/1/2013 12:05:27 PM - Preface xiii

We have developed new pedagogical features to imple-

r Children’s Literature references have been edited and

ment and reinforce the goals discussed above and to

updated. Also, there is additional material offered on

address the many challenges in the course.

the Web site on this topic.

r Check for Understanding have been updated to reflect

the revision of the problem sets.

Summary of Pedagogical Changes

r Mathematical Tasks have been integrated throughout.

to the Tenth Edition

r Author Walk-Throughs videos have been made avail-

able via QR codes on the third page of every chapter.

r Student Page Snapshots have been updated.

r Children’s videos, produced by Blake Peterson and

r Reflection from Research margin notes have been

available via QR codes, have been integrated through-

edited and updated.

out.

Key Features

Problem-Solving Strategies are integrated

Mathematical Structure reveals the mathematical

throughout the book. Six strategies are introduced in

ideas of the book. Main Definitions, Theorems, and

Chapter 1. The last strategy in the strategy box at the

Properties in each section are highlighted in boxes for

top of the second page of each chapter after Chapter l

quick review.

contains a new strategy.

Mathematical Tasks are located in various places

throughout each section. These tasks can be presented to the whole class or small groups to investigate. As the stu-

dents discuss their solutions with

each other and the instructor, the

big mathematical ideas of the sec-

tion emerge.

FMPreface.indd 13

8/1/2013 12:05:28 PM - xiv Preface

Technology Problems appear in the Exercise/Problem sets throughout the

book. These problems rely on and are enriched by the use of technology. The tech-

nology used includes activities from the eManipulaties (virtual manipulatives),

spreadsheets, Geometer’s Sketchpad®, and the TI-34 II MultiView. Most

of these technological resources can be accessed through the accompany-

ing book companion Web site.

Student Page Snapshots have been updated. Each

chapter has a page from an elementary school textbook

relevant to the material being studied.

Exercise/Problem Sets are separated into Part A

(all answers are provided in the back of the book and

all solutions are provided in our supplement Hints and

Solutions for Part A Problems) and Part B (answers are

only provided in the Instructors Resource Manual). In

addition, exercises and problems are distinguished so

that students can learn how they differ.

Analyzing Student Thinking Problems are found

at the end of the Exercise/Problem Sets. These problems

are questions that elementary students might ask their

teachers, and they focus on common misconceptions that

are held by students. These problems give future teachers an

opportunity to think about the concepts they have learned

in the sec-

tion in the

context of

teaching.

Curriculum Standards The NCTM

Standards and Curriculum Focal Points

and the Common Core State Standards

are introduced on the third page of

each chapter. In addition, margin notes

involving these standards are contained

throughout the book.

FMPreface.indd 14

8/1/2013 12:05:32 PM - Preface xv

Reflection from Research Extensive research has

Children’s Literature These margin inserts provide

been done in the mathematics education community that

many examples of books that can be used to connect

focuses on the teaching

reading and mathematics. They should be invaluable to

and learning of elemen-

you when you begin teachig.

tary mathematics. Many

important quotations

from research are given

in the margins to sup-

port the content nearby.

Historical Vignettes open each chapter

and introduce ideas and concepts central to

each chapter.

Mathematical Morsels end every setion

with an interesting historical tidbit. One of

our students referred to these as a reward for

completing the section.

See one Live!

Children’s Videos are author-led videos of children

solving mathematical problems linked to QR codes in the

margin of the book. The codes are placed in locations

where the content being discussed is related to the content

of the problems being solved by the children. These videos

provide a window into how children think mathematically.

Blake E. Peterson

FMPreface.indd 15

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People in Mathematics, a feature

near the end of each chapter, high-

lights many of the giants in mathemat-

ics throughout history.

A Chapter Review is located at the end of each chapter.

A Chapter Test is found at the end of each chapter.

An Epilogue, following Chapter 16, provides a rich eclectic approach to geometry.

Logic and Clock Arithmetic are developed in topic sections near the end of the book.

Supplements for Students

Student Activities Manual with Discussion Questions for the Classroom This activity

manual is designed to enhance student learning as well as to model effective classroom practices. Since

many instructors are working with students to create a personalized journal, this edition of the manual

is shrink-wrapped and three-hole punched for easy customization. This supplement is an extensive revi-

sion of the Student Resoure Handbook that was authored by Karen Swenson and Marcia Swanson for

the first six editions of this book.

ISBN 978-1-118-67904-3

Features Include:

r

Hands-On Activities: Activities that help develop initial understandings at the concrete level.

r

Discussion Questions for the Classroom: Tasks designed to engage students with mathematical

ideas by stimulating communication.

r

Mental Math: Short activities to help develop mental math skills.

r

Exercises: Additional practice for building skills in concepts.

r

Directions in Education: Specially written articles that provide insights into major issues of the day,

including the Standards of the National Council of Teachers of Mathematics.

r

Solutions: Solutions to all items in the handbook to enhance self-study.

r

Two-Dimensional Manipulatives: Cutouts are provided on cardstock.

—Prepared by Lyn Riverstone of Oregon State University

The ETA Cuisenalre® Physical Manipulative Kit A generous assortment of manipulatives

(including blocks, tiles, geoboards, and so forth) has been created to accompany the text as well as the

Student Activity Manual. lt is available to be packaged with the text. Please contact your local Wiley

representative for ordering information.

ISBN 978-1-118-67923-4

Student Hints and Solutions Manual for Part A Problems This manual contains hints and

solutions to all of the Part A problems. It can be used to help students develop problem-solving profi-

ciency in a self-study mode. The features include:

FMPreface.indd 16

8/1/2013 12:05:35 PM - Preface xvii

r Hints: Give students a start on all Part A problems in the text.

r Additional Hints: A second hint is provided for more challenging problems.

r Complete Solutions to Part A Problems: Carefully written-out solutions are provided to model one correct solution.

—Developed by Lynn Trimpe, Vikki Maurer,

and Roger Maurer of Linn-Benton Community College.

ISBN 978-1-118-67925-8

Companion Web site http://www.wiley.com/college/musser

The companion Web site provides a wealth of resources for students.

Resources for Technology Problems

These problems are integrated into the problem sets throughout the book and are denoted by a mouse icon.

r eManipulatives mirror physical manipulatives as well as provide dynamic representations of other mathematical

situations. The goal of using the eManipulatives is to engage learners in a way that will lead to a more in-depth

understanding of the concepts and to give them experience thinking about the mathematics that underlies the

manipulatives.

—Prepared by Lawrence O. Cannon, E. Robert Heal, and Joel Duffin of Utah State University,

Richard Wellman of Westminster College, and Ethalinda K. S. Cannon of A415software.com.

This project is supported by the National Science Foundation.

r The Geometer’s Sketchpad® activities allow students to use the dynamic capabilities of this software to investigate

geometric properties and relationships. They are accessible through a Web browser so having the software is not

necessary.

r The Spreadsheet activities utilize the iterative properties of spreadsheets and the user friendly interface to investigate

problems ranging from graphs of functions to standard deviation to simulations of rolling dice.

Technology Tutorials

r The Geometer’s Sketchpad® tutorial is written for those students who have access to the software and who are

interested in investigating problems of their own choosing. The tutorial gives basic instruction on how to use the

software and includes some sample problems that will help the students gain a better understanding of the software

and the geometry that could be learned by using it.

—Prepared by Armando Martinez-Cruz,

California State University, Fullerton.

r The Spreadsheet Tutorial is written for students who are interested in learning how to use spreadsheets to investi-

gate mathematical problems. The tutorial describes some of the functions of the software and provides exercises for

students to investigate mathematics using the software.

—Prepared by Keith Leatham, Brigham Young University.

Webmodules

r The Algebraic Reasoning Webmodule helps students understand the critical transition from arithmetic to algebra. It

also highlights situations when algebra is, or can be, used. Marginal notes are placed in the text at the appropriate

locations to direct students to the webmodule.

—Prepared by Keith Leatham, Brigham Young University.

r The Children’s Literature Webmodule provides references to many mathematically related examples of children’s

books for each chapter. These references are noted in the margins near the mathematics that corresponds to the

content of the book. The webmodule also contains ideas about using children’s literature in the classroom.

—Prepared by Joan Cohen Jones, Eastern Michigan University.

FMPreface.indd 17

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r The Introduction to Graph Theory Webmodule has been moved from the Topics to the companion Web site to save

space in the book and yet allow professors the flexibility to download it from the Web if they choose to use it.

The companion Web site also includes:

r Links to NCTM Standards

r Links to Common Core Standards

r A Logo and TI-83 graphing calculator tutorial

r Four cumulative tests covering material up to the end of Chapters 4, 9, 12, and 16

r Research Article References: A complete list of references for the research articles that are mentioned in the

Reflection from Research margin notes throughout the book

Guide to Problem Solving This valuable resource, available as a webmodule on the companion Web site, contains

more than 200 creative problems keyed to the problem solving strategies in the textbook and includes:

r Opening Problem: an introductory problem to motivate the need for a strategy.

r Solution/Discussion/Clues: A worked-out solution of the opening problem together with a discussion of the strategy

and some clues on when to select this strategy.

r Practice Problems: A second problem that uses the same strategy together with a worked out solution and two

practice problems.

r Mixed Strategy Practice: Four practice problems that can be solved using one or more of the strategies introduced

to that point.

r Additional Practice Problems and Additional Mixed Strategy Problems: Sections that provide more practice for par-

ticular strategies as well as many problems for which students need to identify appropriate strategies.

—Prepared by Don Miller, who retired as a professor

of mathematics at St. Cloud State University.

Problems for Writing and Discussion are problems that require an analysis of ideas and are good opportunities

to write about the concepts in the book. Most of the Problems for Writing/Discussion that preceded the Chapter Tests

in the Eighth Edition now appear on our Web site.

The Geometer’s Sketchpad© Developed by Key Curriculum Press, this dynamic geometry construction and

exploration tool allows users to create and manipulate precise figures while preserving geometric relationships. This

software is only available when packaged with the text. Please contact your local Wiley representative for further

details.

WileyPLUS WileyPLUS is a powerful online tool that will help you study more effectively, get immediate feedback

when you practice on your own, complete assignments and get help with problem solving, and keep track of how you’re

doing—all at one easy-to-use Web site.

Resources for the Instructor

Companion Web Site

The companion Web site is available to text adopters and provides a wealth of resources including:

r PowerPoint Slides of more than 190 images that include figures from the text and several generic masters for dot

paper, grids, and other formats.

r Instructors also have access to all student Web site features. See above for more details.

Instructor Resource Manual This manual contains chapter-by-chapter discussions of the text material, student

“expectations” (objectives) for each chapter, answers for all Part B exercises and problems, and answers for all of the

even-numbered problems in the Guide to Problem-Solving.

—Prepared by Lyn Riverstone, Oregon State University

ISBN 978-1-118-67924-1

FMPreface.indd 18

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Computerized/Print Test Bank The Computerized/Printed Test Bank includes a collection of over 1,100 open

response, multiple-choice, true/false, and free-response questions, nearly 80% of which are algorithmic.

—Prepared by Mark McKibben, Goucher College

WileyPLUS WileyPLUS is a powerful online tool that provides instructors with an integrated suite of resources,

including an online version of the text, in one easy-to-use Web site. Organized around the essential activities you

perform in class, WileyPLUS allows you to create class presentations, assign homework and quizzes for automatic

grading, and track student progress. Please visit http://edugen.wiley.com or contact your local Wiley representative for

a demonstration and further details.

FMPreface.indd 19

8/1/2013 12:05:35 PM - ACKNOWLEDGMENTS

During the development of Mathematics for Elementary Teach-

Mary Forintos, Ferris State University

ers, Eighth, Ninth, and Tenth Editions, we benefited from

Marie Franzosa, Oregon State University

comments, suggestions, and evaluations from many of our col-

Sonia Goerdt, St. Cloud State University

leagues. We would like to acknowledge the contributions made

Ralph Harris, Fresno Pacific University

by the following people:

George Jennings, California State University, Dominguez Hills

Andy Jones, Prince George’s Community College

Reviewers for the Tenth Edition

Karla Karstens, University of Vermont

Margaret Kidd, California State University, Fullerton

Meg Kiessling, University of Tennessee at Chattanooga

Rebecca Metcalf, Bridgewater State College

Juli Ratheal, University of Texas Permian Basin

Pamela Miller, Arizona State University, West

Marie Franzosa, Oregon State University

Jessica Parsell, Delaware Technical Community College

Mary Beth Rollick, Kent State University

Tuyet Pham, Kent State University

Linda Lefevre, SUNY Oswego

Mary Beth Rollick, Kent State University

Keith Salyer, Central Washington University

Reviewers for the Ninth Edition

Sherry Schulz, College of the Canyons

Carol Steiner, Kent State University

Larry Feldman, Indiana University of Pennsylvania

Abolhassan Tagavy, City College of Chicago

Sarah Greenwald, Appalachian State University

Rick Vaughan, Paradise Valley Community College

Leah Gustin, Miami University of Ohio, Middleton

Demetria White, Tougaloo College

Linda LeFevre, State University of New York, Oswego

John Woods, Southwestern Oklahoma State University

Bethany Noblitt, Northern Kentucky University

Todd Cadwallader Olsker, California State University, Fullerton

In addition, we would like to acknowledge the contributions made

Cynthia Piez, University of Idaho

by colleagues from earlier editions.

Tammy Powell-Kopilak, Dutchess Community College

Edel Reilly, Indiana University of Pennsylvania

Reviewers for the Eighth Edition

Sarah Reznikoff, Kansas State University

Mary Beth Rollick, Kent State University

Seth Armstrong, Southern Utah University

Elayne Bowman, University of Oklahoma

Ninth Edition Interviewees

Anne Brown, Indiana University, South Bend

David C. Buck, Elizabethtown

John Baker, Indiana University of Pennsylvania

Alison Carter, Montgomery College

Paulette Ebert, Northern Kentucky University

Janet Cater, California State University, Bakersfield

Gina Foletta, Northern Kentucky University

Darwyn Cook, Alfred University

Leah Griffith, Rio Hondo College

Christopher Danielson, Minnesota State University, Mankato

Jane Gringauz, Minneapolis Community College

Linda DeGuire, California State University, Long Beach

Alexander Kolesnick, Ventura College

Cristina Domokos, California State University, Sacramento

Gail Laurent, College of DuPage

Scott Fallstrom, University of Oregon

Linda LeFevre, State University of New York, Oswego

Teresa Floyd, Mississippi College

Carol Lucas, University of Central Oklahoma

Rohitha Goonatilake, Texas A&M International University

Melanie Parker, Clarion University of Pennsylvania

Margaret Gruenwald, University of Southern Indiana

Shelle Patterson, Murray State University

Joan Cohen Jones, Eastern Michigan University

Cynthia Piez, University of Idaho

Joe Kemble, Lamar University

Denise Reboli, King’s College

Margaret Kinzel, Boise State University

Edel Reilly, Indiana University of Pennsylvania

J. Lyn Miller, Slippery Rock University

Sarah Reznikoff, Kansas State University

Girija Nair-Hart, Ohio State University, Newark

Nazanin Tootoonchi, Frostburg State University

Sandra Nite, Texas A&M University

Sally Robinson, University of Arkansas, Little Rock

Ninth Edition Focus Group Participants

Nancy Schoolcraft, Indiana University, Bloomington

Karen E. Spike, University of North Carolina, Wilmington

Kaddour Boukkabar, California University of Pennsylvania

Brian Travers, Salem State

Melanie Branca, Southwestern College

Mary Wiest, Minnesota State University, Mankato

Tommy Bryan, Baylor University

Mark A. Zuiker, Minnesota State University, Mankato

Jose Cruz, Palo Alto College

Arlene Dowshen, Widener University

Student Activity Manual Reviewers

Rita Eisele, Eastern Washington University

Mario Flores, University of Texas at San Antonio

Kathleen Almy, Rock Valley College

Heather Foes, Rock Valley College

Margaret Gruenwald, University of Southern Indiana

xx

FMAcknowledgments.indd 20

7/31/2013 12:26:16 PM - Acknowledgments xxi

Kate Riley, California Polytechnic State University

Maura Murray, University of Massachusetts

Robyn Sibley, Montgomery County Public Schools

Kathy Nickell, College of DuPage

Dennis Parker, The University of the Pacific

State Standards Reviewers

William Regonini, California State University, Fresno

James Riley, Western Michigan University

Joanne C. Basta, Niagara University

Kate Riley, California Polytechnic State University

Joyce Bishop, Eastern Illinois University

Eric Rowley, Utah State University

Tom Fox, University of Houston, Clear Lake

Peggy Sacher, University of Delaware

Joan C. Jones, Eastern Michigan University

Janine Scott, Sam Houston State University

Kate Riley, California Polytechnic State University

Lawrence Small, L.A. Pierce College

Janine Scott, Sam Houston State University

Joe K. Smith, Northern Kentucky University

Murray Siegel, Sam Houston State University

J. Phillip Smith, Southern Connecticut State University

Rebecca Wong, West Valley College

Judy Sowder, San Diego State University

Larry Sowder, San Diego State University

Reviewers

Karen Spike, University of Northern Carolina, Wilmington

Debra S. Stokes, East Carolina University

Paul Ache, Kutztown University

Jo Temple, Texas Tech University

Scott Barnett, Henry Ford Community College

Lynn Trimpe, Linn–Benton Community College

Chuck Beals, Hartnell College

Jeannine G. Vigerust, New Mexico State University

Peter Braunfeld, University of Illinois

Bruce Vogeli, Columbia University

Tom Briske, Georgia State University

Kenneth C. Washinger, Shippensburg University

Anne Brown, Indiana University, South Bend

Brad Whitaker, Point Loma Nazarene University

Christine Browning, Western Michigan University

John Wilkins, California State University, Dominguez Hills

Tommy Bryan, Baylor University

Lucille Bullock, University of Texas

Questionnaire Respondents

Thomas Butts, University of Texas, Dallas

Dana S. Craig, University of Central Oklahoma

Mary Alter, University of Maryland

Ann Dinkheller, Xavier University

Dr. J. Altinger, Youngstown State University

John Dossey, Illinois State University

Jamie Whitehead Ashby, Texarkana College

Carol Dyas, University of Texas, San Antonio

Dr. Donald Balka, Saint Mary’s College

Donna Erwin, Salt Lake Community College

Jim Ballard, Montana State University

Sheryl Ettlich, Southern Oregon State College

Jane Baldwin, Capital University

Ruhama Even, Michigan State University

Susan Baniak, Otterbein College

Iris B. Fetta, Clemson University

James Barnard, Western Oregon State College

Marjorie Fitting, San Jose State University

Chuck Beals, Hartnell College

Susan Friel, Math/Science Education Network, University of

Judy Bergman, University of Houston, Clearlake

North Carolina

James Bierden, Rhode Island College

Gerald Gannon, California State University, Fullerton

Neil K. Bishop, The University of Southern Mississippi,

Joyce Rodgers Griffin, Auburn University

Gulf Coast

Jerrold W. Grossman, Oakland University

Jonathan Bodrero, Snow College

Virginia Ellen Hanks, Western Kentucky University

Dianne Bolen, Northeast Mississippi Community College

John G. Harvey, University of Wisconsin, Madison

Peter Braunfeld, University of Illinois

Patricia L. Hayes, Utah State University, Uintah Basin Branch

Harold Brockman, Capital University

Campus

Judith Brower, North Idaho College

Alan Hoffer, University of California, Irvine

Anne E. Brown, Indiana University, South Bend

Barnabas Hughes, California State University, Northridge

Harmon Brown, Harding University

Joan Cohen Jones, Eastern Michigan University

Christine Browning, Western Michigan University

Marilyn L. Keir, University of Utah

Joyce W. Bryant, St. Martin’s College

Joe Kennedy, Miami University

R. Elaine Carbone, Clarion University

Dottie King, Indiana State University

Randall Charles, San Jose State University

Richard Kinson, University of South Alabama

Deann Christianson, University of the Pacific

Margaret Kinzel, Boise State University

Lynn Cleary, University of Maryland

John Koker, University of Wisconsin

Judith Colburn, Lindenwood College

David E. Koslakiewicz, University of Wisconsin, Milwaukee

Sister Marie Condon, Xavier University

Raimundo M. Kovac, Rhode Island College

Lynda Cones, Rend Lake College

Josephine Lane, Eastern Kentucky University

Sister Judith Costello, Regis College

Louise Lataille, Springfield College

H. Coulson, California State University

Roberts S. Matulis, Millersville University

Dana S. Craig, University of Central Oklahoma

Mercedes McGowen, Harper College

Greg Crow, John Carroll University

Flora Alice Metz, Jackson State Community College

Henry A. Culbreth, Southern Arkansas University, El Dorado

J. Lyn Miller, Slippery Rock University

Carl Cuneo, Essex Community College

Barbara Moses, Bowling Green State University

Cynthia Davis, Truckee Meadows Community College

FMAcknowledgments.indd 21

7/31/2013 12:26:16 PM - xxii Acknowledgments

Gregory Davis, University of Wisconsin, Green Bay

Jack Lombard, Harold Washington College

Jennifer Davis, Ulster County Community College

Betty Long, Appalachian State University

Dennis De Jong, Dordt College

Ann Louis, College of the Canyons

Mary De Young, Hop College

C. A. Lubinski, Illinois State University

Louise Deaton, Johnson Community College

Pamela Lundin, Lakeland College

Shobha Deshmukh, College of Saint Benedict/St.

Charles R. Luttrell, Frederick Community College

John’s University

Carl Maneri, Wright State University

Sheila Doran, Xavier University

Nancy Maushak, William Penn College

Randall L. Drum, Texas A&M University

Edith Maxwell, West Georgia College

P. R. Dwarka, Howard University

Jeffery T. McLean, University of St. Thomas

Doris Edwards, Northern State College

George F. Mead, McNeese State University

Roger Engle, Clarion University

Wilbur Mellema, San Jose City College

Kathy Ernie, University of Wisconsin

Clarence E. Miller, Jr. Johns Hopkins University

Ron Falkenstein, Mott Community College

Diane Miller, Middle Tennessee State University

Ann Farrell, Wright State University

Ken Monks, University of Scranton

Francis Fennell, Western Maryland College

Bill Moody, University of Delaware

Joseph Ferrar, Ohio State University

Kent Morris, Cameron University

Chris Ferris, University of Akron

Lisa Morrison, Western Michigan University

Fay Fester, The Pennsylvania State University

Barbara Moses, Bowling Green State University

Marie Franzosa, Oregon State University

Fran Moss, Nicholls State University

Margaret Friar, Grand Valley State College

Mike Mourer, Johnston Community College

Cathey Funk, Valencia Community College

Katherine Muhs, St. Norbert College

Dr. Amy Gaskins, Northwest Missouri State University

Gale Nash, Western State College of Colorado

Judy Gibbs, West Virginia University

T. Neelor, California State University

Daniel Green, Olivet Nazarene University

Jerry Neft, University of Dayton

Anna Mae Greiner, Eisenhower Middle School

Gary Nelson, Central Community College, Columbus Campus

Julie Guelich, Normandale Community College

James A. Nickel, University of Texas, Permian Basin

Ginny Hamilton, Shawnee State University

Kathy Nickell, College of DuPage

Virginia Hanks, Western Kentucky University

Susan Novelli, Kellogg Community College

Dave Hansmire, College of the Mainland

Jon O’Dell, Richland Community College

Brother Joseph Harris, C.S.C., St. Edward’s University

Jane Odell, Richland College

John Harvey, University of Wisconsin

Bill W. Oldham, Harding University

Kathy E. Hays, Anne Arundel Community College

Jim Paige, Wayne State College

Patricia Henry, Weber State College

Wing Park, College of Lake County

Dr. Noal Herbertson, California State University

Susan Patterson, Erskine College (retired)

Ina Lee Herer, Tri-State University

Shahla Peterman, University of Missouri

Linda Hill, Idaho State University

Gary D. Peterson, Pacific Lutheran University

Scott H. Hochwald, University of North Florida

Debra Pharo, Northwestern Michigan College

Susan S. Hollar, Kalamazoo Valley Community College

Tammy Powell-Kopilak, Dutchess Community College

Holly M. Hoover, Montana State University, Billings

Christy Preis, Arkansas State University, Mountain Home

Wei-Shen Hsia, University of Alabama

Robert Preller, Illinois Central College

Sandra Hsieh, Pasadena City College

Dr. William Price, Niagara University

Jo Johnson, Southwestern College

Kim Prichard, University of North Carolina

Patricia Johnson, Ohio State University

Stephen Prothero, Williamette University

Pat Jones, Methodist College

Janice Rech, University of Nebraska

Judy Kasabian, El Camino College

Tom Richard, Bemidji State University

Vincent Kayes, Mt. St. Mary College

Jan Rizzuti, Central Washington University

Julie Keener, Central Oregon Community College

Anne D. Roberts, University of Utah

Joe Kennedy, Miami University

David Roland, University of Mary Hardin–Baylor

Susan Key, Meridien Community College

Frances Rosamond, National University

Mary Kilbridge, Augustana College

Richard Ross, Southeast Community College

Mike Kilgallen, Lincoln Christian College

Albert Roy, Bristol Community College

Judith Koenig, California State University, Dominguez Hills

Bill Rudolph, Iowa State University

Josephine Lane, Eastern Kentucky University

Bernadette Russell, Plymouth State College

Don Larsen, Buena Vista College

Lee K. Sanders, Miami University, Hamilton

Louise Lataille, Westfield State College

Ann Savonen, Monroe County Community College

Vernon Leitch, St. Cloud State University

Rebecca Seaberg, Bethel College

Steven C. Leth, University of Northern Colorado

Karen Sharp, Mott Community College

Lawrence Levy, University of Wisconsin

Marie Sheckels, Mary Washington College

Robert Lewis, Linn-Benton Community College

Melissa Shepard Loe, University of St. Thomas

Lois Linnan, Clarion University

Joseph Shields, St. Mary’s College, MN

FMAcknowledgments.indd 22

7/31/2013 12:26:16 PM - Acknowledgments xxiii

Lawrence Shirley, Towson State University

Maria Zack, Point Loma Nazarene College

Keith Shuert, Oakland Community College

Stanley L. Zehm, Heritage College

B. Signer, St. John’s University

Makia Zimmer, Bethany College

Rick Simon, Idaho State University

James Smart, San Jose State University

Focus Group Participants

Ron Smit, University of Portland

Gayle Smith, Lane Community College

Mara Alagic, Wichita State University

Larry Sowder, San Diego State University

Robin L. Ayers, Western Kentucky University

Raymond E. Spaulding, Radford University

Elaine Carbone, Clarion University of Pennsylvania

William Speer, University of Nevada, Las Vegas

Janis Cimperman, St. Cloud State University

Sister Carol Speigel, BVM, Clarke College

Richard DeCesare, Southern Connecticut State University

Karen E. Spike, University of North Carolina, Wilmington

Maria Diamantis, Southern Connecticut State University

Ruth Ann Stefanussen, University of Utah

Jerrold W. Grossman, Oakland University

Carol Steiner, Kent State University

Richard H. Hudson, University of South Carolina, Columbia

Debbie Stokes, East Carolina University

Carol Kahle, Shippensburg University

Ruthi Sturdevant, Lincoln University, MO

Jane Keiser, Miami University

Viji Sundar, California State University, Stanislaus

Catherine Carroll Kiaie, Cardinal Stritch University

Ann Sweeney, College of St. Catherine, MN

Armando M. Martinez-Cruz, California State University, Fuller-

Karen Swenson, George Fox College

ton

Carla Tayeh, Eastern Michigan University

Cynthia Y. Naples, St. Edward’s University

Janet Thomas, Garrett Community College

David L. Pagni, Fullerton University

S. Thomas, University of Oregon

Melanie Parker, Clarion University of Pennsylvania

Mary Beth Ulrich, Pikeville College

Carol Phillips-Bey, Cleveland State University

Martha Van Cleave, Linfield College

Content Connections Survey Respondents

Dr. Howard Wachtel, Bowie State University

Dr. Mary Wagner-Krankel, St. Mary’s University

Marc Campbell, Daytona Beach Community College

Barbara Walters, Ashland Community College

Porter Coggins, University of Wisconsin–Stevens Point

Bill Weber, Eastern Arizona College

Don Collins, Western Kentucky University

Joyce Wellington, Southeastern Community College

Allan Danuff, Central Florida Community College

Paula White, Marshall University

Birdeena Dapples, Rocky Mountain College

Heide G. Wiegel, University of Georgia

Nancy Drickey, Linfield College

Jane Wilburne, West Chester University

Thea Dunn, University of Wisconsin–River Falls

Jerry Wilkerson, Missouri Western State College

Mark Freitag, East Stroudsberg University

Jack D. Wilkinson, University of Northern Iowa

Paula Gregg, University of South Carolina, Aiken

Carole Williams, Seminole Community College

Brian Karasek, Arizona Western College

Delbert Williams, University of Mary Hardin–Baylor

Chris Kolaczewski, Ferris University of Akron

Chris Wise, University of Southwestern Louisiana

R. Michael Krach, Towson University

John L. Wisthoff, Anne Arundel Community College (retired)

Randa Lee Kress, Idaho State University

Lohra Wolden, Southern Utah University

Marshall Lassak, Eastern Illinois University

Mary Wolfe, University of Rio Grande

Katherine Muhs, St. Norbert College

Vernon E. Wolff, Moorhead State University

Bethany Noblitt, Northern Kentucky University

We would like to acknowledge the following people for their assistance in the preparation of our earlier editions of this book: Ron

Bagwell, Jerry Becker, Julie Borden, Sue Borden, Tommy Bryan, Juli Dixon, Christie Gilliland, Dale Green, Kathleen Seagraves Hig-

don, Hester Lewellen, Roger Maurer, David Metz, Naomi Munton, Tilda Runner, Karen Swenson, Donna Templeton, Lynn Trimpe,

Rosemary Troxel, Virginia Usnick, and Kris Warloe. We thank Robyn Silbey for her expert review of several of the features in our

seventh edition, Dawn Tuescher for her work on the correlation between the content of the book and the common core standards

statements, and Becky Gwilliam for her research contributions to Chapter 10 and the Reflections from Research. Our Mathematical

Morsels artist, Ron Bagwell, who was one of Gary Musser’s exceptional prospective elementary teacher students at Oregon State

University, deserves special recognition for his creativity over all ten editions. We especially appreciate the extensive proofreading and

revision suggestion for the problem sets provided by Jennifer A. Blue for this edition. We also thank Lyn Riverstone, Vikki Maurer,

and Jen Blue for their careful checking of the accuracy of the answers.

We also want to acknowledge Marcia Swanson and Karen Swenson for their creation of and contribution to our Student Resource

Handbook during the first seven editions with a special thanks to Lyn Riverstone for her expert revision of the Student Activity Manual

since. Thanks are also due to Don Miller for his Guide to Problem Solving, to Lyn Trimpe, Roger Maurer, and Vikki Maurer, for their long-

time authorship of our Student Hints and Solutions Manual, to Keith Leathem for the Spreadsheet Tutorial and Algebraic Reasoning

Web Module, Armando Martinez-Cruz for The Geometer’s Sketchpad Tutorial, to Joan Cohen Jones for the Children’s Literature mar-

gin inserts and the associated Webmodule, and to Lawrence O. Cannon, E. Robert Heal, Joel Duffin, Richard Wellman, and Ethalinda

K. S. Cannon for the eManipulatives activities.

FMAcknowledgments.indd 23

7/31/2013 12:26:16 PM - xxiv Acknowledgments

We are very grateful to our publisher, Laurie Rosatone, and our editor, Jennifer Brady, for their commitment and super teamwork;

to our exceptional senior production editor, Kerry Weinstein, for attending to the details we missed; to Elizabeth Chenette, copyedi-

tor, Carol Sawyer, proofreader, and Christine Poolos, freelance editor, for their wonderful help in putting this book together; and

to Melody Englund, our outstanding indexer. Other Wiley staff who helped bring this book and its print and media supplements

to fruition are: Kimberly Kanakes, Marketing Manager; Sesha Bolisetty, Vice President, Production and Manufacturing; Karoline

Luciano, Senior Content Manager; Madelyn Lesure, Senior Designer; Lisa Gee, Senior Photo Editor, and Thomas Kulesa, Senior

Product Designer. They have been uniformly wonderful to work with—John Wiley would have been proud of them.

Finally, we welcome comments from colleagues and students. Please feel free to send suggestions to Gary at glmusser@cox.net

and Blake at peterson@mathed.byu.edu. Please include both of us in any communications.

G.L.M.

B.E.P.

FMAcknowledgments.indd 24

7/31/2013 12:26:16 PM - A NOTE TO OUR STUDENTS

There are many pedagogical elements in our book which are designed to help you as you

learn mathematics. We suggest the following:

1. Begin each chapter by reading the Focus On on the first page of the chapter. This

will give you a mathematical sense of some of the history that underlies the chapter.

2. Try to work the Initial Problem on the second page of the chapter. Since problem

solving is so important in mathematics, you will want to increase your profi-

ciency in solving problems so that you can help your students to learn to solve

problems. Also notice the Problem Solving Strategies box on this second page.

This box grows throughout the book as you learn new strategies to help you

enhance your problem solving ability.

3. The third page of each chapter contains three items. First, the QR code has an

Author Walk-Through narrated by Blake where he will give you a brief preview

of key ideas in the chapter. Next, there is a brief Introduction to the chapter that

will also give you a sense of what is to come. Finally, there are three Lists of

Recommendations that will be covered in the chapter. You will be reminded of

the NCTM Principles and Standards for School Mathematics and the Common

Core Standards in margin notes as you work through the chapter.

4. In addition to the QR code mentioned above, there are many other such codes

throughout the book. These codes lead to brief Children’s Videos where children

are solving problems involving the content near the code. These will give you a

feeling of what it will be like when you are teaching.

5. Each section contains several Mathematical Tasks which are designed to be

solved in groups so you can come to understand the concepts in the section

through your investigation of these mathematical tasks. If these tasks are not

used as part of your classroom instruction, you would benefit from trying them

on your own and discussing your investigation with your peers or instructor.

6. When you finish studying a subsection, work the Set A exercises at the end of

the section that are suggested by the Check for Understanding. This will help you

learn the material in the section in smaller increments which can be a more effec-

tive way to learn. The answers for these exercises are in the back of the book.

7. As you work through each section, take breaks and read through the margin

notes Reflections from Research, NCTM Standards, Common Core, and Algebraic

Reasoning. These should enrich your learning experience. Of course, the Children’s

Literature margin notes should help you begin a list of materials that you can use

when you begin to teach.

8. Be certain to read the Mathematical Morsel at the end of each section. These are

stories that will enrich your learning experience.

9. By the time you arrive at the Exercise/Problem Set, you should have worked all

of the exercises in Set A and checked your answers. This practice should have

helped you learn the knowledge, skill, and understanding of the material in

the section (see our illustrative cube in the Pedagogy section). Next you should

attempt to work all of the Set A problems. These may require slightly deeper

thinking than did the exercises. Once again, the answers to these problems are in

the back of the book. Your teacher may assign some of the Set B exercises and

problems. These do not have answers in this book, so you will have to draw on

what you have learned from the Set A exercises and problems.

10. Finally, when you reach the end of the chapter, carefully work through the

Chapter Review and the Chapter Test.

1

FMANotetoOurStudents.indd 1

8/1/2013 8:39:16 PM - C H A P T E R

1 INTRODUCTION TO

PROBLEM SOLVING

George Pólya—The Father of Modern Problem Solving

George Pólya was born in Hungary in 1887. He into 15 languages, introduced his four-step approach

received his Ph.D. at the University of Budapest.

together with heuristics, or strategies, which are helpful

In 1940 he came to Brown University and then

in solving problems. Other important works by Pólya

joined the faculty at Stanford University in 1942.

are Mathematical Discovery, Volumes 1 and 2, and

Mathematics and Plausible Reasoning, Volumes 1 and 2.

He died in 1985, leaving mathematics with the impor-

tant legacy of teaching problem solving. His “Ten

Commandments for Teachers” are as follows:

1. Be interested in your subject.

2. Know your subject.

3. Try to read the faces of your students; try to see their

expectations and difficulties; put yourself in their

place.

orld Photos

W

4. Realize that the best way to learn anything is to dis-

cover it by yourself.

5. Give your students not only information, but also

©AP/Wide

know-how, mental attitudes, the habit of methodical

In his studies, he became interested in the process of

work.

discovery, which led to his famous four-step process for

6. Let them learn guessing.

solving problems:

7. Let them learn proving.

1. Understand the problem.

8. Look out for such features of the problem at hand as

2. Devise a plan.

may be useful in solving the problems to come—try to

3. Carry out the plan.

disclose the general pattern that lies behind the present

concrete situation.

4. Look back.

9. Do not give away your whole secret at once—let the

Pólya wrote over 250 mathematical papers and three

students guess before you tell it—let them find out by

books that promote problem solving. His most famous

themselves as much as is feasible.

book, How to Solve It, which has been translated

10. Suggest; do not force information down their throats.

2

c01.indd 2

7/30/2013 2:36:04 PM - Problem-Solving

Strategies

1. Guess and Test

Because problem solving is the main goal of mathematics, this chapter introduces the

six strategies listed in the Problem-Solving Strategies box that are helpful in solving

2. Draw a Picture

problems. Then, at the beginning of each chapter, an initial problem is posed that can

3. Use a Variable

be solved by using the strategy introduced in that chapter. As you move through this

book, the Problem-Solving Strategies boxes at the beginning of each chapter expand,

4. Look for a Pattern

as should your ability to solve problems.

5. Make a List

6. Solve a Simpler

Initial Problem

Problem

Place the whole numbers 1 through 9 in the circles in the accompanying triangle so

that the sum of the numbers on each side is 17.

A solution to this Initial Problem is on page 37.

3

c01.indd 3

7/30/2013 2:36:05 PM - AUTHOR

I N T R O D U C T I O N

Once, at an informal meeting, a social scientist asked a mathematics professor, “What’s the main goal

of teaching mathematics?” The reply was “problem solving.” In return, the mathematician asked,

“What is the main goal of teaching the social sciences?” Once more the answer was “problem solving.”

All successful engineers, scientists, social scientists, lawyers, accountants, doctors, business managers,

WALK-THROUGH and so on have to be good problem solvers. Although the problems that people encounter may be very

diverse, there are common elements and an underlying structure that can help to facilitate problem

solving. Because of the universal importance of problem solving, the main professional group in mathematics educa-

tion, the National Council of Teachers of Mathematics (NCTM) recommended in its 1980 Agenda for Actions that

“problem solving be the focus of school mathematics in the 1980s.” The NCTM’s 1989 Curriculum and Evaluation

Standards for School Mathematics called for increased attention to the teaching of problem solving in K-8 mathemat-

ics. Areas of emphasis include word problems, applications, patterns and relationships, open-ended problems, and

problem situations represented verbally, numerically, graphically, geometrically, and symbolically. The NCTM’s

2000 Principles and Standards for School Mathematics identified problem solving as one of the processes by which all

mathematics should be taught.

This chapter introduces a problem-solving process together with six strategies that will aid you in solving problems.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-12–PROBLEM SOLVING

Build new mathematical knowledge through problem solving.

Solve problems that arise in mathematics and in other contexts.

Apply and adapt a variety of appropriate strategies to solve problems.

Monitor and reflect on the process of mathematical problem solving.

Key Concepts from the NCTM Curriculum Focal Points

r KINDERGARTEN: Choose, combine, and apply effective strategies for answering quantitative questions.

r GRADE 1: Develop an understanding of the meanings of addition and subtraction and strategies to solve such

arithmetic problems. Solve problems involving the relative sizes of whole numbers.

r GRADE 3: Apply increasingly sophisticated strategies … to solve multiplication and division problems.

r GRADE 4 AND 5: Select appropriate units, strategies, and tools for solving problems.

r GRADE 6: Solve a wide variety of problems involving ratios and rates.

r GRADE 7: Use ratio and proportionality to solve a wide variety of percent problems.

Key Concepts from the Common Core State Standards for Mathematics

r ALL GRADES

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Mathematical Practice 2: Reason abstractly and quantitatively.

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Mathematical Practice 4: Model with mathematics.

Mathematical Practice 7: Look for and make use of structures.

4

c01.indd 4

7/30/2013 2:36:05 PM - Section 1.1 The Problem-Solving Process and Strategies 5

THE PROBLEM-SOLVING PROCESS AND STRATEGIES

Use any strategy you know to solve the next problem. As you solve this problem, pay

close attention to the thought processes and steps that you use. Write down these strate-

gies and compare them to a classmate’s. Are there any similarities in your approaches to solving this problem?

Lin’s garden has an area of 78 square yards. The length of the garden is 5 less than 3 times its width. What are

the dimensions of Lin’s garden?

Pólya’s Four Steps

In this book we often distinguish between “exercises” and “problems.” Unfortunately,

the distinction cannot be made precise. To solve an exercise, one applies a routine

procedure to arrive at an answer. To solve a problem, one has to pause, reflect, and

perhaps take some original step never taken before to arrive at a solution. This need

for some sort of creative step on the solver’s part, however minor, is what distinguishes

a problem from an exercise. To a young child, finding 3 + 2 might be a problem,

whereas it is a fact for you. For a child in the early grades, the question “How do you

divide 96 pencils equally among 16 children?” might pose a problem, but for you it

suggests the exercise “find 96 ÷ 16.” These two examples illustrate how the distinction

between an exercise and a problem can vary, since it depends on the state of mind of

the person who is to solve it.

Reflection from Research

Doing exercises is a very valuable aid in learning mathematics. Exercises help you

Many children believe that the

to learn concepts, properties, procedures, and so on, which you can then apply when

answer to a word problem can

solving problems. This chapter provides an introduction to the process of problem

always be found by adding, sub-

solving. The techniques that you learn in this chapter should help you to become a

tracting, multiplying, or dividing

better problem solver and should show you how to help others develop their problem-

two numbers. Little thought is

given to understanding the con-

solving skills.

text of the problem (Verschaffel,

A famous mathematician, George Pólya, devoted much of his teaching to helping

De Corte, & Vierstraete, 1999).

students become better problem solvers. His major contribution is what has become

known as Pólya’s four-step process for solving problems.

Common Core – Grades

Step 1 Understand the Problem

K-12 (Mathematical

r Do you understand all the words?

Practice 1)

Mathematically proficient stu-

r Can you restate the problem in your own words?

dents start by explaining to them-

selves the meaning of a problem

r Do you know what is given?

and looking for entry points to its

r Do you know what the goal is?

solution.

r Is there enough information?

r Is there extraneous information?

r Is this problem similar to another problem you have solved?

Common Core – Grades

Step 2 Devise a Plan

K-12 (Mathematical

Can one of the following strategies (heuristics) be used? (A strategy is defi ned as

Practice 1)

an artful means to an end.)

Mathematically proficient stu-

dents analyze givens, constraints,

relationships, and goals. They

1. Guess and test.

8. Use direct reasoning.

make conjectures about the form

2. Draw a picture.

9. Use indirect reasoning.

and meaning of the solution and

plan a solution pathway rather

3. Use a variable.

10. Use properties of numbers.

than simply jumping into a solu-

4. Look for a pattern.

11. Solve an equivalent problem.

tion attempt.

5. Make a list.

12. Work backward.

6. Solve a simpler problem.

13. Use cases.

7. Draw a diagram.

14. Solve an equation.

c01.indd 5

7/30/2013 2:36:05 PM - 6

Chapter 1 Introduction to Problem Solving

15. Look for a formula.

19. Identify subgoals.

16. Do a simulation.

20. Use coordinates.

17. Use a model.

21. Use symmetry.

18. Use dimensional analysis.

The first six strategies are discussed in this chapter; the others are introduced in

subsequent chapters.

Common Core – Grades

Step 3 Carry Out the Plan

K-12 (Mathematical

r Implement the strategy or strategies that you have chosen until the problem is

Practice 1)

solved or until a new course of action is suggested.

Mathematically proficient stu-

dents consider analogous prob-

r Give yourself a reasonable amount of time in which to solve the problem. If

lems and try special cases and

you are not successful, seek hints from others or put the problem aside for a

simpler forms of the original

while. (You may have a flash of insight when you least expect it!)

problem in order to gain insight

into its solution.

r Do not be afraid of starting over. Often, a fresh start and a new strategy will

lead to success.

Common Core – Grades

Step 4 Look Back

K-12 (Mathematical

r Is your solution correct? Does your answer satisfy the statement of the problem?

Practice 1)

Mathematically proficient stu-

r Can you see an easier solution?

dents monitor and evaluate their

progress and change course if

r Can you see how you can extend your solution to a more general case?

necessary.

Usually, a problem is stated in words, either orally or written. Then, to solve the

problem, one translates the words into an equivalent problem using mathematical

symbols, solves this equivalent problem, and then interprets the answer. This process

is summarized in Figure 1.1.

Figure 1.1

Reflection from Research

Learning to utilize Pólya’s four steps and the diagram in Figure 1.1 are first steps

Researchers suggest that teach-

in becoming a good problem solver. In particular, the “Devise a Plan” step is very

ers think aloud when solving

important. In this chapter and throughout the book, you will learn the strategies

problems for the first time in

listed under the “Devise a Plan” step, which in turn help you decide how to proceed to

front of the class. In so doing,

solve problems. However, selecting an appropriate strategy is critical! As we worked

teachers will be modeling suc-

cessful problem-solving behaviors

with students who were successful problem solvers, we asked them to share “clues”

for their students (Schoenfeld,

that they observed in statements of problems that helped them select appropriate

1985).

strategies. Their clues are listed after each corresponding strategy. Thus, in addition

to learning how to use the various strategies herein, these clues can help you decide

when to select an appropriate strategy or combination of strategies. Problem solving

is as much an art as it is a science. Therefore, you will find that with experience you

will develop a feeling for when to use one strategy over another by recognizing certain

clues, perhaps subconsciously. Also, you will find that some problems may be solved

NCTM Standard

in several ways using different strategies.

Instructional programs should

In summary, this initial material on problem solving is a foundation for your

enable all students to apply and

adapt a variety of appropriate

success in problem solving. Review this material on Pólya’s four steps as well as the

strategies to solve problems.

strategies and clues as you continue to develop your expertise in solving problems.

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7/30/2013 2:36:06 PM - From Chapter 6, Lesson “Problem Solving” from My Math, Volume 1 Common Core State Standards, Grade 2, copyright © 2013

by McGraw-Hill Education.

7

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7/30/2013 2:36:09 PM - 8

Chapter 1 Introduction to Problem Solving

Problem-Solving Strategies

The remainder of this chapter is devoted to introducing several problem-solving

strategies.

Guess and Test

Problem

Place the digits 1, 2, 3, 4, 5, 6 in the circles in Figure 1.2 so that the sum of the three

numbers on each side of the triangle is 12.

We will solve the problem in three ways to illustrate three different approaches to

the Guess and Test strategy. As its name suggests, to use the Guess and Test strategy,

you guess at a solution and test whether you are correct. If you are incorrect, you

refine your guess and test again. This process is repeated until you obtain a solution.

Step 1 Understand the Problem

Each number must be used exactly one time when arranging the numbers in the

Figure 1.2

triangle. The sum of the three numbers on each side must be 12.

First Approach: Random Guess and Test

Step 2 Devise a Plan

Tear off six pieces of paper and mark the numbers 1 through 6 on them and then

try combinations until one works.

Step 3 Carry Out the Plan

Arrange the pieces of paper in the shape of an equilateral triangle and check sums.

Keep rearranging until three sums of 12 are found.

Second Approach: Systematic Guess and Test

Step 2 Devise a Plan

Rather than randomly moving the numbers around, begin by placing the smallest

numbers—namely, 1, 2, 3—in the corners. If that does not work, try increasing

the numbers to 1, 2, 4, and so on.

Step 3 Carry Out the Plan

With 1, 2, 3 in the corners, the side sums are too small; similarly with 1, 2, 4. Try

1, 2, 5 and 1, 2, 6. The side sums are still too small. Next try 2, 3, 4, then 2, 3, 5,

and so on, until a solution is found. One also could begin with 4, 5, 6 in the cor-

ners, then try 3, 4, 5, and so on.

Third Approach: Inferential Guess and Test

Figure 1.3

Step 2 Devise a Plan

Start by assuming that 1 must be in a corner and explore the consequences.

Step 3 Carry Out the Plan

If 1 is placed in a corner, we must fi nd two pairs out of the remaining fi ve numbers

whose sum is 11 (Figure 1.3). However, out of 2, 3, 4, 5, and 6, only 6 + 5 = 11.

Thus, we conclude that 1 cannot be in a corner. If 2 is in a corner, there must be

Figure 1.4

two pairs left that add to 10 (Figure 1.4). But only 6 + 4 = 10. Therefore, 2 cannot

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7/30/2013 2:36:10 PM - Section 1.1 The Problem-Solving Process and Strategies 9

be in a corner. Finally, suppose that 3 is in a corner. Then we must satisfy Figure

1.5. However, only 5 + 4 = 9 of the remaining numbers. Thus, if there is a solu-

tion, 4, 5, and 6 will have to be in the corners (Figure 1.6). By placing 1 between 5

and 6, 2 between 4 and 6, and 3 between 4 and 5, we have a solution.

Figure 1.5

Step 4 Look Back

Notice how we have solved this problem in three different ways using Guess

and Test. Random Guess and Test is often used to get started, but it is easy

to lose track of the various trials. Systematic Guess and Test is better because

you develop a scheme to ensure that you have tested all possibilities. Gener-

ally, Inferential Guess and Test is superior to both of the previous methods

because it usually saves time and provides more information regarding possible

solutions.

Figure 1.6

Additional Problems Where the Strategy “Guess and Test”

Is Useful

NCTM Standard

1. In the following cryptarithm—that is, a collection of words where the letters

Instructional programs should

represent numbers—sun and fun represent two three-digit numbers, and swim is

enable all students to monitor

their four-digit sum. Using all of the digits 0, 1, 2, 3, 6, 7, and 9 in place of the

and reflect on the process of

letters where no letter represents two different digits, determine the value of

mathematical problem solving.

each letter.

sun

+ fun

swim

Step 1 Understand the Problem

Each of the letters in sun, fun, and swim must be replaced with the numbers 0,

1, 2, 3, 6, 7, and 9, so that a correct sum results after each letter is replaced with

its associated digit. When the letter n is replaced by one of the digits, then n + n

must be m or 10 + m, where the 1 in the 10 is carried to the tens column. Since

1 + 1 = 2, 3 + 3 = 6, and 6 + 6 = 12, there are three possibilities for n, namely, 1, 3,

or 6. Now we can try various combinations in an attempt to obtain the correct

sum.

Step 2 Devise a Plan

Use Inferential Guess and Test. There are three choices for n. Observe that sun

and fun are three-digit numbers and that swim is a four-digit number. Thus we

have to carry when we add s and f . Therefore, the value for s in swim is 1. This

limits the choices of n to 3 or 6.

Step 3 Carry Out the Plan

Since s = 1 and s + f leads to a two-digit number, f must be 9. Thus there are two

possibilities:

(a)

1u 3

(b)

1u 6

+ 9 u 3

+ 9 u 6

1 w i 6

1 w i 2

In (a), if u = 0, 2, or 7, there is no value possible for i among the remaining digits.

In (b), if u = 3, then u + u plus the carry from 6 + 6 yields i = 7. This leaves w = 0

for a solution.

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7/30/2013 2:36:16 PM - 10 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

The reasoning used here shows that there is one and only one solution to this

problem. When solving problems of this type, one could randomly substitute

digits until a solution is found. However, Inferential Guess and Test simplifi es the

solution process by looking for unique aspects of the problem. Here the natural

places to start are n + n, u + u, and the fact that s + f yields a two-digit number.

2. Use four 4s and some of the symbols +, ×, −, ÷, ( ) to give expressions for the

Figure 1.7

whole numbers from 0 through 9: for example, 5 = (4 × 4 + 4) ÷ 4.

3. For each shape in Figure 1.7, make one straight cut so that each of the two

pieces of the shape can be rearranged to form a square.

(NOTE: Answers for these problems are given after the Solution of the Initial Prob-

lem near the end of this chapter.)

Children’s Literature

Clues

www.wiley.com/college/musser

See “Counting on Frank” by Rod

The Guess and Test strategy may be appropriate when

Clement.

r There is a limited number of possible answers to test.

r You want to gain a better understanding of the problem.

r You have a good idea of what the answer is.

r You can systematically try possible answers.

r Your choices have been narrowed down by the use of other strategies.

r There is no other obvious strategy to try.

Review the preceding three problems to see how these clues may have helped you

select the Guess and Test strategy to solve these problems.

Draw a Picture

Reflection from Research

Often problems involve physical situations. In these situations, drawing a picture can

Training children in the process of

help you better understand the problem so that you can formulate a plan to solve the

using pictures to solve problems

problem. As you proceed to solve the following “pizza” problem, see whether you

results in more improved prob-

can visualize the solution without looking at any pictures first. Then work through

lem-solving performance than

training students in any other

the given solution using pictures to see how helpful they can be.

strategy (Yancey, Thompson, &

Yancey, 1989).

Problem

Can you cut a pizza into 11 pieces with four straight cuts?

Step 1 Understand the Problem

Do the pieces have to be the same size and shape?

Step 2 Devise a Plan

An obvious beginning would be to draw a picture showing how a pizza is usually

cut and to count the pieces. If we do not get 11, we have to try something else

(Figure 1.8). Unfortunately, we get only eight pieces this way.

NCTM Standard

All students should describe,

extend, and make generalizations

about geometric and numeric

patterns.

Figure 1.8

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7/30/2013 2:36:17 PM - Section 1.1 The Problem-Solving Process and Strategies 11

Step 3 Carry Out the Plan

See Figure 1.9

Figure 1.9

Step 4 Look Back

Were you concerned about cutting equal pieces when you started? That is normal.

In the context of cutting a pizza, the focus is usually on trying to cut equal pieces

rather than the number of pieces. Suppose that circular cuts were allowed. Does

it matter whether the pizza is circular or is square? How many pieces can you get

with five straight cuts? n straight cuts?

Additional Problems Where the Strategy “Draw a Picture”

Is Useful

1. A tetromino is a shape made up of four squares where the squares must be

Figure 1.10

joined along an entire side (Figure 1.10). How many different tetromino shapes

are possible?

Step 1 Understand the Problem

The solution of this problem is easier if we make a set of pictures of all possible

arrangements of four squares of the same size.

Step 2 Devise a Plan

Let’s start with the longest and narrowest configuration and work toward the

most compact.

Step 3 Carry Out the Plan

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7/30/2013 2:36:21 PM - 12 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

Many similar problems can be posed using fewer or more squares. The problems

become much more complex as the number of squares increases. Also, new prob-

lems can be posed using patterns of equilateral triangles.

2. If you have a chain saw with a bar 18 inches long, determine whether a 16-foot

log, 8 inches in diameter, can be cut into 4-foot pieces by making only two cuts.

3. It takes 64 cubes to fi ll a cubical box that has no top. How many cubes are not

touching a side or the bottom?

Clues

The Draw a Picture strategy may be appropriate when

r A physical situation is involved.

r Geometric fi gures or measurements are involved.

r You want to gain a better understanding of the problem.

r A visual representation of the problem is possible.

Review the preceding three problems to see how these clues may have helped you

select the Draw a Picture strategy to solve these problems.

NCTM Standard

All students should represent

Use a Variable

the idea of a variable as an

unknown quantity using a letter

Observe how letters were used in place of numbers in the previous “sun + fun = swim”

or a symbol.

cryptarithm. Letters used in place of numbers are called variables or unknowns. The

Use a Variable strategy, which is one of the most useful problem-solving strategies, is

used extensively in algebra and in mathematics that involves algebra.

Problem

What is the greatest number that evenly divides the sum of any three consecutive

whole numbers?

Reflection from Research

By trying several examples, you might guess that 3 is the greatest such number.

Given the proper experiences,

However, it is necessary to use a variable to account for all possible instances of three

children as young as eight and

consecutive numbers.

nine years of age can learn

to comfortably use letters to

represent unknown values and

Step 1 Understand the Problem

can operate on representations

The whole numbers are 0, 1, 2, 3, . . . , so that consecutive whole numbers differ by

involving letters and numbers

while fully realizing that they

1. Thus an example of three consecutive whole numbers is the triple 3, 4, and 5.

did not know the values of the

The sum of three consecutive whole numbers has a factor of 3 if 3 multiplied by

unknowns (Carraher, Schliemann,

another whole number produces the given sum. In the example of 3, 4, and 5, the

Brizuela, & Earnest, 2006).

sum is 12 and 3 × 4 equals 12. Thus 3 + 4 + 5 has a factor of 3.

Step 2 Devise a Plan

Since we can use a variable, say x, to represent any whole number, we can repre-

sent every triple of consecutive whole numbers as follows: x, x

+ ,

1 x

+ .

2 Now we

can discover whether the sum has a factor of 3.

Algebraic Reasoning

In algebra, the letter “x ” is most

Step 3 Carry Out the Plan

commonly used for a variable.

However, any letter (even Greek

The sum of x, x + ,

1 and x + 2 is

letters, for example) can be used

as a variable.

x + (x + )

1 + (x + 2) = x

3 + 3 = 3(x + )

1 .

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7/30/2013 2:36:23 PM - Section 1.1 The Problem-Solving Process and Strategies 13

Thus x + (x + )

1 + (x + 2) is three times x + 1. Therefore, we have shown that the

sum of any three consecutive whole numbers has a factor of 3. The case of x = 0

shows that 3 is the greatest such number.

Step 4 Look Back

Is it also true that the sum of any five consecutive whole numbers has a factor of

5? Or, more generally, will the sum of any n consecutive whole numbers have a

factor of n? Can you think of any other generalizations?

Additional Problems Where the Strategy “Use a Variable”

Is Useful

1. Find the sum of the first 10, 100, and 500 counting numbers.

Step 1 Understand the Problem

Since counting numbers are the numbers 1, 2, 3, 4, . . . , the sum of the first 10 count-

ing numbers would be 1 + 2 + 3 + . . . + 8 + 9 + 10. Similarly, the sum of the first 100

counting numbers would be 1 + 2 + 3 + . . . + 98 + 99 + 100 and the sum of the first

500 counting numbers would be 1 + 2 + 3 + . . . + 498 + 499 + 500.

Step 2 Devise a Plan

Rather than solve three different problems, the “Use a Variable” strategy

can be used to find a general method for computing the sum in all three situa-

tions. Thus, the sum of the first n counting numbers would be expressed as

1 + 2 + 3 + . . . + (n − 2) + (n − 1) + .

n The sum of these numbers can be found by

noticing that the first number 1 added to the last number n is n + 1, which is the

same as (n − )

1 + 2 and (n − 2) + .

3 Adding all such pairs can be done by adding all

of the numbers twice.

Step 3 Carry Out the Plan

1

+ 2 +

3

+ ⋅⋅⋅ + (n − 2) + (n − 1) + n

Reflection from Research

When asked to create their own

+ n

+ (n − 1) + (n − 2) + ⋅⋅⋅ +

3

+ 2 + 1

problems, good problem solvers

(n + 1) + (n + 1) + (n + 1) + ⋅ ⋅ ⋅ + (n + 1) + (n + 1) + (n + 1)

generated problems that were

more mathematically complex

than those of less successful

= n • (n + )

1

problem solvers (Silver & Cai,

1996).

Since each number was added twice, the desired sum is obtained by dividing

n ¥ (n + )

1 by 2 which yields

n ¥ (n + 1)

1 + 2 + 3 + ⋅⋅⋅ + (n − 2) + (n − 1) + n =

2

The numbers 10, 100, and 500 can now replace the variable n to find our desired

solutions:

10 ¥ 10

( +1)

1 + 2 + 3 + ⋅⋅⋅ + 8 + 9 + 10 =

= 55

2

100 ¥ 101

( )

1 + 2 + 3 + ⋅⋅⋅ + 98 + 99 + 100 =

= 5050

2

500 ¥ 501

1 + 2 + 3 + ⋅⋅⋅ + 498 + 499 + 500 =

= 125,250

2

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7/30/2013 2:36:26 PM - 14 Chapter 1 Introduction to Problem Solving

Step 4 Look Back

Since the method for solving this problem is quite unique could it be used to solve

other similar looking problems like:

i. 3 + 6 + 9 + ⋅ ⋅ ⋅ + (3n − 6) + (3n − 3) + 3n

ii. 21 + 25 + 29 + ⋅ ⋅ ⋅ + 113 + 117 + 121

2. Show that the sum of any fi ve consecutive odd whole numbers has a factor of 5.

3. The measure of the largest angle of a triangle is nine times the measure

of the smallest angle. The measure of the third angle is equal to the difference of

the largest and the smallest. What are the measures of the angles? (Recall that

the sum of the measures of the angles in a triangle is 180°.)

Clues

The Use a Variable strategy may be appropriate when

r A phrase similar to “for any number” is present or implied.

r A problem suggests an equation.

r A proof or a general solution is required.

r A problem contains phrases such as “consecutive,” “even,” or “odd” whole

numbers.

r There is a large number of cases.

r There is an unknown quantity related to known quantities.

r There is an infi nite number of numbers involved.

r You are trying to develop a general formula.

Review the preceding three problems to see how these clues may have helped you

select the Use a Variable strategy to solve these problems.

Using Algebra to Solve Problems

To effectively employ the Use a Variable strategy, students need to have a clear

understanding of what a variable is and how to write and simplify equations contain-

ing variables. This subsection addresses these issues in an elementary introduction to

algebra. There will be an expanded treatment of solving equations and inequalities in

Chapter 9 after the real number system has been developed.

NCTM Standard

A common way to introduce the use of variables is to find a general formula

All students should develop an

for a pattern of numbers such as 3, 6, 9, . . . 3 .

n One of the challenges for students is

initial conceptual understanding

to see the role that each number plays in the expression. For example, the pattern

of different uses of variables.

5, 8, 11, . . . is similar to the previous pattern, but it is more difficult to see that each

term is two greater than a multiple of 3 and, thus, can be expressed in general as

3n + 2. Sometimes it is easier for students to use a variable to generalize a geometric

pattern such as the one shown in the following example. This type of example may

be used to introduce seventh-grade students to the concept of a variable. Following

are four typical student solutions.

Describe at least four different ways to count the dots in

Figure 1.11.

S O L U T I O N The obvious method of solution is to count the dots—there are 16.

Figure 1.11

Another student’s method is illustrated in Figure 1.12.

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7/30/2013 2:36:28 PM - Section 1.1 The Problem-Solving Process and Strategies 15

4 × 3 + 4

⎫

⎬ ⎯ →

⎯

4 × (5 − 2) + 4⎭

Figure 1.12

The student counts the number of interior dots on each side, 3, and multiplies by the

number of sides, 4, and then adds the dots in the corners, 4. This method generates the

expression 4 × 3 + 4 = 16. A second way to write this expression is 4 × (5 − 2) + 4 = 16

since the 3 interior dots can be determined by subtracting the two corners from the 5

dots on a side. Both of these methods are shown in Figure 1.12.

A third method is to count all of the dots on a side, 5, and multiply by the number

of sides. Four must then be subtracted because each corner has been counted twice,

once for each side it belongs to. This method is illustrated in Figure 1.13 and generates

the expression shown.

4 × 5 − 4 = 16} ⎯ →

⎯

Figure 1.13

Reflection from Research

In the two previous methods, either corner dots are not counted (so they

Sixth-grade students with no

must be added on) or they are counted twice (so they must be subtracted to avoid

formal instruction in algebra are

double counting). The following fourth method assigns each corner to only one

“generally able to solve prob-

side (Figure 1.14).

lems involving specific cases and

showed remarkable ability to

generalize the problem situations

and to write equations using vari-

ables. However they rarely used

their equations to solve related

4 × 4 = 16

⎫

problems” (Swafford & Langrall,

⎬ ⎯ →

⎯

4

2000).

× (5 − 1) = 16⎭

Figure 1.14

Thus, we encircle 4 dots on each side and multiply by the number of sides. This yields

the expression 4 × 4 = 16. Because the 4 dots on each side come from the 5 total dots

on a side minus 1 corner, this expression could also be written as 4 × (5 − 1) = 16 (see

Figure 1.14).

■

There are many different methods for counting the dots in the previous example

and each method has a geometric interpretation as well as a corresponding arithmetic

expression. Could these methods be generalized to 50, 100, 1000 or even n dots on a

side? The next example discusses how these generalizations can be viewed as well as

displays the generalized solutions of seventh-grade students.

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Suppose the square arrangement of dots in Example 1.1 had

n dots on each side. Write an algebraic expression that would

describe the total number of dots in such a figure (Figure 1.15).

S O L U T I O N It is easier to write a general expression for those in Example 1.1 when

you understand the origins of the numbers in each expression. In all such cases, there

will be 4 corners and 4 sides, so the values that represent corners and sides will stay

Figure 1.15

fi xed at 4. On the other hand, in Figure 1.15, any value that was determined based on

the number of dots on the side will have to refl ect the value of n. Thus, the expressions

Algebraic Reasoning

that represent the total number of dots on a square fi gure with n dots on a side are

Variable is a central concept in

generalized as shown next.

algebra. Students may struggle

with the idea that the letter x

represents many numbers in

4 × 3 + 4

⎫

y = 3x + 4 but only one or two

⎯ →

⎯ 4(n − 2) + 4

⎬

4 × (5 − 2) + 4

numbers in other situations. For

⎭

example, the solution of the

4 × 5 − 4 ⎯ →

⎯ 4n − 4

equation x2 = 9 consists of the

numbers 3 and −3 since each of

4 × 4

⎫

these numbers squared is 9. This

⎬ ⎯ →

⎯ 4(n − )

1

4 × (5 − )

1

■

means, that the equation is true

⎭

whenever x = 3 or x = −3.

Reflection from Research

Since each expression on the right represents the total number of dots in Figure 1.15,

The more an equation varies from

they are all equal to each other. Using properties of numbers and equations, each

the standard a + b = c format, the

equation can be rewritten as the same expression. Learning to simplify expressions

more difficult it is for students

and equations with variables is one of the most important processes in mathematics.

to work with. The most difficult

problems are those with opera-

Traditionally, this topic has represented a substantial portion of an entire course in

tions on both sides of the equal

introductory algebra. An equation is a sentence involving numbers, or symbols repre-

sign (Matthews, Rittle-Johnson,

senting numbers, where the verb is equals (=). There are various types of equations:

McEldoon, & Taylor, 2012).

3 + 4 = 7 True

equation

3 + 4 = 9 False

equation

2x + 5x = 7x Identity

equation

x + 4 = 9

Conditional

equation

A true or false equation needs no explanation, but an identity equation is always

true no matter what numerical value is used for x. A conditional equation is an equa-

tion that is only true for certain values of x. For example, the equation x + 4 = 9 is

true when x = 5, but false when x is any other value. In this chapter, we will restrict

the variables to only whole numbers. For a conditional equation, a value of the vari-

able that makes the equation true is called a solution. To solve an equation means to

find all of the solutions. The following example shows three different ways to solve

equations of the form ax + b = c.

Suppose the square arrangement of dots in Example 1.2 had 84

total dots (Figure 1.16). How many dots are there on each side?

← ⎯

⎯ 84 total dots

Reflection from Research

Figure 1.16

Even 6-year-olds can solve alge-

braic equations when they are

S O L U T I O N From Example 1.2, a square with n dots on a side has 4n − 4 total

rewritten as a story problem,

dots. Thus, we have the equation 4n − 4 = 84. Three elementary methods that can be

logic puzzle, or some other

problem with meaning (Femiano,

used to solve equations such as 4n − 4 = 84 are Guess and Test, Cover Up, and Work

2003).

Backward.

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7/30/2013 2:36:33 PM - Section 1.1 The Problem-Solving Process and Strategies 17

Guess and Test As the name of this method suggests, one guesses values for the

variable in the equation 4n − 4 = 84 and substitutes to see if a true equation results.

Try n = 10: 4 1

( 0) − 4 = 36 ≠ 84

Try n = 25: 4(25) − 4 = 96 ≠ 84

Try n = 22: 4(22) − 4 = 84. Therefore, 22 is the solution of the equation.

Cover Up In this method, we cover up the term with the variable:

h − 4 = 84. To make a true

,

equation the h

must be 88. Thus 4n = 88

Since 4 • 22 = 88, w

e have n = 22.

Work Backward The left side of the equation shows that n is multiplied by 4

and then 4 is subtracted to obtain 84. Thus, working backward, if we add 4 to 84

and divide by 4, we reach the value of n. Here 84 + 4 = 88 and 88 ÷ 4 = 22 so n = 22

Figure 1.17

(Figure 1.17).

■

Using variables in equations and manipulating them are what most people see as

“algebra.” However, algebra is much more than manipulating variables—it includes

the reasoning that underlies those manipulations. In fact, many students solve

algebra-like problems without equations and don’t realize that their reasoning is

algebraic. For example, the Work Backward solution in Example 1.3 can all be done

without really thinking about the variable n. If one just thinks “4 times something

minus 4 is 84,” he can work backward to find the solution. However, the thinking

that is used in the Work Backward method can be mirrored in equations as follows:

4n − 4 = 84

4n = 84 + 4

4n = 88

n = 88 ÷ 4

n = 22

Reflection from Research

Because so much algebra can be done with intuitive reasoning, it is important to help

We are proposing that the

students realize when they are reasoning algebraically. One way to better understand the

teaching and learning of

underlying principles of algebraic reasoning is to look at the Algebraic Reasoning Web

arithmetic be conceived as part

Module on our Web site: www.wiley.com/college/musser/

of the foundation of learning

algebra, not that algebra be

Some researchers say that arithmetic is the foundation of learning algebra.

conceived only as an extension of

Arithmetic typically means computation with different kinds of numbers and the

arithmetic procedures (Carpenter,

underlying properties that make the computation work. Much of what is discussed

Levi, Berman, & Pligge, 2005).

in Chapters 2–9 is about arithmetic and thus contains key parts of the foundation of

algebra. To help you see algebraic ideas in the arithmetic that we study, there will be

places throughout those chapters where these foundational ideas of algebra are called

out in an “Algebraic Reasoning” margin note.

We address algebra in more depth when we talk about solving equations, relations,

and functions in Chapter 9 and then again in Chapter 15 when algebraic ideas are

applied to geometry.

There is a story about Sir Isaac Newton, coinventor of the calculus, who, as

a youngster, was sent out to cut a hole in the barn door for the cats to go

in and out. With great pride he admitted to cutting two holes, a larger one

for the cat and a smaller one for the kittens.

©Ron Bagwell

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7/30/2013 2:36:35 PM - 18 Chapter 1 Introduction to Problem Solving

PROBLEM SET A

1. a. If the diagonals of a square are drawn in, how many

11. Find a set of consecutive counting numbers whose sum

triangles of all sizes are formed?

is each of the following. Each set may consist of 2, 3, 4,

b. Describe how Pólya’s four steps were used to solve

5, or 6 consecutive integers. Use the spreadsheet activity

part a.

Consecutive Integer Sum on our Web site to assist you.

a. 84 b. 213 c. 154

2. Scott and Greg were asked to add two whole numbers.

Instead, Scott subtracted the two numbers and got 10, and

12. Place the digits 1 through 9 so that you can count from 1

Greg multiplied them and got 651. What was the correct

to 9 by following the arrows in the diagram.

sum?

3. The distance around a standard tennis court is 228 feet. If

the length of the court is 6 feet more than twice the width,

find the dimensions of the tennis court.

4. A multiple of 11 I be,

not odd, but even, you see.

My digits, a pair,

when multiplied there,

make a cube and a square

out of me. Who am I?

13. Using a 5-minute and an 8-minute hourglass timer, how

can you measure 1 minute?

5. Show how 9 can be expressed as the sum of two consecutive

numbers. Then decide whether every odd number can be

14. Using the numbers 9, 8, 7, 6, 5, and 4 once each, find the

expressed as the sum of two consecutive counting numbers.

following:

Explain your reasoning.

a. The largest possible sum:

6. Using the symbols +, ,

− ×, and ÷, fill in the following three

blanks to make a true equation. (A symbol may

be used more than once.)

6 6 6 6 = 13

b. The smallest possible (positive) difference:

7. In the accompanying figure (called an arithmogon), the

number that appears in a square is the sum of the numbers

in the circles on each side of it. Determine what numbers

belong in the circles.

15. Using the numbers 1 through 8, place them in the follow-

ing eight squares so that no two consecutive numbers are in

touching squares (touching includes entire sides or simply

one point).

8. Place 10 stools along four walls of a room so that each of

the four walls has the same number of stools.

9. Susan has 10 pockets and 44 dollar bills. She wants to

arrange the money so that there are a different number of

16. Solve this cryptarithm, where each letter represents a digit

dollars in each pocket. Can she do it? Explain.

and no digit represents two different letters:

USSR

10. Arrange the numbers 2, 3, . . . ,10 in the accompanying tri-

angle so that each side sums to 21.

+ USA

PEACE

17. On a balance scale, two spools and one thimble balance

eight buttons. Also, one spool balances one thimble and

one button. How many buttons will balance one spool?

18. Place the numbers 1 through 8 in the circles on the vertices

of the accompanying cube so that the difference of any

two connecting circles is greater than 1.

c01.indd 18

7/30/2013 2:36:38 PM - Section 1.1 The Problem-Solving Process and Strategies 19

23. Using the Chapter 1 eManipulative activity Circle 21

on our Web site, find an arrangement of the numbers 1

through 14 in the 7 circles below so that the sum of the

three numbers in each circle is 21.

19. Think of a number. Add 10. Multiply by 4. Add 200. Divide

by 4. Subtract your original number. Your result should be

60. Why? Show why this would work for any number.

20. The digits 1 through 9 can be used in decreasing

order, with + and − signs, to produce 100 as shown:

98 − 76 + 54 + 3 + 21 = 100. Find two other such combina-

tions that will produce 100.

21. The Indian mathematician Ramanujan observed that the

taxi number 1729 was very interesting because it was the

smallest counting number that could be expressed as the

24. The hexagon below has a total of 126 dots and an equal

sum of cubes in two different ways. Find a, b

, c

, and d

number of dots on each side. How many dots are on

such that a3 + b3 = 1729 and c3 + d 3 = 1729.

each side?

22. Using the Chapter 1 eManipulative activity Number

Puzzles, Exercise 2 on our Web site, arrange the numbers

1, 2, 3, 4, 5, 6, 7, 8, 9 in the following circles so the sum of

the numbers along each line of four is 23.

PROBLEM SET B

1. Find the largest eight-digit number made up of the digits 1,

6. Place numbers 1 through 19 into the 19 circles below so

1, 2, 2, 3, 3, 4, and 4 such that the 1s are separated by one

that any three numbers in a line through the center will

digit, the 2s by two digits, the 3s by three digits, and the 4s

give the same sum.

by four digits.

2. Think of a number. Multiply by 5. Add 8. Multiply by 4.

Add 9. Multiply by 5. Subtract 105. Divide by 100.

Subtract 1. How does your result compare with your

original number? Explain.

3. Carol bought some items at a variety store. All the items

were the same price, and she bought as many items as the

price of each item in cents. (For example, if the items cost

10 cents, she would have bought 10 of them.) Her bill was

$2.25. How many items did Carol buy?

4. You can make one square with four toothpicks. Show how

you can make two squares with seven toothpicks (breaking

toothpicks is not allowed), three squares with 10 tooth-

7. Using three of the symbols +, ,

− ×, and ÷ once each, fill

picks, and five squares with 12 toothpicks.

in the following three blanks to make a true equation.

5. A textbook is opened and the product of the page numbers

(Parentheses are allowed.)

of the two facing pages is 6162. What are the numbers of

the pages?

6 6 6 6 = 66

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7/30/2013 2:36:41 PM - 20 Chapter 1 Introduction to Problem Solving

8. A water main for a street is being laid using a particu-

18. Together, a baseball and a football weigh 1.25 pounds,

lar kind of pipe that comes in either 18-foot sections or

the baseball and a soccer ball weigh 1.35 pounds, and the

20-foot sections. The designer has determined that the

football and the soccer ball weigh 1.9 pounds. How much

water main would require 14 fewer sections of 20-foot

does each of the balls weigh?

pipe than if 18-foot sections were used. Find the total

19. Pick any two consecutive numbers. Add them. Then add 9

length of the water main.

to the sum. Divide by 2. Subtract the smaller of the original

9. Mike said that when he opened his book, the product

numbers from the answer. What did you get? Repeat this

of the page numbers of the two facing pages was 7007.

process with two other consecutive numbers. Make a con-

Without performing any calculations, prove that he was

jecture (educated guess) about the answer, and prove it.

wrong.

20. An additive magic square has the same sum in each row,

10. The Smiths were about to start on an 18,000-mile automo-

column, and diagonal. Find the error in this magic square

bile trip. They had their tires checked and found that each

and correct it.

was good for only 12,000 miles. What is the smallest num-

ber of spares that they will need to take along with them to

47

56

34

22

83

7

24

67

44

26

13

75

make the trip without having to buy a new tire?

29

52

3

99

18

48

11. What is the maximum number of pieces of pizza that can

17

49

89

4

53

37

97

6

3

11

74

28

result from 4 straight cuts?

35

19

46

87

8

54

12. Given: Six arrows arranged as follows:

21. Two points are placed on the same side of a square.

↑ ↑ ↑ ↓ ↓ ↓

A segment is drawn from each of these points to each of

the 2 vertices (corners) on the opposite side of the square.

Goal: By inverting two adjacent arrows at a time, rear-

How many triangles of all sizes are formed?

range to the following:

↑ ↓ ↑ ↓ ↑ ↓

22. Using the triangle in Problem 10 in Set A, determine

whether you can make similar triangles using the digits

Can you find a minimum number of moves?

1, 2, . . . , 9, where the side sums are 18, 19, 20, 21, and 22.

13. Two friends are shopping together when they encounter

23. Using the Chapter 1 eManipulative activity, Number

a special “3 for 2” shoe sale. If they purchase two pairs

Puzzles, Exercise 4 on our Web site, arrange the numbers

of shoes at the regular price, a third pair (of lower or

1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles below so the sum of the

equal value) will be free. Neither friend wants three pairs

numbers along each line of four is 20.

of shoes, but Pat would like to buy a $56 and a $39 pair

while Chris is interested in a $45 pair. If they buy the

shoes together to take advantage of the sale, what is the

fairest share for each to pay?

14. Find digits A, B, C, and D that solve the following

cryptarithm.

ABCD

×

4

24. Using the Chapter 1 eManipulative activity Circle 99

on our Web site, find an arrangement of the numbers

DCBA

provided in the 7 circles below so that the sum of the

three numbers in each circle is 99.

15. If possible, find an odd number that can be expressed as

the sum of four consecutive counting numbers. If impos-

sible, explain why.

16. Five friends were sitting on one side of a table. Gary sat

next to Bill. Mike sat next to Tom. Howard sat in the

third seat from Bill. Gary sat in the third seat from Mike.

Who sat on the other side of Tom?

17. In the following square array on the left, the corner

numbers were given and the boldface numbers were

found by adding the adjacent corner numbers. Following

the same rules, find the corner numbers for the other

square array.

6

19

13

10

25. An arrangement of dots forms the perimeter of an equi-

lateral triangle. There are 87 evenly spaced dots on each

8

14

15

11

side including the dots at the vertices. How many dots are

2

3

1

16

there altogether?

c01.indd 20

7/30/2013 2:36:42 PM - Section 1.2 Three Additional Strategies 21

y

30. When Damian, a second grader, was asked to solve

26. The equation

+ 12 = 23 can be solved by subtracting

5

problems like

+ 3 = 5, he said that he had seen his

y

12 from both sides of the equation to yield

+ 12 − 12 =

older sister working on problems like x + 3 = 5 and

5

wondered if these equations were different. How would

y

23 − 12. Similarly, the resulting equation = 11 can be

you respond?

5

solved by multiplying both sides of the equation by 5 to

31. After the class had found three consecutive odd numbers

obtain y = 55. Explain how this process is related to the

whose sum is 99, Byron tried to find three consecutive

Work Backward method described in Example 1.3.

odd numbers that would add to 96. He said he was

struggling to find a solution. How could you help him

Analyzing Student Thinking

understand the solution to this problem?

27. When the class was asked to solve the equation 3x − 8 = 27,

32. Consider the following problem:

Wesley asked if he could use guess and test. How would

you respond?

The amount of fencing needed to enclose a rectangu-

lar field was 92 yards and the length of the field was 3

28. Rosemary said that she felt the ‘guess and test’ method was

times as long as the width. What were the dimensions

a waste of time; she just wanted to get an answer. What

of the field?

could you tell her about the value of using guess and test?

Vance solved this problem by drawing a picture and

29. Even though you have taught your students how to ‘draw

using guess and test. Jolie set up an equation with x being

a picture’ to solve a problem, Cecelia asks if she has to

the width of the field and solved it. They got the same

draw a picture because she can solve the problems in class

answer but asked you which method was better. How

without it. How would you respond?

would you respond?

THREE ADDITIONAL STRATEGIES

Solve the problem below using Pólya’s four steps and any

strategy. Describe how you used the four steps, focusing on

any new insights that you gained as a result of looking back.

How many rectangles of all shapes and sizes are in the figure at the right?

NCTM Standard

All students should represent,

Look for a Pattern

analyze, and generalize a variety

of patterns with tables, graphs,

When using the Look for a Pattern strategy, one usually lists several specific instances

words, and, when possible, sym-

of a problem and then looks to see whether a pattern emerges that suggests a solu-

bolic rules.

tion to the entire problem. For example, consider the sums produced by adding

consecutive odd numbers starting with 1:1,1 + 3 = 4 (= 2 × 2), 1 + 3 + 5 = 9 (= 3 × 3),

Common Core – Grades

1 + 3 + 5 + 7 = 16 (= 4 × 4), 1 + 3 + 5 + 7 + 9 = 25 (= 5 × 5), and so on. Based on the pat-

K-12 (Mathematical

tern generated by these five examples, one might expect that such a sum will always

Practice 7)

be a perfect square.

Mathematically proficient stu-

dents look closely to discern a

The justification of this pattern is suggested by the following figure.

pattern or structure. Young stu-

dents, for example, might notice

that three and seven more is the

same amount as seven and three

more, or they may sort a collec-

tion of shapes according to how

many sides the shapes have.

Algebraic Reasoning

Recognizing and extending pat-

terns is a common practice in

algebra. Extending patterns to

Each consecutive odd number of dots can be added to the previous square arrange-

the most general case is a natural

place to discuss variables.

ment to form another square. Thus, the sum of the first n odd numbers is n2.

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7/30/2013 2:36:45 PM - 22

Chapter 1 Introduction to Problem Solving

Generalizing patterns, however, must be done with caution because with a

sequence of only 3 or 4 numbers, a case could be made for more than one pattern.

For example, consider the sequence 1, 2, 4, . . . . What are the next 4 numbers in the

sequence? It can be seen that 1 is doubled to get 2 and 2 is doubled to get 4. Following

that pattern, the next four numbers would be 8, 16, 32, 64. If, however, it is noted that

the difference between the first and second term is 1 and the difference between the

second and third term is 2, then a case could be made that the difference is increasing

by one. Thus, the next four terms would be 7, 11, 16, 22. Another case could be made

for the differences alternating between 1 and 2. In that case, the next four terms would

be 5, 7, 8, 10. Thus, from the initial three numbers of 1, 2, 4, at least three different

patterns are possible:

1, 2, 4, 8, 16, 32, 64, . . .

Doubling

1, 2, 4, 7, 11, 16, 22, . . .

Difference increasing by 1

1, 2, 4, 5, 7, 8, 10, . . .

Difference alternating between 1 and 2

Figure 1.18

Problem

How many different downward paths are there from A to B in the grid in Figure 1.18?

A path must travel on the lines.

Step 1 Understand the Problem

What do we mean by different and downward? Figure 1.19 illustrates two paths.

Notice that each such path will be 6 units long. Different means that they are not

exactly the same; that is, some part or parts are different.

Step 2 Devise a Plan

Let’s look at each point of intersection in the grid and see how many different

Figure 1.19

ways we can get to each point. Then perhaps we will notice a pattern (Figure

1.20). For example, there is only one way to reach each of the points on the two

outside edges; there are two ways to reach the middle point in the row of points

labeled 1, 2, 1; and so on. Observe that the point labeled 2 in Figure 1.20 can be

found by adding the two 1s above it.

Step 3 Carry Out the Plan

To see how many paths there are to any point, observe that you need only add the

number of paths required to arrive at the point or points immediately above. To

reach a point beneath the pair 1 and 2, the paths to 1 and 2 are extended down-

Figure 1.20

ward, resulting in 1 + 2 = 3 paths to that point. The resulting number pattern is

shown in Figure 1.21. Notice, for example, that 4 + 6 = 10 and 20 + 15 = 35. (This

pattern is part of what is called Pascal’s triangle. It is used again in Chapter 11.)

The surrounded portion of this pattern applies to the given problem; thus the

answer to the problem is 20.

Step 4 Look Back

Can you see how to solve a similar problem involving a larger square array, say a

Figure 1.21

4 × 4 grid? How about a 10 × 10 grid? How about a rectangular grid?

A pattern of numbers arranged in a particular order is called a number sequence,

Children’s Literature

and the individual numbers in the sequence are called terms of the sequence. The

www.wiley.com/college/musser

counting numbers, 1, 2, 3, 4, . . . , give rise to many sequences. (An ellipsis, the three

See “There Was an Old Lady

Who Swallowed a Fly” by Simms

periods after the 4, means “and so on.”) Several sequences of counting numbers

Taback.

follow.

c01.indd 22

7/30/2013 2:36:47 PM - Section 1.2 Three Additional Strategies 23

SEQUENCE

NAME

2, 4, 6, 8, . . .

The even (counting) numbers

1, 3, 5, 7, . . .

The odd (counting) numbers

1, 4, 9, 16, . . .

The square (counting) numbers

1, 3, 32 33

, , . . .

The powers of three

1, 1, 2, 3, 5, 8, . . .

The Fibonacci sequence (after the two 1s, each

term is the sum of the two preceding terms)

NCTM Standard

Inductive reasoning is used to draw conclusions or make predictions about a large

All students should analyze how

collection of objects or numbers, based on a small representative subcollection. For

both repeating and growing pat-

example, inductive reasoning can be used to find the ones digit of the 400th term of

terns are generated.

the sequence 8, 12, 16, 20, 24, . . . By continuing this sequence for a few more terms,

8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, . . . , one can observe that the ones digit

of every fifth term starting with the term 24 is a four. Thus, the ones digit of the 400th

term must be a four.

Reflection from Research

Additional Problems Where the Strategy “Look for a

In classrooms where problem

Pattern” Is Useful

solving is valued and teachers

have knowledge of children’s

1. Find the ones digit in 399.

mathematical thinking, children

see mathematics as a problem-

solving endeavor in which com-

Step 1 Understand the Problem

municating mathematical thinking

The number 399 is the product of 99 threes. Using the exponent key on one type of

is important (Franke & Carey,

1997).

scientific calculator yields the result 1 71792506547

.

. This shows the first digit, but

not the ones (last) digit, since the 47 indicates that there are 47 places to the right

of the decimal. (See the discussion on scientific notation in Chapter 4 for further

explanation.) Therefore, we will need to use another method.

Step 2 Devise a Plan

Consider 31 32 33 34 35 36 37 38

,

,

,

,

,

,

,

. Perhaps the ones digits of these numbers

form a pattern that can be used to predict the ones digit of 399.

Step 3 Carry Out the Plan

31

3, 32

9, 33

27, 34

81, 35

243, 36

729, 37

2187, 38

=

=

=

=

=

=

=

= 6561. The ones

digits form the sequence 3, 9, 7, 1, 3, 9, 7, 1. Whenever the exponent of the 3 has a

factor of 4, the ones digit is a 1. Since 100 has a factor of 4, 3100 must have a ones

digit of 1. Therefore, the ones digit of 399 must be 7, since 399 precedes 3100 and 7

precedes 1 in the sequence 3, 9, 7, 1.

Step 4 Look Back

Ones digits of other numbers involving exponents might be found in a similar

fashion. Check this for several of the numbers from 4 to 9.

2. Which whole numbers, from 1 to 50, have an odd number of factors? For example,

15 has 1, 3, 5, and 15 as factors, and hence has an even number of factors: four.

3. In the next diagram, the left “H”-shaped array is called the 32-H and the right

array is the 58-H.

a. Find the sums of the numbers in the 32-H. Do the same for the 58-H and the

74-H. What do you observe?

b. Find an H whose sum is 497.

c. Can you predict the sum in any H if you know the middle number? Explain.

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7/30/2013 2:36:49 PM - 24 Chapter 1 Introduction to Problem Solving

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

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68

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71

72

73

74

75

76

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78

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80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

Children’s Literature

Clues

www.wiley.com/college/musser

The Look for a Pattern strategy may be appropriate when

See “Math Appeal” by Greg

Tang.

r A list of data is given.

r A sequence of numbers is involved.

r Listing special cases helps you deal with complex problems.

r You are asked to make a prediction or generalization.

r Information can be expressed and viewed in an organized manner, such as in a table.

Review the preceding three problems to see how these clues may have helped you

select the Look for a Pattern strategy to solve these problems.

Make a List

The Make a List strategy is often combined with the Look for a Pattern strategy to

suggest a solution to a problem. For example, here is a list of all the squares of the

numbers 1 to 20 with their ones digits in boldface.

1,

4,

9,

16,

25,

36,

49,

64,

81,

100,

Reflection from Research

1 1

2 ,

1 4

4 ,

1 9

6 ,

1 6

9 ,

2 5

2 ,

2 6

5 ,

2 9

8 ,

324,

361,

0

40

Problem-solving abililty develops

with age, but the relative dif-

The pattern in this list can be used to see that the ones digits of squares must be one of

ficulty inherent in each problem

0, 1, 4, 5, 6, or 9. This list suggests that a perfect square can never end in a 2, 3, 7, or 8.

is grade independent (Christou &

Philippou, 1998).

Problem

NCTM Standard

The number 10 can be expressed as the sum of four odd numbers in three ways: (i)

Instructional programs should

10 = 7 + 1 + 1 + 1, (ii) 10 = 5 + 3 + 1 + 1, and (iii) 10 = 3 + 3 + 3 + 1. In how many ways

enable all students to build new

can 20 be expressed as the sum of eight odd numbers?

mathematical knowledge through

problem solving.

Step 1 Understand the Problem

Recall that the odd numbers are the numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, . . . Using

the fact that 10 can be expressed as the sum of four odd numbers, we can form

various combinations of those sums to obtain eight odd numbers whose sum

is 20. But does this account for all possibilities?

Step 2 Devise a Plan

Instead, let’s make a list starting with the largest possible odd number in the sum

and work our way down to the smallest.

c01.indd 24

7/30/2013 2:36:49 PM - Section 1.2 Three Additional Strategies 25

Step 3 Carry Out the Plan

20 = 13 + 1 + 1 + 1 + 1 + 1 + 1 + 1

20 = 11 + 3 + 1 + 1 + 1 + 1 + 1 + 1

20 = 9 + 5 + 1 + 1 + 1 + 1 + 1 + 1

20 = 9 + 3 + 3 + 1 + 1 + 1 + 1 + 1

20 = 7 + 7 + 1 + 1 + 1 + 1 + 1 + 1

20 = 7 + 5 + 3 + 1 + 1 + 1 + 1 + 1

20 = 7 + 3 + 3 + 3 + 1 + 1 + 1 + 1

20 = 5 + 5 + 5 + 1 + 1 + 1 + 1 + 1

20 = 5 + 5 + 3 + 3 + 1 + 1 + 1 + 1

20 = 5 + 3 + 3 + 3 + 3 + 1 + 1 + 1

20 = 3 + 3 + 3 + 3 + 3 + 3 + 1 + 1

Reflection from Research

Step 4 Look Back

Correct answers are not a safe

indicator of good thinking.

Could you have used the three sums to 10 to help find these 11 sums to 20? Can

Teachers must examine more

you think of similar problems to solve? For example, an easier one would be to

than answers and must demand

express 8 as the sum of four odd numbers, and a more difficult one would be to

from students more than answers

express 40 as the sum of 16 odd numbers. We could also consider sums of even

(Sowder, Threadgill-Sowder,

numbers, expressing 20 as the sum of six even numbers.

Moyer, & Moyer, 1983).

Additional Problems Where the Strategy “Make a List” Is

Useful

1. In a dart game, three darts are thrown. All hit the target (Figure 1.22). What scores

are possible?

Step 1 Understand the Problem

Assume that all three darts hit the board. Since there are four different numbers

on the board, namely, 0, 1, 4, and 16, three of these numbers, with repetitions

allowed, must be hit.

Step 2 Devise a Plan

We should make a systematic list by beginning with the smallest (or largest) pos-

Figure 1.22

sible sum. In this way we will be more likely to find all sums.

Step 3 Carry Out the Plan

0 + 0 + 0 = 0 0 + 0 + 1 = 1, 0 + 1 + 1 = 2,

1 + 1 + 1 = 3, 0 + 0 + 4 = 4, 0 + 1 + 4 = 5,

1 + 1 + 4 = 6, 0 + 4 + 4 = 8, 1

+ 4 + 4 = 9,

4 + 4 + 4 = 12,

. .

.

, 16

+ 16 + 16 = 48

Step 4 Look Back

Several similar problems could be posed by changing the numbers on the dart-

board, the number of rings, or the number of darts. Also, using geometric prob-

ability, one could ask how to design and label such a game to make it a fair skill

game. That is, what points should be assigned to the various regions to reward

one fairly for hitting that region?

c01.indd 25

7/30/2013 2:36:50 PM - 26 Chapter 1 Introduction to Problem Solving

2. How many squares, of all sizes, are there on an 8 × 8 checkerboard? (See Figure

1.23; the sides of the squares are on the lines.)

3. It takes 1230 numerical characters to number the pages of a book. How many

pages does the book contain?

Clues

Figure 1.23

The Make a List strategy may be appropriate when

r Information can easily be organized and presented.

r Data can easily be generated.

r Listing the results obtained by using Guess and Test.

r Asked “in how many ways” something can be done.

r Trying to learn about a collection of numbers generated by a rule or formula.

Review the preceding three problems to see how these clues may have helped you

select the Make a List strategy to solve these problems.

The problem-solving strategy illustrated next could have been employed in con-

junction with the Make a List strategy in the preceding problem.

Solve a Simpler Problem

Like the Make a List strategy, the Solve a Simpler Problem strategy is frequently used

in conjunction with the Look for a Pattern strategy. The Solve a Simpler Problem

strategy involves reducing the size of the problem at hand and making it more man-

ageable to solve. The simpler problem is then generalized to the original problem.

Problem

In a group of nine coins, eight weigh the same and the ninth is heavier. Assume

that the coins are identical in appearance. Using a pan balance, what is the smallest

number of balancings needed to identify the heavy coin?

Step 1 Understand the Problem

Coins may be placed on both pans. If one side of the balance is lower than the

other, that side contains the heavier coin. If a coin is placed in each pan and the

pans balance, the heavier coin is in the remaining seven. We could continue in this

way, but if we missed the heavier coin each time we tried two more coins, the last

coin would be the heavy one. This would require four balancings. Can we fi nd the

heavier coin in fewer balancings?

Step 2 Devise a Plan

To fi nd a more effi cient method, let’s examine the cases of three coins and fi ve

coins before moving to the case of nine coins.

Step 3 Carry Out the Plan

Figure 1.24

Three coins: Put one coin on each pan (Figure 1.24). If the pans balance, the third

coin is the heavier one. If they don’t, the one in the lower pan is the heavier one.

Thus, it only takes one balancing to fi nd the heavier coin.

Five coins: Put two coins on each pan (Figure 1.25). If the pans balance, the fi fth

coin is the heavier one. If they don’t, the heavier one is in the lower pan. Remove

the two coins in the higher pan and put one of the two coins in the lower pan on

the other pan. In this case, the lower pan will have the heavier coin. Thus, it takes

Figure 1.25

at most two balancings to fi nd the heavier coin.

c01.indd 26

7/30/2013 2:36:50 PM - Section 1.2 Three Additional Strategies 27

Nine coins: At this point, patterns should have been identified that will make this

solution easier. In the three-coin problem, it was seen that a heavy coin can be

found in a group of three as easily as it can in a group of two. From the five-coin

problem, we know that by balancing groups of coins together, we could quickly

Figure 1.26

reduce the number of coins that needed to be examined. These ideas are combined

in the nine-coin problem by breaking the nine coins into three groups of three and

balancing two groups against each other (Figure 1.26). In this first balancing, the

group with the heavy coin is identified. Once the heavy coin has been narrowed to

three choices, then the three-coin balancing described above can be used.

The minimum number of balancings needed to locate the heavy coin out of a set

of nine coins is two.

Step 4 Look Back

In solving this problem by using simpler problems, no numerical patterns

emerged. However, patterns in the balancing process that could be repeated with

a larger number of coins did emerge.

Additional Problems Where the Strategy “Solve a Simpler

Problem” Is Useful

1

1

1

1

1. Find the sum +

+

+ ⋅⋅⋅ +

.

2

22

23

210

Step 1 Understand the Problem

This problem can be solved directly by getting a common denominator, here 210,

and finding the sum of the numerators.

Step 2 Devise a Plan

Instead of doing a direct calculation, let’s combine some previous strategies.

Namely, make a list of the first few sums and look for a pattern.

Step 3 Carry Out the Plan

1

1

1

3

1

1

1

7

1

1

1

1

15

,

+ = ,

+ + = ,

+ + +

=

2

2

4

4

2

4

8

8

2

4

8

16

16

1 3 7 15

The pattern of sums, ,

,

,

, suggests that the sum of the ten fractions is

10

2 4 8 16

2 − 1

1023

or

.

210,

1024

Algebraic Reasoning

Step 4 Look Back

Using a variable, the sum to the

right can be expressed more

This method of combining the strategy of Solve a Simpler Problem with Make a List

generally as follows:

1

1

1

and Look for a Pattern is very useful. For example, what is the sum +

+ ⋅⋅⋅ +

?

1

1

1

2n

1

2

22

2100

+

+ ⋅ ⋅ ⋅ +

.

n =

−

2

22

2

2n

Because of the large denominators, you wouldn’t want to add these fractions directly.

2. Following the arrows in Figure 1.27, how many paths are there from A to B?

Figure 1.27

3. There are 20 people at a party. If each person shakes hands with each other per-

son, how many handshakes will there be?

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Clues

The Solve a Simpler Problem strategy may be appropriate when

r The problem involves complicated computations.

r The problem involves very large or very small numbers.

r A direct solution is too complex.

r You want to gain a better understanding of the problem.

r The problem involves a large array or diagram.

Review the preceding three problems to see how these clues may have helped you

select the Solve a Simpler Problem strategy to solve these problems.

Solve the next problem and pay particular attention to the Devise a Plan step. Which

strategy or strategies did you use? If you used more than one strategy, how were they

used in conjunction with each other?

In the figure below, there are chairs placed around the hexagonal tables. If 27 hexagonal tables were placed in

a similar arrangement, how many chairs would it accommodate?

Combining Strategies to Solve Problems

Reflection from Research

As shown in the previous four-step solution, it is often useful to employ several strate-

The development of a disposition

gies to solve a problem. For example, in Section 1.1, a pizza problem similar to the

toward realistic mathematical

following was posed: What is the maximum number of pieces you can cut a pizza into

modeling and interpreting of

using four straight cuts? This question can be extended to the more general question:

word problems should permeate

the entire curriculum from the

What is the maximum number of pieces you can cut a pizza into using n straight cuts?

outset (Verschaffel & DeCorte,

To answer this, consider the sequence in Figure 1.9: 1, 2, 4, 7, 11. To identify patterns

1997).

in a sequence, observing how successive terms are related can be helpful. In this case,

the second term of 2 can be obtained from the first term of 1 by either adding 1 or

multiplying by 2. The third term, 4, can be obtained from the second term, 2, by add-

ing 2 or multiplying by 2. Although multiplying by 2 appears to be a pattern, it fails

Reflection from Research

as we move from the third term to the fourth term. The fourth term can be found by

Young children, who have had

adding 3 to the third term. Thus, the sequence appears to be the following:

little experience with standard

mathematical problems, can be

taught to search for more than

one solution to a problem and to

employ more than one method

to solve that problem (Tsamir,

Tirosh, Tabach, & Levenson,

Extending the difference sequence, we obtain the following

2010).

Algebraic Reasoning

In order to generalize patterns,

one must first identify what is

staying the same (the initial 1) and

Starting with 1 in the sequence, dropping down to the difference line, then back up to

what is changing (the difference

the number in the sequence line, we find the following:

is increasing by one) with each

step. Secondly, one must notice

1st Term: 1 = 1

the relationship between what is

changing and the step number.

2nd Term: 2 = 1 + 1

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7/30/2013 2:36:53 PM - Section 1.2 Three Additional Strategies 29

3rd Term: 4 = 1 + 1

( + 2)

4th Term: 7 = 1 + 1

( + 2 + 3)

5th Term: 11 = 1 + 1

( + 2 + 3 + 4), and so forth

n(n 1)

Recall that earlier we saw that 1 + 2 + 3 + ⋅⋅⋅ + =

+

n

. Thus, the nth term in the

2

(n 1) n

sequence is 1 + 1 + 2 + ⋅⋅⋅ + n − 1 = 1 +

−

[

(

)]

. Notice that as a check, the eighth

2

7 • 8

term in the sequence is 1 +

= 1 + 28 = 29. Hence, to solve the original problem, we

2

used Draw a Picture, Look for a Pattern, and Use a Variable.

It may be that a pattern does not become obvious after one set of differences. Consider

the following problem where several differences are required to expose the pattern.

Problem

If 10 points are placed on a circle and each pair of points is connected with a segment,

what is the maximum number of regions created by these segments?

Step 1 Understand the Problem

This problem can be better understood by drawing a picture. Since drawing 10

points and all of the joining segments may be overwhelming, looking at a simpler

problem of circles with 1, 2, or 3 points on them may help in further understand-

ing the problem. The first three cases are in Figure 1.28.

Figure 1.28

Step 2 Devise a Plan

So far, the number of regions are 1, 2, and 4 respectively. Here, again, this could

be the start of the pattern, 1, 2, 4, 8, 16, . . . . Let’s draw three more pictures to see if

this is the case. Then the pattern can be generalized to the case of 10 points.

Step 3 Carry Out the Plan

The next three cases are shown in Figure 1.29.

Figure 1.29

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Chapter 1 Introduction to Problem Solving

Making a list of the number of points on the circle and the corresponding number

of regions will help us see the pattern.

Points

1

2

3

4

5

6

Regions

1

2

4

8

16

31

While the pattern for the first five cases makes it appear as if the number of

regions are just doubling with each additional point, the 31 regions with 6 points

ruins this pattern. Consider the differences between the numbers in the pattern

and look for a pattern in the differences.

Because the first, second, and third difference did not indicate a clear pattern, the

fourth difference was computed and revealed a pattern of all ones. This observa-

tion is used to extend the pattern by adding four 1s to the two 1s in the fourth dif-

ference to make a sequence of six 1s. Then we work up until the Regions sequence

has ten numbers as shown next.

By finding successive differences, it can be seen that the solution to our problem is

256 regions.

Step 4 Look Back

By using a combination of Draw a Picture, Solve a Simpler Problem, Make a

List, and Look for a Pattern, the solution was found. It can also be seen that it is

important when looking for such patterns to realize that we may have to look at

many terms and many differences to be able to find the pattern.

Recapitulation

When presenting the problems in this chapter, we took great care in organizing the solu-

tions using Pólya’s four-step approach. However, it is not necessary to label and display

each of the four steps every time you work a problem. On the other hand, it is good

to get into the habit of recalling the four steps as you plan and as you work through a

problem. In this chapter we have introduced several useful problem-solving strategies.

In each of the following chapters, a new problem-solving strategy is introduced. These

strategies will be especially helpful when you are making a plan. As you are planning to

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solve a problem, think of the strategies as a collection of tools. Then an important part

of solving a problem can be viewed as selecting an appropriate tool or strategy.

We end this chapter with a list of suggestions that students who have successfully

completed a course on problem solving felt were helpful tips. Reread this list periodi-

cally as you progress through the book.

Reflection from Research

Suggestions from Successful Problem Solvers

Having children write their own

story problems helps students

r Accept the challenge of solving a problem.

to discern between relevant and

irrelevant attributes in a problem

r Rewrite the problem in your own words.

and to focus on the various parts

r Take time to explore, refl ect, think. . . .

of a problem, such as the known

and unknown quantities (Whitin &

r Talk to yourself. Ask yourself lots of questions.

Whitin, 2008).

r If appropriate, try the problem using simple numbers.

r Many problems require an incubation period. If you get frustrated, do not hesitate

to take a break—your subconscious may take over. But do return to try again.

r Look at the problem in a variety of ways.

r Run through your list of strategies to see whether one (or more) can help you get a

start.

r Many problems can be solved in a variety of ways—you only need to fi nd one

solution to be successful.

Reflection from Research

r Do not be afraid to change your approach, strategy, and so on.

The unrealistic expectations of

r Organization can be helpful in problem solving. Use the Pólya four-step approach

teachers, namely lack of time and

with a variety of strategies.

support, can cause young stu-

dents to struggle with problem

r Experience in problem solving is very valuable. Work lots of problems; your confi -

solving (Buschman, 2002).

dence will grow.

r If you are not making much progress, do not hesitate to go back to make sure that

you really understand the problem. This review process may happen two or three

times in a problem since understanding usually grows as you work toward a solution.

r There is nothing like a breakthrough, a small aha!, as you solve a problem.

r Always, always look back. Try to see precisely what the key step was in your solution.

r Make up and solve problems of your own.

r Write up your solutions neatly and clearly enough so that you will be able to un-

derstand your solution if you reread it in 10 years.

r Develop good problem-solving helper skills when assisting others in solving prob-

lems. Do not give out solutions; instead, provide meaningful hints.

r By helping and giving hints to others, you will fi nd that you will develop many

new insights.

r Enjoy yourself! Solving a problem is a positive experience.

Sophie Germain was born in Paris in 1776, the daughter of a silk merchant. At

the age of 13, she found a book on the history of mathematics in her father’s

library. She became enthralled with the study of mathematics. Even though

her parents disapproved of this pursuit, nothing daunted her—she studied at

night wrapped in a blanket, because her parents had taken her clothing away

from her to keep her from getting up. They also took away her heat and light.

This only hardened her resolve until her father finally gave in and she, at last,

was allowed to study to become a mathematician.

©Ron Bagwell

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PROBLEM SET A

Use any of the six problem-solving strategies introduced thus

5. Look for a pattern in the first two number grids. Then use

far to solve the following.

the pattern you observed to fill in the missing numbers of

the third grid.

1. a. Complete this table and describe the pattern in the

‘Answer’ column.

SUM

ANSWER

1

1

1 + 3

4

6. The triangular numbers are the whole numbers that are

1 + 3 + 5

represented by certain triangular arrays of dots. The first

1 + 3 + 5 + 7

five triangular numbers are shown.

1 + 3 + 5 + 7 + 9

b. How many odd whole numbers would have to be added

to get a sum of 81? Check your guess by adding them.

c. How many odd whole numbers would have to be added

to get a sum of 169? Check your guess by adding them.

a. Complete the following table and describe the pattern in

d. How many odd whole numbers would have to be added

the Number of Dots column.

to get a sum of 529? (You do not need to check.)

NUMBER OF DOTS

2. Find the missing term in each pattern.

NUMBER

(TRIANGULAR NUMBERS)

a. 256, 128, 64, _____, 16, 8

1

1

1 1

1

2

3

b. 1, , , _____,

3 9

81

3

c. 7, 9, 12, 16, _____

4

5

d. 127,863; 12,789; _____; 135; 18

6

3. Sketch a figure that is next in each sequence.

b. Make a sketch to represent the seventh triangular number.

a.

c. How many dots will be in the tenth triangular number?

d. Is there a triangular number that has 91 dots in its

shape? If so, which one?

e. Is there a triangular number that has 150 dots in its

shape? If so, which one?

f. Write a formula for the number of dots in the nth

triangular number.

g. When the famous mathematician Carl Friedrich Gauss

was in fourth grade, his teacher challenged him to add

b.

the first one hundred counting numbers. Find this sum.

1 + 2 + 3 + ⋅⋅⋅ + 100

7. In a group of 12 coins identical in appearance, all weigh

the same except one that is heavier. What is the minimum

number of weighings required to determine the counterfeit

coin? Use the Chapter 1 eManipulative activity Counterfeit

Coin on our Web site for eight or nine coins to better

understand the problem.

8. If 20 points are placed on a circle and every pair of points

4. Consider the following differences. Use your calculator to

is joined with a segment, what is the total number of seg-

verify that the statements are true.

ments drawn?

62 − 52 = 11

9. Find reasonable sixth, seventh, and eighth terms of the

562 − 452 = 1111

following sequences:

5562 − 4452 = 111 111

,

a. 1, 4, 9, 17, 29, _____, _____, _____

b. 3, 7, 13, 21, 31, _____, _____, _____

a. Predict the next line in the sequence of differences. Use

your calculator to check your answer.

10. As mentioned in this section, the square numbers are the

b. What do you think the eighth line will be?

counting numbers 1, 4, 9, 16, 25, 36, . . . . Each square number

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can be represented by a square array of dots as shown

12 + 12 = 1 × 2

in the following figure, where the second square number

12 + 12 + 22 = 2 × 3

has four dots, and so on. The first four square numbers

2

2

2

2

are shown.

1 + 1 + 2 + 3 = 3 × 5

Write out six more terms of the Fibonacci sequence and

use the sequence to predict what 12

12

22

32

. . . 1442

+ +

+

+

+

is without actually computing the sum. Then use your

calculator to check your result.

a. Find two triangular numbers (refer to Problem 6) whose

16. Write out 16 terms of the Fibonacci sequence and observe

sum equals the third square number.

the following pattern:

b. Find two triangular numbers whose sum equals the fifth

+ =

square number.

1 2

3

c. What two triangular numbers have a sum that equals

1 + 2 + 5 = 8

the 10th square number? the 20th square number? the

1 + 2 + 5 + 13 = 21

nth square number?

Use the pattern you observed to predict the sum

d. Find a triangular number that is also a square

number.

1 + 2 + 5 + 13 + ... + 610

e. Find five pairs of square numbers whose difference is a

without actually computing the sum. Then use your calcu-

triangular number.

lator to check your result.

11. Would you rather work for a month (30 days) and get

17. Pascal’s triangle is where each entry other than a 1 is

paid 1 million dollars or be paid 1 cent the first day, 2

obtained by adding the two entries in the row above it.

cents the second day, 4 cents the third day, 8 cents the

fourth day, and so on? Explain.

12. Find the perimeters and then complete the table.

a. Find the sums of the numbers on the diagonals in

Pascal’s triangle as are indicated in the following figure.

number of triangles

1

2

3

4

5

6

10

n

perimeter

40

13. The integers greater than 1 are arranged as shown.

2

3

4

5

9

8

7

6

10

11

12

13

17

16

15

14

•

•

•

a. In which column will 100 fall?

b. Predict the sums along the next three diagonals in

b. In which column will 1000 fall?

Pascal’s triangle without actually adding the entries.

c. How about 1999?

Check your answers by adding entries on your calculator.

d. How about 99,997?

18. Answer the following questions about Pascal’s triangle

14. How many cubes are in the 100th collection of cubes in

(see Problem 17).

this sequence?

a. In the triangle shown here, one number, namely 3, and

the six numbers immediately surrounding it are encircled.

Find the sum of the encircled seven numbers.

15. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, . . . , where

each successive number beginning with 2 is the sum of the

preceding two; for example, 13 = 5 + 8, 21 = 8 + 13, and so

on. Observe the following pattern.

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b. Extend Pascal’s triangle by adding a few rows. Then

d. How many squares would there be in the 10th figure?

draw several more circles anywhere in the triangle like

in the 20th figure? in the 50th figure?

the one shown in part a. Explain how the sums obtained

20. In a dart game, only 4 points or 9 points can be scored

by adding the seven numbers inside the circle are related

on each dart. What is the largest score that it is not

to one of the numbers outside the circle.

possible to obtain? (Assume that you have an unlimited

19. Consider the following sequence of shapes. The sequence

number of darts.)

starts with one square. Then at each step squares are

attached around the outside of the figure, one square per

21. If the following four figures are referred to as stars,

exposed edge in the figure.

the first one is a three-pointed star and the second

one is a six-pointed star. (NOTE: If this pattern of con-

structing a new equilateral triangle on each side of the

existing equilateral triangle is continued indefinitely,

the resulting figure is called the Koch curve or Koch

snowflake.)

a. How many points are there in the third star?

b. How many points are there in the fourth star?

22. Using the Chapter 1 eManipulative activity Color Patterns

a. Draw the next two figures in the sequence.

on our Web site, describe the color patterns for the first

three computer exercises.

b. Make a table listing the number of unit squares in the

figure at each step. Look for a pattern in the number

23. Looking for a pattern can be frustrating if the pattern is

of unit squares. (Hint: Consider the number of squares

not immediately obvious. Create your own sequence of

attached at each step.)

numbers that follows a pattern but that has the capacity

c. Based on the pattern you observed, predict the number

to stump some of your fellow students. Then write an

of squares in the figure at step 7. Draw the figure to

explanation of how they might have been able to discover

check your answer.

your pattern.

PROBLEM SET B

1. Find the missing term in each pattern.

3. The rectangular numbers are whole numbers that are repre-

a. 10, 17, _____, 37, 50, 65

sented by certain rectangular arrays of dots. The first five

3

7

9

rectangular numbers are shown.

b. 1, , _____, ,

2

8 16

c. 243, 324, 405, _____, 567

d. 234; _____; 23,481; 234,819; 2,348,200

2. Sketch a figure that is next in each sequence.

a.

a. Complete the following table and describe the pattern in

the Number of Dots column.

b.

NUMBER OF DOTS

NUMBER

(RECTANGULAR NUMBERS)

1

2

2

6

3

4

5

6

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b. Make a sketch to represent the seventh rectangular number.

7. What is the smallest number that can be expressed as the

c. How many dots will be in the tenth rectangular number?

sum of two squares in two different ways? (You may use

d. Is there a rectangular number that has 380 dots in its

one square twice.)

shape? If so, which one?

8. How many cubes are in the 10th collection of cubes in this

e. Write a formula for the number of dots in the nth rectan-

sequence?

gular number.

f. What is the connection between triangular numbers (see

Problem 6 in Set A) and rectangular numbers?

4. The pentagonal numbers are whole numbers that are rep-

resented by pentagonal shapes. The first four pentagonal

numbers are shown.

⎡4

5⎤

9. The 2 × 2 array of numbers ⎢

⎥ has a sum of 4 × 5, and

5

6

⎣

⎦

⎡6

7

8 ⎤

⎢

⎥

the 3 × 3 array 7

8

9

⎢

⎥ has a sum of 9 × 8.

⎢8

9

10

⎣

⎦⎥

a. What will be the sum of the similar 4 × 4 array starting

with 7?

b. What will be the sum of a similar 100 × 100 array starting

with 100?

a. Complete the following table and describe the pattern in

10. The Fibonacci sequence was defined to be the sequence

. . .

the Number of Dots column.

1, 1, 2, 3, 5, 8, 13, 21,

, where each successive number is the

sum of the preceding two. Observe the following pattern:

NUMBER OF DOTS

1 + 1 = 3 − 1

NUMBER

(PENTAGONAL NUMBERS)

1 + 1 + 2 = 5 − 1

1

1

1 + 1 + 2 + 3 = 8 − 1

2

5

3

1 + 1 + 2 + 3 + 5 = 13 − 1

4

Write out six more terms of the Fibonacci sequence, and

5

use the sequence to predict the answer to

b. Make a sketch to represent the fifth pentagonal number.

1 + 1 + 2 + 3 + 5 + . . . + 144

c. How many dots will be in the ninth pentagonal number?

without actually computing the sum. Then use your calcu-

d. Is there a pentagonal number that has 200 dots in its

lator to check your result.

shape? If so, which one?

e. Write a formula for the number of dots in the nth pen-

11. Write out 16 terms of the Fibonacci sequence.

tagonal number.

a. Notice that the fourth term in the sequence (called F4) is

odd: F = . The sixth term in the sequence (called F

5. Consider the following process:

4

3

6 )

is even: F =

6

8. Look for a pattern in the terms of the

i. Choose a whole number.

sequence, and describe which terms are even and which

ii. Add the squares of the digits of the number to get a new

are odd.

number.

b. Which of the following terms of the Fibonacci sequence

Repeat step 2 several times.

are even and which are odd: F38, F51, F150, F200, F300 ?

a. Apply the procedure described to the numbers 12, 13, 19,

c. Look for a pattern in the terms of the sequence and

21, and 127.

describe which terms are divisible by 3.

b. What pattern do you observe as you repeat the steps

d. Which of the following terms of the Fibonacci

over and over?

sequence are multiples of 3: F48, F75, F196, F379, F1000 ?

c. Check your answer for part b with a number of your choice.

12. Write out 16 terms of the Fibonacci sequence and observe

the following pattern:

6. How many triangles are in the picture?

1 + 3 = 5 − 1

1 + 3 + 8 = 13 − 1

1 + 3 + 8 + 21 = 34 − 1

Use the pattern you observed to predict the answer to

1 + 3 + 8 + 21 + . . . + 377

without actually computing the sum. Then use your

calculator to check your result.

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7/30/2013 2:37:04 PM - 36 Chapter 1 Introduction to Problem Solving

13. Investigate the “Tower of Hanoi” problem on the Chapter

d. How many triangles would there be in the 10th figure?

1 eManipulative activity Tower of Hanoi on our Web site

in the 20th figure? in the 50th figure?

to answer the following questions:

a. Determine the fewest number of moves required when

18. How many equilateral triangles of all sizes are there in the

× ×

you start with two, three, and four disks.

3

3

3 equilateral triangle shown next?

b. Describe the general process to move the disks in the

fewest number of moves.

c. What is the minimum number of moves that it should

take to move six disks?

14. While only 19 years old, Carl Friedrich Gauss proved in

1796 that every positive integer is the sum of at the most

three triangular numbers (see Problem 6 in Set A).

a. Express each of the numbers 25 to 35 as a sum of no

19. Refer to the following figures to answer the questions.

more than three triangular numbers.

(NOTE: If this pattern is continued indefinitely, the resulting

b. Express the numbers 74, 81, and 90 as sums of no more

figure is called the Sierpinski triangle or the Sierpinski

than three triangular numbers.

gasket.)

15. Answer the following for Pascal’s triangle.

a. In the following triangle, six numbers surrounding a

central number, 4, are circled. Compare the products

of alternate numbers moving around the circle; that is,

compare 3 • 1 • 10 and 6 • 1 • 5.

a. How many black triangles are there in the fourth figure?

b. How many white triangles are there in the fourth figure?

c. If the pattern is continued, how many black triangles are

there in the nth figure?

d. If the pattern is continued, how many white triangles are

there in the nth figure?

20. If the pattern illustrated next is continued,

a. find the total number of 1 by 1 squares in the thirtieth

b. Extend Pascal’s triangle by adding a few rows. Then

figure.

draw several more circles like the one shown in part a

b. find the perimeter of the twenty-fifth figure.

anywhere in the triangle. Find the products as described

c. find the total number of toothpicks used to construct

in part a. What patterns do you see in the products?

the twentieth figure.

16. A certain type of gutter comes in 6-foot, 8-foot, and

10-foot sections. How many different lengths can be

formed using three sections of gutter?

17. Consider the sequence of shapes shown in the following

figure. The sequence starts with one triangle. Then at each

step, triangles are attached to the outside of the preceding

21. Find reasonable sixth, seventh, and eighth terms of the

figure, one triangle per exposed edge.

following sequences:

a. 1, 3, 4, 7, 11, _____, _____, _____

b. 0, 1, 4, 12, 29, _____, _____, _____

22. Many board games involve throwing two dice and sum-

ming of the numbers that come up to determine how

many squares to move. Make a list of all the different

sums that can appear. Then write down how many ways

each different sum can be formed. For example, 11 can be

formed in two ways: from a 5 on the first die and a 6 on

the second OR a 6 on the first die and a 5 on the second.

Which sum has the greatest number of combinations?

What conclusion could you draw from that?

a. Draw the next two figures in the sequence.

b. Make a table listing the number of triangles in the

23. There is an old riddle about a frog at the bottom of a

figure at each step. Look for a pattern in the number of

20-foot well. If he climbs up 3 feet each day and slips

triangles. (Hint: Consider the number of triangles added

back 2 feet each night, how many days will it take him

at each step.)

to reach the 20-foot mark and climb out of the well? The

c. Based on the pattern you observed, predict the number

answer isn’t 20. Try doing the problem with a well that is

of triangles in the figure at step 7. Draw the figure to

only 5 feet deep, and keep track of all the frog’s moves.

check your answer.

What strategy are you using?

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7/30/2013 2:37:06 PM - End of Chapter Material 37

Analyzing Student Thinking

can’t see the pattern in 3, 5, 9, 15, 23 because the differ-

ence between the numbers is changing. How could you

24. Marietta extended the pattern 2, 4, 8 to be 2, 4, 8, 16, 32, . . . .

help her see this new pattern?

Pascuel extended the same pattern to be 2, 4, 8, 14, 22, . . ..

They asked you who was correct. How should you respond?

28. Jeremy is asked to find the number of rectangles of all

possible dimensions in the figure below and decides to

25. Eula is asked to find the following sum: 1 + 3 + 5 + 7 + ⋅⋅⋅ +

“solve a simpler problem.”

97 + 99. She decided to “solve a simpler problem” but

doesn’t know where to start. What would you suggest?

26. When Mickey counted the number of outcomes of roll-

ing two 4-sided, tetrahedral dice (see Figure 12.89 for an

example of a tetrahedron), he decided to “make a list”

and got (1,1), (2,2), (3,3), (4,4), (1,2), (3,4), (2,4), (4,1). He

What simpler problem would you suggest he use?

started to get confused about which ones he had listed and

which ones were left. How would you help him create a

29. After solving several simpler problems, Janell commented

more systematic list?

that she often makes a list of solutions of the simpler prob-

27. Bridgette says that she knows how to see patterns like

lems and then looks for a pattern in that list. She asks if it is

3, 7, 11, 15,19, . . . and 5,12,19, 26, 33, . . . because “you are

okay to use more than one type of strategy when solving the

just adding the same amount each time.” However, she

same problem. How should you respond?

END OF CHAPTER MATERIAL

1 = (4 + 4) ÷ (4 + 4)

2 = (4 ÷ 4) + (4 ÷ 4)

Place the whole numbers 1 through 9 in the circles in the

3 = (4 + 4 + 4) ÷ 4

accompanying triangle so that the sum of the numbers on each

4 = 4 + 4 × (4 − 4)

side is 17.

5 = (4 × 4 + 4) ÷ 4

6 = ((4 + 4) ÷ 4) + 4

7 = 4 + 4 − (4 ÷ 4)

8 = ((4 × 4) ÷ 4) + 4

9 = 4 + 4 + (4 ÷ 4)

There are many other possible answers.

3.

Strategy: Guess and Test

Having solved a simpler problem in this chapter, you might

easily be able to conclude that 1, 2, and 3 must be in the cor-

ners. Then the remaining six numbers, 4, 5, 6, 7, 8, and 9, must

Draw a Picture

produce three pairs of numbers whose sums are 12, 13, and 14.

1. 5

The only two possible solutions are as shown.

2. Yes; make one cut, then lay the logs side by side for the sec-

ond cut.

3. 12

Use a Variable

1. 55, 5050, 125,250

2. (2m + )

1 + (2m + 3) + (2m + 5) + (2m + 7) + (2m + 9)

= 10m + 25 = 5(2m + 5)

3. 10°, 80°, 90°

Look for a Pattern

Guess and Test

1. 7

2. Square numbers

1. s = 1, u = 3, n = 6, f = 9, w = 0, i = 7, m = 2

3. a. 224; 406; 518

b. 71

2. 0 = (4 − 4) + (4 − 4)

c. The sum is seven times the middle number.

c01.indd 37

7/30/2013 2:37:09 PM - 38 Chapter 1 Introduction to Problem Solving

Make a List

Solve a Simpler Problem

1. 48, 36, 33, 32, 24, 21, 20, 18, 17, 16, 12, 9, 8, 6, 5, 4, 3,

1023

1.

2, 1, 0

1024

2. 204

2. 377

3. 446

3. 190

ce

Carl Friedrich Gauss

Sophie Germain

(1777–1885)

(1776–1831)

Carl Friedrich Gauss, accord-

Sophie Germain, as a teen-

ing to the historian E. T. Bell,

ager in Paris, discovered math-

“lives everywhere in math-

ematics by reading books from

on/Science Sour

ematics.” His contributions to

her father’s library. At age 18,

Omikr

geometry, number theory, and

Germain wished to attend the

analysis were deep and wide-ranging. Yet he also made

Roger Viollet/Getty Images

prestigious Ecole Polytech-

crucial contributions in applied mathematics. When the

nique in Paris, but women were not admitted. So she

tiny planet Ceres was discovered in 1800, Gauss devel-

studied from classroom notes supplied by sympa-

oped a technique for calculating its orbit, based on mea-

thetic male colleagues, and she began submitting writ-

ger observations of its direction from Earth at several

ten work using the pen name Antoine LeBlanc. This

known times. Gauss contributed to the modern theory

work won her high praise, and eventually she was able

of electricity and magnetism, and with the physicist

to reveal her true identity. Germain is noted for her

W. E. Weber constructed one of the fi rst practical electric

theory of the vibration patterns of elastic plates and

telegraphs. In 1807 he became director of the astronomi-

for her proof of Fermat’s last theorem in some special

cal observatory at Gottingen, where he served until his

cases. Of Sophie Germain, Carl Gauss wrote, “When a

death. At age 18, Gauss devised a method for construct-

woman, because of her sex, encounters infi nitely more

ing a 17-sided regular polygon, using only a compass

obstacles than men . . . yet overcomes these fetters and

and straightedge. Remarkably, he then derived a general

penetrates that which is most hidden, she doubtless

rule that pre dicted which regular polygons are likewise

has the most noble courage, extraordinary talent, and

constructible.

superior genius.”

CHAPTER REVIEW

Review the following terms and problems to determine which require learning or relearning—page numbers are provided for easy

reference.

The Problem-Solving Process and Strategies

Vocabulary/Notation

Exercise 5

Systematic Guess and Test 8

Equation 16

Problem 5

Inferential Guess and Test 8

Solution of an equation 16

Pólya’s four-step process 5

Cryptarithm 9

Solve an equation 16

Strategy 5

Tetromino 11

Random Guess and Test 8

Variable or unknown 12

Problems

For each of the following, (i) determine a reasonable strategy

to use to solve the problem, (ii) state a clue that suggested

the strategy, and (iii) write out a solution using Pólya’s four-

step process.

1. Fill in the circles using the numbers 1 through 9 once each

where the sum along each of the five rows totals 17.

c01.indd 38

7/30/2013 2:37:16 PM - Chapter Test 39

2. In the following arithmagon, the number that appears in a

3. The floor of a square room is covered with square tiles.

square is the product of the numbers in the circles on each

Walking diagonally across the room from corner to corner,

side of it. Determine what numbers belong in the circles.

Susan counted a total of 33 tiles on the two diagonals.

What is the total number of tiles covering the floor of the

room?

Three Additional Strategies

Vocabulary/Notation

Pascal’s triangle 22

Ellipsis 22

Powers 23

Sequence 22

Even numbers 23

Fibonacci sequence 23

Terms 22

Odd numbers 23

Inductive reasoning 23

Counting numbers 22

Square numbers 23

Problems

For each of the following, (i) determine a reasonable strategy

2. a. How many cubes of all sizes are in a 2 × 2 × 2 cube

to use to solve the problem, (ii) state a clue that suggested the

composed of eight 1 × 1 × 1 cubes?

strategy, and (iii) write out a solution using Pólya’s four-step

b. How many cubes of all sizes are in an 8 × 8 × 8 cube com-

process.

posed of 512 1 × 1 × 1 cubes?

1. Consider the following products. Use your calculator to

3. a. What is the smallest number of whole-number gram

verify that the statements are true.

weights needed to weigh any whole-number amount

1 × 1

( ) = 12

from 1 to 12 grams on a scale allowing the weights to be

121 × 1

( + 2 + 1) = 222

placed on either or both sides?

12321 × 1

( + 2 + 3 + 2 + 1) = 3332

b. How about from 1 to 37 grams?

c. What is the most you can weigh using six weights in this

Predict the next line in the sequence of products. Use your

way?

calculator to check your answer.

CHAPTER TEST

Knowledge

5. Given the following problem and its numerical answer,

write the solution in a complete sentence.

1. List the four steps of Pólya’s problem-solving process.

Amanda leaves with a basket of hard-boiled eggs

2. List the six problem-solving strategies you have learned in

to sell. At her first stop she sold half her eggs plus

this chapter.

half an egg. At her second stop she sold half her

eggs plus half an egg. The same thing occurs at her

Skill

third, fourth, and fifth stops. When she finishes,

she has no eggs in her basket. How many eggs did

3. Identify the unneeded information in the following problem.

she start with?

Birgit took her $5 allowance to the bookstore to buy

Answer:

31

some back-to-school supplies. The pencils cost $.10 each,

the erasers cost $.05 each, and the clips cost 2 for $.01. If

she bought 100 items altogether at a total cost of $1, how

Understanding

many of each item did she buy?

6. Explain the difference between an exercise and a problem.

4. Rewrite the following problem in your own words.

7. List at least two characteristics of a problem that would

If you add the square of Ruben’s age to the age of

suggest using the Guess and Test strategy.

Angelita, the sum is 62; but if you add the square of

Angelita’s age to the age of Ruben, the sum is 176. Can

8. List at least two characteristics of a problem that would

you say what the ages of Ruben and Angelita are?

suggest using the Use a Variable strategy.

c01.indd 39

7/30/2013 2:37:16 PM - 40 Chapter 1 Introduction to Problem Solving

Problem Solving/Application

bottles in it? If so, how? (Hint: One row has 6 bottles in it

and the other three rows have 4 bottles in them.)

For each of the following problems, read the problem care-

fully and solve it. Identify the strategy you used.

9. Can you rearrange the 16 numbers in this 4 × 4 array so

that each row, each column, and each of the two diagonals

total 10? How about a 2 × 2 array containing two 1s and

two 2s? How about the corresponding 3 × 3 array?

16. Otis has 12 coins in his pocket worth $1.10. If he only has

nickels, dimes, and quarters, what are all of the possible

coin combinations?

17. Show why 3 always divides evenly into the sum of any

three consecutive whole numbers.

10. In three years, Chad will be three times my present age. I

18. If 14 toothpicks are arranged to form a triangle so none

will then be half as old as he. How old am I now?

of the toothpicks are broken or bent and all 14 toothpicks

are used, how many different-shaped triangles can be

11. There are six baseball teams in a tournament. The teams

formed?

are lettered A through F. Each team plays each of the other

teams twice. How many games are played altogether?

19. Together a baseball and a football weigh 1.25 pounds,

the baseball and a soccer ball weigh 1.35 pounds, and

12. A fish is 30 inches long. The head is as long as the tail.

the football and the soccer ball weigh 1.6 pounds.

If the head was twice as long and the tail was its present

How much does each of the balls weigh? Explain your

length, the body would be 18 inches long. How long is

reasoning.

each portion of the fish?

20. In the figure below, there are 7 chairs arranged around 5

13. The Orchard brothers always plant their apple trees in

tables. How many chairs could be placed around a similar

square arrays, like those illustrated. This year they planted

arrangement of 31 triangular tables?

31 more apple trees in their square orchard than last year.

If the orchard is still square, how many apple trees are

there in the orchard this year?

21. Carlos’ father pays Carlos his allowance each week using

patterns. He pays a different amount each day according

14. Arrange 10 people so that there are five rows each con-

to some pattern. Carlos must identify the pattern in order

taining 4 persons.

to receive his allowance. Help Carlos complete the pattern

for the missing days in each week below.

15. A milk crate holds 24 bottles and is shaped like the one

a. 5¢, 9¢, 16¢, 26¢, _____, _____, _____

shown here. The crate has four rows and six columns. Is it

possible to put 18 bottles of milk in the crate so that each

b. 1¢, 6¢, 15¢, 30¢, 53¢, _____, _____

row and each column of the crate has an even number of

c. 4¢, 8¢, 16¢, 28¢, _____, _____, _____

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7/30/2013 2:37:17 PM - c01.indd 41

7/30/2013 2:37:17 PM - C H A P T E R

2 SETS, WHOLE NUMBERS, AND

NUMERATION

The Mayan Numeration System

The Maya people lived mainly in southeastern The sun, and hence the solar calendar, was very

Mexico, including the Yucatan Peninsula, and in

important to the Maya. They calculated that a year con-

much of northwestern Central America, includ-

sisted of 365.2420 days. (Present calculations measure our

ing Guatemala and parts of Honduras and El Salvador.

year as 365.2422 days long.) Since 360 had convenient

Earliest archaeological evidence of the Maya civiliza-

factors and was close to 365 days in their year and 400 in

tion dates to 9000 B.C.E., with the principal epochs of

their numeration system, they made a place-value system

the Maya cultural development occurring between 2000

where the column values from right to left were 1, 20,

B.C.E. and 1700 C.E.

20 18 (= 360), 202 18 (= 7200), 203

¥

¥

¥18 (= 144,000), and

Knowledge of arithmetic, calendrical, and astronomical

so on. Interestingly, the Maya could record all the days

matters was more highly developed by the ancient Maya

of their history simply by using the place values through

than by any other New World peoples. Their numeration

144,000. The Maya were also able to use larger numbers.

system was simple, yet sophisticated. Their system utilized

One Mayan hieroglyphic text recorded a number equiva-

three basic numerals: a dot, •, to represent 1; a horizontal

lent to 1,841,641,600.

bar, —, to represent 5; and a conch shell,

, to rep-

Finally, the Maya, famous for their hieroglyphic

resent 0. They used these three symbols, in combination,

writing, also used the 20 ideograms pictured here, called

to represent the numbers 0 through 19.

head variants, to represent the numbers 0 1

− 9.

For numbers greater than 19, they initially used a base

twenty system. That is, they grouped in twenties and dis-

played their numerals vertically. Three Mayan numerals

are shown together with their values in our system and the

The Mayan numeration system is studied in this chap-

place values initially used by the Maya.

ter along with other ancient numeration systems.

42

c02.indd 42

7/30/2013 2:39:19 PM - Problem-Solving

Draw a Diagram

Strategies

Often there are problems where, although it is not necessary to draw an actual

1. Guess and Test

picture to represent the problem situation, a diagram that represents the essence

2. Draw a Picture

of the problem is useful. For example, if we wish to determine the number of times

two heads can turn up when we toss two coins, we could literally draw pictures

3. Use a Variable

of all possible arrangements of two coins turning up heads or tails. However, in

4. Look for a Pattern

practice, a simple tree diagram is used like the one shown next.

5. Make a List

6. Solve a Simpler

Problem

7. Draw a Diagram

This diagram shows that there is one way to obtain two heads out of four

possible outcomes. Another type of diagram is helpful in solving the next

problem.

Initial Problem

A survey was taken of 150 college freshmen. Forty of them were majoring in math-

ematics, 30 of them were majoring in English, 20 were majoring in science, 7 had a

double major of mathematics and English, and none had a double (or triple) major

with science. How many students had majors other than mathematics, English, or

science?

Clues

The Draw a Diagram strategy may be appropriate when

r The problem involves sets, ratios, or probabilities.

r An actual picture can be drawn, but a diagram is more effi cient.

r Relationships among quantities are represented.

A solution of this Initial Problem is on page 79.

43

c02.indd 43

7/30/2013 2:39:19 PM - AUTHOR

I N T R O D U C T I O N

Much of elementary school mathematics is devoted to the study of numbers. Children first learn

to count using the natural numbers or counting numbers 1, 2, 3, . . . (the ellipsis, or three periods, means

“and so on”). This chapter develops the ideas that lead to the concepts central to the system of whole

numbers 0, 1, 2, 3, . . . (the counting numbers together with zero) and the symbols that are used to

WALK-THROUGH represent them. First, the notion of a one-to-one correspondence between two sets is shown to be the

idea central to the formation of the concept of number. Then operations on sets are discussed. These

operations form the foundation of addition, subtraction, multiplication, and division of whole numbers. Finally,

the Hindu-Arabic numeration system, our system of symbols that represent numbers, is presented after its various

attributes are introduced by considering other ancient numeration systems.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-2–NUMBER AND OPERATIONS

Count with understanding and recognize “how many” in sets of objects.

Use multiple models to develop initial understandings of place value and the base ten number system.

Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers

and their connections.

Connect number words and numerals to the quantities they represent, using various physical models and representations.

r GRADES 3-5–NUMBER AND OPERATIONS

Understand the place-value structure of the base ten number system and be able to represent and compare whole num-

bers.

Key Concepts from the NCTM Curriculum Focal Points

r PRE-K: Developing an understanding of whole numbers, including concepts of correspondence, counting,

cardinality, and comparison.

r KINDERGARTEN: Representing, comparing, and ordering whole numbers and joining and separating sets.

Ordering objects by measurable attributes.

r GRADE 1: Developing an understanding of whole number relationships, including grouping in tens and ones.

r GRADE 2: Developing an understanding of the base ten numeration system and place-value concepts.

r GRADE 6: Writing, interpreting, and using mathematical expressions and equations.

Key Concepts from the Common Core State Standards for Mathematics

r KINDERGARTEN: Know number names and the count sequence (to 100). Count to tell the number of objects (con-

nect counting to cardinality). Compare numbers (greater than, less than, equal to). Work with numbers 11–19

to gain foundations for place value.

r GRADE 1: Extend the counting sequence (to 120). Understand place value (the two digits of a two-digit number

represent the number of tens and ones).

r GRADE 2: Extend the counting sequence (within 1000) by skip-counting. Understand place value (the three digits

of a three-digit number represent the number of hundreds, tens, and ones).

44

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7/30/2013 2:39:20 PM - Section 2.1 Sets as a Basis for Whole Numbers 45

SETS AS A BASIS FOR WHOLE NUMBERS

When a young child is asked to count the blocks in the set shown below, he will use his mem-

orized cadence of “one, two, three, four . . . ” while pointing to the blocks. The following

three scenarios are common:

Child 1 says “one” while pointing to the red block, “two” while pointing to the green block, and then goes back to the red

block to say “three” before pointing to the yellow block and say, “four.”

Child 2 says “one” while pointing to the yellow block and says “two” while pointing to the red block and is done.

Child 3 points at the green block and says “one,” the red block and says “two,” and finally points to the yellow block while

saying “three.”

What underlying concept do children need to come to understand in order to count correctly?

Sets

Algebraic Reasoning

A collection of objects is called a set and the objects are called elements or members of

When solving algebraic equations,

the set. Sets can be defined in three common ways: (1) a verbal description, (2) a listing

we are looking for all values/

of the elements separated by commas, with braces (“{” and “}”) used to enclose the list

numbers that make an equation

of elements, and (3) set-builder notation. For example, the verbal description “the set of

true. This set of numbers is called

all states in the United States that border the Pacific Ocean” can be represented in the

the solution set.

other two ways as follows:

1. Listing: {Alaska, California, Hawaii, Oregon, Washington}.

2. Set-builder:{x | x is a U.S. state that borders the Pacific Ocean}. (This set-builder nota-

tion is read: “The set of all x such that x is a U.S. state that borders the Pacific Ocean.”)

Sets are usually denoted by capital letters such as A, B, C and so on. The symbols

“∈” and “∉” are used to indicate that an object is or is not an element of a set, respec-

tively. For example, if S represents the set of all U.S. states bordering the Pacific,

then Alaska ∈S and Michigan ∉S. The set without elements is called the empty set

(or null set) and is denoted by { } or the symbol ∅. The set of all U.S. states bordering

Antarctica is the empty set.

Two sets A and B are equal, written A = B, if and only if they have precisely the

same elements. Thus {x | x is a state that borders Lake Michigan} = {Illinois, Indiiana,

Michigan, Wisconsin}. Notice that two sets, A and B, are equal if every element of A

is in B, and vice versa. If A does not equal B, we write A ñ B.

There are two inherent rules regarding sets: (1) The same element is not listed

more than once within a set, and (2) the order of the elements in a set is immate-

rial. Thus, by rule 1, the set { ,

a

,

a

}

b would be written as { ,

a

}

b and by rule 2,

{ ,

a

}

b = { ,

b

}

a , { ,

x

,

y

}

z = { ,

y z, }

x , and so on.

The concept of a 1–1 correspondence, read “one-to-one correspondence,” is need-

ed to formalize the meaning of a whole number.

D E F I N I T I O N 2 . 1

One-to-One Correspondence

A 1-1 correspondence between two sets A and B is a pairing of the elements of A

with the elements of B so that each element of A corresponds to exactly one ele-

ment of B, and vice versa. If there is a 1-1 correspondence between sets A and B,

we write A ~ B and say that A and B are equivalent or matching sets.

c02.indd 45

7/30/2013 2:39:24 PM - 46 Chapter 2 Sets, Whole Numbers, and Numeration

Figure 2.1 shows two possible 1–1 correspondences between two sets, A and B.

There are four other possible 1–1 correspondences between A and B. Notice

that equal sets are always equivalent, since each element can be matched with itself,

but that equivalent sets are not necessarily equal. For example, { ,

1 }

2 ~ { ,

a }

b , but

{ ,

1 }

2 ≠ { ,

a }

b . The two sets A = a

{ , b

} and B = a

{ , b

, c

} are not equivalent. However,

they do satisfy the relationship defined next.

D E F I N I T I O N 2 . 2

Figure 2.1

Subset of a Set: A ⊆ B

Set A is said to be a subset of B, written A ⊆ B, if and only if every element of A

is also an element of B.

The set consisting of New Hampshire is a subset of the set of all New England

states and { ,

a ,

b }

c ⊆ { ,

a ,

b ,

c d

, ,

e f

}. Since every element in a set A is in A, A ⊆ A

for all sets A. Also, { ,

a ,

b }

c ⊆ { ,

a ,

b d

} because c is in the set { ,

a ,

b }

c but not in the

set { ,

a ,

b d

}. Using similar reasoning, you can argue that ∅ ⊆ A for any set A since it

is impossible to find an element in ∅ that is not in A.

If A ⊆ B and B has an element that is not in A, we write A ⊂ B and say that A

is a proper subset of B. Thus { ,

a }

b ⊂ { ,

a ,

b }

c , since { ,

a }

b ⊆ { ,

a ,

b }

c and c is in the

second set but not in the fi rst.

Circles or other closed curves are used in Venn diagrams (named after the English

logician John Venn) to illustrate relationships between sets. These circles are usually

pictured within a rectangle, U , where the rectangle represents the universal set or

universe, the set comprised of all elements being considered in a particular discussion.

Figure 2.2 displays sets A and B inside a universal set U.

Set A is comprised of everything inside circle A, and set B is comprised of

everything inside circle B, including set A. Hence A is a proper subset of B since x ∈B,

but x ∉A. The idea of proper subset will be used later to help establish the meaning

Figure 2.2

of the concept “less than” for whole numbers.

Check for Understanding: Exercise/Problem Set A #1–9

✔

Finite and Infinite Sets

There are two broad categories of sets: finite and infinite. Informally, a set is finite if it is

empty or can have its elements listed (where the list eventually ends), and a set is infinite if

it goes on without end. A little more formally, a set is finite if (1) it is empty or (2) it can be

put into a 1–1 correspondence with a set of the form { ,

1 ,

2 ,

3 . . . , }

n , where n is a counting

number. On the other hand, a set is infinite if it is not finite.

Determine whether the following sets are finite or infinite.

a. { ,

a ,

b }

c

b. { ,

1 ,

2 ,

3 . . .}

c. { ,

2 ,

4 ,

6 . . . , 2 }

0

Reflection from Research

While many students will agree

S O L U T I O N

that two infinite sets such as the

set of counting numbers and the

a. { ,

a ,

b }

c is fi nite since it can be matched with the set { ,

1 ,

2 }

3 .

set of even numbers are equiva-

b. { ,

1 ,

2 ,

3 . . .} is an infi nite set.

lent, the same students will argue

c. { ,

2 ,

4 ,

6 . . . , 2 }

0 is a fi nite set since it can be matched with the set { ,

1 ,

2 ,

3 . . . , 1 }

0 .

that the set of counting numbers

is larger due to its inclusion of the

(Here, the ellipsis means to continue the pattern until the last element is

odd numbers (Wheeler, 1987).

reached.)

■

c02.indd 46

7/30/2013 2:39:30 PM - Section 2.1 Sets as a Basis for Whole Numbers 47

NOTE: The small solid square (■) is used to mark the end of an example or math-

ematical argument.

An interesting property of every infinite set is that it can be matched with a proper

subset of itself. For example, consider the following 1-1 correspondence:

Algebraic Reasoning

Note that B ⊂ A and that each element in A is paired with exactly one element in B,

Due to the nature of infinite sets,

and vice versa. Notice that matching n with 2n indicates that we never “run out” of

a variable is commonly used to

elements from B to match with the elements from set A. Thus an alternative definition

illustrate matching elements in

is that a set is infinite if it is equivalent to a proper subset of itself. In this case, a set is

two sets when showing 1-1 cor-

finite if it is not infinite.

respondences.

Check for Understanding: Exercise/Problem Set A #10–11

✔

After forming a group of students, use a diagram like the one shown to place the names

of each member of your group in the appropriate region. All members of the group of stu-

dents in a class fit in the following rectangle.

r Write a sentence that describes the attributes of a person whose name is in the red region.

r Write a sentence that describes the attributes of a person whose name is not in any of the circles.

r Write a sentence that describes the attributes of a person whose name is in the blue region.

Operations on Sets

Two sets A and B that have no elements in common are called disjoint sets. The sets

{ ,

a ,

b }

c and {d, ,

e f

} are disjoint (Figure 2.3), whereas { ,

x

}

y

and { ,

y

}

z

are not dis-

joint, since y is an element in both sets.

Figure 2.3

There are many ways to construct a new set from two or more sets. The following

operations on sets will be very useful in clarifying our understanding of whole num-

bers and their operations.

c02.indd 47

7/30/2013 2:39:32 PM - 48 Chapter 2 Sets, Whole Numbers, and Numeration

D E F I N I T I O N 2 . 3

Union of Sets: A ∪ B

The union of two sets A and B, written A ∪ B, is the set that consists of all elements

belonging either to A or to B (or to both).

Informally, A ∪ B is formed by putting all the elements of A and B together. The

next example illustrates this definition.

Find the union of the given pairs of sets.

a. { ,

a }

b ∪ { ,

c d

, }

e

b. { ,

1 ,

2 ,

3 ,

4 5} ∪ ∅

c. { ,

m ,

n }

q ∪ { ,

m ,

n }

p

S O L U T I O N

a. { ,

a }

b ∪ { ,

c d

, }

e = { ,

a ,

b ,

c d

, }

e

b. { ,

1 ,

2 ,

3 ,

4 }

5 ∪ ∅ = { ,

1 ,

2 ,

3 ,

4 }

5

c. { ,

m ,

n }

q ∪ { ,

m ,

n }

p = { ,

m ,

n ,

p }

q

■

Notice that although m is a member of both sets in Example 2.2(c), it is listed only

once in the union of the two sets. The union of sets A and B is displayed in a Venn

diagram by shading the portion of the diagram that represents A ∪ B (Figure 2.4).

The notion of set union is the basis for the addition of whole numbers, but only

when disjoint sets are used. Notice how the sets in Example 2.2(a) can be used to

show that 2 + 3 = 5.

Another useful set operation is the intersection of sets.

D E F I N I T I O N 2 . 4

Figure 2.4 Shaded region is

A ∪ B.

Intersection of Sets: A § B

The intersection of sets A and B, written A § B, is the set of all elements common

to sets A and B.

Thus A ∩ B is the set of elements shared by A and B. Example 2.3 illustrates this

definition.

Find the intersection of the given pairs of sets.

a. { ,

a ,

b }

c ∩ { ,

b d

, f

}

b. { ,

a ,

b }

c ∩ { ,

a ,

b }

c

c. { ,

a }

b ∩ { ,

c d

}

S O L U T I O N

a. { ,

a ,

b }

c ∩ { ,

b d

, f

} = { }

b since b is the only element in both sets.

b. { ,

a ,

b }

c ∩ { ,

a ,

b }

c = { ,

a ,

b }

c since a, b

, c

are in both sets.

c. { ,

a }

b ∩ { ,

c d

} = ∅ since there are no elements common to the given two sets.

■

Figure 2.5 displays A ∩ B. Observe that two sets are disjoint if and only if their

intersection is the empty set. Figure 2.3 shows a Venn diagram of two sets whose

intersection is the empty set.

Figure 2.5 Shaded region is

In many situations, instead of considering elements of a set A, it is more productive

A ∩ B.

to consider all elements in the universal set other than those in A. This set is defined next.

c02.indd 48

7/30/2013 2:39:38 PM - Section 2.1 Sets as a Basis for Whole Numbers 49

D E F I N I T I O N 2 . 5

Complement of a Set: A

The complement of a set A, written A, is the set of all elements in the universe, U ,

that are not in A.

The set A is shaded in Figure 2.6.

Find the following sets.

a. A where U = a

{ , b, c, d} and A = a

{ }

b. B where U = { ,

1 ,

2 ,

3 . . .} and B = { ,

2 ,

4 ,

6 . . .}

Figure 2.6 Shaded region is A.

c. A ∪ B and A ∩ B where U = { ,

1 ,

2 ,

3 ,

4 }

5 , A = { ,

1 ,

2 }

3 , and B = { ,

3 }

4

S O L U T I O N

a. A = b

{ , c, d}

b. B = { ,

1 ,

3 ,

5 . . .}

c. A ∪ B = { ,

4 }

5 ∪ { ,

1 ,

2 }

5 = { ,

1 ,

2 ,

4 }

5

A ∩ B = { }

3 = { ,

1 ,

2 ,

4 }

5

■

The next set operation forms the basis for subtraction.

D E F I N I T I O N 2 . 6

Difference of Sets: A - B

The set difference (or relative complement) of set B from set A, written A − B, is the

set of all elements in A that are not in B.

In set-builder notation, A − B = x

{ | x ∈A and x ∉B}. Also, as can be seen in

Figure 2.7, A − B can be viewed as A ∩ B. Example 2.5 provides some examples of

the difference of one set from another.

Find the difference of the given pairs of sets.

a. { ,

a ,

b }

c − { ,

b d

}

Figure 2.7 Shaded region is

A − B.

b. { ,

a ,

b }

c − { }

e

c. { ,

a ,

b ,

c d

} − { ,

b ,

c d

}

S O L U T I O N

a. { ,

a ,

b }

c − { ,

b d

} = { ,

a }

c

b. { ,

a ,

b }

c − { }

e = { ,

a ,

b }

c

c. { ,

a ,

b ,

c d

} − { ,

b ,

c d

} = { }

a

■

Use the set names of A, B, and C as well as the symbols for operations

on sets to describe:

r The blue region in more than one way.

r The red region in more than one way.

c02.indd 49

7/30/2013 2:39:45 PM - 50 Chapter 2 Sets, Whole Numbers, and Numeration

In Example 2.5(c), the second set is a subset of the first. These sets can be used to

show that 4 − 3 = 1.

Another way of combining two sets to form a third set is called the Cartesian prod-

uct. The Cartesian product, named after the French mathematician René Descartes,

forms the basis of whole-number multiplication and is also useful in probability and

geometry. To define the Cartesian product, we need to have the concept of ordered

pair. An ordered pair, written ( ,

a b), is a pair of elements where one of the elements

is designated as first (a in this case) and the other is second (b here). The notion of

an ordered pair differs from that of simply a set of two elements because of the pref-

erence in order. For example, { ,

1 }

2 = { ,

2 }

1 as sets, because they have the same ele-

ments. But (1, 2) ≠ (2, 1) as ordered pairs, since the order of the elements is different.

Two ordered pairs ( ,

a b) and ( ,

c d

) are equal if and only if a = c and b = d.

D E F I N I T I O N 2 . 7

Cartesian Product of Sets: A ë B

The Cartesian product of set A with set B, written A ë B and read “A cross B,” is

the set of all ordered pairs ( ,

a b), where a ∈A and b ∈B.

In set-builder notation, A × B = {(a, b) | a ∈A

and b ∈B}.

Find the Cartesian product of the given pairs of sets.

a. { ,

x ,

y }

z × { ,

m }

n

b. { }

7 × { ,

a ,

b }

c

S O L U T I O N

a. { ,

x ,

y }

z × { ,

m }

n = {( ,

x m), ( ,

x n), ( ,

y m), ( ,

y n), (z, m), (z, n)}

b. { }

7 × { ,

a ,

b }

c = {( ,

7 a), ( ,

7 b), ( ,

7 c)}

■

Notice that when finding a Cartesian product, all possible pairs are formed where

the first element comes from the first and the second element comes from the second

set. Also observe that in Example 2.6 (a), there are three elements in the first set, two

in the second, and six in their Cartesian product, and that 3 × 2 = 6. Similarly, in (b),

these sets can be used to find the whole-number product 1 × 3 = 3.

All of the operations on sets that have been introduced in this subsection result in

a new set. For example, A ∩ B is an operation on two sets that results in the set of

all elements that are common to set A and B. Similarly, C − D is an operation on set

C that results in the set of elements C that are not also in set D. On the other hand,

expressions such as A ⊆ B or x C

∉ are not operations; they are statements that are

either true or false. Table 2.1 lists all such set statements and operations.

TABLE 2.1

STATEMENTS

OPE RATIONS

A ⊆ B

A ∪ B

A ⊂ B

A ∩ B

A ~ B

A

x ∈B

A − B

A − B

A × B

A ⊄ B

x ∉ A

A ≠ B

Check for Understanding: Exercise/Problem Set A #12–32

✔

c02.indd 50

7/30/2013 2:39:51 PM - Section 2.1 Sets as a Basis for Whole Numbers 51

Venn diagrams are often used to solve problems, as shown in Example 2.7.

Thirty elementary teachers were asked which high school cours-

es they appreciated: algebra or geometry. Seventeen appreciated

algebra and 15 appreciated geometry; of these, 5 said that they appreciated both.

How many appreciated neither?

S O L U T I O N Since there are two courses, we draw a Venn diagram with two over-

lapping circles labeled A for algebra and G for geometry [Figure 2.8(a)]. Since 5 teach-

ers appreciated both algebra and geometry, we place a 5 in the intersection of A and

G [Figure 2.8(b)]. Seventeen must be in the A circle; thus 12 must be in the remaining

part of A [Figure 2.8(c)]. Similarly, 10 must be in the G circle outside the intersection

[Figure 2.8(d)]. Thus we have accounted for 12 + 5 + 10 = 27 teachers. This leaves 3 of

the 30 teachers who appreciated neither algebra nor geometry.

■

Figure 2.8

There are several theories concerning the rationale behind the shapes of the

10 digits in our Hindu–Arabic numeration system. One is that the number

represented by each digit is given by the number of angles in the original

digit. Count the “angles” in each digit below. (Here an “angle” is less than

180°.) Of course, zero is round, so it has no angles.

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. Indicate the following sets by the listing method.

c. { ,

1 ,

2 }

3 ⊆ { ,

1 ,

2 }

3

a. Whole numbers between 5 and 9

d. { ,

1 ,

2 }

5 ⊂ { ,

1 ,

2 }

5

b. Whole numbers less than 12 and greater than 2

e. { }

2 ⊆ { ,

1 }

2

c. Even counting numbers less than 15

4. Find four 1-1 correspondences between A and B other than

d. Even counting numbers less than 151

the two given in Figure 2.1.

2. Which of the following sets are equal to { ,

4 ,

5 }

6 ?

5. Determine which of the following sets are equivalent to the

a. { ,

5 }

6

b. { ,

5 ,

4 }

6

set { ,

x ,

y }

z .

c. Whole numbers greater than 3

a. { ,

w ,

x ,

y }

z

b. { ,

1 ,

5 1

}

7

c. { ,

w ,

x }

z

d. Whole numbers less than 7

d. { ,

a ,

b }

a

e. { ,

h ,

n }

s

e. Whole numbers greater than 3 or less than 7

6. Which of the following pairs of sets are equal?

f. Whole numbers greater than 3 and less than 8

a. { ,

a

}

b

and “First two letters of the alphabet”

g. { ,

e f

, g

}

b. { ,

7 ,

8 ,

9 1

}

0 and “Whole numbers greater

3. True or false?

than 7”

a. 7 ∈ 6

{ , 7, 8, 9}

c. { ,

7 ,

a ,

b }

8 and { ,

a ,

b ,

c ,

7 }

8

b. 5 ∉ 2

{ , 3, 4, 6}

d. { ,

x ,

y ,

x }

z and {z, ,

x ,

y }

z

c02.indd 51

7/30/2013 2:40:02 PM - 52 Chapter 2 Sets, Whole Numbers, and Numeration

7. List all the subsets of { ,

a ,

b }

c .

Venn Diagrams on our Web site may help in solving this

problem.

8. List the proper subsets of { ,

s n}.

9. Let A = v

{ , x

, z}, B = w

{ , x, y

}, and C = v

{ , w

, x

, y

, z

}.

In the following, choose all of the possible symbols

(∈, ∉, ⊂ , ,

⊆ ⊆ , ~ , or =) that would make a true statement.

a. z _____ B

b. B _____ C

c. ∅ _____ A

d.

A _____ B e. v _____ C

f. B _____ B

10. Determine which of the following sets are finite. For those

sets that are finite, how many elements are in the set?

a. The set of ears on a typical elephant.

b. { ,

1 ,

2 ,

3 ... , 9 }

9 c. { ,

0 ,

1 ,

2 ,

3 ... , 200}

a. A ∩ (B ∩ C ) b. A − (B ∩ C ) c. A ∪ (B − C )

d. Set of points belonging to a line segment

e. Set of points belonging to a circle

17. Represent the following shaded regions using the symbols

11. Since the set A = { ,

3 ,

6 ,

9 1 ,

2 1 ,

5 . . .} is infinite, there is a

A, B

, C

, ,

∪ ∩, and −. The Chapter 2 eManipulative

proper subset of A that can be matched to A. Find two such

activity Venn Diagrams on our Web site can be used in solv-

proper subsets and describe how each is matched to A.

ing this problem.

12. Given A = { ,

2 ,

4 ,

6 ,

8 1 ,

0 . . .}, and B = { ,

4 ,

8

,

12

,

16

,

20 . . .},

.

a

.

b

answer the following questions.

a. Find the set A ∪ B.

b. Find the set A ∩ B.

c. Is B a subset of A ? Explain.

d. Is A equivalent to B ? Explain.

e. Is A equal to B ? Explain.

f. Is B a proper subset of A ? Explain.

13. True or false?

a. For all sets X and Y , either X ⊆Y or Y ⊆ X.

c.

b. If A is an infinite set and B ⊆ A, then B also is an infinite

set.

c. For all finite sets A and B, if A ∩ B = ∅, then the

number of elements in A ∪ B equals the number of

elements in A plus the number of elements in B.

14. The regions A ∩ B and A ∪ B can each be formed by shad-

ing the set A using lines in one direction and the set B using

lines in another direction. The region with shading in both

directions is A ∩ B and the entire shaded region is A ∪ B.

18. Draw Venn diagrams that represent sets A and B as shown

Use the Chapter 2 eManipulative activity Venn Diagrams

here.

on our Web site to do the following:

a. A ⊂ B b. A ∩ B = ∅ c. A ∪ B = A

a. Find the region (A ∪ B) ∩ C.

19. In the drawing, C is the interior of the circle, T is the

b. Find the region A ∪ (B ∩ C ).

interior of the triangle, and R is the interior of the rect-

c. Based on the results of parts a and b, what conclusions

angle. Copy the drawing on a sheet of paper, then shade

can you draw about the placement of parentheses?

in each of the following regions.

15. Draw a Venn diagram like the following one for each part.

Then shade each Venn diagram to represent each of the

sets indicated.

a. C ∪ T

b. C ∩ R

c. (C ∩ T ) ∪ R

d. (C ∪ R) ∪ T

e. (C ∩ R) ∩ T

f. C ∩ (R ∩ T )

a. S b. S ∩ T c. (S − T ) ∪ (T − S ) d. U − S

20. Let W = {women who have won Nobel Prizes},

A = {Americans who have won Nobel Prizes}, and

16. A Venn diagram can be used to illustrate more than

C = {winners of the Nobel Prize in chemistry}. Describe the

two sets. Shade the regions that represent each of the

elements of the following sets.

following sets. The Chapter 2 eManipulative activity

a. W ∪ A b. W ∩ A c. A ∩ C

c02.indd 52

7/30/2013 2:40:14 PM - Section 2.1 Sets as a Basis for Whole Numbers 53

21. Let A = a

{ , b

, c

}, B

= b

{ , c

}, and C = e

{ }. Find each of the

27. Verify that A ∪ B = A ∩ B in two different ways as follows:

following.

a. Let U = { ,

1 ,

2 ,

3 ,

4 ,

5 }

6 , A = { ,

2 ,

3 }

5 , and B = { ,

1 }

4 .

a. A ∩ B b. B ∪ C c. A − B

List the elements of the sets A ∪ B and A ∩ B. Do the

two sets have the same members?

22. Find each of the following differences.

b. Draw and shade a Venn diagram for each of the sets

a. { ,

s n, ,/ }

h −{ ,

n }

h

A ∪ B and A ∩ B. Do the two Venn diagrams look the

b. { ,

0 ,

1 ,

2 . . .} − { ,

12

,

13

,

14 . . .}

same? NOTE: The equation A ∪ B = A ∩ B is one of two

c. {people} − {married people}

laws called DeMorgan’s laws.

d. { ,

a ,

b ,

c d

} − { }

28. Find the following Cartesian products.

a. { }

a ×{ ,

b }

c

23. Let A = { ,

2 ,

4 ,

6 ,

8 1 ,

0 . . .} and B = {4, 8,12,16, . . .}.

b. { }

5 ×{ ,

a ,

b }

c

a. Find A − B.

c. { ,

a

}

b

×{ ,

1 ,

2 }

3

b. Find B − A.

d. { ,

2 }

3 × { ,

1 }

4

c. Based on your observations in parts a and b, describe a

e. { ,

a ,

b }

c ×{ }

5

general case for which A − B = ∅.

f. { ,

1 ,

2 }

3 ×{ ,

a }

b

24. Given A = { ,

0 ,

1 ,

2 ,

3 ,

4 }

5 , B

= { ,

0 ,

2 ,

4 ,

6 ,

8 1

}

0 , and

29. Determine how many ordered pairs will be in the

C = { ,

0 ,

4 }

8 , find each of the following.

following sets.

a. A ∪ B

b. B ∪ C

c. A ∩ B

a. { ,

1 ,

2 ,

3 }

4 × { ,

a }

b

d. B ∩ C

e. B − C

f. (A ∪ B) − C

b. { ,

m ,

n }

o × { ,

1 ,

2 ,

3 }

4

g. A ∩ ∅

30. If A has two members, how many members does B have

25. Let M = the set of the months of the year,

when A × B has the following number of members? If an

J = {January, June, July}, S = {June, July, August},

answer is not possible, explain why.

W = {December, January, February}. List the members of

a. 4

b. 8

c. 9

each of the following:

d. 50

e. 0

f. 23

a. J ∪ S

b. J ∩W

31. The Cartesian product, A × B, is given in each of the fol-

c. S ∩W

d. J ∩ (S ∪W )

lowing parts. Find A and B.

e. M − (S ∪W )

f. J − S

a. {( ,

a 2), ( ,

a 4), ( ,

a 6)}

26. Let A = { ,

3 ,

6 ,

9 1 ,

2 1 ,

5 1 ,

8 2 ,

1 2 ,

4 . . .} and

b. {( ,

a b

), ( ,

b b), ( ,

b a

), ( ,

a a

)}

B = { ,

6 1 ,

2 1 ,

8 2 ,

4 . . .}.

32. True or false?

a. Is B ⊆ A?

a. {( ,

4 5), ( ,

6 7)} = {( ,

6 7), ( ,

4 5)}

b. Find A ∪ B.

b. {( ,

a b), ( ,

c d )} = {( ,

b a), ( ,

c d )}

c. Find A ∩ B.

c. {( ,

4 5), ( ,

7 6)} = {( ,

7 6), ( ,

5 4)}

d. In general, when B ⊆ A, what is true about A ∪ B ?

About A ∩ B?

d. {( ,

5 6), ( ,

7 4)} ⊆ { ,

4 ,

5 }

6 × { ,

5 ,

7 }

9

PROBLEMS

33. a. If X has five elements and Y has three elements, what is

37. a. When does D ∩ E = D?

the greatest number of elements possible in X ∩Y ?

b. When does D ∪ E = D?

In X ∪Y ?

c. When does D ∩ E = D ∪ E ?

b. If X has x elements and Y has y elements with x greater

than or equal to y, what is the greatest number of ele-

38. Carmen has 8 skirts and 7 blouses. Show how the con-

ments possible in X ∩Y ? In X ∪Y ?

cept of Cartesian product can be used to determine how

many different outfits she has.

34. How many different 1-1 correspondences are possible

39. How many matches are there if 32 participants enter

between A = { ,

1 ,

2 ,

3 }

4 and B = a

{ , b

, c

, d

}?

a single-elimination tennis tournament (one loss elimi-

nates a participant)?

35. How many subsets does a set with the following number

of members have?

40. Can you show a 1-1 correspondence between the points on

a. 0

b. 1

c. 2

base AB of the given triangle and the points on the two sides

d. 3

e. 5

f. n

AC and CB? Explain how you can or why you cannot.

36. If it is possible, give examples of the following. If it is not

possible, explain why.

a. Two sets that are not equal but are equivalent

b. Two sets that are not equivalent but are equal

c02.indd 53

7/30/2013 2:40:24 PM - 54 Chapter 2 Sets, Whole Numbers, and Numeration

41. Can you show a 1-1 correspondence between the points on

a. How many of the 100 people surveyed did not keep up

chord AB and the points on arc ACB? Explain how you

with current events by the three sources listed?

can or why you cannot.

b. How many of the 100 people surveyed read the paper

but did not watch TV news?

c. How many of the 100 people surveyed used only one of

the three sources listed to keep up with current events?

43. At a convention of 375 butchers (B), bakers ( )

A , and

candlestick makers (C ), there were

50 who were both B and A but not C

70 who were B but neither A nor C

60 who were A but neither B nor C

42. A poll of 100 registered voters designed to find out how

40 who were both A and C but not B

voters kept up with current events revealed the following

50 who were both B and C but not A

facts:

80 who were C but neither A nor B

65 watched the news on television.

How many at the convention were A, B

, and C?

39 read the newspaper.

39 listened to radio news.

44. A student says that A − B means you start with all the

20 watched TV news and read the newspaper.

elements of A and you take away all the elements of B.

27 watched TV news and listened to radio news.

So A × B must mean you take all the elements of A

9 read the newspaper and listened to radio news.

and multiply them times all the elements of B. Do you

6 watched TV news, read the newspaper, and listened

agree with the student? How would you explain your

to radio news.

reasoning?

EXERCISE/PROBLEM SET B

EXERCISES

1. Indicate the following sets by the listing method.

b. { ,

0 ,

1 ,

2 }

3 and “whole numbers less than 4’’

a. Whole numbers greater than 8

c. { ,

1 ,

2 ,

1 }

2 and { ,

2 ,

1 ,

2 }

1

b. Odd counting numbers between 2 and 11

d. { ,

1 }

2 and { ,

1 ,

1 }

2 .

c. Odd whole numbers less than 100

7. List all subsets of { ,

s ,

n }

u . Which are proper subsets?

d. Whole numbers less than 0

8. List the proper subsets of { ,

x ,

y }

z

2. Which of the following sets are equal to { ,

0 ,

2 ,

4 6}?

a. { ,

0 ,

1 ,

2 ,

3 ,

4 ,

5 6}

b. { ,

6 ,

2 ,

0 4}

9. Let A = { ,

1 ,

2 ,

3 ,

4 }

5 , B = { ,

3 ,

4 }

5 , and C = { ,

4 ,

5 }

6 .

c. Even counting numbers less than 7

In the following, choose all of the possible symbols

d. Even whole numbers less than 7

( ,

∈ ,

∉ ,

⊂ ,

⊆ ⊆ , ~, or =) that would make a true

statement.

e. Even whole numbers less than 6

a. 2 _____ A

b. B _____ A

c. C _____ B

f. { ,

a ,

c ,

e g}

g. { ,

0 ,

2 ,

4 ,

6 }

0

d. 6 _____ C

e. A _____ A

f. B ∩ C _____ A

3. True or false?

10. Determine which of the following sets are finite. For

a. 2 ∈ 1

{ , 2, 3}

b. 1 ∉ 0

{ ,1, 2}

3

those sets that are finite, how many elements are in

c. { ,

4 }

3 ⊂ { ,

2 ,

3 }

4

d. ∅ ⊆ { }

the set?

e. { ,

1 }

2 ⊆ { }

2

a.

The set of fingers on your hand.

b.

{ ,

2 ,

4 ,

6 ,

8 . . .}

4. Show three different 1-1 correspondences between

c.

{ ,

4 ,

8

,

12 . . . ,

}

100

{ ,

1 ,

2 ,

3 }

4 and { ,

x ,

y z

, }

w .

d.

The set of sides of a polygon.

5. Determine which of the following sets are equivalent to the

11. Show that the following sets are infinite by matching each

set {apple, apple, pear}.

with a proper subset of itself.

a. {apple, pear, apple, pear}

a. { ,

2 ,

4 ,

6 . . . , ,

n . . .}

b. {apple, pear}

b. { ,

50

,

51

,

52

,

53 . . . , ,

n . . .}

c. {squash, eggplant}

d. {pear, apple, apple}

12. Let A = { ,

0 1 ,

0 2 ,

0 3 ,

0 . . .} and B = { ,

5 1 ,

5 2 ,

5 3 ,

5 . . .}. Decide

which of the following are true and which are false.

e. { ,

x

}

y

Explain your answers.

6. Which of the following pairs of sets are equal?

a. A equals B b. B is equivalent to A c. A ∩ B ≠ ∅

a. { ,

0 ,

1 }

2 and “first three counting numbers’’

d. B is equivalent to a proper subset of A.

c02.indd 54

7/30/2013 2:40:34 PM - Section 2.1 Sets as a Basis for Whole Numbers 55

e. There is a proper subset of B that is equivalent to a

c.

proper subset of A.

f. A ∪ B is all multiples of 5.

13. True or false?

a. The empty set is a subset of every set.

b. The set {

,

105

,

110

,

115

,

120 . . .} is an infinite set.

c. If A is an infinite set and B ⊆ A, then B is a finite set.

14. Use Venn diagrams to determine which, if any, of the

following statements are true for all sets A, B

, and C: The

Chapter 2 eManipulative Venn Diagrams on our Web site

18. Draw Venn diagrams that represent sets A and B as

may help in determining which statements are true.

described here.

a. A ∪ (B ∪ C ) = (A ∪ B) ∪ C

a. A ∩ B ≠ ∅

b. A ∪ B = A

b. A ∪ (B ∩ C ) = (A ∪ B) ∩ C

c (A ∩ B) − B = ∅

c. A ∩ (B ∩ C ) = (A ∩ B) ∩ C

d. A ∩ (B ∪ C ) = (A ∩ B) ∪ C

19. Make six copies of the diagram and shade regions in

parts a–f.

15. Draw a Venn diagram like the following for each part.

Then shade each Venn diagram to represent each of the

sets indicated.

a. T − S

b. S ∪ T

c. (S − T ) ∩ (T − S )

16. A Venn diagram can be used to illustrate more than two

a.

(A ∩ B) ∩ C

sets. Shade the regions that represent each of the fol-

b. (C ∩ D) − (A ∪ B)

lowing sets. The Chapter 2 eManipulative activity Venn

c. (A ∩ B) ∩ (C ∩ D)

Diagrams on our Web site may help in solving this

d. (B ∩ C ) ∩ D

problem.

e. (A ∩ D) ∪ (C ∩ B)

f. (A ∩ B) ∪ (C ∩ )

A

20. Let A = a

{ , b

, c

, d

, e

}, B

= c

{ , d

, e

, f

, g

}, and

C = a

{ , e

, f

, h}. List the members of each set.

a. A ∪ B

b. A ∩ B

c. (A ∪ B) ∩ C

d. A ∪ (B ∩ C )

21. If A is the set of all sophomores in a school and B is the

set of students who belong to the orchestra, describe the

following sets in words.

∪

∩

−

−

a. (A ∪ B) ∩ C b. A ∩ (B ∩ C ) c. (A ∪ B) − (B ∩ C )

a. A

B b. A

B c. A

B d. B

A

22. Find each of the following differences.

17. Represent the following shaded regions using the

a. { ,

h i

, j

, k

} − {k}

symbols A, B, C, ,

∪ ,

∩ and −. The Chapter 2

b. { ,

3 1

,

0 1

}

3 − { }

eManipulative activity Venn Diagrams on our Web site

can be used in solving this problem.

c. {two-wheeled vehicles} − {two-wheeled vehicles that are

not bicycles}

a.

b.

d. { ,

0 ,

2 ,

4 ,

6 . . . ,

}

20 − { ,

12

,

14

,

16

,

18

}

20

23. In each of the following cases, find B − A.

a. A ∩ B = ∅ b. A = B c. B ⊆ A

24. Let R = a

{ , b, c}, S = c

{ , d, e, f }, T = x

{ , y, z}. List the

elements of the following sets.

a. R ∪ S b. R ∩ S c. R ∪ T d. R ∩ T

e. S ∪ T f. S ∩ T g. T ∪ ∅

c02.indd 55

7/30/2013 2:40:42 PM - 56 Chapter 2 Sets, Whole Numbers, and Numeration

25. Let A = { ,

50 ,

55

,

60

,

65

,

70

,

75 }

80

29. Determine how many ordered pairs will be in A × B under

B = { ,

50

,

60

,

70

}

80

the following conditions.

C = { ,

60 7

70, 80}

a. A has one member and B has four members.

b. A has two members and B has four members.

D = 55

{ , 65}

c. A has three members and B has seven members.

List the members of each set.

a. A ∪ (B ∩ C ) b. (A ∪ B) ∩ C

30. Find sets A and B so that A × B has the following number

c. (A ∩ C ) ∪ (C ∩ D)

d. (A ∩ C ) ∩ (C ∪ D)

of members.

e. (B − C ) ∩ A f. (A − D) ∩ (B − C )

a. 1

b. 2

c. 3

d. 4

e. 5

f. 6

g. 7

h. 0

26. a. If x ∈X ∩Y , is x ∈X ∪Y ? Justify your answer.

×

b. If x ∈X ∪Y , is x ∈X ∩Y ? Justify your answer.

31. The Cartesian product, X

Y , is given in each of the fol-

lowing parts. Find X and Y .

27. Verify that A ∩ B = A ∪ B in two different ways as follows:

a. {( ,

b c

), ( ,

c c

)}

a. Let U = { ,

2 ,

4 ,

6 ,

8 ,

10 ,

12 ,

14 }

16 , A = { ,

2 ,

4 ,

8 1

}

6 , and

b. {( ,

2 ),

1 ( ,

2 2), ( ,

2 3), ( ,

5 ),

1 ( ,

5 2), ( ,

5 3)}

B = { ,

4 ,

8 ,

12 }

16 . List the elements of the sets A ∩ B and

A ∪ B. Do the two sets have the same members?

32. True or false?

b. Draw and shade a Venn diagram for each of the sets

a. {( ,

3 4), ( ,

5 6)} = {( ,

3 4), ( ,

6 5)}

A ∩ B = A ∪ B. Do the two Venn diagrams look the same?

b. {( ,

a c

), (d, b

)} = { ,

a ,

c d

, }

b

NOTE: The equation A ∩ B = A ∪ B is one of the two laws

c. {( ,

c d ), ( ,

a b

)} = {( ,

c a

), (d, b

)}

called DeMorgan’s laws.

d. {( ,

4 5), ( ,

6 7)} ⊆ { ,

4 ,

5 }

6 × { ,

5 ,

7 }

9

28. Find the following Cartesian products.

a. { ,

a ,

b }

c ×{ }

1

b. { ,

1 }

2 × { ,

p ,

q }

r

c. { ,

p ,

q }

r × { ,

1 }

2

d. { }

a ×{ }

1

PROBLEMS

33. How many 1-1 correspondences are there between the fol-

lowing pairs of sets?

a. Two 2-member sets

b. Two 4-member sets

c. Two 6-member sets

d. Two sets each having m members

34. Find sets (when possible) satisfying each of the following

conditions.

38. A schoolroom has 13 desks and 13 chairs. You want

a. Number of elements in A plus number of elements in B

to arrange the desks and chairs so that each desk

is greater than number of elements in A ∪ B.

has a chair with it. How many such arrangements

b. Number of elements in I plus number of elements in J is

are there?

less than number of elements in I ∪ J .

c. Number of elements in E plus number of elements in F

39. A university professor asked his class of 42 students

equals number of elements in E ∪ F .

when they had studied for his class the previous

d. Number of elements in G plus number of elements in K

weekend. Their responses were as follows:

equals number of elements in G ∩ K .

9 had studied on Friday.

35. Your house can be painted in a choice of 7 exterior colors

18 had studied on Saturday.

and 15 interior colors. Assuming that you choose only 1

30 had studied on Sunday.

color for the exterior and 1 color for the interior, how many

3 had studied on both Friday and Saturday.

different ways of painting your house are there?

10 had studied on both Saturday and Sunday.

6 had studied on both Friday and Sunday.

36. Show a 1-1 correspondence between the points on the

2 had studied on Friday, Saturday, and Sunday.

given circle and the triangle in which it is inscribed.

Explain your procedure.

Assuming that all 42 students responded and answered

honestly, answer the following questions.

a. How many students studied on Sunday but not on

either Friday or Saturday?

b. How many students did all of their studying on

37. Show a 1-1 correspondence between the points on the

one day?

given triangle and the circle that circumscribes it. Explain

c. How many students did not study at all for this class

your procedure.

last weekend?

c02.indd 56

7/30/2013 2:40:48 PM - Section 2.2 Whole Numbers and Numeration 57

40. At an automotive repair shop, 50 cars were inspected.

43. Tonia determines that a set with three elements has as

Suppose that 23 cars needed new brakes and 34 cars need-

many proper subsets as it has nonempty subsets. Is she

ed new exhaust systems.

correct? Explain.

a. What is the least number of cars that could have needed

44. Amanda says that the sets { ,

1 ,

2 ,

3 . . .} and { ,

2 ,

4 ,

6 . . .} can-

both?

not be matched because the first set has many numbers that

b. What is the greatest number of cars that could have

the second one does not. Is she correct? Explain.

needed both?

c. What is the greatest number of cars that could have

45. LeNae says that A ∪ B must always have more elements

needed neither?

than either A or B. Is she correct? Explain.

41. If A is a proper subset of B, and A has 23 elements, how

46. Marshall says that A ∩ B always has fewer elements than

many elements does B have? Explain.

either A or B. Is he correct? Explain.

Analyzing Student Thinking

47. Michael asserts that if A and B are infinite sets, A ∩ B

must be an infinite set. Is he correct? Explain.

42. Kylee says that if two sets are equivalent, they must be

equal. Jake says that if two sets are equal, then they must

48. DeDee says that B must be a subset of A to be able to find

be equivalent. Who is correct? Explain.

the set A − B. How should you respond?

WHOLE NUMBERS AND NUMERATION

When a young child is asked to count the blocks in the set shown below, he will use his

memorized cadence of “one, two, three, four . . . ” while pointing to the blocks. In coming

to understand the concept of number, the following two scenarios are possible:

Child 4 counts out the blocks correctly (points to each block once and only once saying “one, two, three”) and when

asked to show 3 blocks, points to the third block.

Child 5 counts the blocks correctly and then when asked to show 3 blocks, indicates all three blocks.

Identify which student is incorrect and explain the possible rationale behind his reasoning?

Children’s Literature

Numbers and Numerals

www.wiley.com/college/musser

See “George’s Store at the

As mentioned earlier, the study of the set of whole numbers, W = { ,

0 ,

1 ,

2 ,

3 ,

4 . . .}, is

Shore’’ by Francine Bessede.

the foundation of elementary school mathematics. But what precisely do we mean

by the whole number 3? A number is an idea, or an abstraction, that represents a

NCTM Standard

quantity. The symbols that we see, write, or touch when representing numbers are

All students should develop

called numerals. There are three common uses of numbers. The most common use

understanding of the relative

of whole numbers is to describe how many elements are in a finite set. When used

position and magnitude of whole

in this manner, the number is referred to as a cardinal number. A second use is

numbers and of ordinal and

cardinal numbers and their con-

concerned with order. For example, you may be second in line, or your team may

nections.

be fourth in the standings. Numbers used in this way are called ordinal numbers.

Finally, identification numbers are used to name such things as telephone numbers,

Common Core –

bank account numbers, and social security numbers. In this case, the numbers

Kindergarten

are used in a numeral sense in that only the symbols, rather than their values, are

Understand the relationship

important. Before discussing our system of numeration or symbolization, the con-

between numbers and quantities;

connect counting to cardinality.

cept of cardinal number will be considered.

c02.indd 57

7/30/2013 2:40:51 PM - 58

Chapter 2 Sets, Whole Numbers, and Numeration

What is the number 3? What do you think of when you see the sets in Figure 2.9?

Reflection from Research

There is a clear separation of

development in young children

concerning the cardinal and ordi-

nal aspects of numbers. Despite

the fact that they could utilize

the same counting skills, the

understanding of ordinality by

Figure 2.9

young children lags well behind

the understanding of cardinality

First, there are no common elements. One set is made up of letters, one of shapes,

(Bruce & Threlfall, 2004).

one of Greek letters, and so on. Second, each set can be matched with every other set.

Now imagine all the infinitely many sets that can be matched with these sets. Even

though the sets will be made up of various elements, they will all share the common

attribute that they are equivalent to the set { ,

a

,

b

}

c . The common idea that is asso-

ciated with all of these equivalent sets is the number 3. That is, the number 3 is the

attribute common to all sets that match the set { ,

a

,

b

}

c . Similarly, the whole number

2 is the common idea associated with all sets equivalent to the set { ,

a

}

b . All other

nonzero whole numbers can be conceptualized in a similar manner. Zero is the idea,

or number, one imagines when asked: “How many elements are in the empty set?”

Although the preceding discussion regarding the concept of a whole number

may seem routine for you, there are many pitfalls for children who are learning

the concept of numerousness for the first time. Chronologically, children first learn

how to say the counting chant “one, two, three, . . . .” However, saying the chant

and understanding the concept of number are not the same thing. Next, children

must learn how to match the counting chant words they are saying with the objects

they are counting. For example, to count the objects in the set { ,

n ,

s }

u , a child

must correctly assign the words “one, two, three” to the objects in a 1-1 fashion.

Actually, children first learning to count objects fail this task in two ways: (1) They

fail to assign a word to each object, and hence their count is too small; or (2) they

count one or more objects at least twice and end up with a number that is too large.

To reach the final stage in understanding the concept of number, children must be

able to observe several equivalent sets, as in Figure 2.9, and realize that, when they

count each set, they arrive at the same word. Thus this word is used to name the

attribute common to all such sets.

The symbol n(A) is used to represent the number of elements in a finite set A. More

precisely, (1) n(A) = m if A ~ { ,

1 ,

2 . . . , m}, where m is a counting number, and (2)

n(∅) = .

0 Thus

n

a

({ , b, c}) = 3 since { ,

a

,

b

}

c ~ { ,

1

,

2

}

3 , and

n

a

({ , b, c, . . . , z}) = 26 since

{ ,

a ,

b ,

c . . . , }

z ~ { ,

1 ,

2 ,

3 . . . , 2 }

6 ,

and so on, for other finite sets.

Check for Understanding: Exercise/Problem Set A #1–4

✔

Ordering Whole Numbers

NCTM Standard

Children may get their first introduction to ordering whole numbers through the

All students should count with

counting chant “one, two, three, . . . .” For example, “two” is less than “five,” since

understanding and recognize

“two” comes before “five” in the counting chant.

“how many” in sets of objects.

A more meaningful way of comparing two whole numbers is to use 1-1 corre-

spondences. We can say that 2 is less than 5, since any set with two elements matches

a proper subset of any set with five elements (Figure 2.10).

c02.indd 58

7/30/2013 2:40:53 PM - Section 2.2 Whole Numbers and Numeration 59

Common Core –

Kindergarten

Count to answer ``how many?’’

questions about as many as 20

things arranged in a line, a rect-

angular array, or a circle, or as

many as 10 things in a scattered

configuration; given a number

Figure 2.10

from 1-20, count out that many

objects.

The general set formulation of “less than” follows.

Common Core –

Kindergarten

D E F I N I T I O N 2 . 8

Identify whether the number of

objects in one group is greater

Ordering Whole Numbers

than, less than, or equal to the

number of objects in another

Let a = n(A) and b = n(B). Then a < b (read “a is less than b”) or b > a (read “b is

group (e.g., by using matching

greater than a”) if A is equivalent to a proper subset of B.

and counting strategies).

The “greater than” and “less than” signs can be combined with the equal sign to

produce the following symbols: a ≤ b (a is less than or equal to b) and b ≥ a (b is greater

than or equal to a).

Reflection from Research

A third common way of ordering whole numbers is through the use of the whole-

Most counting mistakes made by

number “line” (Figure 2.11). Actually, the whole-number line is a sequence of equally

young children can be attributed

to not keeping track of objects

spaced marks where the numbers represented by the marks begin on the left with 0

that have already been counted

and increase by one each time we move one mark to the right.

and objects that still need to be

counted (Fuson, 1988).

Figure 2.11

Determine the greater of the two numbers 3 and 8 in three dif-

ferent ways.

S O L U T I O N

a. Counting Chant: One, two, three, four, fi ve, six, seven, eight. Since “three” precedes

“eight,” eight is greater than three.

b. Set Method: Since a set with three elements can be matched with a proper

subset of a set with eight elements, 3 < 8 and 8 > 3 [Figure 2.12(a)].

c. Whole-Number Line: Since 3 is to the left of 8 on the number line, 3 is less than 8

and 8 is greater than 3 [Figure 2.12(b)].

■

Figure 2.12

Check for Understanding: Exercise/Problem Set A #5–6

✔

c02.indd 59

7/30/2013 2:40:55 PM - 60 Chapter 2 Sets, Whole Numbers, and Numeration

Today our numeration system has the symbol “2” to represent the number of eyes a person

has. The symbols “1” and “0” combine to represent the number of toes a person has, “10.”

The Roman numeration system used the symbol “X” to represent the number of toes and “C” to represent the number

of years in a century.

Using only the three symbols above, devise your own numeration system and show how you can use your system to represent

all of the quantities 0, 1, 2, 3, 4, . . . , 100. What properties does your numeration system have?

Children’s Literature

Numeration Systems

www.wiley.com/college/musser

See “Count on Your Fingers

To make numbers more useful, systems of symbols, or numerals, have been developed

African Style” by Claudia

to represent numbers. In fact, throughout history, many different numeration systems

Zaslavsky.

have evolved. The following discussion reviews various ancient numeration systems

with an eye toward identifying features of those systems that are incorporated in our

present system, the Hindu–Arabic numeration system.

The Tally Numeration System The tally numeration system is composed

of single strokes, one for each object being counted (Figure 2.13).

Figure 2.13

The next six such tally numerals are

Reflection from Research

An advantage of this system is its simplicity; however, two disadvantages are that (1)

Teaching other numeration

large numbers require many individual symbols, and (2) it is difficult to read the numer-

systems, such as the Chinese

als for such numbers. For example, what number is represented by these tally marks?

system, can reinforce a student’s

conceptual knowledge of place

value (Uy, 2002).

The tally system was improved by the introduction of grouping. In this case, the

fifth tally mark was placed across every four to make a group of five. Thus the last

tally numeral can be written as follows:

Grouping makes it easier to recognize the number being represented; in this case,

there are 37 tally marks.

The Egyptian Numeration System The

Egyptian numeration system, which

developed around 3400 B.C.E., involves grouping by ten. In addition, this system

introduced new symbols for powers of 10 (Figure 2.14).

Figure 2.14

Examples of some Egyptian numerals are shown in Figure 2.15. Notice how this

system required far fewer symbols than the tally system once numbers greater than

c02.indd 60

7/30/2013 2:40:56 PM - Section 2.2 Whole Numbers and Numeration 61

10 were represented. This system is also an additive system, since the values for the

various individual numerals are added together.

Figure 2.15

Notice that the order in which the symbols are written is immaterial. A major dis-

advantage of this system is that computation is cumbersome. Figure 2.16 shows, in

Egyptian numerals, the addition problem that we write as 764 + 598 = 1362. Here 51

individual Egyptian numerals are needed to express this addition problem, whereas

our system requires only 10 numerals!

Figure 2.16

The Roman Numeration System The Roman numeration system, which

developed between 500 B.C.E. and 100 C.E. also uses grouping, additivity, and many

symbols. The basic Roman numerals are listed in Table 2.2.

Children’s Literature

TABLE 2.2

www.wiley.com/college/musser

ROMAN NUMERAL

VALUE

See “Fun with Roman Numerals’’

by David Adler.

I

1

V

5

X

10

L

50

C

100

D

500

M

1000

Roman numerals are made up of combinations of these basic numerals, as

illustrated next.

CCLXXXI (equals 281) MCVIII (equals 1108)

Notice that the values of these Roman numerals are found by adding the values of

the various basic numerals. For example, MCVIII means 1000 + 100 + 5 + 1 + 1 + 1, or

1108. Thus the Roman system is an additive system.

TABLE 2.3

Two new attributes that were introduced by the Roman system were a subtractive

principle and a multiplicative principle. Both of these principles allow the system to

ROMAN NUMERAL

VALUE

use fewer symbols to represent numbers. The Roman numeration system is a sub-

IV

4

tractive system since it permits simplifications using combinations of basic Roman

IX

9

numerals: IV (I to the left of V means five minus one) for 4 rather than using IIII, IX

XL

40

(ten minus one) for 9 instead of VIIII, XL for 40, XC for 90, CD for 400, and CM

XC

90

for 900 (Table 2.3).

CD

400

Thus, when reading from left to right, if the values of the symbols in any pair of

CM

900

symbols increase, group the pair together. The value of this pair, then, is the value of the

larger numeral less the value of the smaller. One may wonder if IC is an acceptable way

to write 99. The answer is no because of the additional restriction that only I’s, X’s, and

C’s may be subtracted, but only from the next two larger numerals in each case. Thus,

the symbols shown in Table 2.3 are the only numerals where the subtraction principle is

applied. To evaluate a complex Roman numeral, one looks to see whether any of these

subtractive pairs are present, groups them together mentally, and then adds values from

left to right. For example,

in MCMXLIV

think M CM XL IV, which is 1000 + 900 + 40 + 4.

c02.indd 61

7/30/2013 2:40:58 PM - 62 Chapter 2 Sets, Whole Numbers, and Numeration

Notice that without the subtractive principle, 14 individual Roman numerals

would be required to represent 1944 instead of the 7 numerals used in MCMXLIV.

Also, because of the subtractive principle, the Roman system is a positional system,

since the position of a numeral can affect the value of the number being represented.

For example, VI is six, whereas IV is four.

Express the following Roman numerals in our numeration system:

NCTM Standard

All students should develop a

a. MCCCXLIV

sense of whole numbers and rep-

resent and use them in flexible

b. MMCMXCIII

ways including relating, compos-

c. CCXLIX

ing, and decomposing numbers.

S O L U T I O N

a. Think: MCCC XL IV, or 1300 + 40 + 4 = 1344

b. Think: MM CM XC III, or 2000 + 900 + 90 + 3 = 2993

c. Think: CC XL IX, or 200 + 40 + 9 = 249

■

The Roman numeration system also utilized a horizontal bar above a numeral to

represent 1000 times the number. For example, V meant 5 times 1000, or 5000; XI meant

11,000; and so on. Thus the Roman system was also a multiplicative system. Although

expressing numbers using the Roman system requires fewer symbols than the Egyptian

system, it still requires many more symbols than our current system and is cumbersome

for doing arithmetic. In fact, the Romans used an abacus to perform calculations instead

of paper/pencil methods as we do (see the Focus On at the beginning of Chapter 3).

The Babylonian Numeration System The Babylonian numeration system,

which evolved between 3000 and 2000 B.C.E., used only two numerals, a one and a

ten (Figure 2.17). For numbers up to 59, the system was simply an additive system.

Figure 2.17

For example, 37 was written using 3 tens and 7 ones (Figure 2.18).

However, even though the Babylonian numeration system was developed about the

same time as the simpler Egyptian system, the Babylonians used the sophisticated notion

of place value, where symbols represent different values depending on the place in which

they were written. The symbol could represent 1 or 1 ¥ 60 or 1 ¥ 60 ¥ 60 depending on

where it is placed. Since the position of a symbol in a place-value system affects its value,

place-value systems are also positional. Thus, the Babylonian numeration system is

another example of a positional system. Figure 2.19 displays three Babylonian numerals

that illustrate this place-value attribute, which is based on 60. Notice the subtle spacing

Figure 2.18

of the numbers in Figure 2.19(a) to assist in understanding that the represents a 1 ¥ 60

and not a 1. Similarly, the symbol,

in Figure 2.19(b) is spaced slightly to the left

to indicate that it represents a 12(60) instead of just 12. Finally, the symbols in Figure

2.19(c) have 2 spaces to indicate is multiplied by 60 ¥ 60 and

is multiplied by 60.

Figure 2.19

Unfortunately, in its earliest development, this system led to some confusion. For

example, as illustrated in Figure 2.20, the numerals representing 74 (= 1 ¥ 60 + 14) and

3614 (= 1 ¥ 60 ¥ 60 + 0 ¥ 60 + 14) differed only in the spacing of symbols. Thus, there was

a chance for misinterpretation. From 300 B.C.E. on, a separate symbol made up of two

small triangles arranged one above the other was used to serve as a placeholder to

indicate a vacant place (Figure 2.21). This removed some of the ambiguity. However,

two Babylonian tens written next to each other could still be interpreted as 20, or 610,

or even 3660. Although their placeholder acts much like our zero, the Babylonians

did not recognize zero as a number.

c02.indd 62

7/30/2013 2:41:00 PM - Section 2.2 Whole Numbers and Numeration 63

Figure 2.20

Figure 2.21

Express the following Babylonian numerals in our numeration

system.

S O L U T I O N

.

a

.

b

c.

■

The Mayan Numeration System The

Mayan numeration system, which developed

between 300 and 900 C.E., was a vertical place-value system, and it introduced a symbol

for zero. The system used only three elementary numerals (Figure 2.22).

Figure 2.22

Reflection from Research

Several Mayan numerals are shown in Figure 2.22 together with their respective val-

Zero is an important number that

should be carefully integrated

ues. The symbol for twenty in Figure 2.23 illustrates the use of place value in that the

into early number experiences. If

“dot” represents one “twenty” and the

represents zero “ones.”

children do not develop a clear

understanding of zero early on,

misconceptions can arise that

might be detrimental to the fur-

ther development of their under-

standing of number properties

Figure 2.23

and subsequent development

of algebraic thinking (Anthony &

Various place values for this system are illustrated in Figure 2.24.

Walshaw, 2004).

Figure 2.24

The bottom section represents the number of ones (3 here), the second section

from the bottom represents the number of 20s (0 here), the third section from the

bottom represents the number of 18 ¥ 20s (6 here), and the top section represents the

number of 18 ¥ 20 ¥ 20s (1 here). Reading from top to bottom, the value of the number

represented is 1 18

(

¥ 20 ¥ 20) + 6 1

( 8 ¥ 20) + 0(20) + 3 1

( ) or 9363. (See the Focus On at

the beginning of this chapter for additional insight into this system.)

c02.indd 63

7/30/2013 2:41:03 PM - 64 Chapter 2 Sets, Whole Numbers, and Numeration

Notice that in the Mayan numeration system, you must take great care in the way

the numbers are spaced. For example, two horizontal bars could represent 5 + 5 as =

or 5 ¥ 20 + 5 as

, depending on how the two bars are spaced. Also notice that the

place-value feature of this system is somewhat irregular. After the ones place comes

the 20s place. Then comes the 18 ¥ 20s place. Thereafter, though, the values of the

places are increased by multiplying by 20 to obtain 18 202, 18 203, 18 204

¥

¥

¥

, and so on.

Express the following Mayan numerals in our numeration

system.

S O L U T I O N

■

Table 2.4 summarizes the attributes of the number systems we have studied.

TABLE 2.4

PLACE

HAS A

SYSTEM

ADDITIVE

SUBTRACTIVE

MULTIPLICATIVE

POSITIONAL

VALUE

ZERO

Tally

Yes

No

No

No

No

No

Egyptian

Yes

No

No

No

No

No

Roman

Yes

Yes

Yes

Yes

No

No

Babylonian

Yes

No

Yes

Yes

Yes

No

Mayan

Yes

No

Yes

Yes

Yes

Yes

Check for Understanding: Exercise/Problem Set A #7–14

✔

When learning the names of two-digit numerals, some children

suffer through a “reversals” stage, where they may write 21 for

twelve or 13 for thirty-one. The following story from the December

5, 1990, Grand Rapids [Michigan] Press stresses the importance of

eliminating reversals. “It was a case of mistaken identity. A trans-

posed address that resulted in a bulldozer blunder. City orders

had called for demolition on Tuesday of a boarded-up house at

451 Fuller Ave. S.E. But when the dust settled, 451 Fuller stood

untouched. Down the street at 415 Fuller Ave. S.E., only a base-

ment remained.”

©Ron Bagwell

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7/30/2013 2:41:05 PM - Section 2.2 Whole Numbers and Numeration 65

EXERCISE/PROBLEM SET A

EXERCISES

1. In the following paragraph, numbers are used in various

10. Change to Mayan numerals.

ways. Specify whether each number is a cardinal number,

a. 17 b. 51 c. 275 d. 401

an ordinal number, or an identification number. Linda

dialed 314-781-9804 to place an order with a popular mail-

11. Change to Hindu-Arabic numerals.

order company. When the sales representative answered,

Linda told her that her customer number was 13905.

Linda then placed an order for 6 cotton T-shirts, stock

d. MCMXCI

e. CMLXXVI

f. MMMCCXLV

number 7814, from page 28 of the catalog. For an extra

$10 charge, Linda could have next-day delivery, but she

chose the regular delivery system and was told her pack-

age would arrive November 20.

2. Identify how the number is used in the sentence: cardinal,

ordinal, or identification.

12. Perform each of the following numeral conversions.

a. Judy came in second in the race.

a. Roman numeral DCCCXXIV to a Babylonian

b. A dog is an animal with four legs.

numeral

c. I live in apartment five.

b. Mayan numeral

to an Egyptian numeral

3. Explain how to count the elements of the set

{ ,

a ,

b ,

c d, ,

e f }.

c. Babylonian numeral

to a Mayan

numeral

4. Which number is larger, 5 or 8? Which numeral is larger?

d. Egyptian numeral

to a

5. Use the set method as illustrated in Figure 2.10 to explain

Roman numeral

why 7 > 3.

13. Imagine representing 246 in the Mayan, Babylonian, and

6. Place < or > in the blanks to make each statement true.

Egyptian numeration systems.

Indicate how you would verify your choice.

a. In which system is the greatest number of symbols

a. 3 _____ 7 b. 11 _____ 9 c. 21 _____ 12

required?

b. In which system is the smallest number of symbols

7. Change to Egyptian numerals.

required?

a. 9 b. 23 c. 453 d. 1231

c. Do your answers for parts a and b hold true for other

8. Change to Roman numerals.

numbers as well?

a. 76 b. 49 c. 192 d. 1741

14. In 2010, the National Football League’s championship

9. Change to Babylonian numerals.

game was Super Bowl XLIV. What was the first year of

a. 47 b. 76 c. 347 d. 4192

the Super Bowl?

PROBLEMS

15. Some children go through a reversal stage; that is,

19. Determine which of the following numbers is larger.

they confuse 13 and 31, 27 and 72, 59 and 95. What

1993 × 1

( + 2 + 3 + 4 + ⋅⋅⋅ + 1994)

numerals would give Roman children similar difficulties?

1994 × 1

( + 2 + 3 + 4 + ⋅⋅⋅ + 1993)

How about Egyptian children?

20. You have five coins that appear to be identical and a balance

16. a. How many Egyptian numerals are needed to

scale. One of these coins is counterfeit and either heavier or

represent the following problems?

lighter than the other four. Explain how the counterfeit coin

i. 59 + 88

ii. 150 − 99

can be identified and whether it is lighter or heavier than the

iii. 7897 + 934

iv. 9698 − 5389

others with only three weighings on the balance scale.

b. State a general rule for determining the number of

(Hint: Solve a simpler problem—given just three coins,

Egyptian numerals needed to represent an addition (or

can you find the counterfeit in two weighings?)

subtraction) problem written in our numeral system.

21. One system of numeration in Greece in about 300 B.C.E.,

17. A newspaper advertisement introduced a new car as follows:

called the Ionian system, was based on letters of the

IV Cams, XXXII Valves, CCLXXX Horsepower, coming

alphabet. The different symbols used for numbers less

December XXVI—the new 1999 Lincoln Mark VII. Write the

than 1000 are as follows:

Roman numeral that represents the model year of the car.

18. Linda pulled out one full page from the Sunday newspa-

per. If the left half was numbered A4 and the right half

was numbered A15, how many pages were in the A section

of the newspaper?

c02.indd 65

7/30/2013 2:41:07 PM - 66 Chapter 2 Sets, Whole Numbers, and Numeration

To represent multiples of 1000 an accent mark was used.

i. mb

ii. cke

iii. gflg

iv. p ′qwa

For example, ′⑀ was used to represent 5000. The accent

b. Express each of the following numerals in the Ionian

mark might be omitted if the size of the number being rep-

numeration system.

resented was clear without it.

i. 85 ii. 744 iii. 2153 iv. 21,534

a. Express the following Ionian numerals in our

numeration system.

c. Was the Ionian system a place-value system?

EXERCISE/PROBLEM SET B

EXERCISES

1. Write sentences that show the number 45 used in each of

10. Change to Mayan numerals.

the following ways.

a. 926

b. 37,865

a. As a cardinal number

11. Express each of the following numerals in our numeration

b. As an ordinal number

system.

c. As an identification number

2. Identify how the number is used in the sentence: cardinal,

ordinal, or identification.

a. What is behind door three?

c. MCCXLVII

b. First turn the key, then open the lock.

c. Mark has five sisters.

12. Complete the following chart expressing the given num-

3. Explain why each of the following sets can or cannot be

bers in the other numeration system.

used to count the number of elements in { ,

a ,

b ,

c d}.

a. { }

4

b. { ,

0 ,

1 ,

2 }

3 c. { ,

1 ,

2 ,

3 }

4

4. Which is more abstract: number or numeral? Explain.

a.

5. Determine the greater of the two numbers 4 and 9 in three

b.

different ways.

6. Place < or > in the blanks to make each statement true.

c.

a. 6_____5 b. 100_____1000 c. 19_____91

d.

7. Change to Egyptian numerals.

a. 2431

b. 10,352

13. Is it possible for a numeration system to be positional and

not place value? Explain why or why not. If possible, give

8. Change to Roman numerals.

an example.

a. 79

b. 3054

14. After the credits for a film, the Roman numeral

9. Change to Babylonian numerals.

MCMLXXXIX appears, representing the year in which the

a. 117

b. 3521

film was made. Express the year in our numeration system.

PROBLEMS

15. What is the largest number that you can enter on your

The numerals are written vertically. Some examples

calculator

follow

a. if you may use the same digit more than once?

b. if you must use a different digit in each place?

16. The following Chinese numerals are part of one of the

oldest numeration systems known.

a. Express each of the following numerals in this

Chinese numeration system: 80, 19, 52, 400, 603,

6031.

b. Is this system a positional system? an additive system?

a multiplicative system?

c02.indd 66

7/30/2013 2:41:09 PM - Section 2.3 The Hindu–Arabic System 67

17. Two hundred persons are positioned in 10 rows, each

r 5IF UPUBM PG BMM PG UIF IFJHIUT JT OFBSMZ IBMG B NJMF *O

containing 20 persons. From each of the 20 columns thus

fact, the sum of the five heights is 2264 feet.

formed, the shortest is selected, and the tallest of these

Find the height of each of the five structures.

20 (short) persons is tagged A. These persons now return

to their initial places. Next, the tallest person in each row

20. Can an 8 × 8 checkerboard with two opposite corner

is selected and from these 10 (tall) persons the shortest is

squares removed be exactly covered (without cutting)

tagged B. Which of the two tagged persons is the taller

by thirty-one 2 × 1 dominoes? Give details.

(if they are different people)?

18. Braille numerals are formed using dots in a two-dot

by three-dot Braille cell. Numerals are preceded by a

backwards “L” dot symbol. The following shows the

basic elements for Braille numerals and two examples.

Analyzing Student Thinking

21. Tammie asserts that if n(A) ≤ n(B) where A and B are

finite sets, then A ⊆ B. Is she correct? Explain.

22. Which way is a child most likely to first learn the order-

ing of whole numbers less than 10: (i) using the counting

chant; (ii) using the set method; (iii) using the number line

method? Explain.

23. Anthony thinks that we should all use the tally numera-

tion system instead of the Hindu–Arabic system because

the Tally system is so easy. How should you respond?

Express these Braille numerals in our numeration system.

24. When asked to express 999 in the Egyptian numeration

system, Misti says that it will require twenty-seven numer-

a.

als and Victor says that it will require a lot more. Which

student is correct? Explain.

b.

25. When teaching the Roman numeration system, you show

your students that it is additive, subtractive, and multipli-

cative. Shalonda asks, “Why can’t you do division in the

19. The heights of five famous human-made structures are

Roman system?” How should you respond?

related as follows:

26. Natalie asks if the symbol comprised of two small tri-

r 5IF IFJHIU PG UIF 4UBUVF PG -JCFSUZ JT GFFU NPSF UIBO

angles, one atop the other, in the Babylonian numeration

half the height of the Great Pyramid at Giza.

system is zero. How should you respond?

r 5IF IFJHIU PG UIF &JGGFM 5PXFS JT GFFU NPSF UIBO UISFF

times the height of Big Ben.

27. Blanca says that since zero means nothing, she doesn’t

r 5IF (SFBU 1ZSBNJE BU (J[B JT GFFU UBMMFS UIBO #JH #FO

have to use a numeral for zero in the Mayan numeration

r 5IF -FBOJOH 5PXFS PG 1JTB JT GFFU TIPSUFS UIBO #JH

system. Instead, she can simply leave a blank space. Is she

Ben.

correct? Explain.

THE HINDU–ARABIC SYSTEM

In Coinland, their money is similar to American money, but in all coins. They have

pennies and dimes (no nickels or quarters), but also have coins for dollars, ten dol-

lars, hundred dollars, and so forth. Martina, who lives in Coinland, likes to carry as few coins as possible. What is the

minimum number of coins Martina could carry for each of the following amounts of money? How many of each coin

would she have in each case?

68¢ 123¢ $4.52 $307.31

If Martina always exchanges her money to have a minimum number of coins, what is the maximum number of any type

of coin that she would have after an exchange? Why?

c02.indd 67

7/30/2013 2:41:10 PM - 68

Chapter 2 Sets, Whole Numbers, and Numeration

The Hindu–Arabic Numeration System

The Hindu–Arabic numeration system that we use today was developed about

800 C.E. The following list features the basic numerals and various attributes of this

system.

1. Digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: These 10 symbols, or digits, can be used in combina-

tion to represent all possible numbers.

Common Core – Grade 1

2. Grouping by tens (decimal system): Grouping into sets of 10 is a basic principle of

Understand that the two digits

this system, probably because we have 10 “digits” on our two hands. (The word

of a two-digit number represent

digit literally means “finger” or “toe.”) Ten ones are replaced by 1 ten, 10 tens are

amounts of tens and ones.

replaced by 1 hundred, 10 hundreds are replaced by 1 thousand, and so on. Figure

2.25 shows how grouping is helpful when representing a collection of objects.

The number of objects grouped together is called the base of the system; thus our

Hindu–Arabic system is a base ten system.

NOTE: Recall that an element is listed only once in a set. Although all the dots in

Figure 2.25 look the same, they are assumed to be unique, individual elements here

and in all such subsequent figures.

Figure 2.25

The following two models are often used to represent multidigit numbers.

a. Bundles of sticks can be any kind of sticks banded together with rubber bands;

10 loose sticks are bound together with a rubber band to represent 10, then 10

bundles of 10 are bound together to represent 100, and so on (Figure 2.26).

Figure 2.26

b. Base ten pieces (also called Dienes blocks) consist of individual cubes, called

“units,” “longs,” made up of 10 units, “flats,” made up of 10 longs, or 100 units

and so on (Figure 2.27). Inexpensive two-dimensional sets of base ten pieces can be

made using grid paper cutouts.

Reflection from Research

Results from research suggest

that base ten blocks should be

used in the development of

students’ place-value concepts

(Fuson, 1990).

Common Core – Grade 2

Figure 2.27

Understand that the three digits

of a three-digit number represent

3. Place value (hence positional): Each of the various places in the numeral 6523, for

amounts of hundreds, tens, and

ones (e.g., 706 equals 7 hun-

example, has its own value.

dreds, 0 tens, and 6 ones).

c02.indd 68

7/30/2013 2:41:12 PM - Section 2.3 The Hindu–Arabic System 69

thousand hundred ten one

6 5 2 3

The 6 represents 6 thousands, the 5 represents 5 hundreds, the 2 represents 2 tens,

and the 3 represents 3 ones due to the place-value attribute of the Hindu–Arabic

system.

The following device is used to represent numbers written in place value. A chip

NCTM Standard

abacus is a piece of paper or cardboard containing lines that form columns, on

All students should use multiple

which chips or markers are used to represent unit values. Conceptually, this model is

models to develop initial under-

more abstract than the previous models because markers represent different values,

standings of place value and the

base ten number system.

depending on the columns in which the markers appear (Figure 2.28).

Figure 2.28

4. Additive and multiplicative: The value of a Hindu–Arabic numeral is found by

multiplying each place value by its corresponding digit and then adding all the

resulting products.

Place values:

thousand

hundred

ten

one

Digits:

6

5

2

3

Numeral value: 6 × 1000 + 5 × 100 + 2 × 10 + 3 × 1

Numeral:

6523

Expressing a numeral as the sum of its digits times their respective place values is

called the numeral’s expanded form or expanded notation. The expanded form of

83,507 is

8 × 10,000 + 3 × 1000 + 5 × 100 + 0 × 10 + 7 × 1.

Because 7 × 1 = 7, we can simply write 7 in place of 7 × 1 when expressing 83,507

in expanded form.

Express the following numbers in expanded form.

Common Core – Grade 2

Read and write numbers to 1000

a. 437

b. 3001

using base ten numerals, number

names, and expanded form.

S O L U T I O N

a. 437 = 4 100

(

) + 3 10

(

) + 7

b. 3001 = 3 1000

(

) + 0 100

(

) + 0 10

(

) + 1 or 3 1000

(

) + 1

■

Notice that our numeration system requires fewer symbols to represent numbers

than did earlier systems. Also, the Hindu–Arabic system is far superior when

performing computations. The computational aspects of the Hindu–Arabic system

will be studied in Chapter 3.

Check for Understanding: Exercise/Problem Set A #1–3

✔

c02.indd 69

7/30/2013 2:41:13 PM - 70

Chapter 2 Sets, Whole Numbers, and Numeration

Naming Hindu–Arabic Numerals

Reflection from Research

Associated with each Hindu–Arabic numeral is a word name. Some of the English

Children are able to recognize

names are as follows:

and read one- and two-digit

numerals prior to being able to

0 zero

10 ten

write them (Baroody, Gannon,

Berent, & Ginsburg, 1983).

1 one

11 eleven

2 two

12 twelve

3 three

13 thirteen (three plus ten)

4 four

14 fourteen (four plus ten)

5 five

21 twenty-one (2 tens plus one)

6 six

87 eighty-seven (8 tens plus seven)

7 seven

205 two hundred five (2 hundreds plus five)

8 eight

1,374 one thousand three hundred seventy-four

9 nine

23,100 twenty-three thousand one hundred

Here are a few observations about the naming procedure:

1. The numbers 0, 1, . . . , 12 all have unique names.

2. The numbers 13, 14, . . . , 19 are the “teens,” and are composed of a combination

of earlier names, with the ones place named first. For example, “thirteen” is short

for “three ten,” which means “ten plus three,” and so on.

3. The numbers 20, . . . , 99 are combinations of earlier names but reversed from the

teens in that the tens place is named first. For example, 57 is “fifty-seven,” which

means “5 tens plus seven,” and so on. The method of naming the numbers from 20

to 90 is better than the way we name the teens, due to the left-to-right agreement

with the way the numerals are written.

4. The numbers 100, . . . , 999 are combinations of hundreds and previous names. For

example, 538 is read “five hundred thirty-eight,” and so on.

NCTM Standard

5. In numerals containing more than three digits, groups of three digits are usually set

All students should connect num-

off by commas. For example, the number

ber words and numerals to the

quantities they represent, using

various physical models and rep-

resentations.

is read “one hundred twenty-three quadrillion four hundred fifty-six trillion seven

hundred eighty-nine billion nine hundred eighty-seven million six hundred fifty-four

thousand three hundred twenty-one.” (Internationally, the commas are omitted and

single spaces are used instead. Also, in some countries, commas are used in place of

decimal points.) Notice that the word and does not appear in any of these names: it is

reserved to separate the decimal portion of a numeral from the whole-number portion.

Figure 2.29 graphically displays the three distinct ideas that children need to learn

in order to understand the Hindu–Arabic numeration system.

Figure 2.29

Check for Understanding: Exercise/Problem Set A #4–7

✔

c02.indd 70

7/30/2013 2:41:14 PM - From Lesson 1-3 “Greater Numbers” from Envision Math Common Core, by Randall I. Charles et al., Grade 3, copyright © by

Pearson Education.

71

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7/30/2013 2:41:16 PM - 72 Chapter 2 Sets, Whole Numbers, and Numeration

The numbers from 0 to 26 are represented using only 3 symbols: ∼, Λ, and ⎡. Explain how

the system works and list the next 10 numbers.

0 = ~

9 = Λ ~~

18 = ~~

1 = Λ

10 = Λ~Λ

19 = ~Λ

2 =

11 = Λ~

20 = ~

3 = Λ~

12 = ΛΛ~

21 = Λ~

4 = ΛΛ

13 = ΛΛΛ

22 = ΛΛ

5 = Λ

14 = ΛΛ

23 = Λ

6 = ~

15 = Λ ~

24 =

~

7 = Λ

16 = Λ Λ

25 =

Λ

8 =

17 = Λ

26 =

How is this system related to the numeration systems we have discussed thus far?

Nondecimal Numeration Systems

Our Hindu–Arabic system is based on grouping by ten. To understand our system better

and to experience some of the difficulties children have when learning our numeration

system, it is instructive to study similar systems, but with different place values. For

example, suppose that a Hindu–Arabic-like system utilized one hand (five digits) instead

of two (ten digits). Then, grouping would be done in groups of five. If sticks were used,

bundles would be made up of five each (Figure 2.30). Here seventeen objects are rep-

resented by three bundles of five each with two left over. This can be expressed by the

equation 17

=

, which is read “seventeen base ten equals three two base five.” (Be

ten

32five

careful not to read 32 as “thirty-two,” because thirty-two means “3 tens and two,” not

five

“3 fives and two.”) The subscript words “ten” and “five” indicate that the grouping was

done in tens and fives, respectively. For simplicity, the subscript “ten” will be omitted;

hence 37 will always mean 37ten. (With this agreement, the numeral 24five could also be

written 245 since the subscript “5” means the usual base ten 5.) The 10 digits 0, 1, 2, . . . ,

9 are used in base ten; however, only the five digits 0, 1, 2, 3, 4 are necessary in base five.

A few examples of base five numerals are illustrated in Figure 2.31.

Reflection from Research

Figure 2.30

Students experience more suc-

cess counting objects that can

Counting using base five names differs from counting in base ten. The first ten base

be moved around than they

five numerals appear in Figure 2.32. Interesting junctures in counting come after a

do counting pictured objects

that cannot be moved (Wang,

number has a 4 in its ones column. For example, what is the next number (written in

Resnick, & Boozer, 1971).

base five) after 24 ? after

? after 44

five

34five

five ? after 444five ?

c02.indd 72

7/30/2013 2:41:20 PM - Section 2.3 The Hindu–Arabic System 73

Figure 2.31

Figure 2.32

Figure 2.33 shows how to find the number after 24five using multibase pieces.

Figure 2.33

Converting numerals from base five to base ten can be done using (1) multibase

pieces or (2) place values and expanded notation.

Express 123five in base ten.

S O L U T I O N

a. Using Base Five Pieces: See Figure 2.34.

Figure 2.34

b. Using Place Value and Expanded Notation

Place values

in base ten ⎯ →

⎯ 25

5

1

123

=

= ( ) + ( ) + ( ) =

five

1

2

3

1 25

2 5

3 1

38

■

Converting from base ten to base five also utilizes place value.

c02.indd 73

7/30/2013 2:41:23 PM - 74 Chapter 2 Sets, Whole Numbers, and Numeration

Convert from base ten to base five.

a. 97

b. 341

S O L U T I O N

a.

25

5

1 ← ⎯

⎯

Base five place values

97 = ?

?

?

expressed using base ten

n numerals

Algebraic Reasoning

Think: How many 25s are in 97? There are three since 3 ¥ 25 = 75 with 22 remain-

The process described for con-

ing. How many 5s in the remainder? There are four since 4 ¥ 5 = 20. Finally, since

verting 97 to a base five number

22 − 20 = 2, there are two 1s.

is really solving the equation

97 = 25x + 5y + z where x, y,

and

97 = 3(25) + 4(5) + 2 = 342five

z are less than 5. The solution is

x = 3, y = 4,

and

z = 2. Although

b. A more systematic method can be used to convert 341 to its base fi ve numeral.

an equation was not used, the

First, fi nd the highest power of 5 that will divide into 341; that is, which is the

reasoning is still algebraic.

greatest among 1, 5, 25, 125, 625, 3125, and so on, that will divide into 341? The

answer in this case is 125. The rest of that procedure uses long division, each time

dividing the remainder by the next smaller place value.

Therefore, 341 = 2 125

(

) + 3(25) + 3(5) + 1, or 2331

=

five. More simply, 341

2331five,

where 2, 3, and 3 are the quotients from left to right and 1 is the fi nal remainder. ■

When expressing place values in various bases, exponents can provide a convenient

shorthand notation. The symbol am represents the product of m factors of a. Thus

53

5 ¥ 5 ¥ 5, 72

7 ¥ 7, 34

=

=

= 3 ¥ 3 ¥ 3 ¥ 3, and so on. Using this exponential notation, the

first several place values of base five, in reverse order, are 1, 5, 52, 53, 54. Although we

have studied only base ten and base five thus far, these same place-value ideas can be

used with any base greater than one. For example, in base two the place values, listed

in reverse order, are 1, 2, 22, 23, 24, . . . ; in base three the place values are 1, 3, 32, 33,

34, . . . . The next two examples illustrate numbers expressed in bases other than five

and their relationship to base ten.

Express the following numbers in base ten.

a. 11011two

b. 1234eight

c. 1ETtwelve

(NOTE: Base twelve has twelve basic numerals: 0 through 9, T for ten, and E for

eleven.)

S O L U T I O N

24

23

22

2

1

a. 11011

=

= ( ) + ( ) + ( ) + ( ) + ( ) =

two

1 16

1 8

0 4

1 2

1 1

27

1

1

0

1

1

83

82

8

1

b. 1234

=

= 1(83) + 2(82 ) + ( ) + ( ) =

+

+

+ =

eight

3 8

4 1

512 128 24 4

668

1

2

3

4

c02.indd 74

7/30/2013 2:41:27 PM - Section 2.3 The Hindu–Arabic System 75

1212

12

1

c. 1

=

= (

) + ( ) + ( ) =

+

+

=

■

twelve

1

122

ET

E 12

T 1

144 132 10

286

1

E

T

Converting from base ten to other bases is accomplished by using grouping just as

we did in base fi ve.

Convert from base ten to the given base.

a. 53 to base two

b. 1982 to base twelve

S O L U T I O N

25

24

23

22

2

1

a. 53 = ?

?

?

?

?

?

Think: What is the largest power of 2 contained in 53?

Answer: 25 = 32. Now we can fi nd the remaining digits by dividing by decreasing

powers of 2.

1

1

0

1

0 1

32)53 16)21

)85 4)5 2)1

32

16

0

4

0

21

5

5

1

1

Therefore, 53 = 110101two.

123(= 1728)

122(= 144)

121(= 12)

1

b. 1982 =

?

?

?

?

1

1

9

2

1

)

728 1982 144)254 12)110 )

1 2

1728

144

108

2

254

110

2

0

Therefore, 1982 = 1192twelve.

■

Check for Understanding: Exercise/Problem Set A #8–25

✔

Consider the three cards shown here. Choose any number from 1 to 7 and note

which cards your number is on. Then add the numbers in the upper right-hand

corner of the cards containing your number. What did you find? This “magic” can

be justified mathematically using the binary (base two) numeration system.

©Ron Bagwell

c02.indd 75

7/30/2013 2:41:28 PM - 76 Chapter 2 Sets, Whole Numbers, and Numeration

EXERCISE/PROBLEM SET A

EXERCISES

1. Write each of the following numbers in expanded notation.

9. Represent each of the following numerals with multi-

a. 70 b. 300

base pieces. Use the Chapter 2 eManipulative activity

c. 984 d. 60,006,060

Multibase Blocks on our Web site to assist you.

a. 134

2. Write each of the following expressions in standard place-

five

b. 1011two c. 3211four

value form.

10. To express 69 with the fewest pieces of base three blocks,

a. 1 1000

(

) + 2 100

(

) + 7

flats, longs, and units, you need _____ blocks, _____ flats,

b. 5 100

(

,000) + 3 100

(

)

_____ longs, and _____ units. The Chapter 2 eManipula-

c. 8 106 + 7 104 + 6 102

(

)

(

)

(

) + 5 1

( )

tive activity Multibase Blocks on our Web site may help in

d. 2 109 + 3 104 + 3 103

(

)

(

)

(

) + 4 10

(

)

the solution.

11. Represent each of the following with bundling sticks and

3. State the place value of the digit 2 in each numeral.

chips on a chip abacus. (The Chapter 2 eManipulative

a. 6234 b. 5142 c. 2168

Chip Abacus on our Web site may help in understanding

4. Words and their roots often suggest numbers. Using this

how the chip abacus works.)

idea, complete the following chart. (Hint: Look for a

a.

24 b. 221five c. 167eight

pattern.)

12. a. Draw a sketch of 62 pennies and trade for nickels

and quarters. Write the corresponding base five

WORD

LATIN ROOT

MEANING OF ROOT

POWER OF 10

numeral.

Billion

bi

2

9

b. Write the base five numeral for 93 and 2173.

Trillion

tri

3

(a)

13. How many different symbols would be necessary for

Quadrillion

quater

(b)

15

a base twenty-three system?

(c)

quintus

5

18

14. What is wrong with the numeral 85

Sextillion

sex

6

21

eight ?

(d)

septem

7

(e)

15. True or false?

Octillion

octo

8

27

a. 7

=

eight

7

Nonillion

novem

(f)

(g)

b. 30

=

four

30

(h)

decem

10

33

c. 200

=

three

200nine

16. a. Write out the base five numerals in order from 1 to

5. Write these numerals in words.

100five.

a. 2,000,000,000 b. 87,000,000,000,000

b. Write out the base two numerals in order from 1 to

c. 52,672,405,123,139

10000two.

c. Write out the base three numerals in order from 1 to

6. The following numbers are written in words. Rewrite each

1000

one using Hindu–Arabic numerals.

three.

d. In base six, write the next four numbers after 254

a. Seven million six hundred three thousand fifty-nine

six.

e. What base four numeral follows 303

b. Two hundred six billion four hundred fifty-three

four ?

thousand

17. Write each of the following base seven numerals in

expanded notation.

7. List three attributes of our Hindu–Arabic numeration

a. 15

system.

seven

b. 123seven c. 5046seven

18. a. What is the largest three-digit base four number?

8. Write a base four numeral for the following set of base

b. What are the five base four numbers that follow it? Give

four pieces. Represent the blocks on the Chapter 2

your answers in base four numeration.

eManipulative activity Multibase Blocks on our Web site

and make all possible trades first.

19. Use the Chapter 2 dynamic spreadsheet Base Converter

on our Web site to convert the base ten numbers 2400 and

2402, which both have four digits, to a base seven number.

What do you notice about the number of digits in the base

seven representations of these numbers? Why is this?

20. Convert each base ten numeral into a numeral in the base

requested.

a. 395 in base eight

b. 748 in base four

c. 54 in base two

c02.indd 76

7/30/2013 2:41:32 PM - Section 2.3 The Hindu–Arabic System 77

21. The base twelve numeration system has the following

23. Write each of the following base ten numerals in base six-

twelve symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E. Change

teen (hexadecimal) numerals.

each of the following numerals to base ten numerals.

a. 375

b. 2941

Use the Chapter 2 dynamic spreadsheet Base Converter on

c. 9520

d. 24,274

our Web site to check your answers.

a. 142

24. Write each of the following numbers in base six and in

twelve

b. 234twelve

c. 503twelve

d. T9

base twelve.

twelve

e. T E

0 twelve

f. ETETtwelve

a. 74 b. 128

22. The hexadecimal numeration system, used in computer

c.

210 d. 2438

programming, is a base sixteen system that uses the

symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

25. Find the missing base.

Change each of the following hexadecimal numerals to

a. 35 = 120 _____

base ten numerals.

b. 41

=

six

27 _____

a. 213

=

sixteen

b. 1C2Bsixteen

c. 52seven

34 _____

PROBLEMS

26. What bases make these equations true?

30. To determine a friend’s birth date, ask him or her to perform

a. 32 = 44 _____

b. 57

=

the following calculations and tell you the result: Multiply the

eight

10 _____

number of the month in which you were born by 4 and add

c. 31

=

=

four

11 _____

d. 15x

30y

13 to the result. Then multiply that answer by 25 and subtract

27. The set of even whole numbers is the set { ,

0 ,

2 ,

4 ,

6 . . .}.

200. Add your birth date (day of month) to that answer and

What can be said about the ones digit of every even num-

then multiply by 2. Subtract 40 from the result and then mul-

ber in the following bases?

tiply by 50. Add the last two digits of your birth year to that

a. 10

b. 4

c. 2

d. 5

answer and, finally, subtract 10,500.

a. Try this sequence of operations with your own birth

28. Mike used 2989 digits to number the pages of a book.

date. How does place value allow you to determine a

How many pages does the book have?

birth date from the final answer? Try the sequence again

29. The sum of the digits in a two-digit number is 12. If the

with a different birth date.

digits are reversed, the new number is 18 greater than the

b. Use expanded notation to explain why this technique

original number. What is the number?

always works.

EXERCISE/PROBLEM SET B

EXERCISES

1. Write each of the following numbers in expanded form.

5. Write these numerals in words.

a. 409

b. 7094

a. 32,090,047

c. 746

d. 840,001

b. 401,002,560,300

c. 98,000,000,000,000,000

2. Write each expression in place-value form.

a. 3 1000

(

) + 7 1

( 0) + 5

6. The following numbers are written in words. Rewrite each

b. 7 10

( ,000) + 6 100

(

)

one using Hindu–Arabic numerals.

c. 6 105 + 3 103

(

)

(

) + 9 1

( )

a. Twenty-seven million sixty-nine thousand fourteen

d. 6 107 + 9 105

(

)

(

)

b. Twelve trillion seventy million three thousand five

3. State the place value of the digit 0 in each numeral.

7. Explain how the Hindu–Arabic numeration system is multi-

plicative and additive.

a. 40,762 b. 9802 c. 0

8. Write a base three numeral for the following set of base

4. The names of the months in the premodern calendar,

three pieces. Represent the blocks on the Chapter 2 eMa-

which had a different number of months than we do

nipulative activity Multibase Blocks on our Web site and

now, was based on the Latin roots for bi, tri, quater, etc.

make all possible trades first.

in Set A Exercise 4. Consider the names of the months

on the premodern calendar. These names suggest the

number of the month. What was September’s number on

the premodern calendar? What was October’s number?

November’s? If December were the last month of the

premodern calendar year, how many months made up

a year?

c02.indd 77

7/30/2013 2:41:34 PM - 78 Chapter 2 Sets, Whole Numbers, and Numeration

9. Represent each of the following numerals with multibase

17. Write each of the following base three numerals in

pieces. Use the Chapter 2 eManipulative Multibase Blocks

expanded notation.

on our Web site to assist you.

a. 22three

a. 221three

b. 212three

b. 122four

c. 12110three

c. 112four

d. List these three numbers from smallest to largest.

18. a. What is the largest six-digit base two number?

e. Explain why you can’t just compare the first digit in

b. What are the next three base two numbers? Give your

the different base representations to determine which

answers in base two numeration.

is larger.

19. Find two different numbers whose base ten representa-

10. To express 651 with the smallest number of base eight

tions have a different number of digits but their base three

pieces (blocks, flats, longs, and units), you need _____

representations each has 4 digits. The Chapter 2 dynamic

blocks, _____ flats, _____ longs, and _____ units.

spreadsheet Base Converter on our Web site can assist in

solving this problem.

11. Represent each of the following with bundling sticks and

chips on a chip abacus. (The Chapter 2 eManipulative

20. Convert each base ten numeral into its numeral in the base

Chip Abacus on our Web site may help in understanding

requested.

how the chip abacus works.)

a. 142 in base twelve

a. 38

b. 52

b. 72 in base two

six

c. 1032five

c. 231 in base eight

12. Suppose that you have ten longs in a set of multibase

pieces in each of the following bases. Make all possible

21. Find the base ten numerals for each of the following. Use

exchanges and write the numeral the pieces represent in

the Chapter 2 dynamic spreadsheet Base Converter on our

that base.

Web site to check your answers.

a. 10 longs in base eight

a. 342

b. TE0

c. 101101

five

twelve

two

b. 10 longs in base six

22. Using the hexidecimal digits described in Set A, Exercise

c. 10 longs in base three

22, rewrite the following hexidecimal numerals as base ten

numerals.

13. If all the letters of the alphabet were used as our single-

a. A4

b. 420E

digit numerals, what would be the name of our base sys-

sixteen

sixteen

tem? If a represented zero, b represented one, and so on,

23. Convert these base two numerals into base eight numerals.

what would be the base ten numeral for the “alphabet”

Can you state a shortcut? [Hint: Look at part d.]

numeral zz?

a. 1001two

b. 110110two

14. What is wrong with the numeral 24

?

c. 10101010two

d. 101111two

three

15. True or false?

24. Convert the following base five numerals into base nine

numerals.

a. 8

=

=

=

nine

8eleven

b. 30five

30six

c. 30eight

40six

a. 12five b. 204five c. 1322five

16. a. Write out the first 20 base four numerals.

25. Find the missing base.

b. How many base four numerals precede 2000four ?

a. 28 = 34 _____

c. Write out the base six numerals in order from 1 to 100six.

b. 28 = 26 _____

d. What base nine numeral follows 888nine ?

c. 23

=

twelve

43 _____

PROBLEMS

26. Under what conditions can this equation be true: a =

This last enlarged number plus the original number equals

b

ba ?

Explain.

2558. What is the original number?

27. Propose new names for the numbers 11, 12, 13, . . . , 19 so

29. What number is twice the product of its two digits?

that the naming scheme is consistent with the numbers 20

30. As described in the Mathematical Morsel, the three cards

and above.

shown here can be used to read minds. Have a person

28. A certain number has four digits, the sum of which is 10.

think of a number from 1 to 7 (say, 6) and tell you what

If you exchange the first and last digits, the new number

card(s) it is on (cards A and B). You determine the per-

will be 2997 larger. If you exchange the middle two digits

son’s number by adding the numbers in the upper right-

of the original number, your new number will be 90 larger.

hand corner (4 + 2 = 6).

c02.indd 78

7/30/2013 2:41:39 PM - End of Chapter Material 79

Analyzing Student Thinking

33. Jason asks why the class studied other numeration systems

like the Roman system and base five. How should you

respond?

34. Brandi asks why you are teaching the Hindu–Arabic

a. How does this work?

numeration system using a chip abacus. How should you

b. Prepare a set of four such magic cards for the numbers

respond?

1–15.

35. Jamie read the number 512 as “five hundred and twelve.”

Is she correct? If not, why not?

31. a. Assuming that you can put weights only on one pan,

show that the gram weights 1, 2, 4, 8, and 16 are suf-

36. The number 43 is read “forty-three” and means “4 tens plus

ficient to weigh each whole-gram weight up to 31 grams

3 ones.” Since 13 is “1 ten and 3 ones,” Juan asks if he could

using a pan balance. (Hint: Base two.)

call the number “onety-three.” How should you respond?

b. If weights can be used on either pan, what would be the

37. After studying base two, Gladys marvels at its simplicity

fewest number of gram weights required to weigh 31

and asks why we don’t use base two rather than base ten.

grams, and what would they weigh?

How should you respond?

32. What is a real-world use for number bases two and

38. After asking Riann to name the numeral 1,000,0002, she

sixteen?

says “one million.” Is she correct? Explain.

END OF CHAPTER MATERIAL

A survey was taken of 150 college freshmen. Forty of them

were majoring in mathematics, 30 of them were majoring in

English, 20 were majoring in science, 7 had a double major of

mathematics and English, and none had a double (or triple)

major with science. How many students had majors other than

mathematics, English, or science?

Additional Problems Where the Strategy “Draw a Diagram” Is

Useful

Strategy: Draw a Diagram

1. A car may be purchased with the following options:

Radio: AM/FM, AM/FM Cassette, AM/FM

Cassette/CD

Sunroof: Pop-up, Sliding

Transmission: Standard, Automatic

How many different cars can a customer select among

these options?

2. One morning a taxi driver travels the following routes: North:

A Venn diagram with three circles, as shown here, is useful

5 blocks; West: 3 blocks; North: 2 blocks; East: 5 blocks; and

in this problem. There are to be 150 students within the rectan-

South: 2 blocks. How far is she from where she started?

gle, 40 in the mathematics circle, 30 in the English circle, 20 in

3. For every 50 cars that arrive at a highway intersection, 25

the science circle, and 7 in the intersection of the mathematics

turn right to go to Allentown, 10 go straight ahead to Bos-

and English circles but outside the science circle.

ton, and the rest turn left to go to Canton. Half of the cars

There are 33 + 7 + 23 + 20, or 83, students accounted for, so

that arrive at Boston turn right to go to Denton, and one of

there must be 150 − 83, or 67, students outside the three circles.

every fi ve that arrive at Allentown turns left to go to Den-

Those 67 students were the ones who did not major in math-

ton. If 10,000 cars arrive at Denton one day, how many cars

ematics, English, or science.

arrived at Canton?

c02.indd 79

7/30/2013 2:41:40 PM - 80 Chapter 2 Sets, Whole Numbers, and Numeration

Emmy Noether (1882−1935)

David Hilbert (1862−1943)

Emmy Noether was born and

David Hilbert attended the

educated in Germany and grad-

&

gymnasium in his home town

rd

uated from the University of

a

rd/CORBIS

of Königsberg and then

a

Erlangen, where her father, Max

went on to the University

Noether, taught mathematics.

of Königsberg where he re-

There were few professional

ceived his doctorate in 1885.

chive Photos/Getty Images

Ar

opportunities for a woman

© Baldwin H. W

Kathryn C. W

He moved on and became a

mathematician, so Noether spent the next eight years

professor of mathematics at the University of Gottin-

doing research at home and teaching for her increas-

gen, where, in 1895, he was appointed to the chair of

ingly disabled father. Her work, which was in algebra, in

mathematics. He spent the rest of his career at Gottingen.

particular, ring theory, attracted the attention of the

Hilbert’s work in geometry has been considered to have

mathematicians Hilbert and Klein, who invited her to

had the greatest infl uence in that area second only to Eu-

the University of Gottingen. Initially, Noether’s lectures

clid. However, Hilbert is perhaps most famous for sur-

were announced under Hilbert’s name, because the uni-

veying the spectrum of unsolved problems in 1900 from

versity refused to admit a woman lecturer. Conditions

which he selected 23 for special attention. He felt these

improved, but in 18 years at Gottingen, she was rou-

23 were crucial for progress in mathematics in the com-

tinely denied the promotions that would have come to a

ing century. Hilbert’s vision was proved to be prophetic.

male mathematician of her ability. When the Nazis came

Mathematicians took up the challenge, and a great deal

to power in 1933, she was dismissed from her position.

of progress resulted from attempts to solve “Hilbert’s

She immigrated to the United States and spent the last

Problems.” Hilbert made important contributions to the

two years of her life at Bryn Mawr College. Upon her

foundations of mathematics and attempted to prove that

death, Albert Einstein wrote in the New York Times that

mathematics was self-consistent. He became one of the

“In the judgment of the most competent living mathema-

most infl uential mathematicians of his time. Yet, when

ticians, Fraulein Noether was the most signifi cant creative

he read or heard new ideas in mathematics, they seemed

mathematical genius thus far produced since the higher

“so diffi cult and practically impossible to understand,”

education of women began.”

until he worked the ideas through for himself.

CHAPTER REVIEW

Review the following terms and problems to determine which require learning or relearning—page numbers are provided for easy

reference.

Sets as a Basis for Whole Numbers

Vocabulary/Notation

Natural numbers 44

Equal sets (=) 45

Finite set 46

Counting numbers 44

Is not equal to (≠) 45

Infinite set 46

Whole numbers 44

1-1 correspondence 45

Disjoint sets 47

Set ({. . .}) 45

Equivalent sets (~) 45

Union (∪) 48

Element (member) 45

Matching sets 45

Intersection (∩) 48

Set-builder notation { . . . | . . . } 45

Subset (⊆) 46

Complement (A) 49

Is an element of ( )

∈ 45

Proper subset (⊂) 46

Set difference (−) 49

Is not an element of ( )

∉ 45

Venn diagram 46

Ordered pair 50

Empty set (∅) or null set 45

Universal set (U ) 46

Cartesian product (×) 50

Exercises

1. Describe three different ways to define a set.

c. { ,

x }

y ⊂ { ,

x ,

y }

z

d.

⊆

2. True or false?

{ ,

a }

b

{ ,

a }

b

e. ∅ = { }

a. 1 ∈{ ,

a

,

b

1

}

f. { ,

a }

b ~ { ,

c d}

b. a ∉{ ,

1 ,

2 }

3

c02.indd 80

7/30/2013 2:41:47 PM - Chapter Review 81

g. { ,

a }

b = { ,

c d

}

4. Explain how you can distinguish between finite sets

h. { ,

1 ,

2 }

3 ∩ { ,

2 ,

3 }

4 = { ,

1 ,

2 ,

3 }

4

and infinite sets.

i. { ,

2 }

3 ∪ { ,

1 ,

3 }

4 = { ,

2 ,

3 }

4

5. A poll at a party having 23 couples revealed that there were

j. { ,

1 }

2 and { ,

2 }

3 are disjoint sets

25 people who liked both country-western and ballroom

k. { ,

4 ,

3 ,

5 }

2 − { ,

2 ,

3 }

4 = { }

5

dancing.

l. { ,

a }

1 × { ,

b }

2 = {( ,

a b

), ( ,

1 2)}

8 who liked only country-western dancing.

3. What set is used to determine the number of elements

6 who liked only ballroom dancing.

in the set { ,

a ,

b ,

c d

, ,

e f

, g

}?

How many did not like either type of dancing?

Whole Numbers and Numeration

Vocabulary/Notation

Number 57

Greater than or equal to (≥) 59

Positional numeration system 62

Numeral 57

Whole-number line 59

Multiplicative numeration system 62

Cardinal number 57

Tally numeration system 60

Babylonian numeration system 62

Ordinal number 57

Grouping 60

Place value 62

Identification number 57

Egyptian numeration system 60

Placeholder 62

Number of a set [ (

n

)

A ] 58

Additive numeration system 61

Mayan numeration system 63

Less than (<), greater than (>) 59

Roman numeration system 61

Zero 63

Less than or equal to (≤) 59

Subtractive numeration system 61

Exercises

1. Is the expression “house number” literally correct? Explain.

f. ∩ ||| = ||| ∩ in the Egyptian system

g. IV = VI in the Roman system

2. Give an example of a situation where each of the following

is useful:

h. =

in the Mayan system

a. cardinal number

4. Express each of the following in our system.

b. ordinal number

c. identification number

a.

b. CXIV

c.

3. True or false?

5. Express 37 in each of the following systems:

a. n a

({ , b

, c

, d

}) = 4

a. Egyptian

b. 7 ≤ 7

b. Roman

c. 3 ≥ 4

c. Mayan

d. 5 < 50

6. Using examples, distinguish between a positional numera-

e. |||| is three in the tally system

tion system and a place-value numeration system.

The Hindu–Arabic System

Vocabulary/Notation

Hindu–Arabic numeration system 68

Bundles of sticks 68

Expanded form 69

Digits 68

Base ten pieces (Dienes blocks) 68

Expanded notation 69

Base 68

Chip abacus 69

Exponents 74

Exercises

1. Explain how each of the following contributes to formulat-

3. True or false?

ing the Hindu–Arabic numeration system.

a. 100 = 212five

a. Digits

b. 172

=

nine

146

b. Grouping by ten

c. 11111

=

two

2222

c. Place value

three

d. Additive and multiplicative attributes

d. 18

=

twelve

11nineteen

2. Explain how the names of 11, 12, . . . , 19 are inconsistent

4. What is the value of learning different number bases?

with the names of 21, 22, . . . , 29.

c02.indd 81

7/30/2013 2:41:51 PM - 82 Chapter 2 Sets, Whole Numbers, and Numeration

CHAPTER TEST

Knowledge

8. Rewrite the base ten number 157 in each of the following

number systems that were described in this chapter.

1. Identify each of the following as always true, sometimes

Babylonian

true, or never true. If it is sometimes true, give one true

Roman

and one false example. (Note: If a statement is sometimes

Egyptian

true, then it is considered to be mathematically false.)

Mayan

a. If A ~ B then A = B.

b. If A ⊂ B, then A ⊆ B.

9. Write a base five numeral for the following set of base five

c. A ∩ B ⊆ A ∪ B.

blocks. (Make all possible trades first.)

d. n a

({ , b

} × x

{ , y

, z

}) = 6.

e. If A ∩ B = ∅, then n(A − B) < n(A).

f. { ,

2 ,

4 ,

6 . . . ,

}

2000000 ~ { ,

4 ,

8

,

12 . . . }.

g. The range of a function is a subset of the codomain of

the function.

h. VI = IV in the Roman numeration system.

i. ÷ represents one hundred six in the Mayan numeration

system.

j. 123 = 321 in the Hindu–Arabic numeration system.

2. How many different symbols would be necessary for a base

nineteen system?

3. Explain what it means for two sets to be disjoint.

10. Represent the shaded region using the appropriate set

Skill

notation.

4. For A = a

{ , b

, c

}, B

= b

{ , c

, d

, e

}, C

= d

{ , e

, f

, g

},

D = e

{ , f

, g

}, find each of the following.

a. A ∪ B

b. A ∩ C

c. A ∩ B

d. A × D

e. C − D

f. (B ∩ D) ∪ (A ∩ C )

5. Write the equivalent Hindu–Arabic base ten numeral for

each of the following numerals.

a. ∩ ∩ ∩ ||(Egyptian) b. CMXLIV (Roman)

c. ... (Mayan)

d.

(Babylonian)

Understanding

e. 10101two

f. ETtwelve

11. Use the Roman and Hindu–Arabic systems to explain the

6. Express the following in expanded form.

difference between a positional numeration system and a

a. 759

b. 7002

c. 1001001

place-value numeration system.

two

7. Shade the region in the following Venn diagram to repre-

12. Determine conditions, if any, on nonempty sets A and B

so that the following equalities will be true.

sent the set A − (B ∪ C ).

a. A ∪ B = B ∪ A

b.

A ∩ B = B ∩ A

c. A − B = B − A

d.

A × B = B × A

13. If ( ,

a b

) and ( ,

c d

) are in A × B, name four elements in B × A.

14. Explain two distinctive features of the Mayan number

system as compared to the other three non-Hindu–Arabic

number systems described in this chapter.

15. Given the universal set of { ,

1 ,

2 ,

3 . . . , 2 }

0 and sets A, B

, and

C as described, place all of the numbers from the universal set

in the appropriate location on the following Venn diagram.

c02.indd 82

7/30/2013 2:41:56 PM - Chapter Test 83

A = { ,

2 ,

4 ,

0 ,

1 ,

3 ,

5 }

6

21. What is the largest Mayan number that can be represented

B = { ,

10 ,

11 ,

12 ,

1 ,

2 ,

7 ,

6 }

14

with exactly 4 symbols? Explain.

C = { ,

18 ,

19 ,

6 1

,

1 1

,

6 1

,

2 ,

9 }

8

22. A third-grade class of 24 students counted the number of

students in their class with the characteristics of brown

eyes, brown hair, and curly hair. The students found the

following:

12 students had brown eyes

19 students had brown hair

12 students had curly hair

5 students had brown eyes and curly hair

10 students had brown eyes and brown hair

7 students had brown, curly hair

3 students had brown eyes with brown, curly hair

16. Let A = x

{ | x is a letter in the alphabet} and

B = { ,

10

,

11

,

12 ... ,

}

40 . Is it possible to have 1-1

To answer the following questions a–c, we will classify all

correspondence between sets A and B? If so, describe

eyes as either brown or blue. Similarly all hair will be either

the correspondence. If not, explain why not.

curly or straight. The color of the hair will be either brown

or non-brown.

17. Represent the numeral 1212three with base three blocks

a. How many students have brown eyes with straight, non-

and chips on a chip abacus.

brown hair?

Problem Solving/Application

b. How many students have curly brown hair and blue

eyes?

c. How many students straight-haired, blue-eyed students

18. If n A

( ) =

,

71 n

B

( ) =

,

53 n

A

( ∩ B) =

,

27 what is n A

( ∪ B)?

are in the class?

19. Find the smallest values for a and b so that 21 =

b

25a.

23. To write the numbers 10, 11, 12, 13, 14, 15 in the Hindu–

20. A number in some base b has two digits. The sum of the

Arabic system requires 12 symbols. If the same numbers

digits is 6 and the difference of the digits is 2. The number

were written in the Babylonian, Mayan, or Roman sys-

is equal to 20 in the base ten. What is the base, b, of the

tems, which system would require the most symbols?

number?

Explain.

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7/30/2013 2:41:59 PM - C H A P T E R

3 WHOLE NUMBERS:

OPERATIONS AND PROPERTIES

Calculation Devices Versus Written Algorithms: A Debate Through Time

The Hindu–Arabic numeration system can be

traced back to 250 B.C.E. However, it wasn’t until

about 800 C.E. when a complete Hindu system

was described in a book by the Persian mathematician

al-Khwarizmi. Although the Hindu–Arabic numerals,

as well as the Roman numeral system, were used to rep-

resent numbers, they were not used for computations,

mainly because of the lack of inexpensive, convenient

writing equipment such as paper and pencil. In fact,

FPO

the Romans used a sophisticated abacus or “sand tray”

made of a board with small pebbles (calculi) that slid in

grooves as their calculator. Another form of the abacus

was a wooden frame with beads sliding on thin rods,

much like those used by the Chinese as shown in the

Focus On for Chapter 4.

J-L Charmet/Science Photo Library/Science Source

From about 1100 to 1500 C.E. there was a great debate

An algorist racing an abacist

among Europeans regarding calculation. Those who

advocated the use of Roman numerals along with the aba-

cus were called the abacists. Those who advocated using

which rendered many forms of written algorithms obso-

the Hindu–Arabic numeration system together with writ-

lete. Yet the debate continues regarding what role the cal-

ten algorithms such as the ones we use today were called

culator should play in arithmetic. Could it be that a debate

algorists. About 1500, the algorists won the argument and

will be renewed between algorists and the modern-day

by the eighteenth century, there was no trace of the abacus

abacists (or “calculatorists”)? Is it possible that we may

in western Europe. However, parts of the world, notably,

someday return to being “abacists” by using our Hindu–

China, Japan, Russia, and some Arabian countries, con-

Arabic system to record numbers while using calculators

tinued to use a form of the abacus.

to perform all but simple mental calculations? Let’s hope

It is interesting, though, that in the 1970s and 1980s,

that it does not take us 400 years to decide the appropriate

technology produced the inexpensive, handheld calculator,

balance between written and electronic calculations!

84

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7/30/2013 2:48:12 PM - Problem-Solving

Use Direct Reasoning

Strategies

1. Guess and Test

The Use Direct Reasoning strategy is used virtually all the time in conjunction

with other strategies when solving problems. Direct reasoning is used to reach a

2. Draw a Picture

valid conclusion from a series of statements. Often, statements involving direct

3. Use a Variable

reasoning are of the form “If A then B.” Once this statement is shown to be

true, statement B will hold whenever statement A does. (An expanded discussion

4. Look for a Pattern

of reasoning is contained in the Logic section near the end of the book.) In the

5. Make a List

following initial problem, no computations are required. That is, a solution can be

obtained merely by using direct reasoning, and perhaps by drawing pictures.

6. Solve a Simpler

Problem

Initial Problem

7. Draw a Diagram

In a group of nine coins, eight weigh the same and the ninth is either heavier or

8. Use Direct

lighter. Assume that the coins are identical in appearance. Using a pan balance, what

Reasoning

is the smallest number of balancings needed to identify the counterfiet coin?

Clues

The Use Direct Reasoning strategy may be appropriate when

r A proof is required.

r A statement of the form “If . . . , then . . .” is involved.

r You see a statement that you want to imply from a collection of known condi-

tions.

A solution of this Initial Problem is on page 124.

85

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7/30/2013 2:48:12 PM - AUTHOR

I N T R O D U C T I O N

The whole-number operations of addition, subtraction, multiplication, and division form the

foundation of arithmetic. Because of their primary importance, this entire chapter is devoted

to the study of these concepts independent of computational procedures. Each of the opera-

tions is introduced independently but the connections between them are carefully highlighted.

WALK-THROUGH The properties of addition and multiplication are discussed as a way to assist in learning basic facts.

Finally, exponents are introduced to simplify multiplication and to serve as a convenient notation for

representing large numbers.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-2–NUMBER AND OPERATIONS

Understand various meanings of addition and subtraction of whole numbers and the relationship between the two

operations.

Understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.

Develop and use strategies for whole-number computations, with a focus on addition and subtraction

r PRE-K-2–ALGEBRA

Illustrate general principles and properties of operations, such as commutativity, using specific numbers.

Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

r GRADES 3-5–NUMBER AND OPERATIONS

Understand various meanings of multiplication and division.

Understand and use properties of operations, such as the distributivity of multiplication over addition.

Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

r GRADES 3-5–ALGEBRA

Identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers.

Key Concepts from the NCTM Curriculum Focal Points

r KINDERGARTEN: Representing, comparing, and ordering whole numbers and joining and separating sets.

r GRADE 1: Developing understandings of addition and subtraction and strategies for basic addition facts and

related subtraction facts.

r GRADE 2: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addi-

tion and subtraction.

r GRADE 3: Developing understandings of multiplication and division and strategies for basic multiplication facts

and related division facts.

r GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole-number

multiplication.

Key Concepts from the Common Core State Standards for Mathematics

r KINDERGARTEN: Understand addition as putting together and adding to, and understand subtraction as taking

apart and taking from.

r GRADE 1: Represent and solve problems involving addition and subtraction. Understand and apply properties of

operations and the relationship between addition and subtraction.

r GRADE 2: Represent and solve problems involving addition and subtraction.

r GRADE 3: Represent and solve problems involving multiplication. Understand properties of multiplication and the

relationship between multiplication and division.

86

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7/30/2013 2:48:12 PM - Section 3.1 Addition and Subtraction 87

ADDITION AND SUBTRACTION

Addition and Its Properties

Finding the sum of two whole numbers is one of the first mathematical ideas a child

encounters after learning the counting chant “one, two, three, four, . . .” and the con-

cept of number. In particular, the question “How many is 3 and 2?” can be answered

using both a set model and a measurement model.

NCTM Standard

Set Model To find “3 + 2” using a set model, one must represent two disjoint sets:

All students should model situ-

one set, A, with three objects and another set, B, with 2 objects. Recall that n(A)

ations that involve the addition

denotes the number of elements in set A. In Figure 3.1, n(A) = 3 and n(B) = 2.

and subtraction of whole num-

bers using objects, pictures, and

symbols.

Figure 3.1

With sets, addition can be viewed as combining the contents of the two sets to form

the union and then counting the number of elements in the union, n(A ∪ B) = 5. In

this case, n(A) + n(B) = n(A ∪ B). The example in Figure 3.1 suggests the following

general definition of addition.

D E F I N I T I O N 3 . 1

Addition of Whole Numbers

Let a and b be any two whole numbers. If A and B are disjoint sets with a = n(A)

and b = n(B), then a + b = n(A ∪ B).

The number a + b, read “a plus b,” is called the sum of a and b, and a and b are

called addends or summands of a + b.

When using sets to discuss addition, care must be taken to use disjoint sets. In

Figure 3.2, the sets A and B are not disjoint because n(A ∩ B) = 1.

This nonempty intersection makes n(A) + n(B) ≠ n(A ∪ B) and so it is not an

example of addition using the set model. However, Figure 3.2 gives rise to a more

general statement about the number of elements in the union of two sets. It is

n(A) + n(B) − n(A ∩ B) = n(A ∪ B). The following example illustrates how to prop-

Figure 3.2

erly use disjoint sets to model addition.

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7/30/2013 2:48:16 PM - 88

Chapter 3 Whole Numbers: Operations and Properties

Use the definition of addition to compute 4 + 5.

S O L U T I O N Let A = a

{ , b, c, d} and B = e

{ , f , g, h, i}. Then n(A) = 4 and n(B) = 5.

Also, A and B have been chosen to be disjoint.

Therefore, 4 + 5 = (

n A ∪ B)

= ({

n

,

a ,

b ,

c d} ∪ { ,

e f , g, ,

h i})

= ({

n

,

a ,

b ,

c d, e, f , g, ,

h i})

= 9.

■

Addition is called a binary operation because two (“bi”) numbers are combined to

produce a unique (one and only one) number. Multiplication is another example of

a binary operation with numbers. Intersection, union, and set difference are binary

operations using sets.

Measurement Model Addition can also be represented on the whole-number

line pictured in Figure 3.3. Even though we have drawn a solid arrow starting at zero

and pointing to the right to indicate that the collection of whole numbers is unending,

the whole numbers are represented by the equally spaced points labeled 0, 1, 2, 3, and

so on. The magnitude of each number is represented by its distance from 0. In later

chapters, the number line will be extended and filled in.

Figure 3.3

Addition of whole numbers is represented by directed arrows of whole-number

lengths. The procedure used to find the sum 3 + 4 using the number line is illustrated

in Figure 3.4. Here the sum, 7, of 3 and 4 is found by placing arrows of lengths 3 and

4 end to end, starting at zero. Notice that the arrows for 3 and 4 are placed end to

end and are disjoint, just as in the set model.

Figure 3.4

Next we examine some fundamental properties of addition of whole numbers that

can be helpful in simplifying computations.

Mentally compute the following and write down what you did. (Don’t convert to base ten

to solve the problems.)

6seven + 4seven

7nine + 4nine + 2nine

By comparing your methods with those of your peers, determine if any similar methods were used.

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7/30/2013 2:48:17 PM - Section 3.1 Addition and Subtraction 89

Properties of Whole-Number Addition The fact that one always obtains a

whole number when adding two whole numbers is summarized by the closure property.

P R O P E R T Y

Closure Property for Whole-Number Addition

The sum of any two whole numbers is a whole number.

In general, when an operation on a set satisfies a closure property, the set is said

to be closed with respect to the given operation. Knowing that a set is closed under

an operation is helpful when checking certain computations. For example, consider

the set of all even whole numbers, { ,

0 ,

2 ,

4 . . . }, and the set of all odd whole numbers,

{ ,

1 ,

3 ,

5 . . . }. The set of even numbers is closed under addition since the sum of two

even numbers is even. Therefore, if one adds a collection of even numbers and obtains

an odd sum, an error has been made. The set of odd numbers is not closed under addi-

tion because the sum 1+ 3 is not an odd number.

Common Core – Grade 1

Many children learn how to add by “counting on.” For example, to find 9 +1, a child

Apply properties of operations

will count on 1 more from 9, namely, think “nine, then ten.” However, if asked to find

as strategies to add and subtract

1+ 9, a child might say “1, then 2, 3, 4, 5, 6, 7, 8, 9, 10.” Not only is this inefficient, but

(e.g., if 8 + 3 = 11 is known, then

also the child might lose track of counting on 9 more from 1. The fact that 1+ 9 = 9 +1

3 + 8 = 11 is also known).

is useful in simplifying this computation and is an instance of the following property.

NCTM Standard

P R O P E R T Y

All students should illustrate gen-

eral principles and properties of

Commutative Property for Whole-Number Addition

operations, such as commutativ-

ity, using specific numbers.

Let a and b be any whole numbers. Then

a + b = b + a.

Problem-Solving Strategy

Note that the root word of commutative is commute, which means “to inter-

Draw a Picture

change.” Figure 3.5 illustrates this property for 3 + 2 and 2 + 3.

Figure 3.5

Now suppose that a child knows all the addition facts through the fives but wants

to find 6 + 3. A simple way to do this is to rewrite 6 + 3 as 5 + 4 by taking one from 6

and adding it to 3. Since the sum 5 + 4 is known to be 9, the sum 6 + 3 is 9. In sum-

mary, this argument shows that 6 + 3 can be thought of as 5 + 4 by following this

reasoning: 6 + 3 = (5 + 1) + 3 = 5 + 1

( + 3) = 5 + 4. The next property is most useful in

simplifying computations in this way.

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7/30/2013 2:48:19 PM - 90 Chapter 3 Whole Numbers: Operations and Properties

Common Core – Grade 1

P R O P E R T Y

Apply properties of operations

as strategies to add and subtract

Associative Property for Whole-Number Addition

(e.g., to add 2 + 6 + 4, the sec-

ond two numbers can be added

Let a, b, and c be any whole numbers. Then

to make a ten, so 2 + 6 + 4 =

2 + 10 = 12).

(a + b) + c = a + (b + c).

Algebraic Reasoning

The root word of associative is associate, which means “to unite” or, in this case,

When solving algebraic equa-

“reunite.” The example in Figure 3.6 illustrates this property.

tions, we often need to “combine

like terms.” The commutative

and associative properties make

this possible. For example,

(3 + 2x) + (4 + 5x) is simplified

by changing the order and

grouping of the terms to be

(3 + 4) + (2x + 5x).

Children’s Literature

www.wiley.com/college/musser

See “The Mission of Addition”

by Brian P. Cleary.

Figure 3.6

Since the empty set has no elements, A ∪ { } = A. A numerical counterpart to this

statement is one such as 7 + 0 = 7. In general, adding zero to any number results in

the same number. This concept is stated in generality in the next property.

P R O P E R T Y

Identity Property for Whole-Number Addition

There is a unique whole number, namely 0, such that for all whole numbers a,

a + 0 = a = 0 + a.

Because of this property, zero is called the additive identity or the identity for

addition.

The previous properties can be applied to help simplify computations. They are

especially useful in learning the basic addition facts (that is, all possible sums of the

digits 0 through 9). Although drilling using flash cards or similar electronic devices is

helpful for learning the facts, an introduction to learning the facts via the following

thinking strategies will pay rich dividends later as students learn to perform multi-

digit addition mentally.

Discuss how a 6-year-old would find the answer to the question “What is 7 + 2?” If the

6-year-old were then asked “What is 2 + 7?”, how would he find the answer to that ques-

tion? Is there a difference? Why or why not?

Thinking Strategies for Learning the Addition Facts The addition table in

Figure 3.7 has 100 empty spaces to be filled. The sum of a + b is placed in the intersection

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7/30/2013 2:48:21 PM - Section 3.1 Addition and Subtraction 91

of the row labeled a and the column labeled b. For example, since 4 + 1 = 5, a 5 appears

in the intersection of the row labeled 4 and the column labeled 1.

Reflection from Research

When students are taught strate-

gies for thinking and working in

mathematics, instead of just basic

facts, their computational accu-

racy, efficiency, and flexibility can

be improved (Crespo, Kyriakides,

& McGee, 2005).

Figure 3.7

Figure 3.8

Figure 3.9

1. Commutativity: Because of commutativity and the symmetry of the table, a child

will automatically know the facts in the shaded region of Figure 3.8 as soon as the

child learns the remaining 55 facts. For example, notice that the sum 4 + 1 is in the

unshaded region, but its corresponding fact 1 + 4 is in the shaded region.

2. Adding zero: The fact that a + 0 = a for all whole numbers fills in 10 of the

remaining blank spaces in the “zero” column (Figure 3.9)—45 spaces to go.

3. Counting on by 1 and 2: Children find sums like 7 + 1, 6 + 2, 3 + 1, and 9 + 2 by

counting on. For example, to find 9 + 2, think 9, then 10, 11. This thinking strategy

fills in 17 more spaces in the columns labeled 1 and 2 (Figure 3.10)—28 facts to go.

Common Core – Grade 1

Relate counting to addition and

subtraction (e.g., by counting on

2 to add 2).

Figure 3.10

Figure 3.11

Common Core –

4. Combinations to ten: Combinations of the ten fingers can be used to find 7 + 3, 6 + 4,

Kindergarten

5 + 5, and so on. Notice that now we begin to have some overlap. There are 25 facts

Decompose numbers less than

left to learn (Figure 3.11).

or equal to 10 into pairs in more

than one way (e.g., 5 = 2 + 3 and

5. Doubles: 1 + 1 = 2, 2 + 2 = 4, 3 + 3 = 6, and so on. These sums, which appear on

5 = 4 + 1).

the main left-to-right downward diagonal, are easily learned as a consequence of

counting by twos: namely, 2, 4, 6, 8, 10, . . . (Figure 3.12). Now there are 19 facts

yet to be determined.

6. Adding ten: When using base ten pieces as a model, adding 10 amounts to laying

down a “long” and saying the new name. For example, 3 + 10 is 3 units and 1 long,

or 13; 7 + 10 is 17, and so on.

7. Associativity: The sum 9 + 5 can be thought of as 10 + 4, or 14, because 9 + 5 =

9 + 1

( + 4) = (9 + 1) + 4. Similarly, 8 + 7 = 10 + 5 = 15, and so on. The rest of the

addition table can be filled using associativity (sometimes called regrouping) com-

bined with adding 10.

8. Doubles ±1 and ±2: This technique overlaps with the others. Many children use it

effectively. For example, 7 + 8 = 7 + 7 +1 = 14 +1 = 15, or 8 + 7 = 8 + 8 − 1 = 15; 5 + 7 =

Figure 3.12

5 + 5 + 2 = 10 + 2 = 12, and so on.

By using thinking strategies 6, 7, and 8, the remaining basic addition facts needed to

complete the table in Figure 3.12 can be determined.

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7/30/2013 2:48:27 PM - 92 Chapter 3 Whole Numbers: Operations and Properties

Use thinking strategies in three different ways to find the sum

Common Core –

of 9 + 7.

Kindergarten

S O L U T I O N

Compose and decompose num-

bers from 11 to 19 into 10 ones

a. 9 + 7 = 9 + 1

( + 6) = (9 + 1) + 6 = 10 + 6 = 16

and some further ones such as

18 = 10 + 8.

b. 9 + 7 = (8 + 1) + 7 = 8 + 1

( + 7) = 8 + 8 = 16

c. 9 + 7 = (2 + 7) + 7 = 2 + (7 + 7) = 2 + 14 = 16

■

NCTM Standard

Thus far we have been adding single-digit numbers. However, thinking strate-

All students should develop

gies can be applied to multidigit addition also. Figure 3.13 illustrates how multidigit

fluency with basic number

addition is an extension of single-digit addition. The only difference is that instead of

combinations for addition and

adding units each time, we might be adding longs, flats, and so on. Mentally combine

subtraction.

similar pieces, and then exchange as necessary.

Figure 3.13

The next example illustrates how thinking strategies can be applied to multidigit

numbers.

Using thinking strategies, find the following sums.

a. 42 + 18

b. 37 + (42 + 13)

c. 51 + 39

S O L U T I O N

a. 42 + 18 = (40 + 2) + 1

( 0 + 8)

Addition

= (40 + 10) + (2 + 8)

Commutativity and associativity

= 50 + 10

Place value and combination to 10

= 60

Addition

b. 37 + (42 + 13) = 37 + 1

( 3 + 42)

= (37 + 13) + 42

= 50 + 42

= 92

c. 51 + 39 = (50 + 1) + 39

= 50 + 1

( + 39)

= 50 + 40

= 90

■

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7/30/2013 2:48:28 PM - Section 3.1 Addition and Subtraction 93

The use of other number bases can help you simulate how these think-

ing strategies are experienced by students when they learn base ten arithmetic.

Perhaps the two most powerful thinking strategies, especially when used together,

are associativity and combinations to the base (base ten above). For ex

ample,

7

+

=

+ (

+

) = (

+

) +

=

+ 4nine =

nine

6nine

7nine

2nine 4nine

7nine 2nine

4nine

10nine

14nine (since the

sum of 7

+

=

+

+

=

nine and 2nine is one of the base in base nine), 4six

5six

3six 1six

5six

3

+

=

+

six

10six

13six (since 1six

5six is one of the base in base six), and so on.

Compute the following sums using thinking strategies.

a. 7

+

+

+

eight

3eight

b. 5seven

4seven

c. 9twelve

9twelve

S O L U T I O N

a. 7

+

=

+ (

+

) = (

+

) +

= 10

+ 2

=

eight

3eight

7eight

1eight

2eight

7eight 1eight

2eight

eight

eight

12eight

b. 5

+

=

+ (

+

) = (

+

) +

= 10

+

seven

4seven

5seven

2seven

2seven

5seven

2seven

2seven

seven

2seven

12seven

c. 9

+

=

+ (

+

) = (

+

twelve

9twelve

9twelve

3twelve

6twelve

9twelve

3twelve ) + 6twelve

10twelve + 6twelve = 16twelve

Notice how associativity and combinations to the base are used.

■

Check for Understanding: Exercise/Problem Set A #1–7

✔

Write a word problem that would have 7 – 2 as its solution. Compare your problem with

those of several peers and determine how these problems are different and how they

are the same from a mathematical perspective. If possible categorize the problems according to the underlying math-

ematical structure.

Subtraction

Reflection from Research

The Take-Away Approach There are two distinct approaches to subtraction.

Students will benefit later in

The take-away approach is often used to introduce children to the concept of subtrac-

their mathematics if they are

tion. The problem “If you have 5 coins and spend 2, how many do you have left?”

taught both the “take-away” and

can be solved with a set model using the take-away approach. Also, the problem “If

“determining the difference”

approaches to subtraction, as

you walk 5 miles from home and turn back to walk 2 miles toward home, how many

well as the inverse relationship

miles are you from home?” can be solved with a measurement model using the take-

between addition and subtraction

away approach (Figure 3.14).

(Selter, Prediger, Nuhrenborger,

Hussmann, 2012).

Figure 3.14

This approach can be stated using sets.

c03.indd 93

7/30/2013 2:48:30 PM - 94

Chapter 3 Whole Numbers: Operations and Properties

D E F I N I T I O N 3 . 2

Subtraction of Whole Numbers: Take-Away Approach

Let a and b be any whole numbers and A and B be sets such that a = n(A), b = n(B),

and B ⊆ A. Then

a − b = n(A − B).

NCTM Standard

The number “a - b” is called the difference and is read “a minus b,” where a is called

All students should understand

the minuend and b the subtrahend. To fi nd 7 − 3 using sets, think of a set with seven

various meanings of addition and

elements, say { ,

a ,

b ,

c d, ,

e f , g}. Then, using set difference, take away a subset of three

subtraction of whole numbers

elements, say { ,

a ,

b }

c . The result is the set {d, ,

e f , g}, so 7 − 3 = 4.

and the relationship between the

two operations.

The Missing-Addend Approach The second method of subtraction, which is

called the missing-addend approach, is often used when making change. For example,

Reflection from Research

if an item costs 76 cents and 1 dollar is tendered, a clerk will often hand back the

Children frequently have difficulty

change by adding up and saying “76 plus four is 80, and twenty is a dollar” as four

with missing-addend problems

pennies and two dimes are returned. This method is illustrated in Figure 3.15.

when they are not related to

word problems. A common

answer to 5 + ? = 8 is 13 (Kamii,

1985).

Figure 3.15

Since 2 + 3 = 5 in each case in Figure 3.15, we know that 5 − 2 = 3.

Common Core – Grade 1

A L T E R N A T I V E D E F I N I T I O N 3 . 3

Understand subtraction as an

unknown-addend problem.

Subtraction of Whole Numbers: Missing-Addend Approach

Let a and b be any whole numbers. Then a − b = c if and only if a = b + c for some

whole number c.

In this alternative definition of subtraction, c is called the missing addend. The missing-

addend approach to subtraction is very useful for learning subtraction facts because it

shows how to relate them to the addition facts via four-fact families (Figure 3.16).

This alternative definition of subtraction does not guarantee that there is an answer

for every whole-number subtraction problem. For example, there is no whole number c

such that 3 = 4 + c, so the problem 3 − 4 has no whole-number answer. Another way of

expressing this idea is to say that the set of whole numbers is not closed under subtraction.

Notice that we have two approaches to subtraction, (i) take away and (ii) missing

addend and that each of these approaches can be modeled in two ways using (i) sets

and (ii) measurement. The combination of these four methods can be visualized in a

Figure 3.16

two-dimensional diagram (Figure 3.17).

c03.indd 94

7/30/2013 2:48:41 PM - Section 3.1 Addition and Subtraction 95

Figure 3.17

Children’s Literature

The reason for learning to add and subtract is to solve problems in the real world.

www.wiley.com/college/musser

For example, consider the next two problems.

See “Ten Sly Piranhas” by William

Wise.

1. Monica was 59" tall last year. She had a growth spurt and is now 66" tall. How

much did she grow during this past year?

Common Core – Grade 1

To solve this problem, we can use the lower right square in Figure 3.17 because

Use addition and subtraction

we are dealing with her height (measurement) and we want to know how many more

within 20 to solve word problems

inches 66 is than 59 (missing addend).

involving situations of adding to,

taking from, putting together,

2. Monica joined a basketball team this year. One of her teammates is 70" tall. How

taking apart, and comparing, with

much taller is that teammate than Monica?

unknowns in all positions.

There is a new aspect to this problem. Instead of considering how much taller

Monica is than she was last year, you are asked to compare her height with another

player. More generally, you might want to compare her height to all the members of

her team. This way of viewing subtraction adds a third dimension, comparison, to the

2-by-2 square in Figure 3.17. (Figure 3.18)

Figure 3.18

c03.indd 95

7/30/2013 2:48:41 PM - 96 Chapter 3 Whole Numbers: Operations and Properties

To solve the problem of comparing Monica’s height with the heights of the rest

of her teammates, we would use the smaller cube in the lower right back of the 2-by-

2-by-2 cube since it uses the measurement model and missing addend approach involv-

ing the comparison with several teammates.

Following is another problem where comparison comes into play: If Larry has $7

and Judy has $3, how much more money does Larry have? Because Larry’s money

and Judy’s money are two distinct sets, a comparison view would be used. To find the

solution, we can mentally match up 3 of Larry’s dollars with 3 of Judy’s dollars and

Figure 3.19

take those matched dollars away from Larry’s (see Figure 3.19).

Thus, this problem would correlate to the smaller cube in the top back left of

Figure 3.18 because it is a set model using take-away approach involving the compari-

son of sets of money. In subtraction situations where there is more than one set or one

measurement situation involved as illustrated by the back row of the cube in Figure

3.18, the subtraction is commonly referred to as using the comparison approach.

Check for Understanding: Exercise/Problem Set A #8–14

✔

Benjamin Franklin was known for his role in politics and as an inven-

tor. One of his mathematical discoveries was an 8-by-8 square made

up of the counting numbers from 1 to 64. Check out the following

properties:

1. All rows and columns total 260.

2. All half-rows and half-columns total 130.

3. The four corners total 130.

4. The sum of the corners in any 4-by-4 or 6-by-6 array is 130.

5. Every 2-by-2 array of four numbers totals 130.

(Note: There are 49 of these 2-by-2 arrays!)

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. a. Draw a figure similar to Figure 3.1 to find 4 + 3.

c. 53 + 47 = 50 + 50

b. Find 3 + 5 using a number line.

d. 1 + 0 = 1

e. 1 + 0 = 0 + 1

2. For which of the following pairs of sets is it true that

f.

n

+

+ =

+

(D) + n(E ) = n(D ∪ E )

(53

48)

7

60

48

? When not true, explain

why not.

5. Use the Chapter 3 eManipulative Number Bars on our

a. D = { ,

1 ,

2 ,

3 }

4 , E = { ,

7 ,

8 ,

9 1 }

0

Web site to model 7 + 2 and 2 + 7 on the same number

b. D = { }, E = { }

1

line. Sketch what is represented on the computer and

c. D = a

{ , b, c, d}, E = d

{ , c, b, a}

describe how the two problems are different. How are

they similar?

3. Which of the following sets are closed under addition?

Why or why not?

6. What property or properties justify that you get the same

a. { ,

0 1 ,

0 2 ,

0 3 ,

0 . . . }

b. { }

0

answer to the following problem whether you add “up”

c. { ,

0 ,

1 }

2

d. { ,

1 }

2

(starting with 9 + 8) or “down” (starting with 3 + 8)?

e. Whole numbers greater than 17

3

4. Identify the property or properties being illustrated.

8

a. 1279 + 3847 must be a whole number.

b. 7 + 5 = 5 + 7

+ 9

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7/30/2013 2:48:47 PM - Section 3.1 Addition and Subtraction 97

7. Addition can be simplified using the associative property of

c.

addition. For example,

26 + 57 = 26 + (4 + 53) = (26 + 4) + 53

= 30 + 53 = 83.

Complete the following statements.

a. 39 + 68 = 40 +

=

12. Using different-shaped boxes for variables provides a

b. 25 + 56 = 30 +

=

transition to alge.bra as well as a means of stating prob-

c. 47 + 23 = 50 +

=

lems. Try some whole numbers in the boxes to determine

whether these properties hold.

8. a. Complete the following addition table in base five.

a.

Is subtraction closed?

Remember to use the thinking strategies.

b.

Is subtraction commutative?

c.

Is subtraction associative?

b.

For each of the following subtraction problems in base

five, rewrite the problem using the missing-addend ap-

proach and find the answer in the table in part a.

d.

Is there an identity element for subtraction?

i. 13

−

−

five

4five

ii. 11five

3five

iii. 12

−

−

five

4five

iv. 10five

2five

9. Complete the following four-fact families in base five.

a. 3

+

=

13. Each situation described next involves a subtraction

five

4five

12five

b. ____________________________________

____________________________________

4

+

=

problem. In each case, briefly name the small cube por-

five

1five

10five

____________________________________

____________________________________

tion of Figure 3.18 that correctly classifies the problem.

____________________________________

____________________________________

Typical answers may be set, take-away, no comparison or

measurement, missing-addend, comparison. Finally, write

c.

____________________________________

an equation to fit the problem.

____________________________________

a.

An elementary teacher started the year with a budget

11 −

=

five

4five

2five

of $200 to be spent on manipulatives. By the end of

____________________________________

December, $120 had been spent. How much money

10. In the following figure, centimeter strips are used to illus-

remained in the budget?

trate 3 + 8 = 11. What two subtraction problems are also

b.

Doreen planted 24 tomato plants in her garden and

being represented?

Justin planted 18 tomato plants in his garden. How

many more plants did Doreen plant?

c.

Tami is saving money for a trip to Hawaii over spring

break. The package tour she is interested in costs $1795.

From her part-time job she has saved $1240 so far.

11. For the following figures, identify the problem being illus-

How much more money must she save?

trated, the model, and the conceptual approach being used.

a.

14. a. State a subtraction word problem involving 8 − 3, the

missing-addend approach, the set model, without

comparison.

b.

State a subtraction word problem involving 8 − 3, the

take-away approach, the measurement model, with

comparison.

b.

c.

Sketch the set model representation of the situation

described in part a.

d.

Sketch the measurement model representation of the

situation described in part b.

PROBLEMS

15. A given set contains the number 1. What other numbers

1 + 2 + 3 − 4 + 5 + 6 + 78 + 9 = 100

must also be in the set if it is closed under addition?

Find a sum of 100 using each of the nine digits and only

16. The number 100 can be expressed using the nine digits

three plus or minus signs.

1, 2, . . . , 9 with plus and minus signs as follows:

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7/30/2013 2:48:57 PM - 98 Chapter 3 Whole Numbers: Operations and Properties

17. Complete the following magic square in which the sum

digits and adding the two numbers has been repeated until

of each row, each column, and each diagonal is the same.

a palindrome is obtained.

When completed, the magic square should contain each

67

of the numbers 10 through 25 exactly once.

+ 76

143

+ 341

484

a.

Try this method with the following numbers.

i. 39

ii. 87

iii. 32

b.

Find a number for which the procedure takes more than

three steps to obtain a palindrome.

20. Mr. Morgan has five daughters. They were all born the

18. Magic squares are not the only magic figures. The

number of years apart as the youngest daughter is old.

following figure is a magic hexagon. What is “magic”

The oldest daughter is 16 years older than the youngest.

about it?

What are the ages of Mr. Morgan’s daughters?

21. Use the Chapter 3 eManipulative activity, Number Puzzles

exercise 2, on our Web site to arrange the numbers 1, 2, 3,

4, 5, 6, 7, 8, 9 in the circles below so the sum of the num-

bers along each line of four is 23.

19. A palindrome is any number that reads the same backward

and forward. For example, 262 and 37673 are palindromes.

In the accompanying example, the process of reversing the

EXERCISE/PROBLEM SET B

EXERCISES

1. Show that 2 + 6 = 8 using two different types of models.

d. (4 + 3) + 6 =

+ (4 + 3)

2. For which of the following pairs of sets is it true that

e. (4 + 3) + 6 = (3 +

) + 6

n(D) + n(E ) = n(D ∪ E )? When not true, explain why not.

f. 2 + 9 is a ______ number.

a. D = a

{ , c, e, g}, E = b

{ , d, f , g}

5. Show that the commutative property of whole-number

b. D = { }, E = { }

addition holds for the following examples in other bases by

c. D = { ,

1 ,

3 ,

5 }

7 , E = { ,

2 ,

4 }

6

using a different number line for each base.

a. 3

+

=

+

+

=

+

five

4five

4five

3five b. 5nine

7nine

7nine

5nine

3. Which of the following sets are closed under addition? Why

or why not?

6. Without performing the addition, determine which sum (if

a. { ,

0 ,

3 ,

6 ,

9 . . . } b.

{1}

either) is larger. Explain how this was accomplished and

c. { ,

1 ,

5 ,

9 1 ,

3 . . . d.

{ ,

8 1 ,

2 1 ,

6 2 ,

0 . . .

what properties were used.

3261

4187

e. Whole numbers less than 17

4287

5291

4. Each of the following is an example of one of the properties

+ 5193

+ 3263

for addition of whole numbers. Fill in the blank to complete

the statement, and identify the property.

7. Look for easy combinations of numbers to compute the

a. 5 +

= 5

following sums mentally. Show and identify the properties

b. 7 + 5 =

+ 7

you used to make the groupings.

c. (4 + 3) + 6 = 4 + (

+ 6)

a.

94 + 27 + 6 + 13

b. 5 + 13 + 25 + 31 + 47

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7/30/2013 2:49:01 PM - Section 3.1 Addition and Subtraction 99

8. a. Complete the following addition table in base six.

c.

Remember to use the thinking strategies.

12. For each of the following, determine whole numbers x, y,

and z that make the statement true.

a.

x − 0 = 0 − x = x

b. x − y = y − x

c.

(x − y) − z = x − (y − z)

b.

For each of the following subtraction problems in base

Which, if any, are true for all whole numbers x, y, and z?

six, rewrite the problem using the missing-addend

approach and find the answer in the table.

13. Each situation described next involves a subtraction

i. 13

−

six

5six

ii. 5 −

problem. In each case, briefly name the small cube por-

six

4six

iii. 12

−

−

tion of Figure 3.18 that correctly classifies the problem.

six

4six

iv. 10six

2six

Typical answers may be set, take-away, no comparison or

9. Using the addition table for base six given in Exercise 8,

measurement, missing-addend, comparison. Finally, write

write the following four-fact families in base six.

an equation to fit the problem.

a. 2

+

=

−

=

a.

Robby has accumulated a collection of 362 sports

six

3six

5six

b. 11six

5six

2six

cards. Chris has a collection of 200 cards. How many

10. Rewrite each of the following subtraction problems as an

more cards than Chris does Robby have?

addition problem.

b.

Jack is driving from St. Louis to Kansas City for a

a.

x − 156 = 279 b. 279 − 156 = x c. 279 − x = 156

meeting, a total distance of 250 miles. After 2 hours he

notices that he has traveled 114 miles. How far is he

11. For the following figures, identify the problem being

from Kansas City at that time?

illustrated, the model, and the conceptual approach

c.

An elementary school library consists of 1095 books.

being used.

As of May 8, 105 books were checked out of the library.

How many books were still available for checkout on

a.

May 8?

14. a. State a subtraction word problem involving 9 − 5, the

missing-addend approach, the measurement model,

without comparison.

b.

b.

State a subtraction word problem involving 9 − 5, the

take-away approach, the set model, with comparison.

c.

Sketch the measurement model representation of the

situation described in part a.

d.

Sketch the set model representation of the situation

described in part b.

PROBLEMS

15. Suppose that S is a set of whole numbers closed under

9 + 4 = 13 and 8 + 3 = 11

addition. S contains 3, 27, and 72.

The circle at the lower right is filled in one of two ways:

a.

List six other elements in S .

b. Why must 24 be in S ?

i. Adding the numbers in the upper circles:

16. A given set contains the number 5. What other numbers

13 + 11 = 24

must also be in the set if it is closed under addition?

ii. Adding across the rows, adding down the columns, and

17. The next figure can provide practice in addition and sub-

then adding the results in each case:

traction. The figure is completed by filling the upper two

+

=

and

+ =

circles with the sums obtained by adding diagonally.

12 12

24

17

7

24

c03.indd 99

7/30/2013 2:49:08 PM - 100 Chapter 3 Whole Numbers: Operations and Properties

a.

Fill in the missing numbers for the following figure.

21. Shown here is a magic triangle discovered by the mental

calculator Marathe. What is its magic?

b.

Fill in the missing numbers for the following figure.

22. The property “If a + c = b + c, then a = b” is called the

additive cancellation property. Is this property true for all

whole numbers? If it is, how would you convince your

students that it is always true? If not, give a counter

example.

Analyzing Student Thinking

23. Kaitlyn doesn’t understand why the definition of Addition

c.

Use variables to show why the sum of the numbers in

of Whole Numbers insists that sets A and B must be dis-

the upper two circles will always be the same as the sum

joint. What can you say to help her?

of the two rows and the sum of the two columns.

24. Enrique says, “I think of closure as a bunch of numbers

18. Arrange numbers 1 to 10 around the outside of the circle

locked in a room and the operation, like addition, comes

shown so that the sum of any two adjacent numbers is the

along and links any two of the numbers together. As long

same as the sum of the two numbers on the other ends

as the link is in the room, the room can be said to be closed

of the spokes. As an example, 6 and 9, 8 and 7 might be

with respect to addition. If the link is found outside of the

placed as shown, since 6 + 9 = 8 + 7.

room, then the set is not closed.” Is his reasoning math-

ematically correct? Explain.

25. Monique prefers using rote memory to learn the addition

facts rather than using thinking strategies. What case can

be made to convince her that there is value in using think-

ing strategies? Explain.

26. Theresa prefers using the number line for addition rather

than the set model. Can she use the number line model

to interpret the properties of whole-number addition?

19. Place the numbers 1 − 16 in the cells of the following magic

Explain.

square so that the sum of each row, column, and diagonal

is the same.

27. Severino says, “When I added 27 and 36, I made the 27 a

30, then I added the 30 and the 36 and got 66, and then

subtracted the 3 from the beginning and got 63.” Patricia

says, “But when I added 27 and 36, I added the 20 and

the 30 and got 50, then I knew 6 plus 6 was 12 so 6 plus 7

was 13, and then I added 50 and 13 and got 63.” Can

you follow the students’ reasonings here? How would

you describe some of their techniques?

20. Use the Chapter 3 eManipulative activity, Number Puzzles

28. Kayla claims that 7

+

=

+ =

eight

3eight

10eight since 7

3 10.

exercise 3, on our Web site to arrange the numbers 1, 2, 3,

How should you respond?

4, 5, 6, 7, 8, 9 in the circles below so the sum of the num-

29. Your classroom has 21 students and there are 9

bers along each line of three is 15.

boys. You ask two students to use these numbers to

determine how many girls are in the classroom.

Conner says, “9 plus 10 is 19, 19 plus 2 is 21, so there

are 10 + 2 = 12 girls.” Chandler says that Conner is

wrong because he did not use “take-away.” How

should you respond?

30. Darren happens to see your copy of this book on your

desk open to Figure 3.18. He says, “What is that used

for?” How should you respond?

c03.indd 100

7/30/2013 2:49:14 PM - Section 3.2 Multiplication and Division 101

MULTIPLICATION AND DIVISION

Write a word problem that would have 3 × 4 as its solution. Compare your problem

with those of several peers and determine how these problems are different and how

they are the same from a mathematical perspective. If possible categorize the problems according to the underlying

mathematical structure.

Children’s Literature

Multiplication and Its Properties

www.wiley.com/college/musser

See “Amanda Bean’s Amazing

There are many ways to view multiplication.

Dream: A Mathematical Story” by

Cindy Neuschwander.

Repeated-Addition Approach Consider the following problems: There are

five children, and each has three silver dollars. How many silver dollars do they

Common Core – Grade 3

have altogether? The silver dollars are about 1 inch wide. If the silver dollars are

Interpret products of whole

laid in a single row with each dollar touching the next, what is the length of the row?

numbers (e.g., 5 × 7) as the total

These problems can be modeled using the set model and the measurement model

number of objects in 5 groups of

7 objects each.

(Figure 3.20).

Reflection from Research

When multiplication is repre-

sented by repeated addition,

students have a great deal of

difficulty keeping track of the two

sets of numbers. For instance,

when considering how many

sets of three there are in fifteen,

students need to keep track of

counting up the threes and how

many sets of three they count

(Steffe, 1988).

Figure 3.20

These models look similar to the ones that we used for addition, since we are

merely adding repeatedly. They show that 3 + 3 + 3 + 3 + 3 = 15, or that 5 × 3 = 15.

D E F I N I T I O N 3 . 4

Children’s Literature

Multiplication of Whole Numbers: Repeated-Addition Approach

www.wiley.com/college/musser

See “One Hundred Hungry Ants”

Let a and b be any whole numbers where a ≠ 0. Then

by Elinor Pinczes.

ab = b + b + ⋅ ⋅ ⋅ + b

a addends

If a = 1, then ab = 1 ¥ b = b; also 0 ¥ b = 0 for all b.

Since multiplication combines two numbers to form a single number, it is a binary

operation. The number ab, read “a times b,” is called the product of a and b. The numbers

a and b are called factors of ab. The product ab can also be written as “a ¥ b” and “a ë b.”

Notice that 0 ¥ b = 0 for all b. That is, the product of zero and any whole number is zero.

Rectangular Array Approach If the silver dollars in the preceding problem are

arranged in a rectangular array, multiplication can be viewed in a slightly different

Figure 3.21

way (Figure 3.21).

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7/30/2013 2:49:18 PM - 102 Chapter 3 Whole Numbers: Operations and Properties

Common Core – Grade 3

A L T E R N A T I V E D E F I N I T I O N 3 . 5

Use multiplication and division

within 100 to solve word prob-

Multiplication of Whole Numbers: Rectangular Array Approach

lems in situations involving equal

groups, arrays, and measurement

Let a and b be any whole numbers. Then ab is the number of elements in a rectan-

quantities (e.g., by using draw-

gular array having a rows and b columns.

ings and equations with a symbol

for the unknown number to rep-

resent the problem).

Cartesian Product Approach A third way of viewing multiplication is an

abstraction of this array approach.

A L T E R N A T I V E D E F I N I T I O N 3 . 6

Multiplication of Whole Numbers: Cartesian Product Approach

Let a and b be any whole numbers. If a = n(A) and b = n(B), then

ab = n(A × B).

For example, to compute 2 ¥ 3, let 2 = n a

({ , b}) and 3 = n( x

{ , y, z}). Then 2 ¥ 3 is

the number of ordered pairs in { ,

a }

b

Figure 3.22

× { ,

x

,

y }

z . Because { ,

a }

b × { ,

x

,

y }

z = {( ,

a x),

( ,

a y), ( ,

a z), ( ,

b x), ( ,

b y), ( ,

b z)} has six ordered pairs, we conclude that 2 ¥ 3 = 6.

Actually, by arranging the pairs in an appropriate row and column configuration,

this approach can also be viewed as the array approach (Figure 3.22).

Tree Diagram Approach Another way of modeling this approach is through

the use of a tree diagram (Figure 3.23). Tree diagrams are especially useful in the field

of probability, which we study in Chapter 11.

Properties of Whole-Number Multiplication You have probably observed

that whenever you multiplied any two whole numbers, your product was always a

Figure 3.23

whole number. This fact is summarized by the following property.

Reflection from Research

P R O P E R T Y

Children can solve a variety of

Closure Property for Multiplication of Whole Numbers

multiplicative problems long

before being formally introduced

The product of two whole numbers is a whole number.

to multiplication and division

(Mulligan & Mitchelmore, 1995).

When two odd whole numbers are multiplied together, the product is odd; thus

the set of odd numbers is closed under multiplication. Closure is a useful idea, since if

we are multiplying two (or more) odd numbers and the product we calculate is even,

we can conclude that our product is incorrect. The set { ,

2 ,

5 ,

8 1 ,

1 1 ,

4 . . . } is not closed

under multiplication, since 2 ¥ 5 = 10 and 10 is not in the set.

The next property can be used to simplify learning the basic multiplication facts. For

example, by the repeated-addition approach, 7 × 2 represents 2 + 2 + 2 + 2 + 2 + 2 + 2 ,

whereas 2 × 7 means 7 + 7. Since 7 + 7 was learned as an addition fact, viewing 7 × 2

as 2 × 7 makes this computation easier.

Common Core – Grade 3

P R O P E R T Y

Apply properties of operations as

strategies to multiply and divide

Commutative Property for Whole-Number Multiplication

(e.g., if 6 × 4 = 24 is known, then

4 × 6 = 24 is also known).

Let a and b be any whole numbers. Then

ab = ba.

c03.indd 102

7/30/2013 2:49:22 PM - From Lesson 4-1 “Multiplication as Repeated Addition” from Envision Math Common Core, by Randall I. Charles et al., Grade 3,

copyright © by Pearson Education.

103

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7/30/2013 2:49:30 PM - 104 Chapter 3 Whole Numbers: Operations and Properties

Problem-Solving Strategy

The example in Figure 3.24 should convince you that the commutative property

Draw a Picture

for multiplication is true.

Reflection from Research

If multiplication is viewed as

computing area, children can see

the commutative property rela-

tively easily, but if multiplication

is viewed as computing the price

of a number of items, the com-

mutative property is not obvious

(Vergnaud, 1981).

Figure 3.24

The product 5 ¥ (2 ¥ 13) is more easily found if it is viewed as (5 ¥ 2) ¥ 13. Regrouping

to put the 5 and 2 together can be done because of the next property.

P R O P E R T Y

Associative Property for Whole-Number Multiplication

Let a, b, and c be any whole numbers. Then

a(bc) = (ab c

) .

Problem-Solving Strategy

To illustrate the validity of the associative property for multiplication, consider

Draw a Picture

the three-dimensional models in Figure 3.25.

Common Core – Grade 3

Apply properties of operations as

strategies to multiply and divide

(e.g., 3 × 5 × 2 can be found by

3 × 5 = 15, then 15 × 2 = 30 or by

5 × 2 = 10, then 3 × 10 = 30).

Figure 3.25

The next property is an immediate consequence of each of our defi nitions of

multiplication.

P R O P E R T Y

Identity Property for Whole-Number Multiplication

The number 1 is the unique whole number such that for every whole number a,

a ¥ 1 = a = 1 ¥ a.

Because of this property, the number 1 is called the multiplicative identity or the

identity for multiplication.

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7/30/2013 2:49:32 PM - Section 3.2 Multiplication and Division 105

There is one other important property of the whole numbers. This property,

distributivity, combines both multiplication and addition. Study the array model in

Figure 3.26. This model shows that the product of a sum, 3(2 + 4), can be expressed as

the sum of products, (3 ¥ 2) + (3 ¥ 4). This relationship holds in general.

Common Core – Grade 3

Apply properties of operations as

strategies to multiply and divide

(e.g., knowing that 8 × 5 = 40 and

8 × 2 = 16, one can find 8 × 7 as

8 × 5

( + 2) = 8

( × 5) + 8

( × 2) =

40 + 16 = 56 ).

Figure 3.26

NCTM Standard

P R O P E R T Y

All students should understand

and use properties of operations

Distributive Property of Multiplication over Addition

such as the distributivity of multi-

plication over addition.

Let a, b, and c be any whole numbers. Then

a(b + c) = ab + ac.

Algebraic Reasoning

Because of commutativity, we can also write (b + c)a = ba + ca. Notice that the

The distributive property is

distributive property “distributes” the a to the b and the c.

commonly used to “combine

like terms” in problems like

3x + 2x = (3 + 2)x = 5x . Becoming

comfortable with the distributive

Rewrite each of the following expressions using the distributive

property with numbers develops

property.

a foundation for applying the

( + )

property in algebra.

a. 3 4 5

b. 5 ¥ 7 + 5 ¥ 3

c. am + an

d. 31 ¥ 76 + 29 ¥ 76

e. a(b + c + d )

S O L U T I O N

a. 3(4 + 5) = 3 ¥ 4 + 3 ¥ 5

b. 5 ¥ 7 + 5 ¥ 3 = 5(7 + 3)

c. am + an = a(m + n)

d. 31 ¥ 76 + 29 ¥ 76 = (31+ 29 7

) 6

e. a(b + c + d ) = a((b + c) + d ) = a(b + c) + ad = ab + ac + ad

■

Let’s summarize the properties of whole-number addition and multiplication.

P R O P E R T Y

Whole-Number Properties

PROPERTY

ADDITION

MULTIPLICATION

Closure

Yes

Yes

Commutativity

Yes

Yes

Associativity

Yes

Yes

Identity

Yes (zero)

Yes (one)

Distributivity of multiplication

Yes

over addition

In addition to these properties, we highlight the following property.

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P R O P E R T Y

Multiplication Property of Zero

For every whole number, a,

a ¥ 0 = 0 ¥ a = 0.

Using the missing-addend approach to subtraction, we will show that a(b − c) =

ab − ac whenever b − c is a whole number. In words, multiplication distributes over

subtraction.

Let

b − c = n

Then

b = c + n

Missing addend

ab = a(c + n)

Multipliction

ab = ac + an

Distributivity

Therefore, ab − ac = an

Missing addend from the ﬁ rst equation

Bu

ut

b − c = n

So, substituting b − c for n, we have

ab − ac = a(b − c).

P R O P E R T Y

Distributive Property of Multiplication over Subtraction

Let a, b, and c be any whole numbers where b ≥ c. Then

a(b − c) = ab − ac.

The following discussion shows how the properties are used to develop thinking

strategies for learning the multiplication facts.

Thinking Strategies for Learning the Multiplication Facts The

multipli-

cation table in Figure 3.27 has 10 × 10 = 100 unfilled spaces.

Figure 3.27

Figure 3.28

Figure 3.29

Reflection from Research

1. Commutativity: As in the addition table, because of commutativity, only 55 facts in

Rote memorization does not

the unshaded region in Figure 3.28 have to be found.

indicate mastery of facts. Instead,

mastery of multiplication facts

2. Multiplication by 0: a ¥ 0 = 0 for all whole numbers a. Thus the first column is all

is indicated by a conceptual

zeros (Figure 3.29).

understanding and computational

fluency (Wallace & Gurganus,

3. Multiplication by 1: 1 ¥ a = a ¥ 1 = a. Thus the column labeled “1” is the same as the

2005).

left-hand column outside the table (Figure 3.29).

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7/30/2013 2:49:37 PM - Section 3.2 Multiplication and Division 107

4. Multiplication by 2: 2 ¥ a = a + a, which are the doubles from addition (Figure 3.29).

We have filled in 27 facts using thinking strategies 1, 2, 3, and 4. Therefore, 28 facts

remain to be found out of our original 55.

5. Multiplication by 5: The counting chant by fives, namely 5, 10, 15, 20, and so on, can

be used to learn these facts (see the column and/or row headed by a 5 in Figure 3.30).

6. Multiplication by 9: The multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, and 81

(Figure 3.30). Notice how the tens digit is one less than the number we are multiply-

ing by 9. For example, the tens digit of 3 ¥ 9 is 2 (one less than 3). Also, the sum of

Figure 3.30

the digits of the multiples of 9 is 9. Thus 3 ¥ 9 = 27 since 2 + 7 = 9. The multiples of

5 and 9 eliminate 13 more facts, so 15 remain.

7. Associativity and distributivity: The remaining facts can be obtained using

these two properties. For example, 8 × 4 = 8 × (2 × 2) = (8 × 2) × 2 = 16 × 2 = 32 or

8 × 4 = 8(2 + 2) = 8 ¥ 2 + 8 ¥ 2 = 16 +16 = 32.

In the next example we consider how knowledge of the basic facts and the proper-

ties can be applied to multiplying a single-digit number by a multidigit number.

NCTM Standard

All students should develop

Compute the following products using thinking strategies.

fluency with basic number com-

×

¥

binations for multiplication and

a. 2

34

b. 5(37 2)

c. 7(25)

division and use combinations to

mentally compute related prob-

S O L U T I O N

lems, such as 30 × 50.

a. 2 × 34 = 2(30 + 4) = 2 ¥ 30 + 2 ¥ 4 = 60 + 8 = 68

b. 5(37 ¥ 2) = 5(2 ¥ 37) = (5 ¥ 2) ¥ 37 = 370

c. 7(25) = (4 + 3 25

)

= 4 ¥ 25 + 3 ¥ 25 = 100 + 75 = 175

■

Check for Understanding: Exercise/Problem Set A #1–10

✔

Write a word problem that would have 12 ÷ 3 as its solution. There are some general types

of division problems. Compare the underlying mathematical structure of your problem

with that of your peers and try to identify and describe some different types of division problems.

Reflection from Research

Division

Multiplication is usually intro-

duced before division and sepa-

Just as with addition, subtraction, and multiplication, we can view division in differ-

rated from it, whereas children

ent ways. Consider these two problems.

spontaneously relate them and

do not necessarily find division

more difficult than multiplication

1. A class of 20 children is to be divided into four teams with the same number of

(Mulligan & Mitchelmore, 1997).

children on each team. How many children are on each team?

2. A class of 20 children is to be divided into teams of four children each. How many

NCTM Standard

teams are there?

All students should understand

various meanings of multiplication

Each of these problems is based on a different conceptual way of viewing division.

and division.

A general description of the first problem is that you have a certain number of objects

that you are dividing into or “sharing” among a specified number of groups and are

asking how many objects are in each group. Because of its sharing nature, this type of

division is referred to as sharing division. A general description of the second problem

is that you have a certain number of objects and you are “measuring out” a specified

number of objects to be in each group and asking how many groups there are. This type

of division is called measurement division (Figure 3.31).

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Chapter 3 Whole Numbers: Operations and Properties

When dealing with whole numbers, the difference between these two types of

division may seem very subtle, but these differences become more apparent when

considering the division of decimals or fractions.

Common Core – Grade 3

Interpret whole-number quotients

of whole numbers (e.g., 56 ÷ 8)

as the number of objects in each

share when 56 objects are par-

titioned equally into 8 shares or

as a number of shares when 56

objects are partitioned into equal

shares of 8 objects each.

Sharing

Figure 3.31

The following examples will help clarify the distinction between sharing and mea-

surement division.

Reflection from Research

Classify each of the following division problems as examples of

Allowing students to write their

either sharing or measurement division.

own division problems helps

students to think about and apply

a. A certain airplane climbs at a rate of 300 feet per second. At this rate, how long

their understanding of division

will it take the plane to reach a cruising altitude of 27,000 feet?

to real-world contexts (Polly &

b. A group of 15 friends pooled equal amounts of money to buy lottery tickets for a

Ruble, 2009).

$1,987,005 jackpot. If they win, how much should each friend receive?

c. Shauna baked 54 cookies to give to her friends. She wants to give each friend a

plate with 6 cookies on it. How many friends can she give cookies to?

Problem-Solving Strategy

S O L U T I O N

Draw a Picture

a. Since every 300 feet can be viewed as a single group corresponding to 1 second, we

are interested in fi nding out how many groups of 300 feet there are in 27,000 feet.

Thus this is a measurement division problem.

NCTM Standard

b. In this case, each friend represents a group and we are interested in how much money

All students should understand

goes to each group. Therefore, this is an example of a sharing division problem.

situations that entail multiplica-

c. Since every group of cookies needs to be of size 6, we need to determine how many

tion and division, such as equal

groups of size 6 there are in 54 cookies. This is an example of a measurement divi-

groupings of objects and sharing

equally.

sion problem.

■

Missing-Factor Approach Figure 3.32 shows that multiplication and division

are related. This suggests the following definition of division.

D E F I N I T I O N 3 . 7

Division of Whole Numbers: Missing-Factor Approach

If a and b are any whole numbers with b ≠ 0, then a ÷ b = c if and only if a = bc for

some whole number c.

Figure 3.32

The symbol a ï b is read “a divided by b.” Also, a is called the dividend, b is called

the divisor, and c is called the quotient or missing factor. The basic facts multiplication

table can be used to learn division facts (Figure 3.32).

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Find the following quotients.

a. 24 ÷ 8 b.

72 ÷ 9 c.

52 ÷ 4 d.

0 ÷ 7

S O L U T I O N

a. 24 ÷ 8 = 3, since 24 = 8 × 3

b. 72 ÷ 9 = 8, since 72 = 9 × 8

c. 52 ÷ 4 = 13, since 52 = 13 × 4

d. 0 ÷ 7 = 0, since 0 = 7 × 0

■

Reflection from Research

Fourth- and fifth-grade students

most frequently defined division

The division problem in Example 3.8(d) can be generalized as follows and verified

as undoing multiplication

(Graeber & Tirosh, 1988).

using the missing-factor approach.

P R O P E R T Y

Division Property of Zero

If a ≠ 0, then 0 ÷ a = 0.

Next, consider the situation of dividing by zero. Suppose that we extend the

missing-factor approach of division to dividing by zero. Then we have the following

two cases.

Algebraic Reasoning

Case 1: a ÷ 0, where a ≠ 0. If a ÷ 0 = c, then a = 0 ¥ c, or a = 0. But a ≠ 0. Therefore,

Notice how variables are used to

a ÷ 0 is undefi ned.

represent any number in order to

show that this explanation works

Case 2: 0 ÷ 0. If 0 ÷ 0 = c, then 0 = 0 ¥ c. But any value can be selected for c, so there

for any number.

is no unique quotient c. Thus division by zero is said to be indeterminate, or unde-

fi ned, here. These two cases are summarized by the following statement.

D I V I S I O N B Y Z E R O

Division by 0 is undefi ned.

Reflection from Research

Second- and third-grade students

tend to use repeated addition to

Now consider the problem 37 ÷ 4. Although 37 ÷ 4 does not have a whole-number

solve simple multiplication AND

answer, there are applications where it is of interest to know how many groups of 4

division problems. In a division

are in 37 with the possibility that there is something left over. For example, if there

problem, such as 15 ÷ 5, they

are 37 fruit slices to be divided among four children so that each child gets the same

will repeatedly add the divisor

number of slices, how many would each child get? We can find as many as 9 fours in

until they reach the quotient

(5 + 5 =

;

10 10 + 5 = 15), often

37 and then have 1 remaining. Thus each child would get nine fruit slices with one left

using their fingers to keep track

undistributed. This way of looking at division of whole numbers, with a remainder,

of the number of times they

is summarized next.

use 5 (Mulligan & Mitchelmore,

1995).

T H E D I V I S I O N A L G O R I T H M

If a and b are any whole numbers with b ≠ 0, then there exist unique whole num-

bers q and r such that a = bq + r, where 0 ≤ r < b.

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Here b is called the divisor, q is called the quotient, and r is the remainder. Notice

that the remainder is always less than the divisor. Also, when the remainder is 0, this

result coincides with the usual defi nition of whole-number division.

Find the quotient and remainder for these problems.

a. 57 ÷ 9 b.

44 ÷ 13 c.

96 ÷ 8

S O L U T I O N

a. 9 × 6 = 54, so 57 = 9 ¥ 6 + 3. The quotient is 6 and the remainder is 3 (Figure 3.33).

Problem-Solving Strategy

Draw a Picture

Figure 3.33

b. 13 × 3 = 39, so 44 = 13 ¥ 3 + 5. The quotient is 3 and the remainder is 5.

c. 8 × 12 = 96, so 96 = 8 ¥ 12 + 0. The quotient is 12 and the remainder is 0.

■

Reflection from Research

Repeated-Subtraction Approach Figure 3.34 suggests alternative ways of

The Dutch approach to written

viewing division.

division calculations involves

repeated subtraction using

increasingly larger chunks. This

approach, which has helped

Dutch students to outperform

students from other nations,

builds progressively on intuitive

strategies (Anghileri, Beishuizen,

& Van Putten, 2002).

Figure 3.34

In Figure 3.34, 13 was subtracted from 44 three successive times until a number

less than 13 was reached, namely 5. Thus 44 divided by 13 has a quotient of 3 and a

remainder of 5. This example shows that division can be viewed as repeated subtrac-

tion. In general, to find a ÷ b using the repeated-subtraction approach, subtract b

successively from a and from the resulting differences until a remainder r is reached,

where r < b. The number of times b is subtracted is the quotient q.

Figure 3.35 provides a visual way to remember the main interconnections among

the four basic whole-number operations. For example, multiplication of whole num-

bers is defined by using the repeated-addition approach, subtraction is defined using

the missing-addend approach, and so on. An important message in this diagram is

that success in subtraction, multiplication, and division begins with a solid founda-

Figure 3.35

tion in addition.

Check for Understanding: Exercise/Problem Set A #11–17

✔

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7/30/2013 2:49:46 PM - Section 3.2 Multiplication and Division 111

The following note appeared in a newspaper.

“What is 241,573,142,393,627,673,576,957,439,048 times 45,994,

811,347,886,846,310,221,728,895,223,034,301,839? The answer is

71 consecutive 1s—one of the biggest numbers a computer has ever

factored. The 71-digit number was factored in 9.5 hours of a Cray

super-computer’s time at Los Alamos National Laboratory in New

Mexico, besting the previous high—69 digits—by two.

Why bother? The feat might affect national security. Some com-

puter systems are guarded by cryptographic codes once thought to

be beyond factoring. The work at Los Alamos could help intelligence

experts break codes.”

©Ron Bagwell

See whether you can find an error in the article and correct it.

EXERCISE/PROBLEM SET A

EXERCISES

1. What multiplication problems are suggested by the follow-

c. A teacher provided three number-2 pencils to each stu-

ing diagrams?

dent taking a standardized test. If a total of 36 students

a.

were taking the test, how many pencils did the teacher

need to have available?

4. The repeated-addition approach can easily be illustrated

b.

c.

using the calculator. For example, 4 × 3 can be found by

pressing the following keys:

3 + 3 + 3 + 3 = 12

d.

or if the calculator has a constant key, by pressing

5

5

5

5

5

5

3 + = = = 12 or

0

5

10

15

20

25

30

3 + + = = = 12

Find the following products using one of these techniques.

2. Illustrate 3 × 2 using the following combinations of models

a. 3 × 12 b.

4 × 17

and approaches.

c. 7 × 93 d.

143 × 6 (Think!)

a. Set model; Cartesian product approach

b. Set model; rectangular array approach

5. Which of the following sets are closed under multiplication?

c. Set model; repeated-addition approach

If the set is not closed, explain why not.

d. Measurement model; rectangular array approach

a. { ,

2 }

4 b.

{ ,

0 ,

2 ,

4 ,

6 . . . }

e. Measurement model; repeated-addition approach

c. { ,

0 }

3 d.

{ ,

0 }

1

3. Each situation described next involves a multiplication

e. { }

1

f. { }

0

problem. In each case tell whether the problem situation

g. { ,

5 ,

7 ,

9 . . . }

is best represented by the repeated-addition approach,

h. { ,

0 ,

7

,

14

,

21 . . . }

the rectangular array approach, or the Cartesian product

. . . , k . . .

approach, and why. Then write an appropriate equation to

i. { ,

0 ,

1 ,

2 ,

4 ,

8 1 ,

6

2 ,

}

fit the situation.

j. Odd whole numbers

a. A rectangular room has square tiles on the floor. Along

6. Identify the property of whole numbers being illustrated.

one wall, Kurt counts 15 tiles and along an adjacent wall he

¥ = ¥

counts 12 tiles. How many tiles cover the floor of the room?

a. 4 5

5 4

b. Jack has three pairs of athletic shorts and eight different

b. 6(3 + 2) = (3 + 2 6

)

T-shirts. How many different combinations of shorts and

c. 5(2 + 9) = 5 ¥ 2 + 5 ¥ 9

T-shirts could he wear to play basketball?

d. 1(x + y) = x + y

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7. Rewrite each of the following expressions using the dis-

iii. ___________________

tributive property for multiplication over addition or for

___________________

multiplication over subtraction.

___________________

a.

4(60 + 37)

13

÷

=

five

2five

4five

b.

(21 + 35) ¥ 6

c.

37 ¥ (60 − 22)

12. Identify each of the following problems as an example

d.

5x + 2x

of either sharing or measurement division. Justify your

e.

5(a + 1) − 3(a + 1)

answers.

a.

Gabriel bought 15 pints of paint to redo all the doors

8. The distributive property of multiplication over addition

in his house. If each door requires 3 pints of paint, how

can be used to perform some calculations mentally. For

many doors can Gabriel paint?

example, to find 13 î 12, you can think

b.

Hideko cooked 12 tarts for her family of 4. If all of the

13 1

( 2) = 13 1

( 0 + 2) = 13 10

(

) + 13(2)

family members receive the same amount, how many

=

tarts will each person receive?

130 + 26 = 156.

c.

Ms. Ivanovich needs to give 3 straws to each student in

How could each of the following products be rewritten

her class for their art activity. If she uses 51 straws, how

using the distributive property so that it is more easily

many students does she have in her class?

computed mentally?

a.

45(11) b.

39(102)

13. a. Write a sharing division problem that would have the

equation 15 ÷ 3 as part of its solution.

c.

23(21) d.

97(101)

b.

Write a measurement division problem that would have

9. a. Compute 374 × 12 without using the 4 key. Explain.

the equation 15 ÷ 3 as part of its solution.

b.

Compute 374 × 12 without using the 2 key. Explain.

14. Rewrite each of the following division problems as a mul-

10. Compute using thinking strategies. Show your reasoning.

tiplication problem.

a.

5(23 × 4) b. 12 × 25

a.

48 ÷ 6 = 8 b.

51 ÷ x = 3 c.

x ÷ 13 = 5

11. a. Complete the following multiplication table in base five.

15. Find the quotient and remainder for the following.

a.

53 ÷ 8 b.

72 ÷ 15 c.

137 ÷ 6

16. Use the Chapter 3 eManipulative, Rectangular Division,

on our Web site to investigate the three problems from

Example 3.9. Explain why the remainder is always smaller

than the divisor in terms of this rectangle division model.

17. In general, each of the following is false if x, y, and z are

whole numbers. Give an example (other than dividing by

b.

Using the table in part a, complete the following four-

zero) where each statement is false.

fact families in base five.

a.

x ÷ y is a whole number.

i. 2

×

=

five

3five

11five

ii. ___________________

b.

x ÷ y = y ÷ x

___________________

___________________

c.

(x ÷ y) ÷ z = x ÷ (y ÷ z)

___________________

22

÷

=

d.

x ÷ y = x = y ÷ x for some y

five

3five

4five

___________________

___________________

e.

x ÷ (y + z) = x ÷ y + x ÷ z

PROBLEMS

18. A square dancing contest has 213 teams of 4 pairs each.

23. Suppose that A is a set of whole numbers closed under

How many dancers are participating in the contest?

addition. Is A necessarily closed under multiplication? (If

you think so, give reasons. If you think not, give a counter

19. A stamp machine dispenses twelve 32¢ stamps. What is the

example, that is, a set A that is closed under addition but

total cost of the twelve stamps?

not multiplication.)

20. If the American dollar is worth 121 Japanese yen, how

24. a. Use the numbers from 1 to 9 once each to complete

many dollars can 300 yen buy?

this magic square. (The row, column, and diagonal

21. An estate valued at $270,000 was left to be split equally

sums are all equal.) (Hint: First determine the sum of

among three heirs. How much did each one get (before

each row.)

taxes)?

22. Shirley meant to add 12349 + 29746 on her calculator.

After entering 12349, she pushed the × button by mistake.

What could she do next to keep from reentering 12349?

What property are you using?

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7/30/2013 2:49:58 PM - Section 3.2 Multiplication and Division 113

b.

Can you make a multiplicative magic square? (The row,

28. Jason, Wendy, Kevin, and Michelle each entered a frog in

column, and diagonal products are equal.) (Note: The

an annual frog-jumping contest. Each of their frogs—Hippy,

numbers 1 through 9 will not work in this case.)

Hoppy, Bounce, and Pounce—placed first, second, or third

in the contest and earned a blue, red, or white ribbon,

respectively. Use the following clues to determine who

entered which frog and the order in which the frogs placed.

a.

Michelle’s frog finished ahead of both Bounce and Hoppy.

b.

Hippy and Hoppy tied for second place.

c.

Kevin and Wendy recaptured Hoppy when he escaped

25. Predict the next three lines in this pattern, and check your

from his owner.

work.

d.

Kevin admired the blue ribbon Pounce received but was

quite happy with the red ribbon his frog received.

1

= 1

3 + 5

= 8

29. A café sold tea at 30 cents a cup and cakes at 50 cents each.

Everyone in a group had the same number of cakes and the

7 + 9 + 11

= 27

same number of cups of tea. (NOTE: This is not to say that

13 + 15 + 17 + 19

= 64

the number of cakes is the same as the number of teas.) The

21 + 23 + 25 + 27 + 29 = 125

bill came to $13.30. How many cups of tea did each have?

30. A creature from Mars lands on Earth. It reproduces itself

26. Using the digits 1 through 9 once each, fill in the boxes to

by dividing into three new creatures each day. How many

make the equations true.

creatures will populate Earth on day 30 if there is one

creature on the first day?

31. There are eight coins and a balance scale. The coins are

alike in appearance, but one of them is counterfeit and

lighter than the other seven. Find the counterfeit coin

using two weighings on the balance scale.

32. Determine whether the property “If ac = bc, then a = b” is

27. Take any number. Add 10, multiply by 2, add 100, divide

true for all whole numbers. If not, give a counter-example.

by 2, and subtract the original number. The answer will be

(Note: This property is called the multiplicative cancellation

the number of minutes in an hour. Why?

property when c ≠ 0.)

EXERCISE/PROBLEM SET B

EXERCISES

1. What multiplication problems are suggested by the follow-

is best represented by the repeated-addition approach,

ing diagrams?

the rectangular array approach, or the Cartesian product

a.

c.

approach, and why. Then write an appropriate equation to

fit the situation.

a. At the student snack bar, three sizes of beverages are

available: small, medium, and large. Five varieties of soft

b.

drinks are available: cola, diet cola, lemon-lime, root

beer, and orange. How many different choices of soft

drink does a student have, including the size that may

be selected?

b. At graduation students file into the auditorium four

abreast. A parent seated near the door counts 72 rows of

students who pass him. How many students participated

2. Illustrate 4 × 6 using the following combinations of models

in the graduation exercise?

and approaches.

c. Kirsten was in charge of the food for an all-school picnic.

a. Set model; rectangular array approach

At the grocery store she purchased 25 eight-packs of hot

b. Measurement model; rectangular array approach

dog buns for 70 cents each. How much did she spend on

c. Set model; repeated-addition approach

the hot dog buns?

d. Measurement model; repeated-addition approach

4. Use a calculator to find the following without using an X

e. Set model; Cartesian product approach

key. Explain your method.

3. Each situation described next involves a multiplication

a. 4 × 39 b.

231 × 3

problem. In each case state whether the problem situation

c. 5 × 172 d.

6 × 843

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7/30/2013 2:50:00 PM - 114 Chapter 3 Whole Numbers: Operations and Properties

5. a. Is the set of whole numbers with 3 removed

b.

Rewrite each of the following division problems in base

i. closed under addition? Why?

eight using the missing-factor approach, and find the

ii. closed under multiplication? Why?

answer in the table.

b.

Answer the same questions for the set of whole numbers

i. 61

÷

÷

eight

7eight

ii. 17eight

3eight

with 7 removed.

iii. 30

÷

÷

eight

6eight

iv. 16eight

2eight

6. Identify the property of whole-number multiplication

v. 44

÷

÷

eight

6eight

vi. 25eight

7eight

being illustrated.

12. Identify each of the following problems as an example

a.

3(5 − 2) = 3 ¥ 5 − 3 ¥ 2

b. 6(7 ¥ 2) = (6 ¥ 7) ¥ 2

of either sharing or measurement division. Justify your

c.

(4 + 7) ¥ 0 = 0

d. (5 + 6) ¥ 3 = 5 ¥ 3 + 6 ¥ 3

answers.

7. Rewrite each of the following expressions using the dis-

a.

For Amberly’s birthday, her mother brought 60

tributive property for multiplication over addition or for

cupcakes to her mathematics class. There were 28

multiplication over subtraction.

students in class that day. If she gives each student

a.

3(29 + 30 + 6) b.

5(x − 2y)

the same number of cupcakes, how many will each

c.

3a + 6a − 4a d.

x(x + 2) + (

3 x + 2)

receive?

e.

37(60 − 22)

b.

Tasha needs 2 cups of flour to make a batch of cookies.

If she has 6 cups of flour, how many batches of cookies

8. The distributive property of multiplication over subtrac-

can she make?

tion can be used to perform some calculations mentally.

c.

Maria spent $45 on three shirts at the store. If the shirts

For example, to find 7(99), you can think

all cost the same, how much did each shirt cost?

7(99) = 7 100

(

− 1) = 7 100

(

) − 7 1

( )

13. a. Write a measurement division problem that would have

= 700 − 7 = 693.

the equation 91 ÷ 7 as part of its solution.

How could each of the following products be rewritten

b.

Write a sharing division problem that would have the

using the distributive property so that it is more easily

equation 91 ÷ 7 as part of its solution.

computed mentally?

14. Rewrite each of the following division problems as a mul-

a.

14(19)

b. 25(38)

tiplication problem.

c. 35(98)

d. 27(999)

a.

24 ÷ x = 12

9. a. Compute 463 × 17 on your calculator without using the

b.

x ÷ 3 = 27

7 key.

c.

a ÷ b = x

b.

Find another way to do it.

15. Find the quotient and remainder for each problem.

c.

Calculate 473 × 17 without using the 7 key.

a.

7 ÷ 3 b.

3 ÷ 7

10. Compute using thinking strategies. Show your reasoning.

c.

7 ÷ 1

d. 1 ÷ 7

a.

8 × 85

b. 12(125)

e.

15 ÷ 5

f. 8 ÷ 12

11. a. Complete the following multiplication table in base

16. How many possible remainders (including zero) are there

eight. Remember to use the thinking strategies.

when dividing by the following numbers? How many pos-

sible quotients are there?

a.

2

b. 12

c.

62

d. 23

17. Which of the following properties hold for division of

whole numbers?

a.

Closure

b.

Commutativity

c.

Associativity

d.

Identity

PROBLEMS

18. A school has 432 students and 9 grades. What is the aver-

20. Compute mentally.

age number of students per grade?

(2348 × ,

7

,

653 214) + (7652 × ,

7

,

653 214)

19. Twelve thousand six hundred people attended a golf

(Hint: Use distributivity.)

tournament. If attendees paid $30 apiece and were dis-

tributed equally among the 18 holes, how much revenue

21. If a subset of the whole numbers is closed under multipli-

is collected per hole?

cation, is it necessarily closed under addition? Discuss.

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7/30/2013 2:50:04 PM - Section 3.2 Multiplication and Division 115

22. Is there a subset of the whole numbers with more than one

A: How old are your three children?

element that is closed under division? Discuss.

B: The product of their ages is 36.

A: That’s not enough information for me to know

23. Complete the pattern and give a justification for your

their ages.

answers. If necessary, check your answers using your

B: The sum of their ages is your house number.

calculator.

A: That’s still not quite enough information.

12,345,679 × 9 = 111 111

,

111

,

B: The oldest child plays the piano.

12,345,679 × 18 = 222,222,222

A: Now I know!

12,345,679 × 27 =

Assume that the ages are whole numbers and that twins

12,345,679 × 63 =

have the same age. How old are the children? (Hint: Make

12,345,679 × 81 =

a list after Bert’s first answer.)

24. Solve this problem posed by this Old English children’s

31. Three boxes contain black and white marbles. One box

rhyme.

has all black marbles, one has all white marbles, and one

has a mixture of black and white. All three boxes are

As I was going to St. Ives

mislabeled. By selecting only one marble, determine how

I met a man with seven wives;

you can correctly label the boxes. (Hint: Notice that “all

Every wife had seven sacks;

black” and “all white” are the “same” in the sense that

Every sack had seven cats;

they are the same color.)

Every cat had seven kits.

Kits, cats, sacks, and wives.

How many were going to St. Ives?

How many wives, sacks, cats, and kits were met?

25. Write down your favorite three-digit number twice to

form a six-digit number (e.g., 587,587). Is your six-digit

32. If a and b are whole numbers and ab = 0, what conclusion

number divisible by 7? How about 11? How about 13?

can you draw about a or b? Defend your conclusion with a

Does this always work? Why? (Hint: Expanded form.)

convincing argument.

26. Find a four-digit whole number equal to the cube of the

Analyzing Student Thinking

sum of its digits.

33. Pascuel claims that the rectangular array approach is the

27. Delete every third counting number starting with 3.

same as the Cartesian product approach. Is there a differ-

1, 2, 4, 5, 7, 8, 1

0, 1

1, 1

3, 1

4, 1

6, 1

7

ence? If so, what is it?

Write down the cumulative sums starting with 1.

34. Phyllis claims that the tree diagram approach for multi-

1, 3, 7, 12

, 19

, 27, 37, 48, 61, 75, 91, 108

plication can only be used to model multiplying two num-

bers, but not more than two. Is she correct? Explain.

Delete every second number from this last sequence, start-

ing with 3. Then write down the sequence of cumulative

35. Maurice rewrites 6(7 ¥ 3) as (6 ¥ 7) × (6 ¥ 3) and says that

sums. Describe the resulting sequence.

he used distributivity. Is he correct in using the distribu-

tive property this way? How should you respond to this

28. Write, side by side, the numeral 1 an even number of

student?

times. Take away from the number thus formed the num-

ber obtained by writing, side by side, a series of 2s half the

36. Before you can respond to Maurice in #35, Taylor

length of the first number. For example,

shouts, “No, that is an application of the associative

property for multiplication.” How should you respond

1111 − 22 = 1089 = 33 × 33.

to this student?

Will you always get a perfect square? Why or why not?

37. Breanne claims that the multiplication property of zero

29. Four men, one of whom committed a crime, said the

should be called the identity property of zero because

following:

when you multiply by zero you get zero. Is her claim

reasonable? Explain.

Bob: Charlie did it.

Charlie: Eric did it.

38. Olga says she can’t see how distributivity can be used as a

Dave: I didn’t do it.

thinking strategy. How should you respond?

Eric: Charlie lied when he said I did it.

39. Ervin says that a + 0 = a and a − 0 = a. Thus, since a × 0 = 0,

a.

If only one of the statements is true, who was guilty?

it must be true that a ÷ 0 = 0. How could you help him

b.

If only one of the statement is false, who was guilty?

better understand this situation?

30. Andrew and Bert met on the street and had the following

40. A student asks you if “4 divided by 12” and “4 divided

conversation:

into 12” mean the same thing. How should you respond?

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7/30/2013 2:50:06 PM - 116 Chapter 3 Whole Numbers: Operations and Properties

ORDERING AND EXPONENTS

NCTM Standard

Ordering and Whole-Number Operations

All students should describe

In Chapter 2, whole numbers were ordered in three different, though equivalent, ways

quantitative change, such as a

student’s growing two inches in

using (1) the counting chant, (2) the whole-number line, and (3) a 1-1 correspondence.

one year.

Now that we have defined whole-number addition, there is another, more useful way

to define “less than.” Notice that 3 < 5 and 3 + 2 = 5, 4 < 9 and 4 + 5 = 9, and 2 < 11 and

2 + 9 = 11. This idea is presented in the next definition of “less than.”

Common Core – Grade 1

D E F I N I T I O N 3 . 8

Compare two 2-digit numbers

based on meanings of the tens

“Less Than” for Whole Numbers

and ones digits, recording the

results of comparisons with the

For any two whole numbers a and b, a < b (or b > a) if and only if there is a nonzero

symbols >, =, and <.

whole number n such that a + n = b.

For example, 7 < 9 since 7 + 2 = 9 and 13 > 8 since 8 + 5 = 13. The symbols “≤” and “≥”

mean “less than or equal to” and “greater than or equal to,” respectively.

One useful property of “less than” is the transitive property.

P R O P E R T Y

Transitive Property of “Less Than” for Whole Numbers

For all whole numbers a, b, and c, if a < b and b < c, then a < c.

The transitive property can be verifi ed using any of our defi nitions of “less than.”

Consider the number line in Figure 3.36.

Problem-Solving Strategy

Draw a Diagram

Figure 3.36

Since a < b, we have a to the left of b, and since b < c, we have b to the left of c. Hence

a is to the left of c, or a < c.

The following is a more formal argument to verify the transitive property. It uses

the definition of “less than” involving addition.

a < b means a + n = b for some nonzero whole number n.

b < c means b + m = c for some nonzero whole number m.

Adding m to a + n and b, we obtain

a + n + m = b + m.

Thus

a + n + m = c since b + m = c.

Therefore, a < c since

a + (n + m) = c and n + m is a nonzero

whole number.

NOTE: The transitive property of “less than” holds if “<” (and “≤”) are replaced

with “greater than” for “>” (and “≥”) throughout.

There are two additional properties involving “less than.” The first involves addi-

tion (or subtraction).

c03.indd 116

7/30/2013 2:50:11 PM - Section 3.3 Ordering and Exponents 117

Algebraic Reasoning

P R O P E R T Y

When finding values of x that

make the expression x − 3 < 6

Less Than and Addition Property for Whole Numbers

true, this property can be used

by adding 3 on both sides as

If a < b, then a + c < b + c.

follows: x − 3 + 3 < 6 + 3. Thus

it can be seen that all values of

x that satisfy x < 9 will make the

As was the case with transitivity, this property can be verifi ed formally using the

original expression true.

defi nition of “less than.” An informal justifi cation using the whole-number line fol-

lows (Figure 3.37).

Problem-Solving Strategy

Draw a Diagram

Figure 3.37

Notice that a < b, since a is to the left of b. Then the same distance, c, is added to each

to obtain a + c and b + c, respectively. Since a + c is left of b + c, we have a + c < b + c.

In the case of “less than” and multiplication, we have to assume that c ≠ 0. The

proof of this property is left for the problem set.

P R O P E R T Y

Less Than and Multiplication Property for Whole Numbers

If a < b and c ≠ 0, then ac < bc.

Since c is a nonzero whole number, it follows that c > 0. In Chapter 8, where neg-

ative numbers are fi rst discussed, this property will have to be modifi ed to include

negatives.

Check for Understanding: Exercise/Problem Set A #1–4

✔

Exponents

Just as multiplication can be defined as repeated addition and division may be viewed

as repeated subtraction, the concept of exponent can be used to simplify situations

involving repeated multiplication.

Children’s Literature

D E F I N I T I O N 3 . 9

www.wiley.com/college/musser

See “The King’s Chessboard” by

Whole-Number Exponent

David Birch.

Let a and m be any two whole numbers where m ≠ 0. Then

am = a ⋅ a…a.

m factors

Given that we define whole-number exponents as am = a ⋅ a …a, discuss with a peer why

each of the following is true.

m factors

am an

am n

¥

= + am bm = a

(

b m

¥

¥ ) (am)n am n

= ¥ am an am n

÷

= −

The number m is called the exponent or power of a, and a is called the base. The

number am is read “a to the power m” or “a to the mth power.” For example, 52, read

c03.indd 117

7/30/2013 2:50:15 PM - 118 Chapter 3 Whole Numbers: Operations and Properties

“5 to the second power” or “5 squared,” is 5 ¥ 5 = 25; 23, read “2 to the third power”

or “2 cubed,” is 2 ¥ 2 ¥ 2 = 8; and 34 = 3 ¥ 3 ¥ 3 ¥ 3 = 81.

There are several properties of exponents that permit us to represent numbers and

to do many calculations quickly.

Rewrite each of the following expressions using a single

exponent.

a. 23 24

¥

b. 35 37

¥

S O L U T I O N

a. 23

24 = (2 2 2) (2 2 2 2) = 2 2 2 2 2 2 2 = 27

¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

b. 35 37

¥

= (3 ¥ 3 ¥ 3 ¥ 3 ¥ 3) ¥ (3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3) = 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 ¥ 3 = 312 ■

In Example 3.10(a) the exponents of the factors were 3 and 4, and the exponent

of the product is 3 + 4 = 7. Also, in (b) the exponents 5 and 7 yielded an exponent of

5 + 7 = 12 in the product.

The fact that exponents are added in this way can be shown to be valid in general.

This result is stated next as a theorem. A theorem is a statement that can be proved

based on known results.

T H E O R E M 3 . 1

Let a, m, and n be any whole numbers where m and n are nonzero. Then

am an

am n

¥

=

+ .

P R O O F

am ¥ an = (a ⋅ a ⋅⋅⋅ a) ¥ (a ⋅ a ⋅⋅⋅ a) = a⋅a ⋅⋅⋅ a = am+n

■

m factors

n factors

m + n factors

The next example illustrates another way of rewriting products of numbers having

the same exponent.

Rewrite the following expressions using a single exponent.

a. 23 53

¥

b. 32 72 112

¥ ¥

S O L U T I O N

a. 23 53 = (2 2 2)(5 5 5) = (2 5) (2 5) (2 5) = (2 5 3

¥

¥ ¥

¥ ¥

¥

¥

¥

¥ )

b. 32 72 112 = (3 3)(7 7) 1

( 1 11) = (3 7 11)(3 7 11) = (3 7 11 2

¥ ¥

¥

¥

¥

¥ ¥

¥ ¥

¥ ¥ )

■

The results in Example 3.11 suggest the following theorem.

T H E O R E M 3 . 2

Let a, b, and m be any whole numbers where m is nonzero. Then

am bm

ab m

¥

= ( ) .

P R O O F

am ¥ bm = a ¥ a ⋅⋅⋅ a ⋅ b ¥ b ⋅⋅⋅ b = (ab)(ab) ⋅ ⋅ ⋅ (ab)= (ab)m

■

m factors

m factors

m pairs of factors

c03.indd 118

7/30/2013 2:50:19 PM - Section 3.3 Ordering and Exponents 119

The next example shows how to simplify expressions of the form (am )n.

Rewrite the following expressions with a single exponent.

a. (53 )2

b. (78 )4

S O L U T I O N

a. (53 )2

53 ¥ 53

53+3

56( 53¥2

=

=

=

=

)

b. (78 )4

78 ¥ 78 ¥ 78 ¥ 78

732( 78¥4

=

=

=

)

■

In general, we have the next theorem.

T H E O R E M 3 . 3

Let a, m, and n be any whole numbers where m and n are nonzero. Then

(am )n

amn

=

.

The proof of this theorem is similar to the proofs of the previous two theorems.

The previous three properties involved exponents and multiplication. However,

notice that (2

3)3

23

33

+

≠

+ , so there is not a corresponding property involving sums

or differences raised to powers.

The next example concerns the division of numbers involving exponents with the

same base number.

Rewrite the following quotients with a single exponent.

a. 57

53

÷ b. 78 75

÷

S O L U T I O N

a. 57

53

54

÷

= , since 57 53 54

= ¥ . Therefore, 57 53 57 3

÷

= − .

b. 78

75

73

÷

= , since 78 75 73

=

¥ . Therefore, 78 75 78 5

÷

= − .

■

You are teaching a unit on exponents and a student asks you what 40 means. One student

volunteers that it is 0 since exponents are a shortcut for multiplication. How do you

respond in a meaningful way?

In general, we have the following result.

T H E O R E M 3 . 4

Let a, m, and n be any whole numbers where m > n and a, m, and n are nonzero. Then

am

an

am n

÷

=

− .

P R O O F

am

an

÷

= c

If and only if am

an

= ¥ c. Since an am n an (m n) am

¥ −

+

−

=

= , we have c am n

=

− . Therefore

am

an

am n

÷

=

− .

■

c03.indd 119

7/30/2013 2:50:23 PM - 120 Chapter 3 Whole Numbers: Operations and Properties

Problem-Solving Strategy

Notice that we have not yet defined a0. Consider the following pattern

Look for a Pattern

Extending this pattern, we see that the following definition is appropriate.

D E F I N I T I O N 3 . 1 0

Zero as an Exponent

a0 = 1 for all whole numbers a ≠ 0.

Notice that 00 is not defi ned. To see why, consider the following two patterns.

Problem-Solving Strategy

PATTERN 1

PATTERN 2

Look for a Pattern

30 = 1

03 = 0

20 = 1

02 = 0

10 = 1

01 = 0

00 = ?

00 = ?

Pattern 1 suggests that 00 should be 1 and pattern 2 suggests that 00 should be 0. Thus

to avoid such an inconsistency, 00 is undefined.

Check for Understanding: Exercise/Problem Set A #5–12

✔

Order of Operations

Now that the operations of addition, subtraction, multiplication, and division as well

as exponents have been introduced, a point of confusion may occur when more than

one operation is in the same expression. For example, does it matter which operation

is performed first in an expression such as 3 + 4 × 5? If the 3 and 4 are added first,

the result is 7 × 5 = 35, but if the 4 and 5 are multiplied first, the result is 3 + 20 = 23.

Since the expression 3 + 5 + 5 + 5 + 5 (= 23) can be written as 3 + 4 × 5, it seems

reasonable to do the multiplication, 4 × 5, first. Similarly, since 4 ¥ 5 ¥ 5 ¥ 5 (= 500) can

be rewritten as 4 53

¥ , it seems reasonable to do exponents before multiplication. Also,

to prevent confusion, parentheses are used to indicate that all operations within

a pair of parentheses are done first. To eliminate any ambiguity, mathematicians

have agreed that the proper order of operations shall be Parentheses, Exponents,

Multiplication and Division, Addition and Subtraction (PEMDAS). Although mul-

tiplication is listed before division, these operations are done left to right in order

of appearance. Similarly, addition and subtraction are done left to right in order of

appearance. The pneumonic device Please Excuse My Dear Aunt Sally is often used

to remember this order.

Use the proper order of operations to simplify the following

expressions.

a. 53

4 ¥ 1

(

2 2

−

+ )

b. 11 − 4 ÷ 2 ¥ 5 + 3

c03.indd 120

7/30/2013 2:50:27 PM - Section 3.3 Ordering and Exponents 121

S O L U T I O N

a. 53

4 ¥ 1

(

2 2

)

53

4 ¥ 32

−

+

=

−

= 125 − 4 ¥ 9

= 125 − 36

= 89

b. 11 − 4 ÷ 2 ¥ 5 + 3 = 11 − 2 ¥ 5 + 3

= 11 − 10 + 3

= 1 + 3

= 4

■

Check for Understanding: Exercise/Problem Set A #13–14

✔

John von Neumann was a brilliant mathematician who made important

contributions to several scientific fields, including the theory and appli-

cation of high-speed computing machines. George Pólya of Stanford

University admitted that “Johnny was the only student I was ever afraid of.

If in the course of a lecture I stated an unsolved problem, the chances were

he’d come to me as soon as the lecture was over, with the complete solu-

tion in a few scribbles on a slip of paper.” At the age of 6, von Neumann

could divide two 8-digit numbers in his head, and when he was 8 he had

mastered the calculus. When he invented his first electronic computer,

someone suggested that he race it. Given a problem like “What is the

smallest power of 2 with the property that its decimal digit fourth from the

right is a 7,” the machine and von Neumann started at the same time and

©Ron Bagwell

von Neumann won!

EXERCISE/PROBLEM SET A

EXERCISES

1. Find the nonzero whole number n in the definition of “less

6. Evaluate each of the following without a calculator and

than” that verifies the following statements.

order them from largest to smallest.

a. 12 < 31

b. 53 > 37

52 43 34 25

,

,

,

2. Using the definitions of < and > given in this section, write

7. Write each of the following expressions in expanded form,

four inequality statements based on the fact that 2 + 8 = 10.

without exponents.

3. The statement a < x < b is equivalent to writing a < x and

a.

3 2 5

x y z b. 7 53

¥ c. (7 5)3

¥

x < b and is called a compound inequality. We often read

a < x < b as “x is between a and b.” For the questions that

8. Rewrite each with a single exponent.

3

4

12

2

follow, assume that a, x , and b are whole numbers.

a.

5 ¥ 5 b.

3 ÷ 3

c.

27 57

¥ d.

8 . 25

If a < x < b and c is a nonzero whole number, is it always

e.

253

52

÷

f. 92 123

¥

¥ 2

true that a + c < x + c < b + c? Try several examples to test

your conjecture.

9. Express 514 in three different ways using the numbers 2,

4. Does the transitive property hold for the following? Explain.

5, 7 and exponents. (You may use a number more than

a. = (equals)

b. ≠ (is not equal to)

once.)

n

n

n

+

=

+

5. Using exponents, rewrite the following expressions in a sim-

10. Let a, b, n W

∈ and n ≥ 1. Determine if (a b)

a

b

pler form.

is always true, sometimes true, or never true. Justify your

conclusion.

a. 3 ¥ 3 ¥ 3 ¥ 3

b. 2 ¥ 2 ¥ 3 ¥ 2 ¥ 3 ¥ 2

c. 6 ¥ 7 ¥ 6 ¥ 7 ¥ 6 d.

x ¥ y ¥ x ¥ y ¥ y ¥ y

11. Find x .

e. a ¥ b ¥ b ¥ a

f. 5 ¥ 6 ¥ 5 ¥ 5 ¥ 6 ¥ 6

a.

37 3

313

¥ x = b.

(3x )4 = 320 c.

3x 2x

6x

¥

=

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7/30/2013 2:50:32 PM - 122 Chapter 3 Whole Numbers: Operations and Properties

12. Use the yx , xy , or ∧ key on your calculator to evaluate

14. For each of the following:

the following.

i. Show that the expression is not equal to the numerical

a.

68 b. (3 5)4

¥

c.

3 54

¥

value.

ii. Insert parentheses in the expression so it will be equal to

13. Simplify each of the following expressions.

the numerical value.

a.

15 − 3(7 − 2) b.

2 52

¥

a. 2 32

¥ − 4; 10

b. 42 − 3 ¥ 4 ÷ 2; 26

6 + 2(33 − 42 2

) + 42

c.

32 4

2(5

3 3

¥ −

− ) d. 6¥(33 − 42)

PROBLEMS

15. Let a, m , and n be whole numbers where a is not zero.

20. Pizzas come in four different sizes, each with or without

Prove the following property: (am )n

am n

=

¥ .

a choice of four ingredients. How many ways are there to

order a pizza?

16. Using properties of exponents, mentally determine the

larger of the following pairs.

21. The perfect square 49 is special in that each of the two

a.

610 and 320 b. 99

320

and

c. 1210

320

and

nonzero digits forming the number is itself a perfect

square.

17. The price of a certain candy bar doubled over a period

a.

Explain why there are no other two-digit squares with

of five years. Suppose that the price continued to double

this property.

every five years and that the candy bar cost 25 cents

b.

What three-digit perfect squares can you find that are

in 2000.

made up of one 1-digit square and one 2-digit square?

a.

What would be the price of the candy bar in the year 2015?

c.

What four-digit perfect squares can you find that are

b.

What would be the price of the candy bar in the year 2040?

made up of two 1-digit squares and one 2-digit square?

c.

Write an expression representing the price of the candy

d.

What four-digit perfect squares can you find that are

bar after n five-year periods.

made up of one 3-digit square and one 1-digit square?

18. Verify the transitive property of “less than” using the 1-1

e.

What four-digit perfect squares can you find that are

correspondence definition.

made up of two 2-digit squares?

f. What four-digit perfect squares can you find that are

19. Observe the following pattern in the sums of consecutive

made up of four 1-digit squares?

whole numbers. Use your calculator to verify that the

statements are true.

22. When asked to find four whole numbers such that the

2 + 3 + 4 = 13 + 23

product of any two of them is one less than the square of

a whole number, one mathematician said, “2, 4, 12, and

5 + 6 + 7 + 8 + 9 = 23 + 33

22.” A second mathematician said, “2, 12, 24, and 2380.”

10 + 11 + 12 + 13 + 14 + 15 + 16 = 33 + 43

Which was correct?

a.

Write the next two lines in this sequence of sums.

23. a. Give a formal proof of the property of less than and

b.

Express 93

103

+

as a sum of consecutive whole numbers.

addition. (Hint: See the proof in the paragraph follow-

c.

Express 123

133

+

as a sum of consecutive whole numbers.

ing Figure 3.37.)

d.

Express n3 + (n

3

+ 1) as a sum of consecutive whole

b.

State and prove the corresponding property for sub-

numbers.

traction.

EXERCISE/PROBLEM SET B

EXERCISES

1. Find the nonzero whole number n in the definition of “less

5. Using exponents, rewrite the following expressions in a

than” that verifies the following statements.

simpler form.

a. 17 < 26 b. 113 > 49

a. 4 ¥ 4 ¥ 4 ¥ 4 ¥ 4 ¥ 4 ¥ 4 b.

4 ¥ 3 ¥ 3 ¥ 4 ¥ 4 ¥ 3 ¥ 3 ¥ 3 ¥ 3

c. y ¥ 2 ¥ x ¥ y ¥ x ¥ x

d. a ¥ b ¥ a ¥ b ¥ a ¥ b ¥ a ¥ b

2. Using the definitions of < and > given in this section, write

four inequality statements based on the fact that 29 + 15 = 44.

6. Evaluate each of the following without a calculator and

arrange them from smallest to largest.

3. If a < x < b and c is a nonzero whole number, is it always

3

2

5

7

true that ac < xc < bc? Try several examples to test your

5 , 6 , 3 , 2

conjecture.

7. Write each of the following expressions in expanded form,

4. Does the transitive property hold for the following? Explain.

without exponents.

a. > (greater than) b. ≤ (less than or equal to)

a. 10

4

ab b. (2 )5

x c. 2 5

x

c03.indd 122

7/30/2013 2:50:37 PM - Section 3.3 Ordering and Exponents 123

8. Write the following with only one exponent.

12. Use the yx , xy , or ∧ key on your calculator to evaluate

a. x3 x6

¥ b.

a15

a3

÷

the following.

3

4

c. x5y5 d.

23 ¥16

9 ¥12

a.

37

24

+ b. 2 56 − 3 23

¥

¥ c.

8

e. 128

23

÷

f. 37 95 272

¥ ¥

3

13. Simplify each of the following expressions.

9. Express 720 in three different ways using the numbers

a.

2 32 − 22

¥

¥ 3

7, 2, 5 and exponents. (You may use a number more

2(4 − 1 3

)

than once.)

b.

15 − 2 ¥ 31(7 + 5 0

)

10. Let a, b, m, n W

∈ and a is not zero. Determine if an am

¥

=

c.

52

42

(6

4 5

)

23

−

+

−

÷

an¥mis always true, sometimes true, or never true. Justify

d.

4

23

64

23 ¥ 32

×

−

÷ (

)

your conclusion.

14. For each of the following:

11. Find the value of x that makes each equation true.

i. Show that the expression is not equal to the numerical value.

a.

2x 25

÷ = 27

ii.

Insert parentheses in the expression so it will be equal to

b.

(6x )2

36x

=

the numerical value.

c.

57 ¥ 3x 15x

=

a. 22 − 2 ¥ 7 − 3; 137

b. 17

2 ¥ 32

+

+ 5 ÷ 5; 8

PROBLEMS

15. a. How does the sum 1 + 2 + 3 compare to the sum

c.

Give a formal proof of the property. (Hint: Use the fact that

13

23

33

+

+ ?

the product of two nonzero whole numbers is nonzero.)

b.

Try several more such examples and determine

d.

State the corresponding property for division.

the relationship between 1 + 2 + 3 + ⋅⋅⋅ + n and

22. Let a, m , and n be whole numbers where m > n and a is

13

23

33

3

+

+ + ⋅⋅⋅ + n .

not zero. Prove the following property: am

an

am n

÷

=

− .

c.

Use the pattern you observed to find the sum of the first

10 cubes, 13

23

33

… 103

+

+

+

+

, without evaluating any

Analyzing Student Thinking

of the cubes.

23. Brooke claims that because “less than and addition for

16. Order these numbers from smallest to largest using prop-

whole numbers” is a property, so too should “less than

erties of exponents and mental methods.

and subtraction for whole numbers” be a property. How

322 414 910 810

,

,

,

should you respond?

17. A pad of 200 sheets of paper is approximately 15 mm

24. A student asks why there is not a property of “less than and

thick. Suppose that one piece of this paper were folded in

division for whole numbers.” How should you respond?

half, then folded in half again, then folded again, and so

25. Jarell was asked to simplify the expression 32 34

¥ and wrote

on. If this folding process was continued until the piece of

32 34

96

¥ = . How would you convince him that this answer

paper had been folded in half 30 times, how thick would

is incorrect? That is, give the correct answer and explain

the folded paper be?

why the student’s method is incorrect.

18. Suppose that you can order a submarine sandwich with or

26. Viridiana observes 0 + 0 = 0 × 0 and 2 + 2 = 2 × 2 and

without each of seven condiments.

assumes that addition and multiplication are the same. Is

a.

How many ways are there to order a sandwich?

she correct? Explain.

b.

How many ways can you order exactly two condiments?

27. Abbi was asked to simplify the expression 85

22

÷ and

c.

Exactly seven?

wrote 85

22

43

÷

= . How would you convince her that this

d.

Exactly one?

answer is incorrect?

e.

Exactly six?

28. Cho rewrote (32 )2 as 3 22

(

) (a form of associativity, per-

19. a. If n2 = 121, what is n?

haps). Was she correct or incorrect? She also rewrote (23 )2

b.

If n2 = 1,234,321, what is n?

as 2 32

(

) . Is this right or wrong? How should you respond?

c.

If n2 = 12,345,654,321, what is n?

29. Ayumi writes am bn

ab m n

¥

=

+

(

)

. Is this statement correct or

d.

If n2 = 123,456,787,654,321, what is n?

incorrect? Explain.

20. 12 is a factor of 102

22

− , 27 is a factor of 202 72

− , and 84

30. Gavin says that because of distributivity, the following

is a factor of 802

42

− . Check to see whether these three

equation is true: (a

b)m

am

bm

+

=

+ . What could you say to

statements are correct. Using a variable, prove why this

help him better understand the situation?

works in general.

31. Bailey wrote the following using “Please Excuse My Dear

21. a. Verify that the property of less than and multiplication

Aunt Sally.” Explain where she may have misinterpreted

holds where a = 3 , b = 7 , and c = 5 .

this acronym.

b.

Show that the property does not hold if c = 0.

(5

3)3

17

53

33

−

+

= − + 17 = 125 − 27 + 17 = 115.

c03.indd 123

7/30/2013 2:50:45 PM - 124

Chapter 3 Whole Numbers: Operations and Properties

END OF CHAPTER MATERIAL

If they balance, we know that group B has the counterfeit

coin. If group A and C do not balance, then group A has

In a group of nine coins, eight weigh the same and the ninth is

the counterfeit coin. We also know whether group A or B

either heavier or lighter. Assume that the coins are identical in

is light or heavy depending on which one went down on the

appearance. Using a pan balance, what is the smallest number

fi rst balancing.

of balancings needed to identify the counterfeit coin?

In two balancings, we know which group of three coins

contains the counterfeit one and whether the counterfeit coin

is heavy or light. The third balancing will exactly identify the

Strategy: Use Direct Reasoning

counterfeit coin.

From the group of three coins that has the counterfeit coin,

Three balancings are sufficient. Separate the coins into three

select two coins and balance them against each other. This will

groups of three coins each.

determine which coin is the counterfeit one. Thus, the counter-

feit coin can be found in three balancings.

Additional Problems Where the Strategy “Use Direct Reasoning”

A

B

C

Is Useful

Balance group A against group B. If they balance, we can

1. The sum of the digits of a three-digit palindrome is odd. De-

deduce that the counterfeit coin is in group C. We then need

termine whether the middle digit is odd or is even.

to determine if the coin is heavier or lighter. Balance group C

2. Jose’s room in a hotel was higher than Michael’s but lower

against group A. Based on whether group C goes up or down,

than Ralph’s. Andre’s room is on a fl oor between Ralph’s

we can deduce whether group C is light or heavy, respectively.

and Jose’s. If it weren’t for Michael, Clyde’s room would be

If the coins in group A do not balance the coins in group

the lowest. List the rooms from lowest to highest.

B, then we can conclude that either A is light or B is heavy

3. Given an 8-liter jug of water and empty 3-liter and 5-liter

(or vice versa). We now balance group A against group C.

jugs, pour the water so that two of the jugs have 4 liters.

John von Neumann

Julia Bowman Robinson

(1903–1957)

(1919–1985)

John von Neumann was one

Julia Bowman Robinson

of the most remarkable math-

spent her early years in Ari-

ematicians of the twentieth

zona, near Phoenix. She

century. His logical power was

said that one of her earliest

A/Photoshot

legendary. It is said that during

memories was of arranging

©UPP

and after World War II the U.S.

Courtesy of George M.Bergman,

University of California, Berkeley.

pebbles in the shadow of a

government reached many scientifi c decisions simply by

giant saguaro—“I’ve always had a basic liking for the

asking von Neumann for his opinion. Paul Halmos, his

natural numbers.” In 1948, Robinson earned her doc-

one-time assistant, said, “The most spectacular thing

torate in mathematics at Berkeley; she went on to con-

about Johnny was not his power as a mathematician,

tribute to the solution of “Hilbert’s tenth problem.”

which was great, but his rapidity; he was very, very

In 1975 she became the first woman mathematician

fast. And like the modern computer, which doesn’t

elected to the prestigious National Academy of Sci-

memorize logarithms, but computes them, Johnny

ences. Robinson also served as president of the Amer-

didn’t bother to memorize things. He computed them.”

ican Mathematical Society, the main professional

Appropriately, von Neumann was one of the fi rst to

organization for research mathematicians. “Rather

realize how a general-purpose computing machine—a

than being remembered as the first woman this or

computer—should be designed. In the 1950s he invent-

that, I would prefer to be remembered simply for

ed a “theory of automata,” the basis for subsequent

the theorems I have proved and the problems I have

work in artifi cial intelligence.

solved.”

c03.indd 124

7/30/2013 2:50:47 PM - Chapter Review 125

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy

reference.

Addition and Subtraction

Vocabulary/Notation

Plus 87

Associativity for addition 90

Minuend 94

Sum 87

Identity for addition 90

Subtrahend 94

Addend 87

Thinking strategies for addition

Missing-addend approach 94

Summand 87

facts 90

Missing addend 94

Binary operation 88

Take-away approach 93

Four-fact families 94

Closure for addition 89

Difference 94

Comparison 95

Commutativity for addition 89

Minus 94

Comparison approach 96

Exercises

1. Show how to find 5 + 4 using

4. Illustrate the following using 7 − 3.

a. a set model.

b. a measurement model.

a. The take-away approach

b. The missing-addend approach

2. Name the property of addition that is used to justify each

of the following equations.

5. Show how the addition table for the facts 1 through 9 can

a. 7 + (3 + 9) = (7 + 3) + 9

be used to solve subtraction problems.

b. 9 + 0 = 9

6. Which of the following properties hold for whole-number

c. 13 + 27 = 27 + 13

subtraction?

d. 7 + 6 is a whole number

a. Closure b.

Commutative

3. Identify and use thinking strategies that can be used to find

c. Associative d.

Identity

the following addition facts.

a. 5 + 6

b. 7 + 9

Multiplication and Division

Vocabulary/Notation

Repeated-addition approach 101

Distributive property of multiplication

Divisor 108

Times 101

over addition 105

Quotient 108

Product 101

Multiplication property of zero 106

Missing factor 108

Factor 101

Distributivity of multiplication over

Division property of zero 109

Rectangular array approach 101

subtraction 106

Division by zero 109

Cartesian product approach 102

Thinking strategies for multiplication

Division algorithm 109

Tree diagram approach 102

facts 106

Divisor 110

Closure for multiplication 102

Sharing division 107

Quotient 110

Commutativity for multiplication 102

Measurement division 107

Remainder 110

Associativity for multiplication 104

Missing-factor approach 108

Repeated-subtraction approach 110

Identity for multiplication 104

Divided by 108

Multiplicative identity 104

Dividend 108

Exercises

1. Illustrate 3 × 5 using each of the following approaches.

2. Name the property of multiplication that is used to justify

a. Repeated addition with the set model

each of the following equations.

b. Repeated addition with the measurement model

a. 37 × 1 = 37

b. 26 × 5 = 5 × 26

c. Rectangular array with the set model

c. 2 × (5 ×17) = (2 × 5) ×17

d. Rectangular array with the measurement model

d. 4 × 9 is a whole number

c03.indd 125

7/30/2013 2:50:48 PM - 126 Chapter 3 Whole Numbers: Operations and Properties

3. Show how the distributive property can be used to simplify

7. Calculate the following if possible. If impossible, explain why.

these calculations.

a.

7 ÷ 0

b. 0 ÷ 7

c. 0 ÷ 0

a. 7 × 27 + 7 × 13

8. Apply the division algorithm and apply it to the calcula-

b. 8 × 17 − 8 × 7

tion 39 ÷ 7.

4. Use and name the thinking strategies that can be used to

find the following addition facts.

9. Which of the following properties hold for whole-number

division?

a. 6 × 7

a.

Closure

b.

Commutative

b. 9 × 7

c.

Associative

d.

Identity

5. Show how 17 ÷ 3 can be found using

10. Label the following diagram and comment on its value.

a.

Repeated subtraction with the set model

b.

Repeated subtraction with the measurement model

6. Show how the multiplication table for the facts 1 through

9 can be used to solve division problems.

Ordering and Exponents

Vocabulary/Notation

Less than 116

Less than and addition property

Power 117

Less than or equal to 116

(subtraction) 117

Base 117

Greater than or equal to 116

Less than and multiplication

Theorem 118

Transitive property of less

property (division) 117

Zero as an exponent 120

than 116

Exponent 117

Order of operations 120

Exercises

1. Describe how addition is used to define “less than” (“great-

3. Rewrite 54 using the definition of exponent.

er than”).

4. Rewrite the following using properties of exponents.

2. For whole numbers a, b, and c, if a < b, what can be said

a.

(73 )4 b. 35

75

× c. 57 53

÷ d. 412 413

×

about

a. a + c and b + c? b. a × c and b × c?

5. Explain how to motivate the definition of zero as an exponent.

CHAPTER TEST

Knowledge

2. Complete the following table for the operations on the set

of whole numbers. Write True in the box if the indicated

1. True or false?

property holds for the indicated operation on whole num-

a. n(A ∪ B) = n(A) + n(B) for all finite sets A and B.

bers and write False if the indicated property does not

b. If B ⊆ A, then n(A − B) = n(A) − n(B) for all finite sets A

hold for the indicated operation on whole numbers.

and B.

c. Commutativity does not hold for subtraction of whole

ADD

SUBTRACT

MULTIPLY

DIVIDE

numbers.

d. Distributivity of multiplication over subtraction does not

Closure

hold in the set of whole numbers.

Commutative

Associative

e. The symbol ma, where m and a are nonzero whole num-

Identity

bers, represents the product of m factors of a.

f. If a is the divisor, b is the dividend, and c is the quotient,

then ab = c.

3. Identify the property of whole numbers being illustrated.

g. The statement “a + b = c if and only if c − b = a” is an

a. a ¥ (b ¥ c) = a ¥ (c ¥ b)

example of the take-away approach to subtraction.

b. (3 + 2) ¥1 = 3 + 2

h. Factors are to multiplication as addends are to addition.

c. (4 + 7) + 8 = 4 + (7 + 8)

i. If n ≠ 0 and b + n = a, then a < b.

d. (2 ¥ 3) ¥ 5 = 2 ¥ (3 ¥ 5)

j. If n(A) = a and n(B) = b, then A × B contains exactly ab

e. (6 − 5) ¥ 2 = 6 ¥ 2 − 5 ¥ 2

ordered pairs.

f. m + (n + p) = m + ( p + n)

c03.indd 126

7/30/2013 2:50:54 PM - Chapter Test 127

Skill

11. a. Using the definition of an exponent, provide an expla-

nation to show that (73 )4

712

=

.

4. Find the following sums and products using thinking strate-

b.

Use the fact that am an

am n

¥

= + to explain why (73)4 712

=

.

gies. Show your work.

a. 39 + 12 b.

47 + 87

12. Show why 3 ÷ 0 is undefined.

c. 5(73 ¥ 2) d.

12 × 33

13. Explain in detail why am bm = (a b m

¥

¥ ) .

5. Find the quotient and remainder when 321 is divided by 5.

14. Explain why (a b)c ≠ a bc

¥

¥ for all a, b, c, W

∈ .

6. Rewrite the following using a single exponent in each case.

a. 37 312

¥ b.

531

57

÷

15. If we make up another operation, say Ω, and we give it a

table using the letters A, B, C, D as the set on which this

c. (73 )5 d.

45 ÷ 28

symbol is operating, determine whether this operation is

e. 712 212 143

¥

¥

f. (128

)4 (36 )2

÷125 ¥

commutative.

7. Perform the following calculations by applying appropriate

Ω

A

B

C

D

properties. Which properties are you using?

A

B

B

B

B

a. 13 ¥ 97 + 13 ¥ 3 b.

(194 + 86) + 6

B

B

C

D

A

c. 7 ¥ 23 + 23 ¥ 3 d.

25 123

(

¥ 8)

C

B

A

C

D

8. Classify each of the following division problems as exam-

D

B

D

A

C

ples of either sharing or measurement division.

a.

Martina has 12 cookies to share between her three friends

Then create your own 4 × 4 table for a new operation @,

and herself. How many cookies will each person receive?

operating on the same four numbers, A, B, C, D. Make

b.

Coach Massey had 56 boys sign up to play intramural

sure your operation is commutative. Can you see some-

basketball. If he puts 7 boys on each team, how many

thing that would always be true in the table of a commuta-

teams will he have?

tive operation?

c.

Eduardo is planning to tile around his bathtub. If he

wants to tile 48 inches up the wall and the individual

Problem Solving/Application

tiles are 4 inches wide, how many rows of tile will

he need?

16. Which is the smallest set of whole numbers that

contains 2 and 3 and is closed under addition and

9. Each of the following situations involves a subtraction

multiplication?

problem. In each case, tell whether the problem is best

represented by the take-away approach or the missing-

17. If the product of two numbers is even and their sum is

addend approach, and why. Then write an equation that

odd, what can you say about the two numbers?

could be used to answer the question.

18. Illustrate the following approaches for 4 × 3 using the

a.

Ovais has 137 basketball cards and Quinn has 163 bas-

measurement model. Please include a written description

ketball cards. How many more cards does Quinn have?

to clarify the illustration.

b.

Regina set a goal of saving money for a $1500 down

a.

Repeated-addition approach

payment on a car. Since she started, she has been able

to save $973. How much more money does she need to

b.

Rectangular array approach

save in order to meet her goal?

19. Use a set model to illustrate that the associative property

c.

Riley was given $5 for his allowance. After he spent

for whole-number addition holds.

$1.43 on candy at the store, how much did he have left

to put into savings for a new bike?

20. Illustrate the missing-addend approach for the subtraction

problem 8 − 5 by using

Understanding

a.

The set model

b. The measurement model

21. Use the rectangular array approach to illustrate that the

10. Using the following table, find A − B.

commutative property for whole-number multiplication

+

A

B

C

holds.

A

C

A

B

22. Find a whole number less than 100 that is both a perfect

B

A

B

C

square and a perfect cube.

C

B

C

A

23. Find two examples where a, b W

∈ and a ¥ b = a + b.

c03.indd 127

7/30/2013 2:50:58 PM - C H A P T E R

4 WHOLE-NUMBER

COMPUTATION—MENTAL,

ELECTRONIC, AND WRITTEN

Computational Devices from the Abacus to the Computer

The abacus was one of the earliest computational devic- In 1671, Leibniz developed his “reckoning machine,”

es. The Chinese abacus, or suan-pan, was composed of

which could also multiply and divide.

a frame together with beads on fixed rods.

ce

SSPL/Getty Images

Will & Demi McIntrye/

Science Sour

In 1812, Babbage built his Difference Engine, which

In 1617, Napier invented lattice rods, called Napier’s

was the first “computer.”

bones. To multiply, appropriate rods were selected,

laid side by side, and then appropriate “columns” were

added—thus only addition was needed to multiply.

© Bettmann/CORBIS

By 1946, ENIAC (Electronic Numerical Integrator

and Computer) was developed. It filled a room, weighed

over 30 tons, and had nearly 20,000 vacuum tubes. Chips

SSPL/Getty Images

permitted the manufacture of microcomputers, such as

About 1594, Napier invented logarithms. Using loga-

the Apple and DOS-based computers in the late 1970s,

rithms, multiplication can be performed by adding respec-

the Apple Macintosh and Windows-based versions in the

tive logarithms. The slide rule, used extensively by engi-

1980s, to laptop computers in the 1990s.

neers and scientists through the 1960s, was designed to use

properties of logarithms.

SSPL/Getty Images

In 1642, Pascal invented the first mechanical adding

machine.

ThinkStock LLC/Index Stock

Chip manufacture also allowed calculators to become

more powerful to the point that the dividing line between

calculator and computer has become blurred.

ce/Getty Images

NYPL/Science

Sour

128

c04.indd 128

7/30/2013 2:52:16 PM - Problem-Solving

Use Indirect Reasoning

Strategies

Occasionally, in mathematics, there are problems that are not easily solved using

1. Guess and Test

direct reasoning. In such cases, indirect reasoning may be the best way to solve

2. Draw a Picture

the problem. A simple way of viewing indirect reasoning is to consider an empty

room with only two entrances, say A and B. If you want to use direct reasoning

3. Use a Variable

to prove that someone entered the room through A, you would watch entrance A.

4. Look for a Pattern

However, you could also prove that someone entered through A by watching

entrance B. If a person got into the room and did not go through B, the person

5. Make a List

had to go through entrance A. In mathematics, to prove that a condition, say

“A,” is true, one assumes that the condition “not A” is true and shows the latter con-

6. Solve a Simpler

dition to be impossible.

Problem

7. Draw a Diagram

Initial Problem

8. Use Direct

The whole numbers 1 through 9 can be used once, each being arranged in a 3 × 3

Reasoning

square array so that the sum of the numbers in each of the rows, columns, and diago-

nals is 15. Show that 1 cannot be in one of the corners.

9. Use Indirect

Reasoning

Clues

The Use Indirect Reasoning strategy may be appropriate when

r Direct reasoning seems too complex or does not lead to a solution.

r Assuming the negation of what you are trying to prove narrows the scope of the

problem.

r A proof is required.

A solution of this Initial Problem is on page 169.

129

c04.indd 129

7/30/2013 2:52:17 PM - AUTHOR

I N T R O D U C T I O N

In the past, much of elementary school mathematics was devoted to learning written methods for

doing addition, subtraction, multiplication, and division. Due to the availability of calculators and

computers, less emphasis is being placed on doing written calculations involving numbers with many

digits. Instead, an emphasis is being placed on developing skills in the use of all three types of com-

WALK-THROUGH putations: mental, written, and electronic. Then, depending on the size of the numbers involved, the

accuracy desired in the answer, and the time available, the appropriate model(s) of calculation will be

selected and employed.

In this chapter, you will study all three forms of computations: mental, electronic, and written.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-2–NUMBER AND OPERATIONS

Develop and use strategies for whole-number computations, with a focus on addition and subtraction.

Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil,

and calculators.

r GRADES 3-5–NUMBER AND OPERATIONS

Develop fluency with basic number combinations for multiplication and division and use these combinations to men-

tally compute related problems, such as 30 × 50.

Develop fluency in adding, subtracting, multiplying, and dividing whole numbers.

Develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of

such results.

Select appropriate methods and tools for computing with whole numbers from among mental computation, estima-

tion, calculators, and paper and pencil according to the context and nature of the computation and use the selected

method or tool.

Key Concepts from the NCTM Curriculum Focal Points

r GRADE 1: Developing an understanding of whole-number relationships, including grouping in tens and ones.

r GRADE 2: Developing quick recall of addition facts and related subtraction facts and fluency with multidigit addi-

tion and subtraction.

r GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole number

multiplication.

r GRADE 5: Developing an understanding of and fluency with division of whole numbers.

r GRADES 4 AND 5: Students select appropriate methods and apply them accurately to estimate products and cal-

culate them mentally, depending on the context and numbers involved.

Key Concepts from the Common Core State Standards for Mathematics

r GRADE 1: Use place value understanding and properties of operations to add and subtract two-digit numbers.

r GRADE 2: Represent and solve problems involving addition and subtraction. Use place value understanding and

properties of operations to add and subtract three-digit numbers. Relate addition and subtraction to length.

r GRADE 3: Represent and solve problems involving multiplication and division within 100. Understand properties

of multiplication and the relationship between multiplication and division. Use place value understanding and

properties of operations to perform multidigit arithmetic.

r GRADE 4: Use the four operations with whole numbers to solve problems. Use place value understanding and

properties of operations to perform multidigit arithmetic. Multiply four-digit whole numbers by one-digit whole

numbers and multiply two two-digit whole numbers. Divide four-digit whole numbers by one-digit divisors.

r GRADE 5: Perform operations with multidigit whole numbers. Specifically, divide four-digit whole numbers by

two-digit divisors.

130

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7/30/2013 2:52:18 PM - Section 4.1 Mental Math, Estimation, and Calculators 131

MENTAL MATH, ESTIMATION, AND CALCULATORS

Compute each of the following mentally and write a sentence for each problem

describing your thought processes. After a discussion with classmates, determine some

common strategies.

32 ¥ 26 − 23 ¥ 32

16

(

× 9) × 25

25 + (39 + 105)

49 + 27

152 − 87

46 × 99

252 ÷ 12

NCTM Standard

Mental Math

All students should select appro-

The availability and widespread use of calculators and computers have permanently

priate methods and tools for

computing with whole numbers

changed the way we compute. Consequently, there is an increasing need to develop

from among mental computation,

students’ skills in estimating answers when checking the reasonableness of results

estimation, calculators, and paper

obtained electronically. Computational estimation, in turn, requires a good working

and pencil according to the con-

knowledge of mental math. Thus this section begins with several techniques for doing

text and nature of the computa-

calculations mentally.

tion and use the selected method

or tool.

In Chapter 3 we saw how the thinking strategies for learning the basic arithmetic

facts could be extended to multidigit numbers, as illustrated next.

Children’s Literature

www.wiley.com/college/musser

Calculate the following mentally.

See “Great Estimations’’ by Bruce

a. 15 + (27 + 25)

b. 21 ¥ 17 − 13 ¥ 21 c. (8 × 7) × 25

Goldstone.

d. 98 + 59

e. 87 + 29 f. 168 ÷ 3

S O L U T I O N

Reflection from Research

Flexibility in mental calculation

a. 15 + (27 + 25) = (27 + 25) + 15 = 27 + (25 + 15) = 27 + 40 = 67. Notice how com-

cannot be taught as a “process

mutativity and associativity play a key role here.

skill” or holistic “strategy.”

b. 21 ¥ 1 7 − 13 ¥ 21 = 21 ¥ 17 − 21 ¥ 13 = 21 1

( 7 − 13) = 21 ¥ 4 = 84. Observe how commuta-

Instead, flexibility can be devel-

tivity and distributivity are useful here.

oped by using the solutions stu-

c. (8 × 7) × 25 = (7 × 8) × 25 = 7 × (8 × 25) = 7 × 200 = 1400. Here commutativity is

dents find in mental calculations

to show how numbers can be

used fi rst; then associativity is used to group the 8 and 25 since their product is

dealt with, be taken apart and

200.

put back together, be rounded

d. 98 + 59 = 98 + (2 + 57) = (98 + 2) + 57 = 157. Associativity is used here to form 100.

and adjusted, etc. (Threlfall,

e. 87 + 29 = 80 + 20 + 7 + 9 = 100 + 16 = 116 using associativity and commutativity.

2002).

f. 168 ÷ 3 = 150

(

÷ 3) + 1

( 8 ÷ 3) = 50 + 6 = 56.

■

Observe that part (f ) makes use of right distributivity of division over addition;

that is, whenever the three quotients are whole numbers, (a + b) ÷ c = (a ÷ c) + (b ÷ c).

Right distributivity of division over subtraction also holds.

The calculations in Example 4.1 illustrate the following important mental tech-

niques.

Common Core – Grade 3

Multiply one-digit whole numbers

Properties Commutativity, associativity, and distributivity play an important

by multiples of 10 in the range

role in simplifying calculations so that they can be performed mentally. Notice how

10–90 (e.g., 9 × 80, 5 × 60) using

useful these properties were in parts (a), (b), (c), (d), and (e) of Example 4.1. Also, the

strategies based on place value

and properties of operations.

solution in part (f ) uses right distributivity.

When Joelle, a fourth grader, was asked to solve 15 − 8, she said, “Well 8 − 5 is 3 and 10 − 3

is 7 so 15 − 8 is 7.” Discuss with a peer what you think Joelle’s mental math methods are

and whether or not you think they are valid.

Compatible Numbers Compatible numbers are numbers whose sums, differ-

ences, products, or quotients are easy to calculate mentally. Examples of compatible

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Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

numbers are 86 and 14 under addition (since 86 + 14 = 100), 25 and 8 under multi-

plication (since 25 × 8 = 200), and 600 and 30 under division (since 600 ÷ 30 = 20).

In part (a) of Example 4.1, adding 15 to 25 produces a number, namely 40, that is

easy to add to 27. Notice that numbers are compatible with respect to an operation.

For example, 86 and 14 are compatible with respect to addition but not with respect

to multiplication.

Calculate the following mentally using properties and/or com-

patible numbers.

a. (4 × 13) × 25

b. 1710 ÷ 9 c. 86 × 15

S O L U T I O N

a. (4 × 13) × 25 = 13 × (4 × 25) = 1300

b. 1710 ÷ 9 = 1800

(

− 90) ÷ 9 = 1800

(

÷ 9) − (90 ÷ 9) = 200 − 10 = 190

Reflection from Research

c. 86 × 15 = (86 × 10) + (86 × 5) = 860 + 430 = 1290 (Notice that 86 × 5 is half of

Recent research shows that com-

86 × 10.)

■

putational fluency and number

sense are intimately related.

These tend to develop together

and facilitate the learning of the

Compensation The sum 43 + (38 + 17) can be viewed as 38 + 60 = 98 using

other (Griffin, 2003).

commutativity, associativity, and the fact that 43 and 17 are compatible numbers.

Finding the answer to 43 + (36 + 19) is not as easy. However, by reformulating the

sum 36 + 19 mentally as 37 + 18, we obtain the sum (43 + 37) + 18 = 80 + 18 = 98.

This process of reformulating a sum, difference, product, or quotient to one that is

more readily obtained mentally is called compensation. Some specific techniques using

compensation are introduced next.

In the computations of Example 4.1(d), 98 was increased by 2 to 100 and then 59

was decreased by 2 to 57 (a compensation was made) to maintain the same sum. This

technique, additive compensation, is an application of associativity. Similarly, additive

compensation is used when 98 + 59 is rewritten as 97 + 60 or 100 + 57. The problem

47 − 29 can be thought of as 48 − 30 (= 18). This use of compensation in subtraction is

called the equal additions method since the same number (here 1) is added to both 47

and 29 to maintain the same difference. This compensation is performed to make the

subtraction easier by subtracting 30 from 48. The product 48 × 5 can be found using

multiplicative compensation as follows: 48 × 5 = 24 × 10 = 240. Here, again, associativity

can be used to justify this method.

Left-to-Right Methods To add 342 and 136, first add the hundreds (300 + 100), then

the tens (40 + 30), and then the ones (2 + 6), to obtain 478. To add 158 and 279, one can

think as follows: 100 + 200 = 300, 300 + 50 + 70 = 420, 420 + 8 + 9 = 437. Alternatively,

158 + 279 can be found as follows: 158 + 200 = 358, 358 + 70 = 428, 428 + 9 = 437.

Subtraction from left to right can be done in a similar manner. Research has found that

people who are excellent mental calculators utilize this left-to-right method to reduce

memory load, instead of mentally picturing the usual right-to-left written method. The

multiplication problem 3 × 123 can be thought of mentally as 3 × 100 + 3 × 20 + 3 × 3

using distributivity. Also, 4 × 253 can be thought of mentally as 800 + 200 + 12 = 1012 or

as 4 × 250 + 4 × 3 = 1000 + 12 = 1012.

Multiplying Powers of 10 These special numbers can be multiplied mentally in

either standard or exponential form. For example, 100 × 1000 = 100,000,104 105 109

×

=

,

20 × 300 = 6000, and 12 000 × 110 000 = 12 × 11 × 107

,

,

= 1,320,000,000.

Multiplying by Special Factors Numbers such as 5, 25, and 99 are regarded

as special factors because they are convenient to use mentally. For example,

since

5 = 10 ÷ 2, we have 38 × 5 = 38 × 10 ÷ 2 = 380 ÷ 2 = 190. Also, since

25 = 100 ÷ 4, 36 × 25 = 3600 ÷ 4 = 900. The product 46 × 99 can be thought of as

46 100

(

− 1) = 4600 − 46 = 4554. Also, dividing by 5 can be viewed as dividing by 10,

then multiplying by 2. Thus 460 ÷ 5 = (460 ÷ 10) × 2 = 46 × 2 = 92.

c04.indd 132

8/2/2013 7:20:46 AM - From Chapter 3, Lesson 1 “Take Apart Tens to Add” from My Math, Volume 1 Common Core State Standards, Grade 2, copyright © 2013

by McGraw-Hill Education,

133

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7/30/2013 2:52:29 PM - 134

Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Calculate mentally using the indicated method.

a. 197 + 248 using additive compensation

b. 125 × 44 using multiplicative compensation

c. 273 − 139 using the equal additions method

d. 321 + 437 using a left-to-right method

e. 3 × 432 using a left-to-right method

f. 456 × 25 using the multiplying by a special factor method

S O L U T I O N

a. 197 + 248 = 197 + 3 + 245 = 200 + 245 = 445

b. 125 × 44 = 125 × 4 × 11 = 500 × 11 = 5500

c. 273 − 139 = 274 − 140 = 134

d. 321 + 437 = 758 [Think: (300 + 400) + (20 + 30) + (1 + 7)]

e. 3 × 432 = 1296 [Think: (3 × 400) + (3 × 30) + (3 × 2)]

f. 456 × 25 = 114 × 100 = 11,400 (Think: 25 × 4 = 100. Thus 456 × 25 = 114 × 4 × 25

= 114 × 100 = 11,400.)

■

Check for Understanding: Exercise/Problem Set A #1–7

✔

Use estimation to answer the following two questions:

1) How much blood does the average heart pump in a day?

2) Is $10 enough to buy items with the following prices? $1.49, $0.99, $2.14, $3.23, $1.33, $0.25.

How is the type of estimation used to solve these two problems different?

NCTM Standard

Computational Estimation

All students should develop and

use strategies to estimate the

The process of estimation takes on various forms. The number of beans in a jar may

results of whole-number compu-

be estimated using no mathematics, simply a “guesstimate.” Also, one may estimate

tations and to judge the reason-

how long a trip will be, based simply on experience. Computational estimation is the

ableness of such results.

process of finding an approximate answer (an estimate) to a computation, often using

mental math. With the use of calculators becoming more commonplace, computa-

Reflection from Research

tional estimation is an essential skill. Next we consider various types of computa-

Students can develop a bet-

tional estimation.

ter understanding of number

through activities involving esti-

Front-End Estimation Three types of front-end estimation will be demonstrated.

mation (Leutzinger, Rathmell, &

Urbatsch, 1986).

Range Estimation

Often it is sufficient to know an interval or range—that is, a low

value and a high value—that will contain an answer. The following example shows

Children’s Literature

how ranges can be obtained in addition and multiplication.

www.wiley.com/college/musser

See “How Much, How Many,

How Far, How Heavy, How Long,

How Tall Is 1000?’’ by Helen

Find a range for answers to these computations by using only

Nolan.

the leading digits.

a.

257 b. 294

+ 576

× 53

S O L U T I O N

a. Sum

Low Estimate

High Estimate

257

200

300

+ 576

+ 500

+ 600

700

900

c04.indd 134

7/30/2013 2:52:31 PM - Section 4.1 Mental Math, Estimation, and Calculators 135

Thus a range for the answer is from 700 to 900. Notice that you have to look at

only the digits having the largest place values (2 + 5 = 7, or 700) to arrive at the low

estimate, and these digits each increased by one (3 + 6 = 9, or 900) to fi nd the high

estimate.

b. Product Low Estimate

High Estimate

294

200

300

× 53

×

50

×

60

10 000

,

18 000

,

Due to the nature of multiplication, this method gives a wide range, here 10,000 to

18,000. Even so, this method will catch many errors.

■

Algebraic Reasoning

One-Column/Two-Column Front-End We can estimate the sum 498 + 251 using

Estimation is a valuable tool of

the one-column front-end estimation method as follows: To estimate 498 + 251,

algebraic reasoning. It can be

think 400 + 200 = 600 (the estimate). Notice that this is simply the low end of the

used to determine if a solution to

range estimate. The one-column front-end estimate always provides low estimates in

an equation seems reasonable for

the situation.

addition problems as well as in multiplication problems. In the case of 376 + 53 + 417,

the one-column estimate is 300 + 400 = 700, since there are no hundreds in 53. The

two-column front-end estimate also provides a low estimate for sums and products.

However, this estimate is closer to the exact answer than one obtained from using

only one column. For example, in the case of 376 + 53 + 417, the two-column front-

end estimation method yields 370 + 50 + 410 = 830, which is closer to the exact answer

842 than the 700 obtained using the one-column method.

Front-End with Adjustment This method enhances the one-column front-end esti-

mation method. For example, to find 498 + 251, think 400 + 200 = 600 and 98 + 51 is

about 150. Thus the estimate is 600 + 150 = 750. Unlike one-column or two-column

front-end estimates, this technique may produce either a low estimate or a high esti-

mate, as in this example.

Keep in mind that one estimates to obtain a “rough” answer, so all of the preced-

ing forms of front-end estimation belong in one’s estimation repertoire.

Estimate using the method indicated.

a. 503 × 813 using one-column front-end

b. 1200 × 35 using range estimation

c. 4376 − 1889 using two-column front-end

d. 3257 + 874 using front-end adjustment

S O L U T I O N

a. To estimate 503 × 813 using the one-column front-end method, think 500 × 800 =

400 000

,

. Using words, think “5 hundreds times 8 hundreds is 400,000.”

b. To estimate a range for 1200 × 35, think 1200 × 30 = 36 000

,

and 1200 × 40 = 48 000

,

.

Thus a range for the answer is from 36,000 to 48,000. One also could use

1000 × 30 = 30 000

,

and 2000 × 40 = 80 000

,

. However, this yields a wider range.

Reflection from Research

c. To estimate 4376 − 1889 using the two-column front-end method, think 4300 −

To be confident and successful

1800 = 2500. You also can think 43 − 18 = 25 and then append two zeros after the

estimators, children need numer-

25 to obtain 2500.

ous opportunities to practice

d. To estimate 3257

estimation and to learn from

+ 874 using front-end with adjustment, think 3000, but since

their experiences. Thus, the skills

257 + 874 is about 1000, adjust to 4000.

■

required by students to develop

a comfort level with holistic, or

range-based, estimation need

Rounding Rounding is perhaps the best-known computational estimation tech-

to be included throughout a

nique. The purpose of rounding is to replace complicated numbers with simpler

young child’s education (Onslow,

numbers. Here, again, since the objective is to obtain an estimate, any of several

Adams, Edmunds, Waters,

Chapple, Healey, & Eady, 2005).

rounding techniques may be used. However, some may be more appropriate than

c04.indd 135

7/30/2013 2:52:34 PM - 136 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

others, depending on the word problem situation. For example, if you are estimating

how much money to take along on a trip, you would round up to be sure that you had

enough. When calculating the amount of gas needed for a trip, one would round the

miles per gallon estimate down to ensure that there would be enough money for gas.

Unlike the previous estimation techniques, rounding is often applied to an answer as

well as to the individual numbers before a computation is performed.

Several different methods of rounding are illustrated next. What is common, how-

ever, is that each method rounds to a particular place. You are asked to formulate

rules for the following methods in the problem set.

Figure 4.1

Round Up (Down) The number 473 rounded up to the nearest tens place is 480

since 473 is between 470 and 480 and 480 is above 473 (Figure 4.1). The number 473

rounded down to the nearest tens place is 470. Rounding down is also called truncat-

ing (truncate means “to cut off”). The number 1276 truncated to the hundreds place

is 1200.

Round a 5 Up The most common rounding technique used in schools is the round

a 5 up method. This method can be motivated using a number line. Suppose that we

wish to round 475 to the nearest ten (Figure 4.2).

Figure 4.2

Since 475 is midway between 470 and 480, we have to make an agreement

concerning whether we round 475 to 470 or to 480. The round a 5 up method always

rounds such numbers up, so 475 rounds to 480. In the case of the numbers 471 to 474,

since they are all nearer 470 than 480, they are rounded to 470 when rounding to the

nearest ten. The numbers 476 to 479 are rounded to 480.

One disadvantage of this method is that estimates obtained when several 5s

are involved tend to be on the high side. For example, the “round a 5 up” to the

nearest ten estimate applied to the addends of the sum 35 + 45 + 55 + 65 yields

40 + 50 + 60 + 70 = 220, which is 20 more than the exact sum 200.

Round to the Nearest Even Rounding to the nearest even can be used to avoid

errors of accumulation in rounding. For example, if 475 + 545(= 1020) is estimated

by rounding up to the tens place or rounding a 5 up, the answer is 480 + 550 = 1030.

By rounding down, the estimate is 470 + 540 = 1010. Since 475 is between 480 and 470

and the 8 in the tens place is even, and 545 is between 550 and 540 and the 4 is even,

round to the nearest even method yields 480 + 540 = 1020.

Estimate using the indicated method. (The symbol “≈” means

“is approximately.”)

a. Estimate 2173 + 4359 by rounding down to the nearest hundreds place.

b. Estimate 3250 − 1850 by rounding to the nearest even hundreds place.

c. Estimate 575 − 398 by rounding a 5 up to the nearest tens place.

S O L U T I O N

a. 2173 + 4359 ≈ 2100 + 4300 = 6400

b. 3250 − 1850 ≈ 3200 − 1800 = 1400

c. 575 − 398 ≈ 580 − 400 = 180

■

Round to Compatible Numbers Another rounding technique can be applied to esti-

mate products such as 26 × 37. A reasonable estimate of 26 × 37 is 25 × 40 = 1000. The

numbers 25 and 40 were selected since they are estimates of 26 and 37, respectively,

and are compatible with respect to multiplication. (Notice that the rounding up tech-

nique would have yielded the considerably higher estimate of 30 × 40 = 1200, whereas

the exact answer is 962.) This round to compatible numbers technique allows one to

round either up or down to compatible numbers to simplify calculations, rather

c04.indd 136

7/30/2013 2:52:36 PM - Section 4.1 Mental Math, Estimation, and Calculators 137

than rounding to specified places. For example, a reasonable estimate of 57 × 98

is 57 × 100 (= 5700). Here, only the 98 needed to be rounded to obtain an estimate

mentally. The division problem 2716 ÷ 75 can be estimated mentally by considering

2800 ÷ 70 (= 40). Here 2716 was rounded up to 2800 and 75 was rounded down to 70

because 2800 ÷ 70 easily leads to a quotient since 2800 and 70 are compatible numbers

with respect to division.

Reflection from Research

Good estimators use three com-

Estimate by rounding to compatible numbers in two different

putational processes. They make

ways.

the number easier to manage

(possibly by rounding), change

a. 43 × 21

b. 256 ÷ 33

the structure of the problem itself

to make it easier to carry out, and

S O L U T I O N

compensate by making adjust-

a. 43 × 21 ≈ 40 × 21 = 840

b. 256 ÷ 33 ≈ 240 ÷ 30 = 8

ments in their estimation after

43 × 21 ≈ 43 × 20 = 860

256 ÷ 33 ≈ 280 ÷ 40 = 7

the problem is carried out (Reys,

Rybolt, Bestgen, & Wyatt, 1982).

(The exact answer is 903.)

(The exact answer is 7 with remainder 25.)

■

Rounding is a most useful and flexible technique. It is important to realize that

the main reasons to round are (1) to simplify calculations while obtaining reasonable

answers and (2) to report numerical results that can be easily understood. Any of the

methods illustrated here may be used to estimate.

The ideas involving mental math and estimation in this section were observed in

children who were facile in working with numbers. The following suggestions should

help develop number sense in all children:

1. Learn the basic facts using thinking strategies, and extend the strategies to multi-

digit numbers.

2. Master the concept of place value.

3. Master the basic addition and multiplication properties of whole numbers.

4. Develop a habit of using the front-end and left-to-right methods.

5. Practice mental calculations often, daily if possible.

6. Accept approximate answers when exact answers are not needed.

7. Estimate prior to doing exact computations.

8. Be flexible by using a variety of mental math and estimation techniques.

Check for Understanding: Exercise/Problem Set A #8–15

✔

Using a Calculator

Although a basic calculator that costs less than $10 is sufficient for most elementary

school students, there are features on $15 to $30 calculators that simplify many

complicated calculations. Also, most smart phones have basic as well as scientific

calculators. When we use the word calculator throughout the book, it may encom-

pass many of these types of calculators. The TI-34 MultiView, manufactured by

Texas Instruments, is shown in Figure 4.3. The TI-34 MultiView, which is designed

especially for elementary and middle schools, performs fraction as well as the usual

decimal calculations, can perform long division with remainders directly, and has the

functions of a scientific calculator. One nice feature of the TI-34 MultiView is that it

has mutiple lines of display, which allows the student to see the input and output at

the same time.

The on key turns the calculator on. The delete key is an abbreviation for “delete”

and allows the user to delete one character at a time from the right if the cursor is at

the end of an expression or delete the character under the cursor. Pressing the clear

key will clear the current entry. The previous entry can be retrieved by pressing the

Vincent LaRussa/John Wiley and Sons

m on the upper right key.

Figure 4.3

Two types of logic are available on common calculators: arithmetic and algebraic.

c04.indd 137

7/30/2013 2:52:40 PM - 138 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Reflection from Research

(NOTE: For ease of reading, we will write numerals without the usual squares

Students in classrooms where

around them to indicate that they are keys.)

calculators are used tend to have

more positive attitudes about

Arithmetic Logic In arithmetic logic, the calculator performs operations in the

mathematics than students in

order they are entered. For example, if 3

classrooms where calculators are

+ 4 × 5 = is entered, the calculations are

not used (Reys & Reys, 1987).

performed as follows: (3 + 4) × 5 = 7 × 5 = 35. That is, the operations are performed

from left to right as they are entered.

Common Core –

Algebraic Logic If your calculator has algebraic logic and the expression 3 +

Mathematical Practices

4 × 5 = is entered, the result is different; here the calculator evaluates expressions

Use appropriate tools strategically.

according to the usual mathematical convention for order of operations.

Proficient students are sufficiently

If a calculator has parentheses, they can be inserted to be sure that the desired

familiar with tools appropriate

for their grade or course to make

operation is performed first. In a calculator using algebraic logic, the calculation

sound decisions about when each

13 − 5 × 4 ÷ 2 + 7 will result in 13 − 10 + 7 = 10. If one wishes to calculate 13 − 5 first,

of these tools might be helpful,

parentheses must be inserted. Thus (13 − 5) × 4 ÷ 2 + 7 = 23.

recognizing both the insight to be

gained and their limitations.

Use the order of operations to mentally calculate the following

and then enter them into a calculator to compare results.

a. (4 + 2 × 5) ÷ 7 + 3

b. 8

22

3

22

÷

+ ×

c. 17 − 4(5 − 2)

Reflection from Research

d. 40

5

23

÷ ×

− 2 × 3

Young people in the workplace

believe that menial tasks, such as

S O L U T I O N

calculations, are low-order tasks

that should be undertaken by

a. (4 + 2 × 5) ÷ 7 + 3 = (4 + 10) ÷ 7 + 3

technology and that their role is

to identify problems and solve

= (14 ÷ 7) + 3

them using technology to sup-

= 2 + 3 = 5

port that solution. Thus, math-

ematics education may need

b. 8

22

3

22

÷

+ ×

= (8 ÷ 4) + (3 × 4)

to reshift its focus from accuracy

= 2 + 12 = 14

and precision relying on arduous

calculations to one that will bet-

c. 17 − 4(5 − 2) = 17 − (4 × 3)

ter fit the contemporary work-

= 17 − 12 = 5

place and life beyond schools

(Zevenbergen, 2004).

d. 40

5

23

2

3

[(40

5)

23

÷ ×

− × =

÷

× ] − (2 × 3)

= 8 × 8 − 2 × 3

= 64 − 6 = 58

■

Now let’s consider some features that make a calculator helpful both as a compu-

tational and a pedagogical device. Several keystroke sequences will be displayed to

simulate the variety of calculator operating systems available.

Parentheses As mentioned earlier when we were discussing algebraic logic, one

must always be attentive to the order of operations when several operations are pres-

ent. For example, the product 2 × (3 + 4) can be found in two ways. First, by using

commutativity, the following keystrokes will yield the correct answer:

3 + 4 = × 2 =

1

4 .

Alternatively, the parentheses keys may be used as follows:

2 × ( 3 + 4 ) =

1

4 .

Parentheses are needed, since pressing the keys 2 × 3 + 4 = on a calculator with

algebraic logic will result in the answer 10. Distributivity may be used to simplify

calculations. For example, 753 ¥ 8 + 753 ¥ 9 can be found using 753 × ( 8 + 9 ) =

instead of 753 × 8 + 753 × 9 = . The enter key on the TI-34 MultiView is often

used in place of an equals key.

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7/30/2013 2:52:43 PM - Section 4.1 Mental Math, Estimation, and Calculators 139

Constant Functions In Chapter 3, multiplication was viewed as repeated addi-

tion; in particular, 5 × 3 = 3 + 3 + 3 + 3 + 3 = 15. Repeated operations are carried out

in different ways depending on the model of calculator. For example, the following

keystroke sequence is used to calculate 5 × 3 on one calculator that has a built-in

constant function:

3 + = = = =

1

5 .

Numbers raised to a whole-number power can be found using a similar technique.

For example, 35 can be calculated by replacing the + with a × in the preceding

examples. A constant function can also be used to do repeated subtraction to find a

quotient and a remainder in a division problem. For example, the following sequence

can be used to find 35 ÷ 8:

35 − 8 = = = = 3 .

The remainder (3 here) is the first number displayed that is less than the divisor (8

here), and the number of times the equal sign was pressed is the quotient (4 here).

Because of the multiple lines of display, the TI-34 MultiView can handle most of

these examples by typing in the entire expression. For example, representing 5 × 3 as

repeated addition is entered into the TI-34 MultiView as

3 + 3 + 3 + 3 + 3 enter

The entire expression of 3 + 3 + 3 + 3 + 3 appears on the first line of display and the

result 15 appears on the end of the same line of display when mode is NORM.

Exponent Keys There are three common types of exponent keys: x2 y2

,

, and ∧ .

The x2 key is used to find squares in one of two ways:

3

2

= 9 or 3 2

x

x

9 .

The yx and ∧ keys are used to find more general powers and have similar key-

strokes. For example, 73 may be found as follows:

7 yx 3 = 343

or 7 ∧ 3 343 .

Memory Functions Many calculators have a memory function designated by

the keys M+ , M − , MR , or STO , RCL , SUM . Your calculator’s display will

probably show an “M” to remind you that there is a nonzero number in the memory.

The problem 5 × 9 + 7 × 8 may be found as follows using the memory keys:

5 × 9 =

+

M

7 × 8 = M+ MR

101

or

5 × 9 = SUM 7 × 8 = SUM RCL

101

.

It is a good practice to clear the memory for each new problem using the all clear key.

The TI-34 MultiView has seven memory locations—x, y, z, t, a, b, c—which can be

accessed by pressing the sto c key and then pressing the yzt

x

key multiple times to

abc

select the desired variable. The following keystrokes are used to evaluate the expres-

sion above.

5 × 9 stoc

yzt

yzt

yzt

x

enter

and 7 × 8 stoc x

x

enter

abc

abc

abc

The above keys will store the value 45 in the memory location x and the value 56 in the

memory location y. The values x and y can then be added together as follows.

2nd recall enter + 2nd recall . enter enter

This will show 45 + 56 on the left of the calculator display and 101 on the right of the

display.

Additional special keys will be introduced throughout the book as the need arises.

Scientific Notation Input and output of a calculator are limited by the number

of places in the display (generally 8, 10, or 12). Two basic responses are given when

c04.indd 139

7/30/2013 2:52:46 PM - 140 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

a number is too large to fit in the display. Simple calculators either provide a par-

tial answer with an “E” (for “error”), or the word “ERROR” is displayed. Many

scientific calculators automatically express the answer in scientific notation (that is,

as the product of a decimal number greater than or equal to 1 but less than 10, and

the appropriate power of 10). For example, on the TI-34 MultiView, the product

of 123,456,789 and 987 is displayed 1 218518507 1011

.

×

(NOTE: Scientific notation

is discussed in more detail in Chapter 9 after decimals and negative numbers have

been studied.) If an exact answer is needed, the use of the calculator can be com-

bined with paper and pencil and distributivity as follows:

123, 456,789 × 987 = 123,456,789 × 900 + 123,456,789

× 80 + 123,456,789 × 7

= 111 111

,

110

,

100

,

+ 9,876,543 120

,

+ 864 197

,

,523.

Now we can obtain the product by adding:

111 111

,

110

,

100

,

9,876,543 120

,

+ 864 197

,

,523

121,851,850,74

43

Calculations with numbers having three or more digits will probably be performed

on a calculator (or computer) to save time and increase accuracy. Even so, it is pru-

dent to estimate your answer when using your calculator.

Check for Understanding: Exercise/Problem Set A #16–24

✔

Shakuntala Devi (1929-2013), an Indian mathematical wizard, was dubbed

‘the human computer’ because she could calculate swiftly mentally. At six,

she performed at circuses and other road shows and became the source

of support for her family. In 1977, she found the 23rd root of a 201-digit

number in 50 seconds while a Univac computer required 62 seconds. In

1982, she earned a place in Guinness Book of World Records for mentally

multiplying two 13 digit numbers in 28 seconds. She also had careers in

astrology and as a cookbook author and a novelist.

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. Calculate mentally using properties.

3. Calculate mentally left to right.

a. (37 + 25) + 43 b. 47 ¥15 + 47 ¥ 85

a. 123 + 456

b. 342 + 561

c. (4 × 13) × 25 d. 26 ¥ 24 − 21¥ 24

c. 587 − 372 d. 467 − 134

2. Find each of these differences mentally using equal

4. Calculate mentally using the indicated method.

additions. Write out the steps that you thought

a. 198 + 387 (additive compensation)

through.

b. 84 × 5 (multiplicative compensation)

a. 43 − 17 b. 62 − 39

c. 99 × 53 (special factor)

c. 132 − 96 d. 250 − 167

d. 4125 ÷ 25 (special factor)

c04.indd 140

7/30/2013 2:52:49 PM - Section 4.1 Mental Math, Estimation, and Calculators 141

5. Calculate mentally.

14. Here are four ways to estimate 26 × 12:

a. 58 000

,

× 5 000

,

000

,

26 × 10 = 260 30 × 12 = 360

b. 7 105

×

× 21000

,

25 × 12 = 300 30 × 10 = 300

c. 13 000

,

× 7 000

,

000

,

Estimate the following in four ways.

d. 4 105

3 106

7 103

×

× ×

× ×

a. 31 × 23

b. 35 × 46

e. 5 103

7 107

4 105

×

× ×

× ×

c. 48 × 27

d. 76 × 12

f. 17 000

,

000

,

× 6 000

,

000

,

000

,

15. Estimate the following values and check with a calculator.

6. The sum 26 + 38 + 55 can be found mentally as follows:

a. 656 × 74 is between _____ 000 and _____ 000.

26 + 30 = 56, 56 + 8 = 64, 64 + 50 = 114, 114 + 5 = 119. Find

b. 491 × 3172 is between _____ 00000 and _____ 00000.

the following sums mentally using this technique.

c. 1432 is between _____ 0000 and _____ 0000.

a. 32 + 29 + 56

b. 54 + 28 + 67

16. Guess what whole numbers can be used to fill in the

c. 19 + 66 + 49

blanks. Use your calculator to check.

d. 62 + 84 + 27 + 81

a. _____ 6 = 4096

b. _____ 4 = 28,561

7. In division you can sometimes simplify the problem by

multiplying or dividing both the divisor and dividend by

17. Guess which is larger. Check with your calculator.

the same number. This is called division compensation. For

a. 54 or 45 ?

b. 73 or 37 ?

example,

c. 74 or 47 ?

d. 63 or 36 ?

72 ÷ 12 = (72 ÷ 2) ÷ 1

( 2 ÷ 2) = 36 ÷ 6 = 6 and

145 ÷ 5 = 145

(

× 2) ÷ (5 × 2) = 290 ÷ 10 = 29

18. Some products can be found most easily using a

combination of mental math and a calculator. For

Calculate the following mentally using this technique.

example, the product 20 × 47 × 139 × 5 can be found by

a. 84 ÷ 14 b. 234 ÷ 26 c. 120 ÷ 15 d. 168 ÷ 14

calculating 47 × 139 on a calculator and then multiplying

your result by 100 mentally (20 × 5 = 100). Calculate the

8. Before granting an operating license, a scientist has to

following using a combination of

estimate the amount of pollutants that should be allowed

mental math and a calculator.

to be discharged from an industrial chimney. Should she

overestimate or underestimate? Explain.

a. 17 × 25 × 817 × 4

b. 98 × 2 × 673 × 5

9. Estimate each of the following using the four front-end

c. 674 × 50 × 889 × 4

methods: (i) range, (ii) one-column, (iii) two-column, and

(iv) with adjustment.

d. 783 × 8 × 79 × 125

a. 3741

b. 1591

c. 2347

19. Compute the quotient and remainder (a whole number)

+ 1252

346

58

for the following problems on a calculator without using

589

192

repeated subtraction. Describe the procedure you used.

+ 163

+ 5783

a. )

8 103

b. 17)543

c. 1

)

23 849

d. 894)107,214

10. Find a range estimate for these products.

20. 1233 122

332

=

+

and 8833

882

332

=

+

. How about 10,100

a. 37 × 24 b. 157 × 231 c. 491 × 8

and 5,882,353? (Hint: Think 588 2353.)

11. Estimate using compatible number estimation.

a. 63 × 97

b. 51 × 212

c. 3112 ÷ 62

21. Determine whether the following equation is true for

n = 1, 2, or 3.

d. 103 × 87

e. 62 × 58

f. 4254 ÷ 68

1n

6n

8n

2n

4n

9n

+

+

=

+

+

12. Round as specified.

3

3

3

a. 373 to the nearest tens place

22. Notice that 153 = 1 + 5 + 3 . Determine which of the fol-

b. 650 using round a 5 up method to the hundreds place

lowing numbers have the same property.

c. 1123 up to the tens place

a. 370

b. 371

c. 407

d. 457 to the nearest tens place

23. Determine whether the following equation is true when n

e. 3457 to the nearest thousands place

is 1, 2, or 3.

n

n

n

n

n

n

13. Cluster estimation is used to estimate sums and products

1 + 4 + 5 + 5 + 6 + 9 =

when several numbers cluster near one number. For

2n + 3n + 3n + 7n + 7n + 8n

example, the addends in 789 + 810 + 792 cluster around

800. Thus 3 × 800 = 2400 is a good estimate of the sum.

24. Simplify the following expressions using a calculator.

Estimate the following using cluster estimation.

a. 135 − 7(48 − 33)

a. 347 + 362 + 354 + 336

b.

32 512

¥

b. 61 × 62 × 58

c. 132 34

3(45

37 3

¥

−

− )

c. 489 × 475 × 523 × 498

26 + 4(313 − 1722 2

) + 2 ¥ 401

d.

d. 782 + 791 + 834 + 812 + 777

26 ¥ (313 − 1722 )

c04.indd 141

7/30/2013 2:52:56 PM - 142 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

PROBLEMS

25. Five of the following six numbers were rounded to

The answer is 71 consecutive 1s—one of the biggest num-

the nearest thousand, then added to produce an estimated

bers a computer has ever factored. This factorization

sum of 87,000. Which number was not included?

bested the previous high, the factorization of a 69-digit

number. Find a mistake here, and suggest a correction.

5228 14,286 7782 19,628 9168 39,228

34. Some mental calculators use the following fact:

26. True or false?

(a + b)(a − b) = a2 − b2. For example, 43 × 37 =

2

2

493 827 1562

,

,

= 246,913,578 × 987,654,312

(40 + 3)(40 − 3) = 40 − 3 = 1600 − 9 = 1591. Apply this

technique to find the following products mentally.

27. Place a multiplication sign or signs so that the product in

a. 54 × 46

each problem is correct; for example, in 1 2 3 4 5 6 =

b. 81 × 79

41,472, the multiplication sign should be between the

c. 122 × 118

2 and the 3 since 12 × 3456 = 41,472.

d. 1210 × 1190

a. 1 3 5 7 9 0 = 122,130

b. 6 6 6 6 6 6 = 439,956

35. Fermat claimed that

c. 7 8 9 3 4 5 6 = 3,307,824

100,895,598 169

,

= 898,423 × 112,303.

d. 1 2 3 4 5 6 7 = 370,845

Check this on your calculator.

28. Develop a method for finding a range for subtraction

36. Show how to find 439,268 × 6852 using a calculator that

problems for three-digit numbers.

displays only eight digits.

29. Find 13 333 3332

,

,

.

37. Insert parentheses (if necessary) to obtain the following

results.

30. Calculate 99 ¥ 36 and 99 ¥ 23 and look for a pattern. Then

a. 76 × 54 + 97 = 11,476

predict 99 ¥ 57 and 99 ¥ 63 mentally and check your predic-

2

tions with a calculator.

b. 4 × 13 = 2704

c. 13

592

+

× 47 = 163,620

31. a. Calculate 252 352 452

,

,

, and 552 and look for a

d. 79

43

2 172

−

÷ +

= 307

pattern. Then find 652 752

,

, and 952 mentally and

check your answers.

38. a. Find a shortcut.

b. Using a variable, prove that your result holds for squar-

24 × 26 = 624

ing numbers that have a 5 as their ones digit.

62 × 68 = 4216

32. George Bidder was a calculating prodigy in England dur-

73 × 77 = 5621

ing the nineteenth century. As a nine-year-old, he was

41 × 49 = 2009

asked: If the moon were 123,256 miles from the Earth and

86 × 84 = 7224

sound traveled at the rate of 4 miles a minute, how long

would it be before inhabitants of the moon could hear

57 × 53 =

the battle of Waterloo? His answer—21 days, 9 hours, 34

b. Prove that your result works in general.

minutes—was given in 1 minute. Was he correct? Try to

c. How is this problem related to Problem 31?

do this calculation in less than 1 minute using a calculator.

(NOTE: The moon is about 240,000 miles from Earth and

39. Develop a set of rules for the round a 5 up method.

sound travels about 12.5 miles per second.)

40. There are eight consecutive odd numbers that when multi-

plied together yield 34,459,425. What are they?

33. Found in a newspaper article: What is

41. Jill goes to get some water. She has a 5-liter pail and a

241, 573, 142, 393, 627, 673, 576, 957, 439, 048 × 45, 994,

3-liter pail and is supposed to bring exactly 1 liter back.

811, 347, 886, 846, 310, 221, 728, 895, 223, 034, 301, 839?

How can she do this?

EXERCISE/PROBLEM SET B

EXERCISES

1. Calculate mentally using properties.

3. Calculate mentally using the left-to-right method.

a. 52 ¥14 − 52 ¥ 4

b. (5 × 37) × 20

a. 246 + 352

b. 49 + 252

c. (56 + 37) + 44

d. 23 ¥ 4 + 23 ¥ 5 + 7 ¥ 9

c. 842 − 521

d. 751 − 647

2. Find each of these differences mentally using

4. Calculate mentally using the method indicated.

equal additions. Write out the steps that you thought

a. 359 + 596 (additive compensation)

through.

b. 76 × 25 (multiplicative compensation)

a. 56 − 29

b. 83 − 37

c. 37 × 98 (special factor)

c. 214 − 86

d. 542 − 279

d. 1240 ÷ 5 (special factor)

c04.indd 142

7/30/2013 2:53:01 PM - Section 4.1 Mental Math, Estimation, and Calculators 143

5. Calculate mentally.

b. 5714 × 13 is between _____ 0000 and _____ 0000.

a. 32 000

,

× 400

b. 6000 × 12 000

,

c. 2563 is between _____ 000000 and _____ 000000.

c. 4000 × 5000 × 70 d. 5 104

30 105

×

×

×

16. Guess what whole numbers can be used to fill in the

e. 12 000

4 107

,

× ×

f. 23 000

,

000

,

× 5 000

,

000

,

blanks. Use your calculator to check.

6. Often subtraction can be done more easily in steps.

a. _____ 4 = 6561

For example, 43 − 37 can be found as follows:

b. _____ 5 = 16,807

43 − 37 = (43 − 30) − 7 = 13 − 7 = 6. Find the following

differences using this technique.

17. Guess which is larger. Check with your calculator.

a. 52 − 35

b. 173 − 96

a. 64 or 55 ?

b. 38 or 46 ?

c. 241 − 159

d. 83 − 55

c. 54 or 93 ?

d. 85 or 66 ?

7. The halving and doubling method can be used to multi-

18. Compute the following products using a combination

ply two numbers when one factor is a power of 2. For

of mental math and a calculator. Explain your method.

example, to find 8 × 17, find 4 × 34 or 2 × 68 = 136. Find the

a. 20 × 14 × 39 × 5

following products using this method.

b. 40 × 27 × 25 × 23

a. 16 × 21 b. 4 × 72 c. 8 × 123 d. 16 × 211

c. 647 × 50 × 200 × 89

d. 25 × 91 × 2 × 173 × 2

8. In determining an evacuation zone, a scientist must esti-

mate the distance that lava from an erupting volcano will

19. Find the quotient and remainder using a calculator. Check

flow. Should she overestimate or underestimate? Explain.

your answers.

,

÷

9. Estimate each of the following using the four front-end

a. 18 114

37

b. 381,271 ÷ 147

methods: (i) one-column, (ii) range, (iii) two-column, and

c. 9,346,870 ÷ 1349

d. 817,293 ÷ 749

(iv) with adjustment.

20. Check to see that 1634 14

64

34

44

= +

+

+ . Then determine

a.

4652

b. 2659

c. 15923

which of the following four numbers satisfy the same

+ 8134

3752

672

property.

79

2341

a. 8208

b. 9474

+ 143

+ 251

c. 1138

d. 2178

2

2

2

2

2

2

10. Find a range estimate for these products.

21. It is easy to show that 3 + 4 = 5 , and 5 + 12 = 13 .

a. 57 × 1924

b. 1349 × 45

c. 547 × 73,951

However, in 1966 two mathematicians claimed the following:

275

845 1105 1335

1445

+

+

+

=

.

11. Estimate using compatible number estimation.

a. 84 × 49

b. 5527 ÷ 82

c. 2315 ÷ 59

True or false?

d. 78 × 81

e. 207 × 73

f. 6401 ÷ 93

22. Verify the following patterns.

12. Round as specified.

32 + 42 = 52

a. 257 down to the nearest tens place

2

2

2

2

2

b. 650 to the nearest even hundreds place

10 + 11 + 12 = 13 + 14

c. 593 to the nearest tens place

212 + 222 + 232 + 242 = 252 + 262 + 272

d. 4157 to the nearest hundreds place

e. 7126 to the nearest thousands place

23. For which of the values n = 1, 2, 3, 4 is the following true?

13. Estimate using cluster estimation.

1n + 5n + 8n + 12n + 18n + 19n =

a. 547 + 562 + 554 + 556 b. 31 × 32 × 35 × 28

2n + 3n + 9n + 13n + 16n + 20n

c. 189 + 175 + 193 + 173 d. 562 × 591 × 634

24. Simplify each of the following expressions using a calculator.

14. Estimate the following products in two different ways

a. 42 632 − 522

¥

¥ 23

and explain each method.

a. 52 × 39

b. 17 × 74

72(43 − 31 3

)

b.

c. 88 × 11

d. 26 × 42

162 − 2 ¥ 32(27 + 115 0

)

2

2

5

3

15. Estimate the following values and check with a calculator.

c. 35 − 24 + (26 − 14) ÷ 2

a. 324 × 56 is between _____ 000 and _____ 000.

d. 14

233

364

123 32

×

−

÷

¥

PROBLEMS

25. Notice how by starting with 55 and continuing to raise the dig-

Check to see whether this phenomenon is also true for

its to the third power and adding, 55 reoccurs in three steps.

these three numbers:

a. 136

b. 160

c. 919

55 ⎯ →

⎯ 53 + 53 = 250 ⎯ →

⎯ 23 + 53

26. What is interesting about the quotient obtained by divid-

= 133 ⎯ →

⎯ 13 + 33 + 33 = 55

ing 987,654,312 by 8? (Do this mentally.)

c04.indd 143

7/30/2013 2:53:08 PM - 144 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

27. Using distributivity, show that (a − b)2 = a2 − ab + b2

2

.

37. a. Find a pattern for multiplying the following pairs.

How can this idea be used to compute the following

32 × 72 = 2304 43 × 63 = 2709

squares mentally? (Hint: 99 = 100 − 1.)

73 × 33 = 2409

a. 992

b. 9992

c. 99992

Try finding these products mentally.

28. Fill in the empty squares to produce true

17 × 97 56 × 56 42 × 62

equations.

b. Prove why your method works.

38. Have you always wanted to be a calculating genius?

Amaze yourself with the following problems.

a. To multiply 4,109,589,041,096 by 83, simply put the 3

in front of it and the 8 at the end of it. Now check your

answer.

b. After you have patted yourself on the back, see whether

you can find a fast way to multiply

7,894,736,842 105

,

,263 158

,

by 86.

(NOTE: This works only in special cases.)

39. Develop a set of rules for the round to the nearest even

method.

29. Discuss the similarities/differences of using (i) special fac-

40. Find the ones digits.

tors and (ii) multiplicative compensation when calculating

a. 210 b. 43210 c. 36 d. 2936

36 × 5 mentally.

41. A magician had his subject hide an odd number of coins

30. Find 166 666 6662

,

,

.

in one hand and an even number of coins in another. He

then told the subject to multiply the number of coins in

31. Megan tried to multiply 712,000 by 864,000 on her

the right hand by 2 and the number of coins in the left

calculator and got an error message. Explain how she can

hand by 3 and to announce the sum of these two numbers.

use her calculator (and a little thought) to find the exact

The magician immediately then correctly stated which

product.

hand had the odd number of coins. How did he do it?

32. What is the product of 777,777,777 and 999,999,999?

Analyzing Student Thinking

33. Explain how you could calculate 342 × 143 even if the

42. Jessica calculated 84 − 28 as 84 − 30 = 54 and 54 + 2 = 56.

3 and 4 keys did not work.

Is her method valid? Explain.

34. Find the missing products by completing the pattern.

43. Kalil tells you that multiplying by 5 is a lot like dividing

Check your answers with a calculator.

by 2. For example, 48 × 5 = 240, but it is easier just to go

11 × 11 = 121

48 ÷ 2 = 24 and then affix a zero at the end. Will his meth-

111 × 111 = 12321

od always work? Explain.

1111 × 1111 =

44. Kalil also says that dividing by 5 is the same as mul-

11111 × 111111 =

tiplying by 2 and “dropping” a zero. Does this work?

111111 × 111111 =

Explain.

45. To work the problem 125 ÷ 5, Paula did the following:

35. Use your calculator to find the following products.

(100 ÷ 5) + (25 ÷ 5) = 20 + 5 = 2 .

5 She said that she used a

12 × 11 24 × 11 35 × 11

form of distributivity. Do you agree? Why or why not?

Look at the middle digit of each product and at the first

46. Alyssa was asked to find a reasonable range for the sum

and last digits. Write a rule that you can use to multiply

158 + 547. She said 600 to 700. How should you respond?

by 11. Now try these problems using your rule and check

47. Jared was asked which method would give the best esti-

your answers with your calculator.

mate for addition of numbers with at least 3 digits: one-

54 × 11 62 × 11 36 × 11

column front-end or two-column front-end. He said that

it didn’t matter because an estimate is an estimate. Is he

Adapt your rule to handle

correct? Explain.

37 × 11 59 × 11 76 × 11.

48. Nicole multiplied 3472 and 259 on her calculator and she

wrote down the answer of 89,248. How can you see imme-

36. When asked to multiply 987,654,321 by 123,456,789,

diately using estimation that she made an error? Can you

one mental calculator replied, “I saw in a flash that

see how she made the error?

987,654,321 × 81 = 80,000,000,001, so I multiplied

123,456,789 by 80,000,000,001 and divided by 81.”

49. Kysha was asked to calculate the product of 987,654,321

Determine whether his reasoning was correct. If it was, see

and 123 on her 10 digit calculator. She said that it was

whether you can find the answer using your calculator.

impossible. How should you respond?

c04.indd 144

7/30/2013 2:53:11 PM - Section 4.2 Written Algorithms for Whole-Number Operations 145

WRITTEN ALGORITHMS FOR WHOLE-NUMBER

OPERATIONS

Section 4.1 was devoted to mental and calculator computation. This section presents the

common written algorithms as well as some alternative ones that have historical interest

and can be used to help students better understand how their algorithms work.

When a class of students was given the problem 48 + 35, the following three responses were

typical of what the students did.

Nick

Trevor

Courtney

48

I combined 40 and

1

8 + 5 is 13

48

48 plus 30 is 78.

48

+ 35

30 to get 70

Carry the 1

+ 35

Now I add the 5

+ 35

70

8 + 5 = 13.

write down 3

83

and get 83.

83

13

70 and 13 is 83

4 + 3 + 1 = 8

83

write down 8

Which of these methods demonstrates the best (least) understanding of place value? Which method is the best? Justify.

Common Core – Grade 4

Algorithms for the Addition of Whole Numbers

Fluently add and subtract mul-

tidigit whole numbers using the

An algorithm is a systematic, step-by-step procedure used to find an answer, usu-

standard algorithm.

ally to a computation. The common written algorithm for addition involves two

main procedures: (1) adding single digits (thus using the basic facts) and (2) carrying

(regrouping or exchanging).

A development of our standard addition algorithm is used in Figure 4.4 to find the

sum 134 + 325.

+

+

Figure 4.4

Observe how the left-to-right sequence in Figure 4.4 becomes progressively more

abstract. When one views the base ten pieces (1), the hundreds, tens, and ones are

Reflection from Research

distinguishable due to their sizes and the number of each type of piece. In the chip

Students in upper grades can

develop shortcut strategies

abacus (2), the chips all look the same. However, representations are distinguished

for addition and subtraction,

by the number of chips in each column and by the column containing the chips (i.e.,

but younger students need

place value). In the place-value representation (3), the numbers are distinguished by

careful instruction to support

the digits and the place values of their respective columns. Representation (4) is the

the discovery of such shortcut

common “add in columns” algorithm. The place-value method can be justified using

strategies (Torbeyns, De Smedt,

Ghesquiere, & Vershaffel, 2009).

expanded form and properties of whole-number addition as follows.

c04.indd 145

7/30/2013 2:53:12 PM - 146

Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Common Core – Grade 1

134 + 325 = 1

( ¥ 102 + 3 ¥ 10 + 4) + (3 ¥ 102 + 2 ¥ 10 + 5)

Understand that in adding two-

Expande

ed form

= 1

( ¥ 102 + 3 ¥ 102 ) + (3 ¥ 10 + 2 ¥ 10)

digit numbers, one adds tens and

Associativity and

tens, ones and ones; and some-

+ (4 + 5)

commutativity

times it is necessary to compose

= 1

( + 3 102

)

+ (3 + 2 10

)

+ (4 + 5)

Distributivity

a ten.

= 4 ¥ 102 + 5 ¥ 10 + 9

Addition

n

Simplified form

= 459

Note that 134 + 325 can be found working from left to right (add the hundreds first,

etc.) or from right to left (add the ones first, etc.).

An addition problem when regrouping is required is illustrated in Figure 4.5 to

find the sum 37 + 46. Notice that grouping 10 units together and exchanging them for

a long with the base ten pieces is not as abstract as exchanging 10 ones for 1 ten in the

chip abacus. The abstraction occurs because the physical size of the 10 units is main-

tained with the 1 long of the base ten pieces but is not maintained when we exchange

10 dots in the ones column for only 1 dot in the tens column of the chip abacus.

Figure 4.5

The procedure illustrated in the place-value representation can be refined in a

series of steps to lead to our standard carrying algorithm for addition. Intermediate

algorithms that build on the physical models of base ten pieces and place-value repre-

sentations lead to our standard addition algorithm and are illustrated next.

Reflection from Research

(a) INTERMEDIATE

(b) INTERMEDIATE

(c) STANDARD

Students who initially used

ALGORITHM 1

ALGORITHM 2

ALGORITHM

invented strategies for addition

and subtraction demonstrated

568

568

1 1

knowledge of base ten number

568

concepts before students who

+ 394

+ 394

+ 394

relied primarily on standard

12

sum of ones

12

algorithms (Carpenter, Franke,

962

Jacobs, Fennema, & Empson,

150

sum of tens

15

1997).

sum of hundreds

800

8

final sum

962

962

The preceding intermediate algorithms are easier to understand than the stan-

dard algorithm. However, they are less efficient and generally require more time

and space.

Throughout history many other algorithms have been used for addition. One of

these, the lattice method for addition, is illustrated next.

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7/30/2013 2:53:13 PM - Section 4.2 Written Algorithms for Whole-Number Operations 147

Notice how the lattice method is very much like intermediate algorithm 2. Other

interesting algorithms are contained in the problem sets.

Check for Understanding: Exercise/Problem Set A #1–11

✔

Use base ten pieces to solve the problem 146 – 25. On a piece of paper, sketch the actions

used with the pieces to solve the problem. How do the actions with the blocks relate to the

subtraction algorithms?

NCTM Standard

Algorithms for the Subtraction of Whole Numbers

All students should develop and

use strategies for whole-number

The common algorithm for subtraction involves two main procedures: (1) subtract-

computations, with a focus on

ing numbers that are determined by the addition facts table and (2) exchanging or

addition and subtraction.

regrouping (the reverse of the carrying process for addition). Although this exchang-

ing procedure is commonly called “borrowing,” we choose to avoid this term because

the numbers that are borrowed are not paid back. Hence the word borrow does not

represent to children the actual underlying process of exchanging.

A development of our standard subtraction algorithm is used in Figure 4.6 to find

the difference 357 − 123.

Figure 4.6

Notice that in the example in Figure 4.6, the answer will be the same whether we

subtract from left to right or from right to left. In either case, the base ten blocks are

used by first representing 357 and then taking away 123. Since there are 7 units from

which 3 can be taken and there are 5 longs from which 2 can be removed, the use of the

blocks is straightforward. The problem 423 − 157 is done differently because we cannot

take 7 units away from 3 units directly. In this problem, a long is broken into 10 units to

create 13 units and a flat is exchanged from 10 longs and combined with the remaining

long to create 11 longs (see Figure 4.7). Once these exchanges have been made, 157 can

be taken away to leave 266.

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Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

In Figure 4.7, the representations of the base ten blocks, chip abacus, and place-

value model become more and more abstract. The place-value procedure is finally

shortened to produce our standard subtraction algorithm. Even though the stan-

dard algorithm is abstract, a connection with the base ten blocks can be seen when

exchanging 1 long for 10 units because this action is equivalent to a “borrow.”

3 11 13

4 2 3

423 make

exchanges

+ 1 5 7

− 157 ⎯→

2 6 6

One nontraditional algorithm that is especially effective in any base is called the

subtract-from-the-base algorithm. This algorithm is illustrated in Figure 4.8 using base

ten pieces to find 323 − 64.

Figure 4.7

Figure 4.8

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7/30/2013 2:53:15 PM - Section 4.2 Written Algorithms for Whole-Number Operations 149

In (1), observe that the 4 is subtracted from the 10 (instead of finding 13 − 4, as in

the standard algorithm). The difference, 10 − 4 = 6, is then combined with the 3 units

in (2) to form 9 units. Then the 6 longs are subtracted from 1 flat (instead of find-

ing 11 − 6). The difference, 10 − 6 = 4 longs, is then combined with the 1 long in (3)

to obtain 5 longs (or 50). Thus, since 2 flats remain, 323 − 64 = 259. The following

illustrates this process symbolically.

Reflection from Research

The advantage of this algorithm is that we only need to know the addition facts

Having students create their own

and differences from 10 (as opposed to differences from all the teens). As you will see

computational algorithms for

later in this chapter, this method can be used in any base (hence the name subtract-

large number addition and sub-

traction is a worthwhile activity

from-the-base). After a little practice, you may find this algorithm to be easier and

(Cobb, Yackel, & Wood, 1988).

faster than our usual algorithm for subtraction.

Check for Understanding: Exercise/Problem Set A #12–19

✔

1. How could the base ten pieces be used to model 13 × 24?

2. What might an intermediate algorithm for 13 × 24 look like?

Discuss how the work with the blocks in number 1 is related to the intermediate algorithm in number 2.

Common Core – Grade 5

Algorithms for the Multiplication of Whole Numbers

Fluently multiply multidigit whole

numbers using the standard algo-

The standard multiplication algorithm involves the multiplication facts, distributivity,

rithm.

and a thorough understanding of place value. A development of our standard multipli-

cation algorithm is used in Figure 4.9 to find the product 3 × 213.

Place-Value Representations

Figure 4.9

Reflection from Research

Next, the product of a two-digit number times a two-digit number is found.

Teachers should help students

Calculate 34 × 12.

to develop a conceptual under-

standing of how multidigit mul-

34 × 12 = 34 1

( 0 + 2)

tiplication and division relates to

and builds upon place value and

=

Expanded form

34 ¥10 + 34 ¥ 2

Distributivity

basic multiplication combinations

= (30 + 4)10 + (30 + 4 2

)

Expanded form

m

(Fuson, 2003).

= 30 ¥10 + 4 ¥10 + 30 ¥ 2 + 4 ¥ 2

Distributivity

= 300 + 40 + 60 + 8

Multiplication

=

Addition

408

The product 34 × 12 also can be represented pictorially (Figure 4.10).

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Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Figure 4.10

It is worthwhile to note how the Intermediate Algorithm 1 closely connects to the

base block representation of 34 × 12. The numbers in the algorithm have been color

coded with the regions in the rectangular array in Figure 4.10 to make the connection

more apparent. The intermediate algorithms assist in the transition from the concrete

blocks to the abstract standard algorithm.

Algebraic Reasoning

(A) INTERMEDIATE

(B) INTERMEDIATE

(C) STANDARD

In Figure 4.10, the model of a

ALGORITHM 1

ALGORITHM 2

ALGORITHM

rectangular array can be used to

multiply expressions such as x + 2

34

34

and x + 3. Tiles that look similar

× 12

× 12

to base ten blocks are placed

with an x + 2 down the side and

8

68 Think 2 × 34

x + 3 across the top. When the

60

rectangle is filled in, the product

of (x + 2) (x + 3) is seen to be

40

340 Think 10 × 34

x2 + 5x + 6.

300

408

408

One final complexity in the standard multiplication algorithm is illustrated next.

Calculate 857 × 9.

(A) INTERMEDIATE ALGORITHM

(B) STANDARD ALGORITHM

Notice the complexity involved in explaining the standard algorithm!

The lattice method for multiplication is an example of an extremely simple multipli-

cation algorithm that is no longer used, perhaps because we do not use paper with

lattice markings on it and it is too time-consuming to draw such lines.

To calculate 35 × 4967, begin with a blank lattice and find products of the digits

in intersecting rows and columns. The 18 in the completed lattice was obtained by

multiplying its row value, 3, by its column value, 6 (Figure 4.11). The other values are

filled in similarly. Then the numbers are added down the diagonals as in lattice addi-

tion. The numbers in the “6’’ diagonal add to 12, but 1 (actually 10) is carried from

Figure 4.11

the previous “7’’ diagonal that yields 13. Then the 1 (actually 10) from 13 is carried

to the “9’’ diagonal to yield 2 + 2 + 2 + 1 = 7. The answer, read counterclockwise from

left to right, is 173,845.

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7/30/2013 2:53:19 PM - Section 4.2 Written Algorithms for Whole-Number Operations 151

Check for Understanding: Exercise/Problem Set A #20–28

✔

Use base ten pieces to solve the problem 432 ÷ 3. On a piece of paper, sketch the actions

used with the pieces to solve the problem. Next solve the problem using the standard

long division algorithm. There are several actions with the blocks that relate to specific steps in the algorithm. Explain

those connections.

Algorithms for the Division of Whole Numbers

Reflection from Research

The long-division algorithm is the most complicated procedure in the elementary

A weak understanding of the

mathematics curriculum. Because of calculators, the importance of written long divi-

place-value system and a

sion using multidigit numbers has greatly diminished. However, the long-division

procedure-oriented performance

algorithm involving one- and perhaps two-digit divisors continues to have common

of algorithms cause students to

applications. The main idea behind the long-division algorithm is the division algo-

struggle to understand the

traditional long division algorithm

rithm, which was given in Section 3.2. It states that if a and b are any whole numbers

(Lee, 2007).

with b ≠ 0, there exist unique whole numbers q and r such that a = bq + r, where

0 ≤ r < b. For example, if a = 17 and b = 5, then q = 3 and r = 2 since 17 = 5 ¥ 3 + 2. The

Children’s Literature

purpose of the long division algorithm is to find the quotient, q, and remainder, r, for

www.wiley.com/college/musser

any given divisor, b, and dividend, a.

See “The Doorbell Rang” by Pat

To gain an understanding of the division algorithm, we will use base ten blocks

Hutchins.

and a fundamental definition of division. Find the quotient and remainder of 461

divided by 3. This can be thought of as 461 divided into groups of size 3. As you read

Common Core – Grade 6

this example, notice how the manipulation of the base ten blocks parallels the written

Fluently divide multidigit numbers

algorithm. The following illustrates long division using base ten blocks.

using the standard algorithm.

Thought One

Thought Two

Think: One group of 3 flats, which is 100 groups of 3,

Think: Convert the 1 leftover flat to 10 longs and add

leaves 1 flat left over.

it to the existing 6 longs to make 16 longs.

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7/30/2013 2:53:21 PM - 152 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Thought Three

Thought Four

Think: Five groups of 3 longs, which is 50 groups of 3,

Think: Convert the 1 leftover long into 10 units and add

which leaves 1 long left over.

it to the existing 1 unit to make 11 units.

Thought Five

Think: Three groups of 3 units each leaves 2 units left over. Since there is one group

of flats (hundreds), five groups of longs (tens), and three groups of units (ones) with

2 left over, the quotient is 153 with a remainder of 2.

We will arrive at the final form of the long-division algorithm by working through

various levels of complexity to illustrate how one can gain an understanding of the

algorithm by progressing in small steps.

Find the quotient and remainder for 5739 ÷ 31.

The scaffold method is a good one to use first.

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7/30/2013 2:53:22 PM - Section 4.2 Written Algorithms for Whole-Number Operations 153

3 )

1 5739

How many 31s in 5739? Guess:100

Reflection from Research

When solving arithmetic prob-

− 3100

100(31)

lems, students who move directly

2639

How many 31s in 26

639? Guess: 50

from using concrete strategies

to algorithms, without being

− 1550

50(31)

allowed to generate their own

1089

How many 31s in 1089? Guess: 30

abstract strategies, are less likely

to develop a conceptual under-

− 930

30(31)

standing of multidigit numbers

(Ambrose, 2002).

159

How many 31s in

n 159? Guess: 5

155

+

5(31)

Since 4 < 31, we stop and add 100 + 50 + 30 + 5.

4

185(31)

Therefore, the quotient is 185 and remainder is 4.

Check: 31 ¥ 185 + 4 = 5735 + 4 = 5739.

As just shown, various multiples of 31 are subtracted successively from 5739

(or the resulting difference) until a remainder less than 31 is found. The key to this

method is how well one can estimate the appropriate multiples of 31. In the scaffold

method, it is better to estimate too low rather than too high, as in the case of the 50.

However, 80 would have been the optimal guess at that point. Thus, although the

quotient and remainder can be obtained using this method, it can be an inefficient

application of the Guess and Test strategy.

The next example illustrates how division by a single digit can be done more efficiently.

Find the quotient and remainder for 3159 ÷ 7.

INTERMEDIATE ALGORITHM

451

1

50

Start here:

400

7)3159

Think: How many 7s in 3100

0? 400

−2800

359

Think: How many 7s in 350? 50

−350

9

Think: How many 7s in 9

9? 1

−7

2

Therefore, the quotient is the sum 400 + 50 + 1, or 451, and the remainder is 2. Check:

7 ¥ 451 + 2 = 3157 + 2 = 3159.

Now consider division by a two-digit divisor. Find the quotient and remainder for

1976 ÷ 32.

INTERMEDIATE ALGORITHM

61

1

60

32)1976

Think: How many 32s in 1976? 60

−1920

56

Th

hink: How many 32s in 56? 1

−32

24

Therefore, the quotient is 61 and the remainder is 24.

Check: 32 ¥ 61 + 24 = 1952 + 24 = 1976.

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7/30/2013 2:53:23 PM - 154 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Next we will employ rounding to help estimate the appropriate quotients. Find the

quotient and remainder of 4238 ÷ 56.

5

70

56)4238

Think: How many 60s in 420

00? 70

−3920

70 × 56 = 3920

318

Think: How many 60s in 310? 5

−280

5 × 56 = 280

38

Therefore, the quotient is 70 + 5 = 75 and the remainder is 38.

Check: 56 ¥ 75 + 38 = 4200 + 38 = 4238.

Observe how we rounded the divisor up to 60 and the dividend down to 4200 in

Algebraic Reasoning

the first step. This up/down rounding assures us that the quotient at each step will

Solving equations like

x3 − x2

2

− 3x + 6 = 0 can be done

not be too large.

by factoring. One way to factor

This algorithm can be simplified further to our standard algorithm for division

is a type of algebraic long

by reducing the “think” steps to divisions with single-digit divisors. For example,

division. The standard division

in place of “How many 60s in 4200?” one could ask equivalently, “How many 6s

algorithm provides a foundation

in 420?” Even easier, “How many 6s in 42?” Notice also that in general, the divisor

for this type of algebraic

reasoning.

should be rounded up and the dividend should be rounded down.

Find the quotient and remainder for 4238 ÷ 56.

S O L U T I O N

Think

⎧

: How many 6s in 42? 7

75

⎪

56)4238 ⎪Put the 7 above thee 3 since we are actually finding

⎨4230 ÷ 56.

392

⎪

⎪The 392 is 7

⎩

¥ 56.

318

Think

⎧

: How many 6s in 31? 5

−

⎪

280

Put the 5 above the 8 ssince we are finding 318

⎨

÷ 56.

38

The 280 is 5 56.

⎩⎪

¥

■

The quotient and remainder for 4238 ÷ 56 can be found using a calculator. The

TI-34 MultiView does long division with remainder directly using 4238 2nd int÷ 56

75r38

enter . The result for this problem is shown in Figure 4.12 as it appears on the second

line of the calculator’s display. To find the quotient and remainder using a standard

Figure 4.12

calculator, press these keys: 4238 ÷ 56 = . Your display should read 75.678571

(perhaps with fewer or more decimal places). Thus the whole-number quotient is 75.

From the relationship a = bq + r, we see that r = a − bq is the remainder. In this case

r = 4238 − 56 ¥ 75, or 38.

As the preceding calculator example illustrates, calculators virtually eliminate the

need for becoming skilled in performing involved long divisions and other tedious

calculations.

In this section, each of the four basic operations was illustrated by different algo-

rithms. The problem set contains examples of many other algorithms. Even today,

other countries use algorithms different from our “standard” algorithms. Also, students

may even invent “new” algorithms. Since computational algorithms are aids for simpli-

fying calculations, it is important to understand that all correct algorithms are accept-

able. Clearly, computation in the future will rely less and less on written calculations

and more and more on mental and electronic calculations.

Check for Understanding: Exercise/Problem Set A #29–34

✔

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7/30/2013 2:53:25 PM - Section 4.2 Written Algorithms for Whole-Number Operations 155

Have someone write down a three-digit number using three different

digits hidden from your view. Then have the person form all of the other

five 3-digit numbers that can be obtained by rearranging his or her three

digits. Add these six numbers together with a seventh number, which is

any other one of the six. The person tells you the sum, and then you, in

turn, tell him or her the seventh number.

Here is how. Add the thousands digit of the sum to the remaining

three-digit number (if the sum was 3635, you form 635 + 3 = 638). Then

take the remainder upon division of this new number by nine (638 ÷ 9

leaves a remainder of 8). Multiply the remainder by 111 8

( × 111 = 888)

and add this to the previous number (638 + 888 = 15 )

26 . Finally, add the

thousands digit (if there is one) to the remaining number (1526 yields

©Ron Bagwell

526 + 1 = 527, the seventh number!).

section 4.2 EXERCISE/PROBLEM SET A

EXERCISES

1. Using the Chapter 4 eManipulative activity Base Blocks—

Use this expanded form of the addition algorithm to com-

Addition on our Web site, model the following addition

pute the following sums.

problems using base ten blocks. Sketch how the base ten

a.

351

b. 564

blocks would be used.

+ 635

+ 345

a. 327 + 61

b. 347 + 86

2. The physical models of base ten blocks and the chip

5. Use the Intermediate Algorithm 1 to compute the following

abacus have been used to demonstrate addition. Bundling

sums.

sticks can also be used. Sketch 15 + 32 using the following

a. 598 + 396

b. 322 + 799 + 572

models.

6. An alternative algorithm for addition, called scratch addi-

a. Chip abacus

b. Bundling sticks

tion, is shown next. Using it, students can do more compli-

3. Give a reason or reasons for each of the following steps to

cated additions by doing a series of single-digit additions.

justify the addition process.

This method is sometimes more effective with students hav-

ing trouble with the standard algorithm. For example, to

17 + 21 = (4¥10 + 7) + (2 ¥10 + 1)

compute 78 + 56 + 38:

= 1

( ¥10 + 2 ¥10) + (7 + 1)

= 3 ¥10 + 8

7

8 Add the number in the units place start-

=

ing at the top. When the sum is ten or

38

5

6

more, scratch a line through the last num-

3

84

4. There are many ways of providing intermediate steps

+

ber added and write down the unit. The

between the models for computing sums (base ten blocks,

scratch represents 10.

chip abacus, etc.) and the algorithm for addition. One of

2

Continue adding units (adding

7

8

these is to represent numbers in their expanded forms.

4 and 8). Write the last number of units

Consider the following examples:

5

6

below the line. Count the number of

3

8 4

+

scratches, and write above the second

246 = 2 hundreds + 4 tens + 6

2

column.

+

352 = 3 hundreds + 5

tens + 2

2

Repeat the procedure for each column.

= 5 hundreds + 9 tens + 8

7

8

= 598

5

6

34

8 4

547 = 5 10 2

(

) + 4(10) + 7

+

+

1

7

2

296 = 2(10)2 + 9(10) + 6

7(10)2 + 13(10) + 13 ⎫

Compute the following additions using the scratch

← ⎯

⎯⎯⎯

⎪

algorithm.

2

⎪

7 1

( 0) + 14 1

( 0) + 3

regroup

ping

⎬

← ⎯

⎯

⎪

a. 734

b. 1364

8 1

( 0 2

) + 4 10

(

) + 3

⎭⎪

468

7257

= 843

+ 27

+ 4813

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7/30/2013 2:53:29 PM - 156 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

7. Compute the following sums using the lattice method.

732

700 + 30 + 2

a. 482

b. 567

−

or

378

− (300 + 70 + 8)

+ 269

+ 765

8. Give an advantage and a disadvantage of each of the

700 + 20 + 12

600 + 120 + 12

following methods for addition.

− (300 + 70 + 8)

or

− (300 + 70 + 8)

a. Intermediate algorithm

b. Lattice

300 + 50 + 4 = 354

9. Add the following numbers. Then turn this page upside

Use expanded form with regrouping as necessary

down and add it again. What did you find? What feature

to perform the following subtractions.

of the numerals 1, 6, 8, 9 accounts for this?

a. 652 − 175 b. 923 − 147 c. 8257 − 6439

986

818

17. The cashier’s algorithm for subtraction is closely related

969

to the missing-addend approach to subtraction; that is,

989

a − b = c if and only if a = b + c. For example, you buy $23

696

of school supplies and give the cashier a $50 bill. While

handing you the change, the cashier would say “$23, $24,

616

$25, $30, $40, $50.” How much change did you receive?

10. Without performing the addition, tell which sum, if either,

What cashier said: $23, $24, $25, $30, $40, $50

is greater.

Money received: 0, $1, $1, $5, $10, $10

23,456

20,002

Now you are the cashier. The customer owes you

23,400

32

$62 and gives you a $100 bill. What will you say

23,000

432

to the customer, and how much will you give back?

+ 20,002 + 65,432

18. Subtraction by adding the complement goes as follows:

The complement of 476 is 523

11. Without calculating the actual sums, select the smallest sum,

619

619 since 4476 + 523 = 999.

the middle sum, and the largest sum in each group. Mark

− 476

+ 523 (The sum in each place is 9.)

them A, B, and C, respectively. Use your estimating powers.

1142 Cross out the lea

ading digit.

a. _____ 283 + 109 _____ 161 + 369 _____ 403 + 277

+

1

Add 1.

b. _____ 629 + 677 _____ 723 + 239 _____ 275 + 631

143 Answer = 143

Find the following differences using this algorithm.

Now check your estimate with a calculator.

a. 537

b. 86 124

,

12. Using the Chapter 4 eManipulative activity Base Blocks—

−

Subtraction on our Web site, model

− 179

38,759

the following subtraction problems using base

ten blocks. Sketch how the base ten blocks would

19. A subtraction algorithm, popular in the past, is called the

be used.

equal-additions algorithm. Consider this example.

a. 87

b. 483

−

2

35

− 57

436

4 ¥10 + 3 ¥ 10 + 6

− 282

− (2 ¥102 + 8 ¥ 10 + 2)

13. Sketch solutions to the following problems, using

bundling sticks and a chip abacus

13

2

a.

¥

+ 13 ¥ 10 +

57

b. 34

4 10

6

4 3 6

3

−

− (3 ¥ 102 + 8 ¥10 + 2)

37

− 29

− 2 82

1 ¥102 + 5 ¥10 + 4 = 154

15 4

14. 9342 is usually thought of as 9 thousands, 3 hundreds, 4

tens, and 2 ones, but in subtracting 6457 from 9342 using

To

subtract

8 ¥10 from 3 ¥10, add 10 tens to the minuend

the customary algorithm, we regroup and think of 9342 as

and 1 102

¥

(or 10 tens) to the subtrahend. The problem

is changed, but the answer is the same. Use the

_____ thousands, _____ hundreds, _____ tens, and _____

equal-additions algorithm to find the following

ones.

differences.

15. Order these computations from easiest to hardest.

a. 421 − 286 b. 92,863 − 75,387

a. 809

b. 8

c. 82

20. Consider the product of 27 × 33.

− 306

− 3

− 67

a. Sketch how base ten blocks can be used in a rectangular

16. To perform some subtractions, it is necessary to reverse

array to model this product.

the rename and regroup process of addition. Perform

b. Find the product using the Intermediate Algorithm 1.

the following subtraction in expanded form and follow

c. Describe the relationship between the solutions in parts

regrouping steps.

a and b.

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7/30/2013 2:53:34 PM - Section 4.2 Written Algorithms for Whole-Number Operations 157

21. Show how to find 28 × 34 on this grid paper.

26. The Russian peasant algorithm for multiplying 27 × 51

is illustrated as follows:

Notice that the numbers in the first column are halved

(disregarding any remainder) and that the numbers in

the second column are doubled. When 1 is reached in the

halving column, the process is stopped. Next, each row

with an even number in the halving column is crossed out

and the remaining numbers in the doubling column are

added. Thus

27 × 51 = 51 + 102 + 408 + 816 = 1377.

Use the Russian peasant algorithm to compute the

22. The pictorial representation of multiplication can be adapt-

following products.

ed as follows to perform 23 × 16.

a. 68 × 35 b. 38 × 62

27. The use of finger numbers and systems of finger com-

putation has been widespread through the years. One

such system for multiplication uses the finger positions

shown for computing the products of numbers from 6

through 10.

Use this method to find the following products.

a. 15 × 36

b. 62 × 35

23. Justify each step in the following proof that 72 × 10 = 720.

The two numbers to be multiplied are each represented

on a different hand. The sum of the raised fingers is the

72 × 10 = (70 + 2) × 10

number of tens, and the product of the closed fingers is

= 70 ×10 + 2 ×10

the number of ones. For example, 1 + 3 = 4 fingers raised,

= (7 ×10) ×10 + 2 ×10

and 4 × 2 = 8 fingers down.

= 7 × 1

( 0 × 10) + 2 × 10

= 7 ×100 + 2 ×10

= 700 + 20

= 720

24. Solve the following problems using (i) the lattice method

for multiplication and (ii) an intermediate algorithm.

Use this method to compute the following products.

a. 23 × 62 b. 17 × 45

a. 7 × 8 b. 6 × 7 c. 6 × 10

25. Study the pattern in the following left-to-right

28. The duplication algorithm for multiplication combines a

multiplication.

succession of doubling operations, followed by addition.

731

This algorithm depends on the fact that any number

can be written as the sum of numbers that are powers

× 238

of 2. To compute 28 × 36, the 36 is repeatedly doubled as

1462

(2 ¥ 731)

shown.

2193

(3 ¥ 731)

1 × 36 = 36

5848

(8 ¥ 731)

2 × 36 = 72

173978

→ 4 × 36 = 144

Use this algorithm to do the following computations.

→ 8 × 36 = 288

a. 75 × 47 b. 364 × 421

→ 16

× 36 = 576

↑

powers of 2

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7/30/2013 2:53:38 PM - 158 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

This process stops when the next power of 2 in the list is

31. Without using the divide key, find the quotient for each of

greater than the number by which you are multiplying.

the following problems.

Here we want 28 of the 36s, and since 28 = 1

( 6 + 8 + 4), the

a. 24 ÷ 4 b. 56 ÷ 7

product of 28 × 36 = 1

( 6 + 8 + 4) ¥ 36. From the last column,

c. Describe how the calculator was used to find the quo-

you add 144 + 288 + 576 = 1008.

tients.

Use the duplication algorithm to compute the following

products.

32. When asked to find the quotient and remainder of

a. 25 × 62 b. 35 × 58 c. 73 × 104

431 ÷ 17, one student did the following with a calculator:

Display

29. Use the scaffold method in the Chapter 4 dynamic

spreadsheet Scaffold Division on our Web site to find

431 ÷ 17 = 25 35294

.

−

the following quotients. Write down how the scaffold

method was used.

Display

a. 899 ÷ 13 b. 5697 ÷ 23

25 = × 17 = 6 0000000

.

30. A third-grade teacher prepared her students for division

this way:

a. Try this method on the following pairs.

(i) 1379 ÷ 87

20 ÷ 4

20

(ii) 69,431 ÷ 139

− 4 ✔

(iii)

1 111

,

111

,

÷ 333

16

b. Does this method always work? Explain.

− 4 ✔

33. Sketch how base ten blocks can be used to model 762 ÷ 5.

12

− 4 ✔ 20 ÷ 4 = 5

34. When performing the division problem 2137 ÷ 14 using

8

the standard algorithm, the first few steps look like what

is shown here. The next step is to “bring down” the 3.

− 4 ✔

Explain how that process of “bringing down” the 3 is

4

modeled using base ten blocks.

− 4 ✔

1

0

14)2137

14

How would her students find 42 ÷ 6 ?

7

PROBLEMS

35. Larry, Curly, and Moe each add incorrectly as follows.

39. Given next is an addition problem.

2

a. Replace seven digits with 0s so that the sum of the num-

Larry:

29

Curly:

29

Moe:

29

bers is 1111.

+ 83

+ 83

+ 83

999

1012

121

102

777

How would you explain their mistakes to each of them?

555

333

36. Use the digits 1 to 9 to make an addition problem and

answer. Use each digit only once.

+ 111

b. Do the same problem by replacing (i) eight digits with

0s, (ii) nine digits with 0s, (iii) ten digits with 0s.

40. Consider the following array:

37. Place the digits 2, 3, 4, 6, 7, 8 in the boxes to obtain the

following sums.

a. The greatest sum b. The least sum

Compare the pair 26, 37 with the pair 36, 27.

38. Arrange the digits 1, 2, 3, 4, 5, 6, 7 such that they add up

a. Add: 26 + 37 = _____. 36 + 27 = _____.

to 100. (For example, 12 + 34 + 56 + 7 = 109.)

What do you notice about the answers?

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7/30/2013 2:53:44 PM - Section 4.2 Written Algorithms for Whole-Number Operations 159

b. Subtract: 37 − 26 =_____. 36 − 27 =_____.

What do you notice about the answers?

c. Are your findings true for any two such pairs?

d. What similar patterns can you find?

41. The x’s in half of each figure can be counted in two ways.

a. Draw a similar figure for 1 + 2 + 3 + 4.

b. Express that sum in a similar way. Is the sum correct?

c. Use this idea to find the sum of whole numbers from 1

What instructional procedures might you use to help each

to 50 and from 1 to 75.

of these students?

42. In the following problems, each letter represents a differ-

ent digit and any of the digits 0 through 9 can be used.

44. The following is an example of the German low-stress

×

However, both additions have the same result. What are

algorithm. Find 5314

79 using this method and explain

the problems?

how it works.

4967 × 35

ZZZ

ZZZ

KKK

PPP

21

+ LLL

+ QQQ

1835

RSTU

RSTU

2730

1245

43. Following are some problems worked out by students. Each

20

student has a particular error pattern. Find that error, and

17384

45

tell what answer that student will get for the last problem.

45. Select any four-digit number. Arrange the digits to form the

largest possible number and the smallest possible number.

Subtract the smallest number from the largest number. Use

the digits in the difference and start the process over again.

Keep repeating the process. What do you discover?

EXERCISE/PROBLEM SET B

EXERCISES

1. Sketch how base ten blocks could be used to solve the fol-

= (3 ¥10 + 5 ¥10) + 1¥10 + 4

lowing problems. You may may find the Chapter 4 eMa-

= (3 ¥ 5 + 1) + 1¥10 + 4

nipulative activity Base Blocks—Addition on our Web site

to be useful in this process.

= 9 ¥10 + 4

a. 258 b. 627

= 94

+ 149

+ 485

4. Use the expanded form of the addition algorithm (see Set

A, Exercise 4) to compute the following sums.

2. Sketch the solution to 46 + 55 using the following models.

a.

b.

a. Bundling sticks b. Chip abacus

478

1965

+ 269

+ 857

3. Give a reason for each of the following steps to justify the

addition process.

5. Use the Intermediate Algorithm 2 to compute the

following sums.

38 + 56 = (3 ¥10 + 8) + (5 ¥10 + 6)

+

+

=

a.

347

679 b. 3538

784

(3 ¥10 + 5 ¥10) + (8 + 6)

c. Describe the advantages and disadvantages of

= (3 ¥10 + 5 ¥10) + 14

Inter-mediate Algorithm 2 versus Intermediate

= (3 ¥10 + 5 ¥10) + 1¥10 + 4

Algorithm 1.

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7/30/2013 2:53:46 PM - 160 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

6. Another scratch method involves adding from left to right,

each of the following cases. How much change does the

as shown in the following example.

customer receive?

a. Customer owes: $28; Cashier receives: $40

987

9 8 7

9 8 7

987

b. Customer owes: $33; Cashier receives: $100

+ 356 + 3 5 6 + 3 5 6 + 356

18. Use the adding the complement algorithm described in Set

12

1 2 3

1 2 3 3

1343

A, Exercise 18 to find the differences in parts a and b.

3

3

4

a. 3479

First the hundreds column was added. Then the tens

− 2175

column was added and, because of carrying, the 2 was

scratched out and replaced by a 3. The process continued

b. 6,002,005

until the sum was complete. Apply this method to com-

− 4 187

,

,269

pute the following sums.

c. Explain how the method works with three-digit numbers.

a. 475 b. 856 c. 179

+

19. Use the equal-additions algorithm described in Set A,

381

+ 907

+ 356

Exercise 19 to find the differences in parts a and b.

7. Compute the following sums using the lattice method.

a. 3476 − 558

a. 982 b. 4698

b. 50,004 − 36,289

+ 659

+ 5487

c. Will this algorithm work in general? Why or why not?

20. Consider the product of 42 × 24.

8. Give an advantage and a disadvantage of each of the

following methods for addition.

a. Sketch how base ten blocks can be used in a rectangular

array to model this product.

a. Expanded form

b. Find the product using the Intermediate Algorithm 1.

b. Standard algorithm

c. Describe the relationship between the solutions in parts

9. In Set A, Exercise 9, numbers were listed that add to the

a and b.

same sum whether they were read right side up or upside

down. Find another 6 three-digit numbers that have the

21. Show how to find 17 × 26 on this grid paper.

same sum whether read right side up or upside down.

10. Gerald added 39,642 and 43,728 on his calculator and got

44,020 as the answer. How could he tell, mentally, that his

sum is wrong?

11. Without calculating the actual sums, select the smallest

sum, the middle sum, and the largest sum in each group.

Mark them A, B, and C, respectively.

a. _____ 284 + 625 _____ 593 + 237 _____ 304 + 980

b. _____ 427 + 424 _____ 748 + 611 _____ 272 + 505

12. Sketch how base ten blocks could be used to solve the fol-

lowing problems. You may find the Chapter 4

eManipulative activity Base Blocks—Subtraction on

our Web site to be useful in this process.

a. 536 − 54

22. Use the pictorial representation shown in Set A, Exercise

b. 625 − 138

22 to find the products in parts a and b.

13. Sketch the solution to the following problems using

a. 23 × 48

bundling sticks and a chip abacus.

b. 34 × 52

a. 42 − 27

b. 324 − 68

c. How do the numbers within the grid compare with the

steps of Intermediate Algorithm 1?

14. To subtract 999 from 1111, regroup and think of 1111 as

_____ hundreds, _____ tens, and _____ ones.

23. Justify each step in the following proof that shows

573 × 100 = 57,300.

15. Order these computations from easiest to hardest.

a. 81 b. 80 c. 8819

573 × 100 = (500 + 70 + 3) × 100

− 36

− 30

− 3604

= 500 ×100 + 70 ×100 + 3 ×100

= (5 ×100) ×100 + (7 ×10) ×100

16. Use the expanded form with regrouping (see Set A,

+ 3 ×

Exercise 16) to perform the following subtractions.

100

a. 455 − 278

= 5 × 100

(

× 100) + 7 × 1

( 0 × 100)

b. 503 − 147

+ 3 × 100

c. 3426 − 652

= 5 × 10,000 + 7 × 10000 + 300

= ,

+

+

17. Use the cashiers algorithm (see Set A, Exercise 17) to

50 000

7000

300

describe what the cashier would say to the customer in

= 57,300

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7/30/2013 2:53:50 PM - Section 4.2 Written Algorithms for Whole-Number Operations 161

24. Solve the following problems using (i) the lattice method for

31. Without using the divide key, use a calculator to find

multiplication and (ii) an intermediate algorithm.

the quotient and remainder for each of the following

a. 237 × 48 b. 617 × 896

problems.

a. )

3 39 b. )

8 89 c. 6)75

25. Use the left-to-right multiplication algorithm shown in Set

A, Exercise 25 to find the following products.

d. Describe how the calculator was used to find the re-

mainders.

a. 276 × 43 b. 768 × 891

32. Find the quotient and remainder to the problems in parts

26. Use the Russian peasant algorithm for multiplication

a, b, and c using a calculator and the method illustrated

shown in Set A, Exercise 26 to find the following products.

in Set A, Exercise 32. Describe how the calculator was

a. 44 × 83 b. 31 × 54

used.

27. Use the finger multiplication method described in Set A,

a. 18 114

,

÷ 37

Exercise 27 to find the following products. Explain how

b. 381,271 ÷ 47

the algorithm was used in each case.

c. 9,346,870 ÷ 349

a. 9 × 6 b. 7 × 9 c. 8 × 8

d. Does this method always work? Explain.

28. Use the duplication algorithm for multiplication shown in

33. Sketch how to use base ten blocks to model the operation

Set A, Exercise 28 to find the products in parts a, b, and c.

673 ÷ 4.

a. 14 × 43

b. 21 × 67

c. 43 × 73

34. When finding the quotient for 527 ÷ 3 using the standard

d. Which property justifies the algorithm?

division algorithm, the first few steps are shown here.

29. Use the scaffold method to find the following quotients.

1

Show how the scaffold method was used.

)3527

a. 749 ÷ 22

b. 3251 ÷ 14

− 3

2

30. a. Use the method described in Set A, Exercise 30 to find the

quotient 63 ÷ 9 and describe how the method was used.

Explain how the process of subtracting 3 from 5 is

b. Which approach to division does this method illustrate?

modeled with base ten blocks.

PROBLEMS

35. Peter, Jeff, and John each perform subtraction incorrectly

If each letter in this message represents a different digit,

as follows:

how much MONEY (in cents) is he asking for?

3

9

b. The father, considering the request, decides to send

4

10 13

4

10 13

Peter: 503 Jeff: 5 0 3 John: 5 0 3

some money along with some important advice.

− 269

− 2 6 9

− 2 6 9

SAVE

366

2 4 4

1 3 4

+ MORE

How would you explain their mistakes to each of them?

MONEY

36. Let A, B,C, and D represent four consecutive whole

However, the father had misplaced the request and

numbers. Find the values for A, B,C, and D if the four boxes

could not recall the amount. If he sent the largest

are replaced with

amount of MONEY (in cents) represented by this sum,

A, B, C, and D in an unknown order.

how much did the college student receive?

39. Consider the sums

1 + 11 =

1 + 11 + 111 =

1 + 11 + 111 + 1111 =

37. Place the digits 3, 5, 6, 2, 4, 8 in the boxes to obtain the

a. What is the pattern?

following differences.

b. How many addends are there the first time the pattern

a. The greatest difference

b. The least difference

no longer works?

40. Select any three-digit number whose first and third

digits are different. Reverse the digits and find the

difference between the two numbers. By knowing only the

38. a. A college student, short of funds and in desperate need,

hundreds digit in this difference, it is possible to deter-

writes the following note to his father:

mine the other two digits. How? Explain how the trick

works.

SEND

+

41. Choose any four-digit number, reverse its digits, and add

MORE

the two numbers. Is the sum divisible by 11? Will this

MONEY

always be true?

c04.indd 161

7/30/2013 2:53:54 PM - 162 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

42. Select any number larger than 100 and multiply it

45. Show that a perfect square is obtained by adding 1 to the

by 9. Select one of the digits of this result as the “missing

product of two whole numbers that differ by 2. For exam-

digit.” Find the sum of the remaining digits. Continue

ple, 8 10 + 1 = 92

(

)

, 11 13 1 122

×

+ =

, etc.

adding digits in resulting sums until you have a one-digit

number. Subtract that number from 9. Is that your miss-

Analyzing Student Thinking

ing digit? Try it again with another number. Determine

which missing digits this procedure will find.

46. A student’s father was curious about why his daughter

was learning addition using base ten pieces when he

43. Following are some division exercises done by students.

had not learned it that way. How should you respond?

47. Is it okay for a student to use the lattice method for addi-

tion or must all of your students eventually use the stan-

dard algorithm? Explain.

48. When finding the difference 123 − 67, why might the sub-

tract-from-the-base algorithm be easier for some

students?

49. What is one advantage and one disadvantage of having

your students use base ten blocks to model the standard

multiplication algorithm? Explain.

50. After two subtraction algorithms were shared in class,

Nicholas asked why they needed two different methods

and which of them he was supposed to learn. How should

you respond?

51. A student found 4005 − 37 as follows:

4005

− 37

2078

Determine the error pattern and tell what each student will

Diagnose what the student was doing wrong.

get for the last problem.

What instructional procedures might you use to help

52. Natasha asked what the advantages of the lattice method

each of these students?

are as compared to the standard multiplication algorithm.

How would you respond?

44. Three businesswomen traveling together stopped at a motel

53. Andres found the product of 46 and 53 horizontally as

to get a room for the night. They were charged $30 for the

follows: (40 + 6)(50 + 3) = 40 × 50 + 40 × 3 + 6 × 50 +

room and agreed to split the cost equally. The manager

6 × 3 = 2000 + 120 + 300 + 18 = 2438. Will this method

later realized that he had overcharged them by $5. He gave

always work? Explain.

the refund to his son to deliver to the women. This smart

son, realizing it would be difficult to split $5 equally, gave

54. When using the standard division algorithm to solve

the women $3 and kept $2 for himself. Thus, it cost each

621 ÷ 3, Dylan found the quotient to be 27 instead of 207.

woman $9 for the room. Therefore, they spent $27 for the

How could base ten pieces be used to clarify this misun-

room plus the $2 “tip.” What happened to the other dollar?

derstanding?

ALGORITHMS IN OTHER BASES

All of the algorithms you have learned can be used in any base. In this section we

apply the algorithms in base five and then let you try them in other bases in the

problem set. The purpose for doing this is to help you see where, how, and why your

future students might have difficulties in learning the standard algorithms.

Use any of the algorithms learned in Section 4.2 and your knowledge of base five

numbersto solve the following problems. Compare and contrast your strategies with

those of your peers.

132

241

34

five

five

five

+

3five)

433

− 143

433five

five

five

× 24five

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7/30/2013 2:53:56 PM - Section 4.3 Algorithms in Other Bases 163

Operations in Base Five

Addition Addition in base five is facilitated using the thinking strategies in base

five. These thinking strategies may be aided by referring to the base five number line

shown in Figure 4.13, in which 2

+

=

five

4five

11five is illustrated. This number line also

provides a representation of counting in base five. (All numerals on the number line

are written in base five with the subscripts omitted.)

Figure 4.13

Find 342

+

five

134five using the following methods.

a. Lattice method

b. Intermediate algorithm

c. Standard algorithm

S O L U T I O N

a. Lattice Method b. Intermediate c. Standard

Algorithm Algorithm

■

Be sure to use thinking strategies when adding in base five. It is helpful to find sums

to five first; for example, think of 4

+

+

+

=

+

+

five

3five as 4five

1

( five

2five )

(4five

1five )

2

=

five

12five and so on.

Check for Understanding: Exercise/Problem Set A #1–6

✔

Subtraction There are two ways to apply a subtraction algorithm successfully.

One is to know the addition facts table forward and backward. The other is to use the

missing-addend approach repeatedly. For example, to find 12

−

five

4five, think “What

number plus 4five is 12five?” To answer this, one could count up “10five, 11five, 12five.” Thus

12

−

=

five

4five

3five (one for each of 10five, 1

1five, and 12five). A base five number line can

also illustrate 12

−

=

five

4five

3five. The missing-addend approach is illustrated in Figure

4.14 and the take-away approach is illustrated in Figure 4.15.

Figure 4.14

Figure 4.15

c04.indd 163

7/30/2013 2:53:58 PM - 164 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

A copy of the addition table for base five is included in Figure 4.16 to assist you in

working through Example 4.11. (All numerals in Figure 4.16 are written in base five

with the subscripts omitted.)

Figure 4.16

Calculate 412

−

five

143five using the following methods.

S O L U T I O N

a. Standard Algorithm

b. Subtract-from-the-Base

Think:

Think:

■

Notice that to do subtraction in base five using the subtract-from-the-base

algorithm, you only need to know two addition combinations to five, namely 1

+

five

4

=

five

10fiveand 2

+

=

five

3five

10five. These two, in turn, lead to the four subtraction facts

you need to know, namely 10

−

=

−

=

,

−

=

five

4five

1five, 10five 1five

4five 1

0five

3five

2five, and

10

−

=

five

2five

3five.

Check for Understanding: Exercise/Problem Set A #7–11

✔

Algebraic Reasoning

Multiplication To perform multiplication efficiently, one must know the multipli-

To find 3 ¥ 4 in base five, one

cation facts. The multiplication facts for base five are displayed in Figure 4.17. The

must solve 12 = 5a + b where

entries in this multiplication table can be visualized by referring to the number line

a, b < 5. Although this is typically

shown in Figure 4.18.

done mentally, algebraic reason-

ing is required to complete this

base five multiplication table.

Figure 4.17

The number line includes a representation of 4

×

=

five

3five

22five using the repeated-

addition approach. (All numerals in the multiplication table and number line are

written in base five with subscripts omitted.)

c04.indd 164

7/30/2013 2:54:01 PM - Section 4.3 Algorithms in Other Bases 165

Figure 4.18

Calculate 43

×

five

123five using the following methods.

S O L U T I O N

a. Lattice Method b. Intermediate c. Standard

Algorithm Algorithm

■

Notice how efficient the lattice method is. Also, instead of using the multiplica-

tion table, you could find single-digit products using repeated addition and thinking

strategies. For example, you could find 4

×

five

2five as follows.

4

×

=

×

=

+

=

+ ( +

five

2five

2five

4five

4five

4five

4five

1 3)five

= (4 + 1) + 3 =

fiive

five

1 f

3 ive

Although this may look like it would take a lot of time, it would go quickly mentally,

especially if you imagine base five pieces.

Check for Understanding: Exercise/Problem Set A #12–14

✔

Division As shown in Section 3.2, division can be displayed using a number

line. Figure 4.19 displays 23

÷

five

4five using the repeated subtraction approach.

(All numerals below the number line without subscripts are assumed to be base

five.)

Figure 4.19

Figure 4.19 shows that 23

÷

five

4five has a quotient of 3five with a remainder of 1five.

Doing long division in other bases points out the difficulties of learning this algo-

rithm and, especially, the need to become proficient at approximating multiples of

numbers.

c04.indd 165

7/30/2013 2:54:02 PM - 166 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

Find the quotient and remainder for 1443

÷

five

34five using the

following methods.

S O L U T I O N

a. Scaffold Method b. Standard Long-Division Algorithm

34

23five

five )1443five

34five)1443

−

five

1230 20five

− 123 (2 × 34 )

213

five

five

213

− 212 3five

− 212 (

×

five

3

34five )

1 23five

1

Quotient: 23five

Remainder: 1five

In the scaffold method, the fi rst estimate, namely 20five, was selected because

2

×

five

3five is 11five, which is less than 14five.

■

In summary, doing computations in other number bases can provide insights into

how computational difficulties arise, whereas our own familiarity and competence with

our algorithms in base ten tend to mask the trouble spots that children face.

Check for Understanding: Exercise/Problem Set A #15–18

✔

George Parker Bidder (1806–1878), who lived in Devonshire, England, was blessed with

an incredible memory as well as being a calculating prodigy. When he was 10, he was

read a number backward and he immediately gave the number back in its correct form.

An hour later, he repeated the original number, which was

2,563,721,987,653,461,598,746,231,905,607,541,127,975,231.

Furthermore, his brother memorized the entire Bible and could give the chapter and

verse of any quoted text. Also, one of Bidder’s sons could multiply 15-digit numbers in

©Ron Bagwell

his head.

EXERCISE/PROBLEM SET A

EXERCISES

1. Create a number line to illustrate the following operations.

4. Write out a base six addition table. Use your table and the

a. 12

+

+

intermediate algorithm to compute the following sums.

four

3four b. 4six

15six

a. 32

+

+

six

23 b. 45six

34

2. Use multibase blocks to illustrate each of the following

six

six

c. 145

+

+

six

541 d. 355six

211

base four addition problems. The Chapter 4 eManipulative

six

six

activity Multibase Blocks on our Web site may be helpful in

5. Use the lattice method to compute the following sums.

the solution process.

a. 46

+

+

seven

13seven

b. 13four

23four

a. 1

+

+

four

2four b. 11four

23four

6. Use the standard algorithm to compute the following sums.

c. 212

+

+

four

113four

d. 2023four

3330four

a. 213

+

+

five

433five

b. 716eight

657eight

3. Use bundling sticks or chip abacus for the appropriate base

to illustrate the following sums.

7. Create a number line to illustrate the following

operations.

a. 41

+

+

six

33six b. 555seven

66seven

a. 12

−

−

four

3

b. 14six

5

c. 3030

+

four

six

four

322four

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7/30/2013 2:54:05 PM - Section 4.3 Algorithms in Other Bases 167

8. Use multibase blocks to illustrate each of the following

14. Solve the following problems using the lattice method, an

base four subtraction problems. The Chapter 4 eManip-

intermediate algorithm, and the standard algorithm.

ulative activity Multibase Blocks on our Web site may be

a. 31four b. 43

c. 312four

helpful in the solution process.

five

× 2

× 3

× 3

a. 31

−

−

four

five

four

four

12four

b. 123four

32four

c. 1102

−

four

333four

15. Create a number line to illustrate the following

9. Use bundling sticks or chip abacus for the appropriate

operations.

base to illustrate the following differences.

a. 22

÷

÷

four

3four

b. 24six

5six

a. 41

−

−

six

33six

b. 555seven

66seven

16. Use the scaffold method of division to compute the

c. 3030

−

four

102four

following numbers. (Hint: Write out a multiplication table

10. Solve the following problems using both the standard

in the appropriate base to help you out.)

algorithm and the subtract-from-the-base algorithm.

a. 22

÷

six

2six

a. 36

−

−

eight

17eight

b. 1010two

101two

b. 4044

÷

seven

51seven

c. 32

−

four

13four

c. 13002

÷

four

33four

11. Find 10201

−

three

2122three using “adding the complement.”

17. Solve the following problems using the missing-factor defi-

What is the complement of a base three number?

nition of division. (Hint: Use a multiplication table for the

12. Create a number line to illustrate the following operations.

appropriate base.)

a. 2

×

×

a. 21

÷

÷

÷

four

3

b. 23six

3

c. 24eight

5

four

13four b. 3six

5six

four

six

eight

13. Sketch the rectangular array of multibase pieces and find

18. Sketch how to use base seven blocks to illustrate the oper-

the products of the following problems.

ation 534

÷

seven

4seven.

a. 23

×

×

four

3four

b. 23five

12five

PROBLEMS

19. 345 _____ + 122 _____ = 511 _____ is an addition problem

22. What single number can be added separately to 100 and

done in base _____.

164 to make them both perfect square numbers?

20. Jane has $10 more than Bill, Bill has $17 more than Tricia,

23. What is wrong with the following problem in base four?

and Tricia has $21 more than Steve. If the total amount of

Explain.

all their money is $115, how much money does each have?

1022four

21. Without using a calculator, determine which of the five

+

four

413

numbers is a perfect square. There is exactly one.

24. Write out the steps you would go through to mentally

39,037,066,087

subtract 234five from 421five using the standard subtraction

39,037,066,084

algorithm.

39,037,066,082

38,336,073,623

38,

,

414

,

432 028

EXERCISE/PROBLEM SET B

EXERCISES

1. Create a number line to illustrate the following operations.

3. Use bundling sticks or chip abacus for the appropriate base

a. 12

+

+

to illustrate the following problems.

seven

6seven b. 4eight

16eight

a. 32

+

four

33four

2. Use multibase blocks to illustrate each of the following base

b. 54

+

eight

55eight

five addition problems. The Chapter 4 eManipulative activ-

c. 265

+

nine

566

ity Multibase Blocks on our Web site may be helpful in the

nine

solution process.

4. Use an intermediate algorithm to compute the following

a. 12

+

+

five

20five b. 24five

13five

sums.

c. 342

+

+

five

13five d. 2134five

1330five

a. 78

+

+

nine

65nine b. TEtwelve

EEtwelve

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7/30/2013 2:54:09 PM - 168 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

5. Use the lattice method to compute the following sums.

12. Create a number line to illustrate the following operations.

a. 54

+

×

×

six

34six

a. 2seven

13seven b. 3eight

5eight

b. 1 8

T

+

eleven

499eleven

13. Sketch the rectangular array of multibase pieces and find

6. Use the standard algorithm to compute the following sums.

the products of the following problems.

a. 79

+

a. 32

×

×

six

13 b. 34seven

21

twelve

85twelve

six

seven

b. T

+

eleven

1

99eleven

14. Solve the following problems using the lattice method, an

7. Create a number line to illustrate the following operations.

intermediate algorithm, and the standard algorithm.

a. 12

−

−

seven

3seven b. 14eight

5eight

a. 11011two b. 43twelve c. 66seven

×

×

×

twelve

23

66seven

8. Use multibase blocks to illustrate each of the following base

two

1101

five subtraction problems. The Chapter 4 eManipulative

15. Create a number line to illustrate the following operations.

activity Multibase Blocks on our Web site may be helpful in

a. 22

÷

÷

seven

3

b. 23eight

5

the solution process.

seven

eight

a. 32

−

−

five

14five

b. 214five

32five

16. Use the scaffold method of division to compute the fol-

c. 301

−

five

243five

lowing numbers. (Hint: Write out a multiplication table in

the appropriate base to help you out.)

9. Use bundling sticks or chip abacus for the appropriate

a. 14

÷

÷

five

3 b. 2134six

14

base to illustrate the following problems.

five

six

÷

a. 123

−

−

c. 61245seven

354seven

five

24five b. 253eight

76eight

c. 1001

−

two

110two

17. Solve the following problems using the missing-factor defi-

10. Solve the following problems using both the standard

nition of division. (Hint: Use a multiplication table for the

algorithm and the subtract-from-the-base algorithm.

appropriate base.)

a. 45

−

−

a. 42

÷

÷

÷

seven

5

b. 62nine

7

c. 92

E

seven

36seven b. 99twelve

7 twelve

T

seven

nine

twelve

twelve

c. 100

−

eight

77eight

18. Sketch how to use base four blocks to illustrate the opera-

11. Find 1001010

−

tion 3021

÷

four

11

two

111001two by “adding the complement.”

four.

PROBLEMS

19. 320 _____ − 42 _____ = 256 _____ is a correct subtraction

1022four

problem in what base?

−

four

413

20. Betty has three times as much money as her brother Tom.

Is this a reasonable base four subtraction problem?

If each of them spends $1.50 to see a movie, Betty will have

Explain.

nine times as much money left over as Tom. How much

money does each have before going to the movie?

26. When doing a multiplication problem in base five,

21. To stimulate his son in the pursuit of mathematics, a math

Veronica uses the intermediate algorithm. Geoff doesn’t

professor offered to pay his son $8 for every equation cor-

want to use it because it has too many steps. How should

rectly solved and to fine him $5 for every incorrect solu-

you respond in this situation?

tion. At the end of 26 problems, neither owed any money

27. One of your students says that she is having difficulty

to the other. How many did the boy solve correctly?

doing division in base five using the standard algorithm.

22. Prove: If n is a whole number and n2 is odd, then n is odd.

Suggest how you might help the student.

(Hint: Use indirect reasoning. Either n is even or it is odd.

28. When Sherry did the subtraction problem 342

−

six

135

Assume that n is even and reach a contradiction.)

six ,

she “borrowed” from the 4 to get 12 in the ones columns

as shown.

Analyzing Student Thinking

3

3 4 1 2

23. Antolina asks why they have to be doing algorithms in

six

other bases. How should you respond?

− 1 3 5six

24. Ralph asks if it is okay to use the base five addition table

when subtracting in base five or does he have to memorize

She got confused when she subtracted 12 − 5 and got 7

the table first?

because the digit “7” is not in base six. However, when

she converted 7 to 11six she didn’t know what to do with

25. You ask a student to make up a subtraction problem in

both ones. Where is Sherry’s misunderstanding and how

base four. The student writes the following:

would you help her clarify her thinking?

c04.indd 168

7/30/2013 2:54:15 PM - End of Chapter Material 169

END OF CHAPTER MATERIAL

The whole numbers 1 through 9 can be used once, each being

arranged in a 3 × 3 square array so that the sum of the numbers

in each of the rows, columns, and diagonals is 15. Show that 1

cannot be in one of the corners.

Strategy: Use Indirect Reasoning

Additional Problems Where the Strategy “Use Indirect Rea-

soning” Is Useful

Suppose that 1 could be in a corner as shown in the following

figure. Each row, column, and diagonal containing 1 must

2 +

+

have a sum of 15. This means that there must be three pairs of

1. If x represents a whole number and x

2x 1 is even,

numbers among 2 through 9 whose sum is 14. Hence the sum

prove that x cannot be even.

of all three pairs is 42. However, the largest six numbers—9, 8,

2. If n is a whole number and n4 is even, then prove that n is

7, 6, 5, and 4—have a sum of 39, so that it is impossible to find

even.

three pairs whose sum is 14. Therefore, it is impossible to have

3. For whole numbers x and y, if x2

y2

+ is a square, then prove

1 in a corner.

that x and y cannot both be odd.

Grace Brewster Murray

John Kemeny (1926–1992)

Hopper (1906–1992)

John Kemeny was born in

Grace Brewster Murray Hop-

Budapest, Hungary. His

per recalled that as a young

family immigrated to the

girl, she disassembled one

United States, and he at-

of her family’s alarm clocks.

tended high school in New

When she was unable to reas-

York City. He entered

© Bettmann/CORBIS

semble the parts, she disman-

Photo by John Prieto/

The Denver Post via Getty

Images

Princeton where he stud-

tled another clock to see how those parts fi t together.

ied mathematics and philosophy. As an undergraduate,

This process continued until she had seven clocks in

he took a year off to work on the Manhattan Project.

pieces. This attitude of exploration foreshadowed her

While he and Tom Kurtz were teaching in the mathemat-

innovations in computer programming. Trained in

ics department at Dartmouth in the mid-1960s, they cre-

mathematics, Hopper worked with some of the fi rst

ated BASIC, an elementary computer language. Keme-

computers and invented the business language CO-

ny made Dartmouth a leader in the educational uses of

BOL. After World War II she was active in the Navy,

computers in his era. Kemeny’s fi rst faculty position was

where her experience in computing and programming

in philosophy. “The only good job offer I got was from

began. In 1985 she was promoted to the rank of rear

the Princeton philosophy department,” he said, explain-

admiral. Known for her common sense and spirit of

ing that his degree was in logic, and he studied philoso-

invention, she kept a clock on her desk that ran (and

phy as a hobby in college. Kemeny had the distinction of

kept time) counterclockwise. Whenever someone ar-

having served as Einstein’s mathematical assistant while

gued that a job must be done in the traditional man-

still a young graduate student at Princeton. Einstein’s as-

ner, she just pointed to the clock. Also, to encourage

sistants were always mathematicians. Contrary to popu-

risk taking, she once said, “I do have a maxim—I

lar belief, Einstein did need help in mathematics. He was

teach it to all youngsters: A ship in port is safe, but

very good at it, but he was not an up-to-date research

that’s not what ships are built for.”

mathematician.

c04.indd 169

7/30/2013 2:54:21 PM - 170 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy

reference.

Mental Math, Estimation, and Calculators

Vocabulary/Notation

Right distributivity 131

Front-end estimation 134

Round a 5 up 136

Compatible numbers 131

Range estimation 134

Round to the nearest even 136

Compensation 132

Range 134

Round to compatible numbers 136

Additive compensation 132

One-column front-end estimation

Arithmetic logic 138

Equal additions method 132

method 135

Algebraic logic 138

Multiplicative compensation 132

Two-column front-end estimation

Parentheses 138

Left-to-right methods 132

method 135

Constant function 139

Powers of 10 132

Front-end with adjustment 135

Exponent keys 139

Special factors 132

Round up/down 136

Memory functions 139

Computational estimation 134

Truncate 136

Scientific notation 139

Exercises

1. Calculate the following mentally, and name the

4. Insert parentheses (wherever necessary) to produce the indi-

property(ies) you used.

cated results.

a. 97 + 78

b. 267 ÷ 3

a. 3 + 7 × 5 = 38

c. (16 × 7) × 25

d. 16 × 9 − 6 × 9

b. 7 × 5 − 2 + 3 = 24

e. 92 × 15

f. 17 × 99

c. 15 + 48 ÷ 3 × 4 = 19

g. 720 ÷ 5

h. 81 − 39

5. Fill in the following without using a calculator.

2. Estimate using the techniques given.

a. 5 + 2 × 3 =

a. Range: 157 + 371

b. 8 − 6 ÷ 3 =

b. One-column front-end: 847 × 989

c. Front-end with adjustment: 753 + 639

c. 2 × ( 3 + 2 ) =

d. Compatible numbers: 23 × 56

d. 2 yx 3 =

3. Round as indicated.

e. 3 STO 7 + RCL =

a. Up to the nearest 100: 47,943

f. 3 × 5 = STO 8 ÷ 2 = + RCL =

b. To the nearest 10: 4751

c. Down to the nearest 10: 576

Written Algorithms for Whole-Number Operations

Vocabulary/Notation

Algorithm 145

Subtract-from-the-base algorithm 148

Long division using base ten

Standard addition algorithm 145

Standard multiplication

blocks 151

Lattice method for addition 146

algorithm 149

Scaffold method for division 152

Standard subtraction algorithm 147

Lattice method for multiplication 150

Standard algorithm for division 154

Exercises

1. Find 837 + 145 using

3. Find 72 × 43 using

a. the lattice method.

a. an intermediate algorithm.

b. an intermediate algorithm.

b. the standard algorithm.

c. the standard algorithm.

c. the lattice method.

2. Find 451 − 279 using

4. Find 253 ÷ 27 using

a. the standard algorithm.

a. the scaffold method.

b. an intermediate algorithm.

b. a nonstandard algorithm.

c. the standard algorithm. d. a calculator.

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7/30/2013 2:54:24 PM - Chapter Test 171

Algorithms in Other Bases

Vocabulary/Notation

Operations in base five 163

Exercises

1. Find 413

+

×

six

254six using

3. Find 21four

32four using

a. the lattice method.

a. an intermediate algorithm.

b. an intermediate algorithm.

b. the standard algorithm.

c. the standard algorithm.

c. the lattice method.

2. Find 234

−

÷

seven

65seven using

4. Find 213five

41five using

a. the standard algorithm.

a. repeated subtraction

b. the scaffold method.

b. a nonstandard algorithm.

c. the standard algorithm.

CHAPTER TEST

Knowledge

8. In the standard multiplication algorithm, why do we “shift

over one to the left” as illustrated in the 642 in the follow-

1. True or false?

ing problem?

a. An algorithm is a technique that is used exclusively for

doing algebra.

321

b. Intermediate algorithms are helpful because they require

× 23

less writing than their corresponding standard algo-

963

rithms.

+ 642

c. There is only one computational algorithm for each of

the four operations: addition, subtraction, multiplication,

9. To check an addition problem where one has “added down

and division.

the column,” one can “add up the column.” Which prop-

d. Approximating answers to computations by rounding is

erties guarantee that the sum should be the same in both

useful because it increases the speed of computation.

directions?

Skill

10. Sketch how a chip abacus could be used to perform the

following operation.

2. Compute each of the following using an intermediate algo-

rithm.

374 + 267

a. 376 + 594

b. 56 × 73

11. Use an appropriate intermediate algorithm to compute the

3. Compute the following using the lattice method.

following quotient and remainder.

a. 568 + 493

b. 37 × 196

7261 ÷ 43

4. Compute the following mentally. Then, explain how you

did it.

12. Sketch how base ten blocks could be used to find the quo-

a. 54 + 93 + 16 + 47

b. 9223 − 1998

tient and remainder of the following.

c. 3497 − 1362

d. 25 × 52

538 ÷ 4

5. Find 7496 ÷ 32 using the standard division algorithm, and

check your results using a calculator.

13. Sketch how base four blocks could be used to find the fol-

lowing difference.

6. Estimate the following using (i) one-column front-end, (ii)

range estimation, (iii) front-end with adjustment, and (iv)

32

−

four

13four

rounding to the nearest 100.

14. Sketch how a chip abacus could be used to find the fol-

a. 546 + 971 + 837 + 320

lowing sum.

b. 731 × 589

278

+

nine

37

Understanding

nine

15. Sketch how base ten blocks could be used to model the

7. Compute 32 × 21 using expanded form; that is, continue the

following operations and explain how the manipulations

following.

of the blocks relate to the standard algorithm.

32 × 21 = 3

( 0 + 2)(20 +1) = ...

a. 357 + 46 b. 253 − 68 c. 789 ÷ 5

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7/30/2013 2:54:26 PM - 172 Chapter 4 Whole-Number Computation—Mental, Electronic, and Written

16. Compute 492 × 37 using an intermediate algorithm and a

Problem Solving/Application

standard algorithm. Explain how the distributive property

20. If each different letter represents a different digit, find

is used in each of these algorithms.

the number “HE” such that (HE)2 = SHE. (Note: “HE”

17. State some of the advantages and disadvantages of the

means 10 ¥ H + E due to place value.)

standard algorithm versus the lattice method for multipli-

21. Find values for a, b, and c in the lattice multiplication

cation.

problem shown. Also find the product.

18. State some of the advantages and disadvantages of the

standard subtraction algorithm versus the subtract-from

the-base algorithm.

19. Show how to find 17 × 23 on the following grid paper and

explain how the solution on the grid paper can be related

to the intermediate algorithm for multiplication.

22. Find digits represented by A, B, C, and D so that the fol-

lowing operation is correct.

ABA

+ BAB

CDDC

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7/30/2013 2:54:27 PM - c04.indd 173

7/30/2013 2:54:27 PM - C H A P T E R

5 NUMBER THEORY

Famous Unsolved Problems

Number theory provides a rich source of intrigu- The following list contains several such problems that

ing problems. Interestingly, many problems in

are still unsolved. If you can solve any of them, you will

number theory are easily understood, but still

surely become famous, at least among mathematicians.

have never been solved. Most of these problems are

statements or conjectures that have never been proven

1. Goldbach’s conjecture. Every even number greater than

right or wrong. The most famous “unsolved” problem,

4 can be expressed as the sum of two odd primes. For

known as Fermat’s Last Theorem, is named after Pierre

example, 6 = 3 + 3, 8 = 3 + 5, 1

0 = 5 + 5, 1

2 = 5 + 7, and

de Fermat who is pictured below. It states “There are no

so on. It is interesting to note that if Goldbach’s conjec-

nonzero whole numbers a, b

, c

, where an

bn

cn

+

= , for n

ture is true, then every odd number greater than 7 can

a whole number greater than two.”

be written as the sum of three odd primes.

2. Twin prime conjecture. There is an infinite number of

pairs of primes whose difference is two. For example, (3,

5), (5, 7), and (11, 13) are such prime pairs. Notice that

3, 5, and 7 are three prime numbers where 5 − 3 = 2 and

7 − 5 = 2. It can easily be shown that this is the only such

triple of primes.

3. Odd perfect number conjecture. There is no odd perfect

number; that is, there is no odd number that is the sum

of its proper factors (factors less than the number). For

example, 6 = 1 + 2 + 3; hence 6 is a perfect number. It

has been shown that the even perfect numbers are all of

the form 2p − 1 (2 p − 1), where 2p −1 is a prime.

© Lebrecht Authors/Lebrecht

Music & Arts/Corbis

4. Ulam’s conjecture. If a nonzero whole number is even,

Fermat left a note in the margin of a book saying that he

divide it by 2. If a nonzero whole number is odd, multi-

did not have room to write up a proof of what is now called

ply it by 3 and add 1. If this process is applied repeat-

Fermat’s Last Theorem. However, it remained an unsolved

edly to each answer, eventually you will arrive at 1. For

problem for over 350 years because mathematicians were

example, the number 7 yields this sequence of numbers:

unable to prove it. In 1993, Andrew Wiles, an English math-

7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

ematician on the Princeton faculty, presented a “proof ” at

Interestingly, there is a whole number less than 30 that

a conference at Cambridge University. However, there was

requires at least 100 steps before it arrives at 1. It can

a hole in his proof. Happily, Wiles and Richard Taylor

be seen that 2n requires n steps to arrive at 1. Hence one

produced a valid proof in 1995, which followed from work

can find numbers with as many steps (finitely many) as

done by Serre, Mazur, and Ribet beginning in 1985.

one wishes.

174

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7/30/2013 2:55:53 PM - Problem-Solving

Use Properties of Numbers

Strategies

Understanding the intrinsic nature of numbers is often helpful in solving problems.

1. Guess and Test

For example, knowing that the sum of two even numbers is even and that an odd

2. Draw a Picture

number squared is odd may simplify checking some computations. The solution of

the initial problem will seem to be impossible to a naive problem solver who attempts

3. Use a Variable

to solve it using, say, the Guess and Test strategy. On the other hand, the solution is

4. Look for a

immediate for one who understands the concept of divisibility of numbers.

Pattern

5. Make a List

Initial Problem

6. Solve a Simpler

A major fast-food chain held a contest to promote sales. With each purchase a cus-

Problem

tomer was given a card with a whole number less than 100 on it. A $100 prize was

given to any person who presented cards whose numbers totaled 100. The following

7. Draw a Diagram

are several typical cards. Can you find a winning combination?

8. Use Direct

3

9

12

15

18

27

51

72

84

Reasoning

Can you suggest how the contest could be structured so that there would be at most

9. Use Indirect

1000 winners throughout the country? (Hint: What whole number divides evenly in

Reasoning

each sum?)

10. Use Properties of

Numbers

Clues

The Use Properties of Numbers strategy may be appropriate when

r Special types of numbers, such as odds, evens, primes, and so on, are involved.

r A problem can be simplifi ed by using certain properties.

r A problem involves lots of computation.

A solution of this Initial Problem is on page 202.

175

c05.indd 175

7/30/2013 2:55:53 PM - AUTHOR

I N T R O D U C T I O N

Number theory is a branch of mathematics that is devoted primarily to the study of the set of counting

numbers. The aspects of the counting numbers central to the elementary curriculum that are covered

in this chapter include primes, composites, and divisibility tests as well as the notions of greatest com-

mon factor and least common multiple. Many of these topics are useful in other areas of mathematics

WALK-THROUGH such as fractions (Chapter 6) and algebra. Being able to find common factors and greatest common

factors between pairs of numbers is useful when simplifying fractions. Understanding least common

multiples is useful for finding a common denominator when adding and subtracting fractions. The principles of factor-

ing numbers studied in this chapter generalize into similar situations in algebra when factoring expressions.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-2–NUMBER AND OPERATIONS

Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and

decomposing numbers.

r GRADES 3-5–NUMBER AND OPERATIONS

Recognize equivalent representations for the same number and generate them by decomposing and composing num-

bers.

Describe classes of numbers according to characteristics such as the nature of their factors.

r GRADES 6-8–NUMBER AND OPERATIONS

Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.

Key Concepts from the NCTM Curriculum Focal Points

r GRADE 3: Developing understandings of multiplication and division and strategies for basic multiplication facts

and related division facts.

r GRADE 4: Developing quick recall of multiplication facts and related division facts and fluency with whole-number

multiplication.

Key Concepts from the Common Core State Standards for Mathematics

r GRADE 4: Gain familiarity with factors and multiples.

r GRADE 6: Find common factors and multiples. Apply and extend previous understandings of arithmetic to alge-

braic expressions.

176

c05.indd 176

7/30/2013 2:55:54 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 177

PRIMES, COMPOSITES, AND TESTS FOR DIVISIBILITY

On a piece of paper, sketch all of the possible rectangles that can be

made up of exactly 10 squares. An example of a rectangle consisting of

6 squares is shown at the right.

Repeat these sketches for 11 squares and 12 squares. How are the dimensions of the rectangles related

to the number of squares? For each number of squares, the number of rectangles that can be con-

structed varies. Why?

Primes and Composites

Prime numbers are building blocks for the counting numbers 1, 2, 3, 4, . . . .

D E F I N I T I O N 5 . 1

Prime and Composite Numbers

A counting number with exactly two different factors (itself and 1) is called a prime

Common Core – Grade 4

Determine whether a given whole

number, or a prime. A counting number with more than two factors is called a

number in the range 1–100 is

composite number, or a composite.

prime or composite.

NCTM Standard

For example, 2, 3, 5, 7, 11 are primes, since they have only themselves and 1 as fac-

All students should use factors,

tors; 4, 6, 8, 9, 10 are composites, since they each have more than two factors; 1 is

multiples, prime factorization,

neither prime nor composite, since 1 is its only factor.

and relatively prime numbers to

An algorithm used to find primes is called the Sieve of Eratosthenes (Figure 5.1).

solve problems.

Reflection from Research

Talking, drawing, and writing

about types of numbers such as

factors, primes, and composites,

allows students to explore and

explain their ideas about general-

izations and patterns dealing with

these types of numbers (Whitin &

Whitin, 2002).

Figure 5.1

The directions for using this procedure are as follows: Skip the number 1. Circle 2

and cross out every second number after 2. Circle 3 and cross out every third number

after 3 (even if it had been crossed out before). Continue this procedure with 5, 7,

and each succeeding number that is not crossed out. The circled numbers will be the

primes and the crossed-out numbers will be the composites, since prime factors cause

them to be crossed out. Again, notice that 1 is neither prime nor composite.

Composite numbers have more than two factors and can be expressed as the prod-

uct of two smaller numbers. Figure 5.2 shows how a composite can be expressed as

the product of smaller numbers using factor trees.

Figure 5.2

c05.indd 177

7/30/2013 2:55:54 PM - 178

Chapter 5 Number Theory

Notice that 60 was expressed as the product of two factors in several different

ways. However, when we kept factoring until we reached primes, each method led us

to the same prime factorization, namely 60 = 2 ¥ 2 ¥ 3 ¥ 5. This example illustrates the

following important result.

T H E O R E M 5 . 1

Fundamental Theorem of Arithmetic

Each composite number can be expressed as the product of primes in exactly one

way (except for the order of the factors).

Algebraic Reasoning

To factor an expression such as

Express each number as the product of primes.

x2 − 5x − 24 it is important to

a. 84 b. 180 c. 324

be able to factor 24 into pairs of

factors.

S O L U T I O N

a. 84

4

21

2 ¥ 2 ¥ 3 ¥ 7

22

= ×

=

=

¥ 3 ¥ 7

b. 180

10 18

2 ¥ 5 ¥ 2 ¥ 3 ¥ 3 22 ¥ 32

=

×

=

=

¥ 5

c. 324

4

81

2 ¥ 2 ¥ 3 ¥ 3 ¥ 3 ¥ 3 22 ¥ 34

= ×

=

=

■

Next, we will study shortcuts that will help us find prime factors. When division

yields a zero remainder, as in the case of 15 ÷ 3, for example, we say that 15 is divisible

by 3, 3 is a divisor of 15, or 3 divides 15. In general, we have the following definition.

Children’s Literature

www.wiley.com/college/musser

D E F I N I T I O N 5 . 2

See “A Remainder of One” by

Elinor Pinczes.

Divides

Let a and b be any whole numbers with a ≠ 0. We say that a divides b, and write

a | b, if and only if there is a whole number x such that ax = b. The symbol a Ը b

means that a does not divide b.

Common Core – Grade 4

Recognize that a whole number

In words, a divides b if and only if a is a factor of b. When a divides b, we can also

is a multiple of each of its fac-

say that a is a divisor of b, a is a factor of b, b is a multiple of a, and b is divisible by a.

tors. Determine whether a given

We can also say that a | b if b objects can be arranged in a rectangular array with

whole number in the range 1–100

is a multiple of a given one-digit

a rows. For example, 4 | 12 because 12 dots can be placed in a rectangular array with

number.

4 rows, as shown in Figure 5.3(a). On the other hand, 5 Ը12 because if 12 dots are

placed in an array with 5 rows, a rectangular array cannot be formed [Figure 5.3(b)].

Algebraic Reasoning

Figure 5.3

Each time we ask ourselves if a | b,

we are solving the equation ax = b.

For example, we use algebraic

reasoning when we decide if

Determine whether the following are true or false. Explain.

a statement like 7 | 91 is true

because we are looking for a

a. 3 | 12.

b. 8 is a divisor of 96.

whole number x that will make

c. 216 is a multiple of 6.

d. 51 is divisible by 17.

7x = 91 true.

e. 7 divides 34.

f. (22 3) | (23 32

¥

¥ ¥ 5).

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McGraw-Hill Education.

179

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7/30/2013 2:56:10 PM - 180 Chapter 5 Number Theory

S O L U T I O N

a. True. 3 | 12, since 3 ¥ 4 = 12.

b. True. 8 is a divisor of 96, since 8 ¥ 12 = 96.

c. True. 216 is a multiple of 6, since 6 ¥ 36 = 216.

d. True. 51 is divisible by 17, since 17 ¥ 3 = 51.

e. False. 7 Ը 34, since there is no whole number x such that 7x = 34.

f. True. (22 3) | (23 32

⋅

⋅ ⋅5), since (22 3)(2 3 5) = (23 32

¥

¥ ¥

¥ ¥ 5).

■

Check for Understanding: Exercise/Problem Set A #1–6

✔

Reflection from Research

Tests for Divisibility

When students are allowed to

use calculators to generate data

Some simple tests can be employed to help determine the factors of numbers. For

and are encouraged to examine

example, which of the numbers 27, 45, 38, 70, and 111,110 are divisible by 2, 5, or 10?

the data for patterns, they often

discover divisibility rules on their

If your answers were found simply by looking at the ones digits, you were applying

own (Bezuszka, 1985).

tests for divisibility. The tests for divisibility by 2, 5, and 10 are stated next.

Children’s Literature

T H E O R E M 5 . 2

www.wiley.com/college/musser

See “The Great Divide” by Dayle

Tests for Divisibility by 2, 5, and 10

Ann Dodds.

A number is divisible by 2 if and only if its ones digit is 0, 2, 4, 6, or 8.

A number is divisible by 5 if and only if its ones digit is 0 or 5.

A number is divisible by 10 if and only if its ones digit is 0.

Now notice that 3 | 27 and 3 | 9. It is also true that 3 | (27 + 9) and that 3 | (27 − 9).

This is an instance of the following theorem.

T H E O R E M 5 . 3

Let a, m, n, and k be whole numbers where a ≠ 0.

a. If a | m and a | n, then a | (m + n).

b. If a | m and a | n, then a | (m − n) for m ≥ n.

c. If a | m, then a | km.

P R O O F

a. If a | m, then ax = m for some whole number x.

If a | n, then ay = n for some whole number y.

Therefore, adding the respective sides of the two equations, we have ax + ay =

m + n, or

a(x + y) = m + n.

Since x + y is a whole number, this last equation implies that a | (m + n). Part (b) can

be proved simply by replacing the plus signs with minus signs in this discussion. The

proof of (c) follows from the definition of divides.

■

Part (a) in the preceding theorem can also be illustrated using the rectangular array

description of divides. In Figure 5 4

. , 3 | 9 is represented by a rectangle with 3 rows of

3 blue dots. 3 | 12 is represented by a rectangle of 3 rows of 4 black dots. By placing

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7/30/2013 2:56:15 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 181

the 9 blue dots and the 12 black dots together, there are (9 + 12) dots arranged in 3

rows, so 3 | (9 + 12).

Figure 5.4

Algebraic Reasoning

Using this result, we can verify the tests for divisibility by 2, 5, and 10. The main

To the right, notice how the vari-

idea of the proof of the test for 2 is now given for an arbitrary three-digit number (the

ables a, b, and c represent the

same idea holds for any number of digits).

digits 0, 1, 2, . . . , 9. In this way,

all three-digit numbers may be

represented (where a ≠ 0). Then,

Let r = a ¥ 102 + b ¥ 10 + c be any three-digit number.

rearranging sums using proper-

Observe that a ¥ 102 + b ¥ 10 = 10(a ¥ 10 + b).

ties and techniques of algebra

shows that the digit c is the one

Since 2 | 10, it follows that 2 | 10(a ¥ 10 + b) or 2 a ⋅102

| (

+ b ⋅10) for any digits a

that determines whether a num-

and b.

ber is even or not.

Thus if 2 | c (where c is the ones digit), then 2 | 10

[ (a 10

⋅ + b) + c].

Thus 2 a ¥ 102

| (

+ b ¥10 + c), or 2 | .r

Conversely, let 2 a ⋅102

| (

+ b ⋅10 + c). Since 2 a ¥102

| (

+ b ¥10), it follows that

2

a ¥ 102 + b ¥ 10 + c − a ¥ 102

| [(

)

(

+ b ¥10)] or 2 | c.

Therefore, we have shown that 2 divides a number if and only if 2 divides the num-

ber’s ones digit. One can apply similar reasoning to see why the tests for divisibility

for 5 and 10 hold.

The next two tests for divisibility can be verified using arguments similar to the test

for 2. Their verifications are left for the problem set.

T H E O R E M 5 . 4

Tests for Divisibility by 4 and 8

A number is divisible by 4 if and only if the number represented by its last two

digits is divisible by 4.

A number is divisible by 8 if and only if the number represented by its last three

digits is divisible by 8.

Notice that the test for 4 involves two digits and 22 = 4. Also, the test for 8 requires

that one consider the last three digits and 23 = 8.

Determine whether the following are true or false. Explain.

a. 4 | 1432

b. 8 | 4204

c. 4 | 2,345,678

d. 8 | 98,765, 432

S O L U T I O N

a. True. 4 | 1432, since 4 | 32.

b. False. 8 Ը 4204, since 8 Ը 204.

c. False. 4 Ը 2,345,678, since 4 Ը 78.

d. True. 8 | 98,765, 432, since 8 | 432.

■

The next two tests for divisibility provide a simple way to test for factors of 3 or 9.

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7/30/2013 2:56:21 PM - 182 Chapter 5 Number Theory

T H E O R E M 5 . 5

Tests for Divisibility by 3 and 9

A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

A number is divisible by 9 if and only if the sum of its digits is divisible by 9.

Determine whether the following are true or false. Explain.

a. 3 | 12,345

b. 9 | 12,345 c. 9 | 6543

S O L U T I O N

a. True. 3 | 12,345, since 1 + 2 + 3 + 4 + 5 = 15 and 3 |15.

b. False. 9 Ը12,345, since 1 + 2 + 3 + 4 + 5 = 15 and 9 Ը15.

c. True. 9 | 6543, since 9 | (6 + 5 + 4 + 3).

■

The following justification of the test for divisibility by 3 in the case of a three-

digit number can be extended to prove that this test holds for any whole number.

Let r = a ¥ 102 + b ¥ 10 + c be any three-digit number. We will show that if

3 | (a + b + c), then 3 | r. Rewrite r as follows:

r = a ¥ (99 + )

1 + b ¥ (9 + )

1 + c

= a ¥ 99 + a ¥1 + b ¥ 9 + b ¥1 + c

= a ¥ 99 + b ¥ 9 + a + b + c

= (a ¥111 + b)9 + a + b + c.

Since 3 | 9, it follows that 3 | (a ⋅11 + b 9

) . Thus if 3 | (a + b + c), where a + b + c

is the sum of the digits of r, then 3 | r since 3 | [(a ¥ 11 + b 9

) + (a + b + c)].

On the other hand, if 3 | r, then 3 | (a + b + c), since 3 | [r − (a ¥ 11 + b 9

) ] and

r − (a ¥ 11 + b)9 = a + b + c.

The test for divisibility by 9 can be justified in a similar manner.

The following is a test for divisibility by 11.

T H E O R E M 5 . 6

Test for Divisibility by 11

A number is divisible by 11 if and only if 11 divides the difference of the sum of

the digits whose place values are odd powers of 10 and the sum of the digits whose

place values are even powers of 10.

When using the divisibility test for 11, first compute the sums of the two different sets

of digits, and then subtract the smaller sum from the larger sum.

Determine whether the following are true or false. Explain.

a. 11 | 5346

b. 11 | 909,381 c. 11 | 76,543

S O L U T I O N

a. True. 11 | 5346, since 5 + 4 = 9, 3 + 6 = 9, 9 − 9 = 0, and 11 | 0.

b. True. 11 | 909,381, since 0 + 3 + 1 = 4, 9 + 9 + 8 = 26, 26 − 4 = 22, and 11 | 22.

c. False. 11Ը 76,543, since 6 + 4 = 10, 7 + 5 + 3 = 15, 15

− 10 = 5, and 11Ը 5.

■

c05.indd 182

7/30/2013 2:56:25 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 183

The justification of this test for divisibility by 11 is left for Problem 44 in the

Exercise/Problem Set A. Also, a test for divisibility by 7 is given in Exercise 10 in the

Exercise/Problem Set A.

One can test for divisibility by 6 by applying the tests for 2 and 3.

T H E O R E M 5 . 7

Test for Divisibility by 6

A number is divisible by 6 if and only if both of the tests for divisibility by 2 and

3 hold.

This technique of applying two tests simultaneously can be used in other cases

also. For example, the test for 10 can be thought of as applying the tests for 2 and 5

simultaneously. By the test for 2, the ones digit must be 0, 2, 4, 6, or 8, and by the test

for 5 the ones digit must be 0 or 5. Thus a number is divisible by 10 if and only if its

ones digit is zero. Testing for divisibility by applying two tests can be done in general.

T H E O R E M 5 . 8

A number is divisible by the product, ab, of two nonzero whole numbers a and b

if it is divisible by both a and b, and a and b have only the number 1 as a common

factor.

According to this theorem, a test for divisibility by 36 would be to test for 4 and

test for 9, since 4 and 9 both divide 36 and 4 and 9 have only 1 as a common factor.

However, the test “a number is divisible by 24 if and only if it is divisible by 4 and 6”

is not valid, since 4 and 6 have a common factor of 2. For example, 4 | 36 and 6 | 36,

but 24 Ը 36. The next example shows how to use tests for divisibility to find the prime

factorization of a number.

Find the prime factorization of 5148.

S O L U T I O N First, since the sum of the digits of 5148 is 18 (which is a mul-

tiple of 9), we know that 5148 = 9 ¥ 572. Next, since 4 | 72, we know that 4 | 572. Thus

5148

9 ¥ 572 9 ¥ 4 ¥143 32 ¥ 22

=

=

=

¥143. Finally, since in 143, 1 + 3 − 4 = 0 is divisible by 11,

the number 143 is divisible by 11, so 5148 22 32

=

¥ ¥11¥13.

■

Determine a process that could be used to determine if a number such as 211 is prime.

Write a brief description of that process that could easily be generalized to determine if

any given number is prime.

We can also use divisibility tests to help decide whether a particular counting

number is prime. For example, we can determine whether 137 is prime or composite

by determining if it has any prime factors less than 137. None of the numbers 2, 3,

or 5 divides 137. How about 7? 11? 13? How many prime factors must be considered

before we know whether 137 is a prime? Consider the following example.

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7/30/2013 2:56:42 PM - 184

Chapter 5 Number Theory

Determine whether 137 is a prime.

S O L U T I O N First, by the tests for divisibility, none of the numbers 2, 3, or 5 is a

factor of 137. Next try 7, 11, 13, and so on.

Notice that the numbers in column 1 form an increasing list of primes and the

numbers in column 2 are decreasing. Also, the numbers in the two columns “cross

over” between 11 and 13. Thus, if there is a prime factor of 137, it will appear in col-

umn 1 fi rst and reappear later as a factor of a number in column 2. Thus, as soon as

the crossover is reached, there is no need to look any further for prime factors. Since

the crossover point was passed in testing 137 and no prime factor of 137 was found,

we conclude that 137 is prime.

■

Example 5.7 suggests that to determine whether a number n is prime, we need

only search for prime factors p, where p2 ≤ n. Recall that y = x (read “the

square root of x”) means that y2 = x where y ≥ 0. For example, 25 = 5 since

52 = 25. Not all whole numbers have whole-number square roots. For example,

using a calculator, 27 ≈ 5 196

.

, since 5 1962

.

≈ 27. (A more complete discussion of

the square root is contained in Chapter 9.) Thus the search for prime factors of

a number n by considering only those primes p where p2 ≤ n can be simplified even

further by using the x key on a calculator and checking only those primes p

where p ≤ n.

T H E O R E M 5 . 9

Prime Factor Test

To test for prime factors of a number n, one need only search for prime factors p

of n, where p2 ≤ n (or p ≤ n).

Determine whether the following numbers are prime or composite.

a. 299 b. 401

S O L U T I O N

a. Only the prime factors 2 through 17 need to be checked, since 172

299 192

<

<

(check this on your calculator). None of the numbers 2, 3, 5, 7, or 11 is a factor,

but since 299 = 13 ¥ 23, the number 299 is composite.

b. Only primes 2 through 19 need to be checked, since 401 ≈ 20. Since none of the

primes 2 through 19 are factors of 401, we know that 401 is a prime. (The tests for

divisibility show that 2, 3, 5, and 11 are not factors of 401. A calculator, tests for

divisibility, or long division can be used to check 7, 13, 17, and 19.)

■

Check for Understanding: Exercise/Problem Set A #7–14

✔

c05.indd 184

7/30/2013 2:56:54 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 185

Finding large primes is a favorite pastime of some mathematicians.

Before the advent of calculators and computers, this was certainly

a time-consuming endeavor. Three anecdotes about large primes

follow.

t Euler once announced that 1,000,009 was prime. However, he

later found that it was the product of 293 and 3413. At the time

of this discovery, Euler was 70 and blind.

t Fermat was once asked whether 100,895,598,169 was prime. He

replied shortly that it had two factors, 898,423 and 112,303.

67

t For more than 200 years the Mersenne number 2

− 1 was

thought to be prime. In 1903, Frank Nelson Cole, in a speech to

the American Mathematical Society, went to the blackboard and

without uttering a word, raised 2 to the power 67 (by hand, using

our usual multiplication algorithm!) and subtracted 1. He then

©Ron Bagwell

multiplied 193,707,721 by 761,838,257,287 (also by hand). The

two numbers agreed! When asked how long it took him to crack

the number, he said, “Three years of Sundays.”

EXERCISE/PROBLEM SET A

EXERCISES

1. Using the Chapter 5 eManipulative activity Sieve of

6. If the variables represent counting numbers, determine

Eratosthenes on our Web site, find all primes less than 100.

whether each of the following is true or false.

a. If x Ը y and x Ը z, then x Ը (y + z).

2. Find a factor tree for each of the following numbers.

b. If 2 | a and 3 | a, then 6 | a.

a. 36 b. 54 c. 102 d. 1000

3. A factor tree is not the only way to find the prime factor-

7. a. Show that 8 |123 152

,

using the test for divisibility by 8.

ization of a composite number. Another method is to divide

b. Show that 8 |123 152

,

by finding x such that 8x = 123 152

,

.

the number first by 2 if possible, then divide the quotient

c. Is the x that you found in part b a divisor of 123,152?

by 2 if possible, then by 3, then by 5, and so on, until all

Prove it.

possible divisions by prime numbers have been performed.

8. Which of the following are multiples of 3? of 4? of 9?

For example, to find the prime factorization of 108, you

a. 123,452 b. 1,114,500

might organize your work as follows to conclude that

108

22

33

=

× .

9. Use the test for divisibility by 11 to determine which of the

3

following numbers are divisible by 11.

3 9

a. 2838 b. 71,992 c. 172,425

3 27

10. A test for divisibility by 7 is illustrated as follows. Does 7

2 54

divide 17,276?

2 108

Use this method to find the prime factorization of the

Test:

17276

following numbers.

− 12

Subtract 2 × 6 from 1727

a. 216 b. 2940 c. 825 d. 198,198

1715

− 10

Subtract 2 × 5 frrom 171

4. Determine which of the following are true. If true, illustrate

161

it with a rectangular array. If false, explain.

− 2

×

a. 3 | 9

b. 12 | 6

Subtract 2 1 from 16

c. 3 is a divisor of 21.

d. 6 is a factor of 3.

14

e. 4 is a factor of 16.

f. 0 | 5

Since 7 |14, we also have 7 |17,276. Use this test to see

g. 11 |11

h. 48 is a multiple of 16.

whether the following numbers are divisible by 7.

5. If 21 divides m, what else must divide m?

a. 8659 b. 46,187 c. 864,197,523

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7/30/2013 2:56:58 PM - 186 Chapter 5 Number Theory

11. True or false? Explain.

a. 6 | 80

b. 15 |10,000

a. If a counting number is divisible by 9, it must be divis-

c. 4 |15,000

d. 12 | 32,304

ible by 3.

b. If a counting number is divisible by 3 and 11, it must be

13. Which of the following numbers are composite? Why?

divisible by 33.

a. 12 b. 123 c. 1234 d. 12,345

12. Decide whether the following are true or false using

14. To determine if 467 is prime, we must check to see if it is

only divisibility ideas given in this section (do not use

divisible by any numbers other than 1 and itself. List all

long division or a calculator). Give a reason for your

of the numbers that must be checked as possible factors to

answers.

see if 467 is prime.

PROBLEMS

15. a. Write 36 in prime factorization form.

23. Which of the following numbers can be written as the sum

b. List the divisors of 36.

of two primes, and why?

c. Write each divisor of 36 in prime factorization

form.

7,17, 27, 37, 47, . . .

d. What relationship exists between your answer to part a

24. One of Fermat’s theorems states that every prime of the

and each of your answers to part c?

form 4x + 1 is the sum of two square numbers in one and

e. Let n = 132 × 295. If m divides n, what can you conclude

only one way. For example, 13 = 4(3) + 1, and 13 = 4 + 9,

about the prime factorization of m?

where 4 and 9 are square numbers.

16. Justify the tests for divisibility by 5 and 10 for any three-

a. List the primes less than 100 that are of the form 4x + 1,

digit number by reasoning by analogy from the test for

where x is a whole number.

divisibility by 2.

b. Express each of these primes as the sum of two square

numbers.

17. The symbol 4! is called four factorial and means

4 × 3 × 2 × 1; thus 4! = 24. Which of the following state-

25. The primes 2 and 3 are consecutive whole numbers. Is

ments are true?

there another such pair of consecutive primes? Justify your

a. 6 | 6!

b. 5 | 6!

c. 11 | 6!

answer.

d. 30 | 30!

e. 40 | 30!

f. 30 | (30! + 1)

(Do not multiply out parts d to f.)

26. Two primes that differ by 2 are called twin primes. For

example, 5 and 7, 11 and 13, 29 and 31 are twin primes.

18. a. Does 8 | 7!? b. Does 7 | 6!?

Using the Chapter 5 eManipulative activity Sieve of

c. For what counting numbers n will n divide (n − )!

1 ?

Eratosthenes on our Web site to display all primes less

than 200, find all twin primes less than 200.

19. There is one composite number in this set: 331, 3331,

33,331, 333,331, 3,333,331, 33,333,331, 333,333,331.

27. One result that mathematicians have been unable to prove

Which one is it? (Hint: It has a factor less than 30.)

true or false is called Goldbach’s conjecture. It claims that

each even number greater than 2 can be expressed as the

20. Show that the formula p(n) = n2 + n + 17 yields primes for

sum of two primes. For example.

n = 0,1, 2, and 3. Find the smallest whole number n for

which p(n) = n2 + n + 17 is not a prime.

4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5,

21. a. Compute n2 + n + 41, where n = 0,1, 2, . . . ,10, and deter-

10 = 5 + 5, 12

= 5 + 7.

mine which of these numbers are prime.

b. On another piece of paper, continue the following

a. Verify that Goldbach’s conjecture holds for even num-

spiral pattern until you reach 151. What do you notice

bers through 40.

about the main upper left to lower right diagonal?

b. Assuming that Goldbach’s conjecture is true, show how

each odd whole number greater than 6 is the sum of

etc.

three primes.

53

52

51

50

28. For the numbers greater than 5 and less than 50, are there

43

42

49

at least two primes between every number and its double?

44

41

48

If not, for which number does this not hold?

45

46

47

29. Find two whole numbers with the smallest possible dif-

ference between them that when multiplied together will

22. In his book The Canterbury Puzzles (1907), Dudeney men-

produce 1,234,567,890.

tioned that 11 was the only number consisting entirely of

30. Find the largest counting number that divides every num-

ones that was known to be prime. In 1918, Oscar Hoppe

ber in the following sets.

proved that the number 1,111,111,111,111,111,111 (19

a. {1 ¥ 2 ¥ ,

3 2 ¥ 3 ¥ ,

4 3 ¥ 4 ¥ ,

5 . . .}

ones) was prime. Later it was shown that the number

¥ ¥ ¥ ¥ ¥ ¥

made up of 23 ones was also prime. Now see how many of

b. {1 3

,

5 2 4

,

6 3 5

,

7 . . .}

these “repunit” numbers up to 19 ones you can factor.

Can you explain your answer in each case?

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7/30/2013 2:57:02 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 187

31. Find the smallest counting number that is divisible by the

41. The annual sales for certain calculators were $2567 one

numbers 2 through 10.

year and $4267 the next. Assuming that the price of the

calculators was the same each of the two years, how many

32. What is the smallest counting number divisible by 2, 4, 5,

calculators were sold in each of the two years?

6, and 12?

33. Fill in the blank. The sum of three consecutive counting

42. Observe that 7 divides 2149. Also check to see that 7

numbers always has a divisor (other than 1) of _____. Prove.

divides 149,002. Try this pattern on another four-digit

number using 7. If it works again, try a third. If that one

34. Choose any two numbers, say 5 and 7. Form a sequence

also works, formulate a conjecture based on your three

of numbers as follows: 5, 7, 12, 19, and so on, where each

examples and prove it. (Hint: 7 |1001.)

new term is the sum of the two preceding numbers until you

have 10 numbers. Add the 10 numbers. Is the seventh num-

43. How long does this list continue to yield primes?

ber a factor of the sum? Repeat several times, starting with

17 + 2 = 19

a different pair of numbers each time. What do you notice?

+ =

Prove that your observation is always true.

19

4

23

23 + 6 = 29

35. a. 5! = 5 · 4 · 3 · 2 · 1 is divisible by 2, 3, 4, and 5. Prove that

+ =

5! + 2, 5! + 3, 5! + 4, and 5! + 5 are all composite.

29

8

37

b.

Find 1000 consecutive numbers that are composite.

44. Justify the test for divisibility by 11 for four-digit numbers

3

2

36. The customer said to the cashier, “I have 5 apples at 27

by completing the following: Let a ¥10 + b ¥10 + c ¥10 + d

cents each and 2 pounds of potatoes at 78 cents per pound.

be any four-digit number. Then

I also have 3 cantaloupes and 6 lemons, but I don’t remem-

3

2

ber the price for each.” The cashier said, “That will be

a ¥10 + b ¥10 + c ¥10 + d

$3.52.” The customer said, “You must have made a mis-

= a 1001

(

−1) + b(99 +1) + c 11

( −1) + d

take.” The cashier checked and the customer was correct.

= ⋅⋅⋅⋅

How did the customer catch the mistake?

37. There is a three-digit number with the following property:

45. If p is a prime greater than 5, then the number 111. . .1,

If you subtract 7 from it, the difference is divisible by 7; if

consisting of p − 1 ones, is divisible by p. For example,

× ,

=

you subtract 8 from it, the difference is divisible by 8; and

7 |111 111

,

, since 7 15 873 111 111

,

. Verify the initial sen-

if you subtract 9 from it, the difference is divisible by 9.

tence for the next three primes.

What is the number?

46. a. Find the largest n such that 3n | 24!.

38. Paula and Ricardo are serving cupcakes at a school party.

b. Find the smallest n such that 36 | n!.

If they arrange the cupcakes in groups of 2, 3, 4, 5, or 6,

c. Find the largest n such that 12n | 24!.

they always have exactly one cupcake left over. What is the

47. Do Problem 20 using a spreadsheet to create a table of

smallest number of cupcakes they could have?

values, n and p(n), for n = 1, 2, . . . 20. Once the smallest

39. Prove that all six-place numbers of the form abcabc (e.g.,

value of n is found, evaluate p(n + )

1 and p(n + 2). Are

416,416) are divisible by 13. What other two numbers are

they prime or composite?

always factors of a number of this form?

48. In Exercise 10, 2 times the ones digit was subtracted from

40. a. Prove that all four-digit palindromes are divisible by 11.

the remaining number repeatedly as part of the divisibility

b. Is this also true for every palindrome with an even num-

test for 7. Use this same technique to develop a test for

ber of digits? Prove or disprove.

divisibility by 11. (Hint: Use something other than 2.)

EXERCISE/PROBLEM SET B

EXERCISES

1. An efficient way to find all the primes up to 100 is to

43

44

45

46

47

48

arrange the numbers from 1 to 100 in six columns. As with

49

50

51

52

53

54

the Sieve of Eratosthenes, cross out the multiples of 2, 3, 5,

55

56

57

58

59

60

and 7. What pattern do you notice? (Hint: Look at the col-

umns and diagonals.)

61

62

63

64

65

66

67

68

69

70

71

72

1

2

3

4

5

6

73

74

75

76

77

78

7

8

9

10

11

12

79

80

81

82

83

84

13

14

15

16

17

18

85

86

87

88

89

90

19

20

21

22

23

24

91

92

93

94

95

96

25

26

27

28

29

30

97

98

99

100

31

32

33

34

35

3 6

2. Find a factor tree for each of the following numbers.

37

38

39

40

41

42

a. 192 b. 380 c. 1593 d. 3741

c05.indd 187

7/30/2013 2:57:07 PM - 188 Chapter 5 Number Theory

3. Factor each of the following numbers into primes.

9. Use the test for divisibility by 11 to determine which of the

a. 39

b. 1131

c. 55

following numbers are divisible by 11.

d. 935

e. 3289

f. 5889

a. 945,142 b. 6,247,251 c. 385,627

4. Use the definition of divides to show that each of the

10. There is a test for divisibility by 17 similar to the one for

following is true. (Hint: Find x that satisfies the definition

7 as shown in Set A, Exercise 10. Rather than using 2

of divides.)

times the ones digit, however, one must use 5 in place of

a. 7 | 49

2 before subtracting. Determine which of the following

b. 21 | 210

numbers are divisible by 17 using that test.

c. 3 | (9 × 18)

a. 2963 b. 37,142 c. 95,757,923

d. 2 22

| (

× 5 × 7)

11. True or false? Explain.

e. 6 24 × 32 × 73 × 135

| (

)

a. If a counting number is divisible by 6 and 8, it must be

f. 100,000 | (27

39

511

178

×

×

×

)

divisible by 48.

g. 6000 221 × 317 × 589 × 2937

| (

)

b. If a counting number is divisible by 4, it must be divis-

h. 22 | 121

(

× 4)

ible by 8.

i. p3q5r

p5q13r7s t

2 27

| (

)

j. 7 | (5 × 21 + 14)

12. Decide whether the following are true or false using only

5. If 24 divides b, what else must divide b?

divisibility ideas given in this section (do not use long divi-

sion or a calculator). Give a reason for your answers.

6. If the variables represent counting numbers, determine

a. 24 | 325,608

b. 45 |13,075

whether each of the following is true or false.

c. 40 |1,732,800

d. 36 | 677,916

a. If 2 | a and 6 | a, then 12 | a.

b. 6 | xy, then 6 | x or 6 | y.

13. Which of the following numbers are composite? Why?

a. 123,456

b. 1,234,567

c. 123,456,789

7. a. Prove in two different ways that 2 divides 114.

b. Prove in two different ways that 3 | 336.

14. To determine if 769 is prime, we must check to see if it is

divisible by any numbers other than 1 and itself. List all

8. Which of the following are multiples of 3? of 4? of 9?

of the numbers that must be checked as possible factors to

a. 2,199,456 b. 31,020,417

see if 769 is prime.

PROBLEMS

15. A calculator may be used to test for divisibility of one

21. It is claimed that the formula n2 − n + 41 yields a prime for

number by another, where n and d represent counting

all whole-number values for n. Decide whether this state-

numbers.

ment is true or false.

a. If n ÷ d gives the answer 176, is it necessarily true that

d | n?

22. In 1644, the French mathematician Mersenne asserted that

n

b. If n ÷ d gives the answer 56.3, is it possible that d | n?

2 − 1 was prime only when n = 2, 3, 5, 7,13,17,19, 31, 67,127,

and 257. As it turned out, when n = 67 and n = 257, 2n − 1

16. Justify the test for divisibility by 9 for any four-digit number.

was a composite, and 2n − 1 was also prime when n = 89

(Hint: Reason by analogy from the test for divisibility by 3.)

and n = 107. Show that Mersenne’s assertion was correct

17. Justify the tests for divisibility by 4 and 8.

concerning n = 3, 5, 7, and 13.

18. Find the first composite number in this list.

23. It is claimed that every prime greater than 3 is either one

3! − 2! + 1! = 5 Prime

more or one less than a multiple of 6. Investigate. If it

seems true, prove it. If it does not, find a counterexample.

4! − 3! + 2! − 1! = 19 Prime

5! − 4! + 3! − 2! + 1! = 101 Prim

me

24. Is it possible for the sum of two odd prime numbers to be

a prime number? Why or why not?

Continue this pattern. (Hint: The first composite comes

within the first 10 such numbers.)

25. Mathematician D. H. Lehmer found that there are

209 consecutive composites between 20,831,323 and

19. In 1845, the French mathematician Bertrand made the

20,831,533. Pick two numbers at random between

following conjecture: Between any whole number greater

20,831,323 and 20,831,533 and prove that they are com-

than 1 and its double there is at least one prime. In 1911,

posite.

the Russian mathematician Tchebyshev proved the con-

jecture true. Using the Chapter 5 eManipulative activity

26. Prime triples are consecutive primes whose difference is 2.

Sieve of Eratosthenes on our Web site to display all primes

One such triple is 3, 5, 7. Find more or prove that there

less than 200, find three primes between each of the fol-

cannot be any more.

lowing numbers and its double.

27. A seventh-grade student named Arthur Hamann made the

a. 30 b. 50 c. 100

following conjecture: Every even number is the difference

20. The numbers 2, 3, 5, 7, 11, and 13 are not factors of 211.

of two primes. Express the following even numbers as the

Can we conclude that 211 is prime without checking for

difference of two primes.

more factors? Why or why not?

a. 12 b. 20 c. 28

c05.indd 188

7/30/2013 2:57:13 PM - Section 5.1 Primes, Composites, and Tests for Divisibility 189

28. The numbers 1, 7, 13, 31, 37, 43, 61, 67, and 73 form a

42. A merchant marked down some pads of paper from $2

3 × 3 additive magic square. (An additive magic square has

and sold the entire lot. If the gross received from the sale

the same sum in all three rows, three columns, and two

was $603.77, how many pads did she sell?

main diagonals.) Find it.

43. How many prime numbers less than 100 can be written

29. Can you find whole numbers a and b such that 3a

5b

= ?

using the digits 1, 2, 3, 4, 5 if

Why or why not?

a. no digit is used more than once?

b. a digit may be used twice?

30. I’m a two-digit number less than 40. I’m divisible by only

one prime number. The sum of my digits is a prime, and

44. Which of the numbers in the set

the difference between my digits is another prime. What

{ ,

9 9 ,

9

,

999

,

9999 . . .}

number could I be?

are divisible by 7?

31. What is the smallest counting number divisible by 2, 4, 6,

8, 10, 12, and 14?

45. Two digits of this number were erased: 273*49*5. However,

we know that 9 and 11 divide the number. What is it?

32. What is the smallest counting number divisible by the

numbers 1, 2, 3, 4, . . . 24, 25 ? (Hint: Give your answer in

46. This problem appeared on a Russian mathematics

prime factorization form.)

exam: Show that all the numbers in the sequence 100001,

10000100001, 1000010000100001

, . . . . are composite.

33. The sum of five consecutive counting numbers has a divi-

Show that 11 divides the first, third, fifth numbers in this

sor (other than 1) of _____. Prove.

sequence, and so on, and that 111 divides the second. An

American engineer claimed that the fourth number was

34. Take any number with an even number of digits. Reverse

the product of

the digits and add the two numbers. Does 11 divide your

result? If yes, try to explain why.

21401 and 4672725574038601.

35. Take a number. Reverse its digits and subtract the smaller

Was he correct?

of the two numbers from the larger. Determine what num-

47. The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, . . . , is formed

ber always divides such differences for the following types

by adding any two consecutive numbers to find the next

of numbers.

term. Prove or disprove: The sum of any ten consecutive

a. A two-digit number b. A three-digit number

Fibonacci numbers is a multiple of 11.

c. A four-digit number

48. Do Set A, Problem 21 (a) using a spreadsheet to create a

36. Choose any three digits. Arrange them three ways to

table of values, n and p(n), for n = 1, 2, . . . 20.

form three numbers. Claim: The sum of your three

49. In Exercise 10 of Set A, 2 times the ones digit was sub-

numbers has a factor of 3. True or false?

tracted from the remaining number repeatedly as part

Example:

371

of the divisibility test for 7. Use this same technique to

137

develop a test for divisibility by 13. (Hint: Use something

+ 713

other than 2.).

1221

Analyzing Student Thinking

and 1221 = 3 × 407

50. Courtney asserts that if the Sieve of Eratosthenes can be

37. Someone spilled ink on a bill for 36 sweatshirts. If only

used to find all the primes, then it should also be able to

the first and last digits were covered and the other three

find all the composite numbers. Is she correct? Explain.

digits were, in order, 8, 3, 9 as in ?83.9?, how much did

51. One of the theorems in this section states “If a | m and a | n,

each cost?

then a | (m + n) when a is nonzero.” A student suggests

+

38. Determine how many zeros are at the end of the numerals

that “If a | (m

n), then a | m or a | n.” Is the student cor-

for the following numbers in base ten.

rect? Explain.

a. 10! b. 100! c. 1000!

52. Christa asserts that the test for divisibility by 9 is similar

to the test for divisibility by 3 because every power of 10 is

39. Find the smallest number n with the following property:

one more than a multiple of 9. Is she correct? Explain.

If n is divided by 3, the quotient is the number obtained

by moving the last digit (ones digit) of n to become the

53. Brooklyn says to you that to find all of the prime factors

first digit. All of the remaining digits are shifted one to the

of 113, she only has to check to see if any of 2, 3, 5, 7 are

right.

a factors of 113. How do you respond?

40. A man and his grandson have the same birthday. If for

54. A student says that she is confused by the difference

six consecutive birthdays the man is a whole number of

between a | b, which means “a divides b,” and a b

/ , which

times as old as his grandson, how old is each at the sixth

is read “a divided by b.” How would you help her resolve

birthday?

her confusion?

41. Let m be any odd whole number. Then m is a divisor

55. Kelby notices that if 2 | 36 and 9 | 36, then 18 | 36 since

of the sum of any m consecutive whole numbers. True

18 = 2 × 9. He also notices that 4 | 36 and 6 | 36 but 24 Ը 36.

or false? If true, prove. If false, provide a counter-

How could you help him understand why one case works

example.

and the other does not?

c05.indd 189

7/30/2013 2:57:17 PM - 190 Chapter 5 Number Theory

56. Lorena claims that since a number is divisible by 5

57. Suppose a student said that the sum of the digits of the

if its ones digit is a 5, then the number 357 is divisible

number 354 is 12 and therefore 354 is divisible by any

by 7 because the last digit is a 7. How would you

number that divides into 12, like 2, 3, 4, and 6. Would you

respond?

agree with the student? Explain.

COUNTING FACTORS, GREATEST COMMON FACTOR,

AND LEAST COMMON MULTIPLE

Following recess, the 1000 students of Wilson School lined up for the following activity:

The first student opened all of the 1000 lockers in the school. The second student closed

all lockers with even numbers. The third student “changed” all lockers that were numbered with multiples of 3 by closing

those that were open and opening those that were closed. The fourth student changed each locker whose number was

a multiple of 4, and so on. After all 1000 students had completed the activity, which lockers were open? Why?

Problem-Solving Strategy

Counting Factors

Look for a Pattern

In addition to finding prime factors, it is sometimes useful to be able to find how

many factors (not just prime factors) a number has. The fundamental theorem of

Common Core – Grade 4

arithmetic is helpful in this regard. For example, to find all the factors of 12, consider

Find all factor pairs for a whole

its prime factorization 12

22 31

=

¥ . All factors of 12 must be made up of products of at

number in the range 1–100.

most 2 twos and 1 three. All such combinations are contained in the table. Therefore,

12 has six factors, namely, 1, 2, 3, 4, 6, and 12.

EXPONENT OF 2

EXPONENT OF 3

FACTOR

0

0

20 30

¥ = 1

Children’s Literature

www.wiley.com/college/

1

0

21 30

¥ = 2

musser See “You Can Count

2

0

22 30

¥ = 4

on Monsters” by Richard Evan

0

1

20 31

¥ = 3

Schwartz.

1

1

21 31

¥ = 6

2

1

22 31

¥ = 12

The technique used in this table can be used with any whole number that is

expressed as the product of primes with their respective exponents. To find the num-

ber of factors of 23 52

¥ , a similar list could be constructed. The exponents of 2 would

range from 0 to 3 (four possibilities), and the exponents of 5 would range from 0 to

2 (three possibilities). In all there would be 4 ¥ 3 combinations, or 12 factors of 23 52

¥ ,

as shown in the following table.

EXPONENTS OF

5

2

0

1

2

0

2050 = 1

2051 = 5

2052 = 25

1

2150 = 2

2151 = 10

2152 = 50

2

2250 = 4

2251 = 20

2252 = 100

3

2350 = 8

2351 = 40

2352 = 200

This method for finding the number of factors of any number can be summarized

as follows.

T H E O R E M 5 . 1 0

Suppose that a counting number n is expressed as a product of distinct primes with

their respective exponents, say n

( pn1

2

1 )( pn

2 )

( pnm

=

⋅⋅⋅ m ). Then the number of factors

of n is the product (n +

¥

+ ⋅⋅⋅

+

1

)

1 (n2

)

1

(nm

)

1 .

c05.indd 190

7/30/2013 2:57:20 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 191

Find the number of factors.

a. 144 b. 23 57 74

¥ ¥ c. 95 112

¥

S O L U T I O N

a. 144

24 32

=

¥ . So, the number of factors of 144 is (4 + )(

1 2 + )

1 = 1 .

5

b. 23 57 74

¥ ¥ has (3 + )(

1 7 + )

1 (4 + )

1 = 160 factors.

c. 95 112 = 310 112

¥

¥

has (10 + )(

1 2 + )

1 = 33 factors. (NOTE: 95 had to be rewritten as 310,

since 9 was not prime.)

■

Notice that the number of factors does not depend on the prime factors, but rather

on their respective exponents.

Check for Understanding: Exercise/Problem Set A #1–2

✔

Greatest Common Factor

The concept of greatest common factor is useful when simplifying fractions.

Common Core – Grade 6

Find the greatest common factor

D E F I N I T I O N 5 . 3

of two whole numbers less than

or equal to 100.

Greatest Common Factor

The greatest common factor (GCF) of two (or more) nonzero whole numbers is the

largest whole number that is a factor of both (all) of the numbers. The GCF of a

and b is written GCF(a, b).

Use your understanding of the greatest common factor to find GCF(27, 45). Write a

detailed description of the method used to find it.

Algebraic Reasoning

By definition, GCF( ,

a a) = a.

The concept of greatest common

There are two elementary ways to find the greatest common factor of two num-

factor (applied to 6, 15, and 12

bers: the set intersection method and the prime factorization method. The GCF(24,

in 6x − 15y + 12) is useful when

36) is found next using these two methods.

factoring or simplifying such

expressions.

Set Intersection Method

Step 1

Find all factors of 24 and 36. Since 24

23

=

¥ 3, there are 4 ¥ 2 = 8 factors of 24, and since

36

22 32

=

¥ , there are 3 ¥ 3 = 9 factors of 36. The set of factors of 24 is {1, 2, 3, 4, 6, 8,

12, 24}, and the set of factors of 36 is {1, 2, 3, 4, 6, 9, 12, 18, 36}.

Step 2

Find all common factors of 24 and 36 by taking the intersection of the two sets in

step 1.

{ ,

1

,

2

,

3

,

4

,

6

,

8 1 ,

2 2 }

4 ∩ { ,

1

,

2

,

3

,

4

,

6

,

9 1 ,

2 1 ,

8 3 }

6 = { ,

1

,

2

,

3

,

4

,

6 1 }

2

Step 3

Find the largest number in the set of common factors in step 2. The largest number

in {1, 2, 3, 4, 6, 12} is 12. Therefore, 12 is the GCF of 24 and 36. (NOTE: The set

intersection method can also be used to find the GCF of more than two numbers in

a similar manner.)

c05.indd 191

7/30/2013 2:57:23 PM - 192 Chapter 5 Number Theory

While the set intersection method can be cumbersome and at times less efficient

than other methods, it is conceptually a natural way to think of the greatest common

factor. The “common” part of the greatest common factor is the “intersection” part

of the set intersection method. Once students have a good understanding of what the

greatest common factor is by using the set intersection method, the prime factoriza-

tion method, which is often more efficient, can be introduced.

Prime Factorization Method

Step 1

Express the numbers 24 and 36 in their prime factor exponential form: 24

23

=

¥ 3 and

36

22 32

=

¥ .

Step 2

The GCF will be the number 2 3

m n where m is the smaller of the exponents of the 2s

and n is the smaller of the exponents of the 3s. For 23 ¥ 3 and 22 32

¥ , m is the smaller

of 3 and 2, and n is the smaller of 1 and 2. Therefore, the GCF of 23 31

¥ and 22 32

¥ is

22 31

¥ = 12. Review this method so that you see why it always yields the largest number

that is a factor of both of the given numbers.

Find GCF(42, 24) in two ways.

S O L U T I O N

Set Intersection Method

42 = 2 ¥ 3 ¥ 7, so 42 has 2 ¥ 2 ¥ 2 = 8 factors.

24

23

=

¥ 3, so 24 has 4 ¥ 2 = 8 factors.

Factors of 42: {1, 2, 3, 6, 7, 14, 21, 42}.

Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}.

Common factors: {1, 2, 3, 6}.

GCF(42, 24) = 6.

Prime Factorization Method

42 = 2 ¥ 3 ¥ 7 and 24 23

=

¥ 3.

GCF(

,

42 24) = 2 ¥ 3 = .

6

Notice that only the common primes (2 and 3) are used, since the exponent on the 7

is zero in the prime factorization of 24.

■

Earlier in this chapter we obtained the following result: If a | m, a | n, and m ≥ n,

then a | (m − n). In words, if a number divides each of two numbers, then it divides

their difference. Hence, if c is a common factor of a and b, where a ≥ b, then c is also a

common factor of b and a − b. Since every common factor of a and b is also a common

factor of b and a − b, the pairs (a, b) and (a − ,

b b

) have the same common factors. So

GCF(a, b) and GCF(a − ,

b b

) must also be the same.

T H E O R E M 5 . 1 1

If a and b are whole numbers, with a ≥ b, then

GCF(a, b) = GCF(a − b, b).

The usefulness of this result is illustrated in the next example.

c05.indd 192

7/30/2013 2:57:27 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 193

Find the GCF(546, 390).

S O L U T I O N

GCF(546, )

390 = GCF(546 − 390, 390)

= GCF(156, 390)

= GCF(390 − 156,156)

= GCF(234,156)

= GCF(234 − 156,156)

= GCF(78 ,156)

= GCF(156 − 78, 78)

= GCF(78, 78)

= 78

■

Using a calculator, we can find the GCF(546, 390) as follows:

546 − 390 = 156

390 − 156 = 234

234 − 156 = 78

156 − 78 = 78

Therefore, since the last two numbers in the last line are equal, then the GCF(546,

390) = 78. Notice that this procedure may be shortened by storing 156 in the calcula-

tor’s memory.

This calculator method can be refined for very large numbers or in exceptional

cases. For example, to find GCF(1417, 26), 26 must be subtracted many times to

produce a number that is less than (or equal to) 26. Since division can be viewed as

repeated subtraction, long division can be used to shorten this process as follows:

54 R 13

26)1417

Here 26 was “subtracted” from 1417 a total of 54 times to produce a remainder of 13.

Thus GCF(1417, 26) = GCF(13, 26). Next, divide 13 into 26.

2 R 0

1 )

3 26

Thus GCF(13, 26) = 13, so GCF(1417, 26) = 13. Each step of this method can be

justified by the following theorem.

T H E O R E M 5 . 1 2

If a and b are whole numbers with a ≥ b and a = bq + r, where r < b, then

GCF(a, b) = GCF(r, b).

Thus, to find the GCF of any two numbers, this theorem can be applied repeat-

edly until a remainder of zero is obtained. The final divisor that leads to the zero

remainder is the GCF of the two numbers. This method is called the Euclidean

algorithm.

c05.indd 193

7/30/2013 2:57:28 PM - 194 Chapter 5 Number Theory

Find the GCF(840, 3432).

S O L U T I O N

4 R 72

840)3432

11 R 48

72)840

) 1 R 24

48 72

2 R 0

24)48

Therefore, GCF(840, 3432) = 24.

■

A calculator also can be used to calculate the GCF(3432, 840) using the Euclidean

algorithm.

3432 ÷ 840 = 4 085714286

.

3432 − 4 × 840 =

72.

840 ÷ 72 = 11 66666667

.

840 − 11 × 72 =

48.

72 ÷ 48 =

1 5

.

72 − 1 × 48 =

24.

48 ÷ 24 =

2.

Therefore, 24 is the GCF(3432, 840). Notice how this method parallels the one in

Example 5.12.

Check for Understanding: Exercise/Problem Set A #3–8

✔

Reﬂ ection from Research

Least Common Multiple

Possibly because students often

confuse factors and multiples, the

The least common multiple is useful when adding or subtracting fractions.

greatest common factor and the

least common multiple are dif-

ficult topics for students to grasp

(Graviss & Greaver, 1992).

D E F I N I T I O N 5 . 4

Least Common Multiple

Common Core – Grade 6

Find the least common multiple

The least common multiple (LCM) of two (or more) nonzero whole numbers is the

of two whole numbers less than

smallest nonzero whole number that is a multiple of each (all) of the numbers. The

or equal to 12.

LCM of a and b is written LCM(a, b).

Use your understanding of the least common multiple to find LCM(18, 24). Write

a detailed description of the method used to find it.

By definition, LCM(a, a) = a.

c05.indd 194

7/30/2013 2:57:30 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 195

Algebraic Reasoning

For the LCM, there are three elementary ways to find the least common multiple

Finding a Least Common Multiple

of two numbers: the set intersection method, the prime factorization method, and the

is really solving a system of equa-

build-up method. The LCM(24, 36) is found next using these three methods.

tions. When searching for the

LCM(24, 36), we are searching for

whole numbers x , y , and z that

Set Intersection Method

will make the equations 24x = z

and 36y = z true.

Step 1

List the first several nonzero multiples of 24 and 36. The set of nonzero mul-

tiples of 24 is { ,

24

,

48

,

72

,

96

,

120

,

144 . . .}, and the set of nonzero multiples of 36 is

{ ,

36

,

72

,

108

,

144 . . .}. (NOTE: The set of multiples of any nonzero whole number is an

infinite set.)

Step 2

Find the first several common multiples of 24 and 36 by taking the intersection of the

two sets in step 1:

{ ,

24

,

48

,

72

,

96

,

120

,

144 . . .} ∩ { ,

36

,

72

,

108

,

144 . . .} = { ,

72

,

144 . . .}.

Step 3

Find the smallest number in the set of common multiples in step 2. The smallest

number in { ,

72

,

144 . . .} is 72. Therefore, 72 is the LCM of 24 and 36 (Figure 5.5).

Similar to the methods for finding the greatest common factor, the “intersection”

part of the set intersection method illustrates the “common” part of the least com-

mon multiple. Thus, the set intersection method is a natural way to introduce the least

common multiple.

Figure 5.5

Prime Factorization Method

Step 1

Express the numbers 24 and 36 in their prime factor exponential form: 24

23

=

¥ 3 and

36

22 32

=

¥ .

Step 2

The LCM will be the number 2r3s, where r is the larger of the exponents of the 2s

and s is the larger of the exponents of the 3s. For 23 31

¥ and 22 32

¥ , r is the larger of

3 and 2 and s is the larger of 1 and 2. That is, the LCM of 23 31

¥ and 22 32

¥ is 23 32

¥ ,

or 72. Review this procedure to see why it always yields the smallest number that is a

multiple of both of the given numbers.

Build-up Method

Step 1

As in the prime factorization method, express the numbers 24 and 36 in their prime

factor exponential form: 24

23

=

¥ 3 and 36 22 32

=

¥ .

Step 2

Select the prime factorization of one of the numbers and build the LCM from that as

follows. Beginning with 24

23

=

⋅ 3, compare it to the prime factorization of 36 22 32

=

⋅ .

Because 22 32

⋅ has more factors of three than 23 31

⋅ , build up the 23 31

⋅ to have the

same number of factors of three as 22 32

¥ , making the LCM 23 32

⋅ . If there are more

than two numbers for which the LCM is to be found, continue to compare and build

with each subsequent number.

c05.indd 195

7/30/2013 2:57:33 PM - 196 Chapter 5 Number Theory

Find the LCM(42, 24) in three ways.

S O L U T I O N

Set Intersection Method

Multiples of 42: { ,

42

,

84

,

126

,

168 . . . .}.

Multiples of 24: { ,

24

,

48

,

72

,

96

,

120

,

144

,

168 . . . .}.

Common multiples: {

,

168 . . .}.

LCM(42, 24) = 168.

Prime Factorization Method

42 = 2 ¥ 3 ¥ 7 and 24 23

=

¥ 3.

LCM(

,

42 24)

23

=

¥ 3 ¥ 7 = 168.

Build-up Method

42 = 2 ¥ 3 ¥ 7 and 24 23

=

¥ 3.

Beginning with 24

23

=

¥ 3, compare to 2 ¥ 3 ¥ 7 and build 23 ¥ 3 up to 23 ¥ 3 ¥ 7.

LCM(

,

42 24)

23

=

¥ 3 ¥ 7 = 168.

Notice that all primes from either number are used when forming the least common

multiple.

■

Check for Understanding: Exercise/Problem Set A #9–11

✔

In the diagram at the right, the circle labeled F

F

F

48 contains

48

72

the prime factors of 48 and the circle labeled F contains

72

the prime factors of 72. Place all the prime factors of 48 and 72 in the appropriate location

in the Venn diagram at the right.

How are the GCF(48, 72) and the LCM(48, 72) related to the placement of the numbers in

the diagram?

Extending the Concepts of GCF and LCM These methods can also be

applied to find the GCF and LCM of several numbers.

Find the (a) GCF and (b) LCM of the three numbers

25 32 57, 24 34 53

¥ ¥

¥ ¥ ¥ 7, and 2 36 54 132

¥ ¥ ¥

.

S O L U T I O N

a. The GCF is 21 32 53

¥ ¥ (use the common primes and the smallest respective

exponents).

b. Using the build-up method, begin with 25 32 57

¥ ¥ . Then compare it to 24 34 53

¥ ¥ ¥ 7

and build up the LCM to 25 34 57

¥ ¥ ¥ 7. Now compare with 2 36 54 132

¥ ¥ ¥

and build

up 25 34 57 7 to 25 36 57 7 132

¥ ¥ ¥

¥ ¥ ¥ ¥

.

■

If you are trying to find the GCF of several numbers that are not in prime-factored

exponential form, as in Example 5.14, you may want to use a computer program. By

considering examples in exponential notation, one can observe that the GCF of a, b,

and c can be found by finding GCF(a, b) first and then GCF(GCF(a, b), c). This idea

can be extended to as many numbers as you wish. Thus one can use the Euclidean

algorithm by finding GCFs of numbers, two at a time. For example, to find GCF(24,

36, 160), find GCF(24, 36), which is 12, and then find GCF(12, 160), which is 4.

Finally, there is a very useful connection between the GCF and LCM of two num-

bers, as illustrated in the next example.

c05.indd 196

7/30/2013 2:57:36 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 197

Find the GCF and LCM of a and b, for the numbers

a = 25 37 52

¥ ¥ ¥ 7 and b = 23 32 56

¥ ¥ ¥11.

S O L U T I O N Notice in the following solution that the products of the factors of a

and b, which are in bold type, make up the GCF, and the products of the remaining

factors, which are circled, make up the LCM.

Hence

GCF(a, b) × LCM(a, b) = (23 ¥ 32 ¥ 52 )(25 ¥ 37 ¥ 56 ¥ 7 ¥1 )

1

= (25 ¥ 37 ¥ 52 ¥ 7) ¥ (23 32 56

¥ ¥ ¥11)

= a × .

b

■

Example 5.15 illustrates that all of the prime factors and their exponents from the

original number are accounted for in the GCF and LCM. This relationship is stated next.

T H E O R E M 5 . 1 3

Let a and b be any two whole numbers. Then

GCF(a, b) × LCM(a, b) = ab.

ab

Also, LCM(a, b) =

is a consequence of this theorem. So if the GCF of two

GC (

F a, b)

numbers is known, the LCM can be found using the GCF.

Find the LCM(36, 56).

36 ¥ 56

S O L U T I O N GCF(36, 56) = 4. Therefore, LCM =

= 9 ¥ 56 = 504.

■

4

This technique applies only to the case of finding the GCF and LCM of two num-

bers.

We end this chapter with an important result regarding the primes by proving that

there is an infinite number of primes.

T H E O R E M 5 . 1 4

There is an infinite number of primes.

P R O O F

Problem-Solving Strategy

Either there is a finite number of primes or there is an infinite number of primes. We

Use Indirect Reasoning

will use indirect reasoning. Let us assume that there is only a finite number of primes,

say 2, 3, 5, 7,11, . . . , ,

p where p is the greatest prime. Let N = (2 ¥ 3 ¥ 5 ¥ 7 ¥11. . . p) + 1.

This number, N , must be 1, prime, or composite. Clearly, N is greater than 1. Also,

N is greater than any prime. But then, if N is composite, it must have a prime factor.

Yet whenever N is divided by a prime, the remainder is always 1 (think about this)!

Therefore, N is neither 1, nor a prime, nor a composite. But that is impossible. Using

indirect reasoning, we conclude that there must be an infinite number of primes. ■

c05.indd 197

7/30/2013 2:57:38 PM - 198 Chapter 5 Number Theory

There are also infinitely many composite numbers (for example, the even numbers

greater than 2).

Check for Understanding: Exercise/Problem Set A #12–17

✔

On January 25, 2013, a new large Mersenne prime was found. Mersenne

primes are special primes that take on a certain form, 2n– 1. This largest

known prime is 257885161 – 1 and has 17,425,170 digits. The second largest

known prime was found on August 23, 2008 the prime 243112609 – 1 was

found and verified to have 12,978,189 digits. Typically each new prime

that is found is larger than the previous ones. However, between 2008

and 2013, two other large primes were found but neither was as large

as the prime found in August 2008. If the largest prime, found in 2013,

were written out in a typical newsprint size, it would fill about 900 pages

of a newspaper. Why search for such large primes? One reason is that it

requires trillions of calculations and hence can be used to test computer

speed and reliability. Also, it is important in writing messages in code.

Besides, as a computer expert put it, “it's like Mount Everest; why do

people climb mountains?” To keep up on the ongoing race to find a big-

ger prime, visit the Web site www.mersenne.org.

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. How many factors do each of the following numbers have?

8. Two counting numbers are relatively prime if the greatest

a. 22 × 3

b. 33

52

×

c. 52

73

114

×

×

common factor of the two numbers is 1. Which of the fol-

lowing pairs of numbers are relatively prime?

2. a. Factor 36 into primes.

a. 4 and 9

b. 24 and 123

c. 12 and 45

b. Factor each divisor of 36 into primes.

c. What relationship exists between the prime factors

9. Use the set intersection method to find the following

in part b and the prime factors in part a?

LCMs.

d. Let x = 74 × 172. If n is a divisor of x, what can you say

a. LCM(24, 30)

about the prime factorization of n?

b. LCM(42, 28)

e. How many factors does x = 74 172

¥

have? List them.

c. LCM(12, 14)

3. Use the set intersection method to find the GCFs.

10. Use the (i) prime factorization method and the (ii) build-

a. GCF(12, 18)

b. GCF(42, 28)

c. GCF(60, 84)

up method to find the following LCMs.

a. LCM(6, 8)

b. LCM(4, 10)

4. Use the prime factorization method to find the GCFs.

c. LCM(7, 9)

d. LCM(8, 10)

a. GCF(8, 18) b. GCF(36, 42)

c. GCF(24, 66)

5. Using a calculator method, find the following.

11. Find the following LCMs using any method.

a. GCF(138, 102) b. GCF(484, 363)

a. LCM(60, 72)

b. LCM(35, 110)

c. LCM(45, 27)

c. GCF(297, 204) d. GCF(222, 2222)

12. Use any method to find the following GCFs.

6. Using the Euclidean algorithm, find the following GCFs.

a. GCF(12, 60, 90)

b. GCF(55, 75, 245)

a. GCF(24, 54)

c. GCF(1105, 1729, 3289)

d. GCF(1421, 1827, 2523)

b. GCF(39, 91)

e. GCF(33 55 119, 34 53 117, 37 57 117, 311 56 115

¥ ¥

¥ ¥

¥ ¥

¥ ¥

)

c. GCF(72, 160)

f. GCF(23 34 112

,

13 22 36 7 132, 24 35 53

¥ ¥

¥

¥ ¥ ¥

¥ ¥ ¥13)

d. GCF(5291, 11951)

13. Use any method to find the following LCMs.

7. Use any method to find the following GCFs.

a. LCM(2, 3, 5)

b. LCM(8, 12, 18)

a. GCF(51, 85)

b. GCF(45, 75)

c. LCM(33 55 119, 311 56 115, 24 35 53

¥ ¥

¥ ¥

¥ ¥ ¥13)

c. GCF(42, 385)

d. GCF(117, 195)

d. LCM(23 34 112

,

13 22 36 7 132, 37 57 117

¥ ¥

¥

¥ ¥ ¥

¥ ¥

)

c05.indd 198

7/30/2013 2:57:40 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 199

14. Another method of finding the LCM of two or more num-

16. For each of the pairs of numbers in parts a − c below

bers is shown. Find the LCM(27, 36, 45, 60).

i. sketch a Venn diagram with the prime factors of a and b

Use this method to find the following LCMs.

in the appropriate locations.

2 27

36

45

60

Divide all even numbers by 2. If not

2 27

18

45

30

divisible by 2, bring down. Repeat

3 27

9

45

15

until none are divisible by 2.

3

9

3

15

5

Proceed to the next prime number

3

3

1

5

5

and repeat the process.

5

1

1

5

5

Continue until the last row is all ones.

1

1

1

1

LCM = 22 · 33 · 5 (see the left column)

a. LCM(21, 24, 63, 70)

ii. find GCF(a, b) and LCM(a, b).

b. LCM(20, 36, 42, 33)

c. LCM(15, 35, 42, 80)

Use the Chapter 5 eManipulative Factor Tree on our Web

site to better understand the use of the Venn diagram.

15. a. For a = 91, b = 39 find GCF(a, b) and a ¥ b. Use these

a. a = 63, b = 90

two values to find LCM(a, b).

b. a = 48, b = 40

b. For a = 36 73 115, b = 23 34 75

¥ ¥

¥ ¥ find LCM(a, b) and a ¥ b.

c. a = 16, b = 49

Use these values to find

GCF(a, b).

17. Which is larger, GCF(12, 18) or LCM(12, 18)?

PROBLEMS

18. The factors of a number that are less than the number

Determine which of the following pairs are betrothed.

itself are called proper factors. The Pythagoreans classified

a. (140, 195)

b. (1575, 1648)

c. (2024, 2295)

numbers as deficient, abundant, or perfect, depending on

the sum of their proper factors.

21. In the following problems, you are given three pieces of

a. A number is deficient if the sum of its proper factors is less

information. Use them to answer the question.

than the number. For example, the proper factors of 4 are

a. GCF( ,

a b) = 2 × ,

3

2

3

1 and 2. Since 1 + 2 = 3 < 4, 4 is a deficient number. What

LCM(a, b) = 2 × 3 × ,

5

other numbers less than 25 are deficient?

b = 22 × 3 × 5. What is a?

b. A number is abundant if the sum of its proper factors is

b. GCF( ,

a b) = 22 × 7 × 1 ,

1

greater than the number. Which numbers less than 25

LCM( ,

a b) = 25 × 32 × 5 × 73 × 112,

are abundant?

b = 25 × 32 × 5 × 7 × 11. What is a?

c. A number is perfect if the sum of its proper factors is

equal to the number. Which number less than 25 is

22. What is the smallest whole number having exactly the

perfect?

following number of divisors?

a. 1

b. 2

c. 3

d. 4

19. A pair of whole numbers is called amicable if each is the

e. 5

f. 6

g. 7

h. 8

sum of the proper divisors of the other. For example, 284

and 220 are amicable, since the proper divisors of 220 are 1,

23. Find six examples of whole numbers that have the follow-

2, 4, 5, 10, 11, 20, 22, 44, 55, 110, which sum to 284, whose

ing number of factors. Then try to characterize the set of

proper divisors are 1, 2, 4, 71, 142, which sum to 220.

numbers you found in each case.

a. 2

b. 3

c. 4

d. 5

Determine which of the following pairs are amicable.

n − 1

n

a. 1184 and 1210

24. Euclid (300 B.C.E.) proved that 2

(2 − 1) produced a

n −

b. 1254 and 1832

perfect number [see Exercise 13(c)] whenever 2 1 is prime,

c. 5020 and 5564

where n = 1, 2, 3, . . . . Find the first four such perfect num-

bers. (Note: Some 2000 years later, Euler proved that this

20. Two numbers are said to be betrothed if the sum of all

formula produces all even perfect numbers.)

proper factors greater than 1 of one number equals

25. Find all whole numbers x such that GCF(24, x) = 1 and

the other, and vice versa. For example, 48 and 75 are

1 ≤ x ≤ 24.

betrothed, since

48 = 3 + 5 + 15 + 25,

26. George made enough money by selling candy bars at 15

cents each to buy several cans of pop at 48 cents each. If

proper factors of 75 except for 1,

he had no money left over, what is the fewest number of

and

candy bars he could have sold?

75 = 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24,

27. Three chickens and one duck sold for as much as two

proper factors of 48 except for 1.

geese, whereas one chicken, two ducks, and three geese

c05.indd 199

7/30/2013 2:57:45 PM - 200 Chapter 5 Number Theory

were sold together for $25. What was the price of each

32. What is the least number of cards that could satisfy the

bird in an exact number of dollars?

following three conditions?

28. Which, if any, of the numbers in the set

If all the cards are put in two equal piles, there is one

{ ,

10

,

20

,

40

,

80

,

160 . . .} is a perfect square?

card left over.

29. What is the largest three-digit prime all of whose digits are

If all the cards are put in three equal piles, there is one

prime?

card left over.

30. Take any four-digit palindrome whose digits are all non-

If all the cards are put in five equal piles, there is one

zero and not all the same. Form a new palindrome by

card left over.

interchanging the unlike digits. Add these two numbers.

33. Show that the number 343 is divisible by 7. Then

Example: 8,448

prove or disprove: Any three-digit number of the form

+

+

+

+ =

4,884

100a

10b

a, where a

b

7, is divisible by 7.

13,332

34. In the set {18, 96, 54, 27, 42}, find the pair(s) of numbers

with the greatest GCF and the pair(s) with the smallest

a. Find a whole number greater than 1 that divides every

LCM.

such sum.

b. Find the largest such whole number.

35. Using the Chapter 5 eManipulative Fill ’n Pour on our

Web site, determine how to use container A and container

31. Fill in the following 4 × 4 additive magic square, which is

B to measure the described target amount.

comprised entirely of primes.

a. Container A = 8 ounces

3

61

19

37

Container B = 12 ounces

Target = 4 ounces

43

31

5

—

b. Container A = 7 ounces

—

—

—

29

Container B = 11 ounces

—

—

23

—

Target = 1 ounce

EXERCISE/PROBLEM SET B

EXERCISES

1. How many factors do each of the following numbers have?

7. Use any method to find the following GCFs.

a. 22

32

× b. 73 113

×

a. GCF(38, 68)

b. GCF(60, 126)

c. 711

196

7923

×

×

d. 124

c. GCF(56, 120)

d. GCF(338, 507)

e. How many factors does x = 115 133

¥

have? List them.

8. a. Show that 83,154,367 and 4 are relatively prime.

2. a. Factor 120 into primes.

b. Show that 165,342,985 and 13 are relatively prime.

b. Factor each divisor of 120 into primes.

c. Show that 165,342,985 and 33 are relatively prime.

c. What relationship exists between the prime factors in

9. Use the set intersection method to find the following

part b and the prime factors in part a?

LCMs.

d. Let x = 115 × 133. If n is a divisor of x, what can you say

a. LCM(15, 18) b. LCM(26, 39) c. LCM(36, 45)

about the prime factorization of n?

3. Use the set intersection method to find the GCFs.

10. Use the (i) prime factorization method and the (ii) build-

a. GCF(24, 16) b. GCF(48, 64) c. GCF(54, 72)

up method to find the following LCMs.

a. LCM(15, 21)

b. LCM(14, 35)

4. Use the prime factorization method to find the following

c. LCM(75, 100)

d. LCM(130, 182)

GCFs.

a. GCF(36, 54)

11. Find the following LCMs using any method.

b. GCF(16, 51)

a. LCM(21, 51)

c. GCF(136, 153)

b. LCM(111, 39)

c. LCM(125, 225)

5. Using a calculator method, find the following.

a. GCF(276, 54)

b. GCF(111, 111111)

12. Use any method to find the following GCFs.

c. GCF(399, 102)

d. GCF(12345, 54323)

a. GCF(15, 35, 42)

6. Using the Euclidean algorithm and your calculator, find the

b. GCF(28, 98, 154)

following GCFs.

c. GCF(1449, 1311, 1587)

a. GCF(2244, 418)

d. GCF(2737, 3553, 3757)

b. GCF(963, 657)

e. GCF(54 73 136, 56 75 112 134, 54 77 1311, 53 76 138

¥ ¥

¥ ¥

¥

¥ ¥

¥ ¥

)

c. GCF(7286, 1684)

f. GCF(34 5 74 112, 22 33 77 133, 35 72 11 132

¥ ¥ ¥

¥ ¥ ¥

¥ ¥ ¥

)

c05.indd 200

7/30/2013 2:57:47 PM - Section 5.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 201

13. Use any method to find the following LCMs.

a. LCM(4, 5, 6)

b. LCM(9, 15, 25)

c. LCM(33 55 74, 52 76 114, 34 73 113

¥ ¥

¥

¥ ¥

)

d. LCM(23 54 113, 38 72 137, 24 52 75 133

¥ ¥

¥ ¥

¥ ¥ ¥

)

14. Use the method described in Set A, Exercise 14 to find the

following LCMs.

a. LCM(12, 14, 45, 35)

b. LCM(54, 40, 44, 50)

c. LCM(39, 36, 77, 28)

ii. find GCF(a, b) and LCM(a, b).

15. a. For a = 49, b = 84 find GCF(a, b) and a ¥ b. Use these

Use the Chapter 5 eManipulative Factor Tree on our Web

two values to find LCM(a, b).

site to better understand the use of the Venn diagram.

b. Given LCM( ,

a b) = 27 54 75 114

¥ ¥ ¥

and

a. a = 30, b = 24

a ¥ b = 27 ¥ 57 ¥ 79 ¥114, find GCF(a, b).

b. a = 4, b = 27

c. Find two different candidates for a and b in part b.

c. a = 18, b = 45

16. For each of the pairs of numbers in parts a − c below

17. Let a and b represent two nonzero whole numbers. Which

i. sketch a Venn diagram with the prime factors of a and b

is larger, GCF(a, b) or LCM(a, b)?

in the appropriate locations.

PROBLEMS

18. Identify the following numbers as deficient, abundant, or

22. Let the letters p, q, and r represent different primes. Then

perfect. [See Set A, Problem 18(a)]

p2qr3 has 24 divisors. So would p23. Use p, q, and r to

a. 36 b. 28

describe all whole numbers having exactly the following

c. 60 d. 51

number of divisors.

a. 2 b. 3 c. 4 d. 5 e. 6 f. 12

19. Determine if the following pairs of numbers are amicable.

(See Set A, Problem 19)

23. Let a and b represent whole numbers. State the conditions

a. 1648, 1576

on a and b that make the following statements true.

b. 2620, 2924

a. GCF(a, b) = a

b. LCM(a, b) = a

c. If 17,296 is one of a pair of amicable numbers, what is

c. GCF( ,

a b) = a × b

d. LCM( ,

a b) = a × b

the other one? Be sure to check your work.

24. If GCF(x, y) = 1, what is GCF(x2, y2 )? Justify your

20. Determine if the following pairs of numbers are betrothed.

answer.

(See Set A, Problem 20).

25. It is claimed that every perfect number greater than 6 is

a. (248, 231)

the sum of consecutive odd cubes beginning with 1. For

b. (1050, 1925)

example, 28 13

33

= + . Determine whether the preceding

c. (1575, 1648)

statement is true for the perfect numbers 496 and 8128.

21. a. Complete the following table by listing the factors for

26. Plato supposedly guessed (and may have proved) that

the given numbers. Include 1 and the number itself as

there are only four relatively prime whole numbers that

factors.

satisfy both of the following equations simultaneously.

b. What kind of numbers have only two factors?

x2

y2

z2

and

x3

y3

z3

w3

+

=

+

+

=

c. What kind of numbers have an odd number of factors?

If x = 3 and y = 4 are two of the numbers, what are z

NUMBER

FACTORS

NUMBER OF FACTORS

and w?

1

1

1

27. Tilda’s car gets 34 miles per gallon and Naomi’s gets

2

1, 2

2

3

8 miles per gallon. When traveling from Washington,

4

D.C., to Philadelphia, they both used a whole number

5

of gallons of gasoline. How far is it from Philadelphia to

6

Washington, D.C.?

7

8

28. Three neighborhood dogs barked consistently last night.

9

Spot, Patches, and Lady began with a simultaneous bark

10

at 11 P.M. Then Spot barked every 4 minutes, Patches

11

every 3 minutes, and Lady every 5 minutes. Why did Mr.

12

Jones suddenly awaken at midnight?

13

14

29. The numbers 2, 5, and 9 are factors of my locker number

15

and there are 12 factors in all. What is my locker number,

16

and why?

c05.indd 201

7/30/2013 2:57:53 PM - 202 Chapter 5 Number Theory

30. Which number less than 70 has the greatest number of

Euclidean algorithm of at least 10 steps to find the GCF.

factors?

(Hint: The numbers are not necessarily large, but they can

be found by thinking of doing the algorithm backward.)

31. The theory of biorhythm states that there are three

“cycles” to your life:

Analyzing Student Thinking

3

4

The physical cycle: 23 days long

37. Colby used the ideas in this section to claim that 8 ¥ 9 has

The emotional cycle: 28 days long

(3 + )(

1 4 + )

1 = 20 factors. Is he correct? Explain.

The intellectual cycle: 33 days long

38. Eva is confused by the terms LCM and GCF. She says

If your cycles are together one day, in how many days will

that the LCM of two numbers is greater than the GCF

they be together again?

of two numbers. How can you help her understand this

apparent contradiction?

32. Show that the number 494 is divisible by 13. Then

prove or disprove: Any three-digit number of the form

39. A student noticed the terms LCD and GCD in another

100a + 10b + a, where a + b = 13, is divisible by 13.

math book. What do you think these abbreviations stand

for?

33. A Smith number is a counting number the sum of whose

digits is equal to the sum of all the digits of its prime

40. Brett knows that all primes have exactly two factors, but

factors. Prove that 4,937,775 (which was discovered by

does not think that there is anything special about whole

Harold Smith) is a Smith number.

numbers that have exactly three factors. How should you

respond?

34. a. Draw a 2 × 3 rectangular array of squares. If one diago-

nal is drawn in, how many squares will the diagonal go

41. Vivian says that her older brother told her that the GCF

through?

and LCM are useful when studying fractions. Is the older

b. Repeat for a 4 × 6 rectangular array.

brother correct in both cases? Explain.

c. Generalize this problem to an m × n array of

42. Cecilia says that she prefers to use the Set Intersection

squares.

method to find the GCF of two numbers, but Mandy

35. Ramanujan observed that 1729 was the smallest number

prefers the Prime Factorization Method. How can you

that was the sum of two cubes in two ways. Express 1729

help the students see the value of learning both methods?

as the sum of two cubes in two ways.

43. In the theorem proving that there is an infinite number

36. The Euclidean algorithm is an iterative process of find-

of primes, it considers (2 ¥ 3 ¥ 5 ¥ 7 ¥11¥ ⋅⋅⋅ ¥ p) + .

1 One of

ing the remainder over and over again in order to find

your students says that 2 ¥ 3 ¥ 5 ¥ 7 ¥11¥13 + 1 is not a prime

the GCF. Use the dynamic spreadsheet Euclidean on our

because it is equal to 59 ¥ 509. Is the student correct? Does

Web site to identify two numbers that will require the

this invalidate the theorem? Explain.

END OF CHAPTER MATERIAL

control the number of winners, a simple way is to include only

1000 cards with the number 1 on them.

A major fast-food chain held a contest to promote sales. With

Additional Problems Where the Strategy “Use Properties of

each purchase a customer was given a card with a whole num-

Numbers” Is Useful

ber less than 100 on it. A $100 prize was given to any person

who presented cards whose numbers totaled 100. The following

1. How old is Mary?

are several typical cards. Can you find a winning combination?

r She is younger than 75 years old.

3

9

12

15

18

27

51

72

84

r Her age is an odd number.

Can you suggest how the contest could be structured so that

r The sum of the digits of her age is 8.

there would be at most 1000 winners throughout the country?

r Her age is a prime number.

Strategy: Use Properties of Numbers

r She has three great-grandchildren.

Perhaps you noticed something interesting about the num-

2. A folding machine folds letters at a rate of 45 per minute

bers that were on sample cards—they are all multiples of 3.

and a stamping machine stamps folded letters at a rate of

From work in this chapter, we know that the sum of two (hence

60 per minute. What is the fewest number of each machine

any number of) multiples of 3 is a multiple of 3. Therefore, any

required so that all machines are kept busy?

combination of the given numbers will produce a sum that is a

multiple of 3. Since 100 is not a multiple of 3, it is impossible to

3. Find infi nitely many natural numbers each of which has

win with the given numbers. Although there are several ways to

exactly 91 factors.

c05.indd 202

7/30/2013 2:57:55 PM - Chapter Review 203

Srinivasa Ramanujan

Constance Bowman

(1887–1920)

Reid (1918–2010)

e

Srinivasa Ramanujan devel-

Constance Bowman

c

oped a passion for mathemat-

Reid, Julia Bowman

ics when he was a young man

Robinson’s older sister,

in India. Working from nu-

became a high school

, The University of

merical examples, he arrived

English and journalism

SPL/Science Sour

at astounding results in num-

Paul Halmos Collection,

e_ph_0249_01 , The Dolph

Briscoe Center for American

History

Texas at Austin

teacher. She gave up

ber theory. Yet he had little formal training beyond high

teaching after marriage to become a freelance writer. She

school and had only vague notions of the principles of

coauthored Slacks and Calluses, a book about two young

mathematical proof. In 1913 he sent some of his results

teachers who, in 1943, decided to help the war effort dur-

to the English mathematician George Hardy, who was

ing their summer vacation by working on a B-24 bomber

astounded at the raw genius of the work. Hardy ar-

production line. Reid became fascinated with number

ranged for the poverty-stricken young man to come to

theory because of discussions with her mathematician

England. Hardy became his mentor and teacher but

sister. Her fi rst popular exposition of mathematics in a

later remarked, “I learned from him much more than

1952 Scientifi c American article was about fi nding per-

he learned from me.” After several years in England,

fect numbers using a computer. One of the readers com-

Ramanujan’s health declined. He went home to India

plained that Scientifi c American articles should be written

in 1919 and died of tuberculosis the following year. On

by recognized authorities, not by housewives! Reid has

one occasion, Ramanujan was ill in bed. Hardy went to

written the popular books From Zero to Infi nity, A Long

visit, arriving in taxicab number 1729. He remarked to

Way from Euclid, and Introduction to Higher Mathemat-

Ramanujan that the number seemed rather dull, and he

ics as well as biographies of mathematicians. She also

hoped it wasn’t a bad omen. “No,” said Ramanujan, “it

wrote Julie: A Life of Mathematics, which contains an

is a very interesting number; it is the smallest number

autobiography of Julia Robinson and three articles about

expressible as a sum of cubes in two different ways.”

Julia’s work by outstanding mathematical colleagues.

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy

reference.

Primes, Composites, and Tests for Divisibility

Vocabulary/Notation

Prime number 177

Fundamental Theorem of

Factor 178

Composite number 177

Arithmetic 178

Multiple 178

Sieve of Eratosthenes 177

Divides (a | b) 178

Is divisible by 178

Factor tree 177

Does not divide (a Ը b) 178

Square root 184

Prime factorization 178

Divisor 178

Prime Factor Test 184

Exercises

1. Find the prime factorization of 17,017.

e. 21 is a factor of 63.

2. Find all the composite numbers between 90 and 100

f. 123 is a multiple of 3.

inclusive.

g. 81 is divisible by 27.

h. 169 = 13.

3. True or false?`

a. 51 is a prime number.

4. Using tests for divisibility, determine whether 2, 3, 4, 5, 6,

b. 101 is a composite number.

8, 9, 10, or 11 are factors of 3,963,960.

c. 7 | 91.

d. 24 is a divisor of 36.

5. Invent a test for divisibility by 25.

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7/30/2013 2:58:02 PM - 204 Chapter 5 Number Theory

Counting Factors, Greatest Common Factor, and Least

Common Multiple

Vocabulary/Notation

Greatest common factor

Euclidean algorithm 193

Least common multiple

[GCF (a, b)] 191

[LCM (a, b)] 194

Exercises

1. How many factors does 35 73

¥ have?

5. Given that there is an infinite number of primes, show that

there is an infinite number of composite numbers.

2. Find GCF(144, 108) using the prime factorization method.

6. Explain how the four numbers 81, 135, GCF(81, 135), and

3. Find GCF(54, 189) using the Euclidean algorithm.

LCM(81, 135) are related.

4. Find LCM(144, 108).

CHAPTER TEST

Knowledge

Understanding

1. True or false?

9. Explain how the Sieve of Eratosthenes can be used to find

a. Every prime number is odd.

composite numbers.

b. The Sieve of Eratosthenes is used to find primes.

10. Is it possible to find nonzero whole numbers x and y such

c. A number is divisible by 6 if it is divisible by 2 and 3.

that 7x = 11y ? Why or why not?

d. A number is divisible by 8 if it is divisible by 4 and 2.

11. Show that the sum of any four consecutive counting num-

e. If a ≠ b, then GCF(a, b) < LCM(a, b).

bers must have a factor of 2.

f. The number of factors of n can be determined by the

exponents in its prime factorization.

12. a. Show why the following statement is not true. “If 4 | m

g. The prime factorization of a and b can be used to find

and 6 | m then 24 | m.”

the GCF and LCM of a and b.

b. Devise a divisibility test for 18.

h. The larger a number, the more prime factors it has.

13. Given that a ¥ b = 270 and GCF(a, b) = 3, find LCM(a, b).

i. The number 12 is a multiple of 36.

14. Use rectangular arrays to illustrate why 4 | 8 but 3 Ը 8.

j. Every counting number has more multiples than factors.

15. If n = 2 ¥ 3 ¥ 2 ¥ 7 ¥ 2 ¥ 3 and m = 2 22 32

¥ ¥ ¥ 7, how are m and n

2. Write a complete sentence that conveys the meaning of and

related? Justify your answer.

correctly uses each of the following terms or phrases.

a. divided into b. divided by c. divides

Problem Solving/Application

Skill

16. Find the smallest number that has factors of 2, 3, 4, 5, 6,

7, 8, 9, and 10.

3. Find the prime factorization of each of the following numbers.

17. The primes 2 and 5 differ by 3. Prove that this is the only

a. 120 b. 10,800 c. 819

pair of primes whose difference is 3.

4. Test the following for divisibility by 2, 3, 4, 5, 6, 8, 9, 10,

18. If a = 22 33

¥ and the LCM(a, b) is 1080, what is the (a)

and 11.

smallest and (b) the largest that b can be?

a. 11,223,344 b. 6,543,210 c. 23 34 56

¥ ¥

19. Find the longest string of consecutive composite numbers

5. Determine the number of factors for each of the following

between 1 and 50. What are those numbers?

numbers.

20. Identify all numbers between 1 and 20 that have an odd

a. 360 b. 216 c. 900

number of divisors. How are these numbers related?

6. Find the GCF and LCM of each of the following pairs of

21. What is the maximum value of n that makes the statement

numbers.

2n |15 ¥14 ¥13 ¥12 ⋅⋅⋅ 3 ¥ 2 ¥1 true?

a. 144, 120

b. 147, 70

c. 23 35 57, 27 34 53

¥ ¥

¥ ¥

d. 2419, 2173

22. Find two pairs of numbers a and b such that

e. 45, 175, 42, 60

GCF(a, b) = 15 and LCM(a, b) = 180.

23. When Alpesh sorts his marbles, he notices that if he puts

7. Use the Euclidean algorithm to find GCF(6273, 1025).

them into groups of 5, he has 1 left over. When he puts

8. Find the LCM(18, 24) by using the (i) set intersection

them in groups of 7, he also has 1 left over, but in groups

method, (ii) prime factorization method, and (iii) build-up

of 6, he has none left over. What is the smallest number of

method.

marbles that he could have?

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7/30/2013 2:58:06 PM - c05.indd 205

7/30/2013 2:58:06 PM - C H A P T E R

6 FRACTIONS

Fractions—A Historical Sketch

The first extensive treatment of fractions known to Although the symbol looks similar to the

us appears in the Ahmes (or Rhind) Mathematical

zero,

, in the Mayan number system, the Egyptians

Papyrus (1600 B.C.E.), which contains the work of

Egyptian mathematicians.

used it to denote the unit fraction with the denominator of

the fraction written below the

.

Because of its common use, the Egyptians did not write

2 as the sum 1

1

+ , but rather wrote it as a unit fraction

3

2

6

with a denominator of 3 . Therefore, 2 was written as the

2

3

reciprocal of 3 with the special symbol shown here.

2

Our present way of expressing fractions is probably due

onoz/NewsCom

to the Hindus. Brahmagupta (circa 630 C.E.) wrote the sym-

bol 2 (with no bar) to represent “two-thirds.” The Arabs

3

lbum/Or

A

introduced the “bar” to separate the two parts of a frac-

tion, but this first attempt did not catch on. Later, due to

The Egyptians expressed fractions as unit fractions (that

typesetting constraints, the bar was omitted and, at times,

is, fractions in which the numerator is 1). Thus, if they

the fraction “two-thirds” was written as 2/3.

wanted to describe how much fish each person would

The name fraction comes from the Latin word fractus,

get if they were dividing 5 fish among 8 people, they

which means “to break.” The term numerator comes

wouldn’t write it as 5 but would express it as 1

1

+ .

8

2

8

from the Latin word numerare, which means “to num-

Of course, the Egyptians used hieroglyphics to represent

ber” or to count and the name denominator comes from

these unit fractions.

nomen or name. Therefore, in the fraction two-thirds

2

( ), the denominator tells the reader the name of the

3

objects (thirds) and the numerator tells the number of

the objects (two).

206

c06.indd 206

7/30/2013 2:58:37 PM - Problem-Solving

Solve an Equivalent Problem

Strategies

One’s point of view or interpretation of a problem can often change a seemingly

1. Guess and Test

difficult problem into one that is easily solvable. One way to solve the next problem

2. Draw a Picture

is by drawing a picture or, perhaps, by actually finding some representative blocks

to try various combinations. On the other hand, another approach is to see whether

3. Use a Variable

the problem can be restated in an equivalent form, say, using numbers. Then if the

4. Look for a

equivalent problem can be solved, the solution can be interpreted to yield an answer

Pattern

to the original problem.

5. Make a List

Initial Problem

6. Solve a Simpler

A child has a set of 10 cubical blocks. The lengths of the edges are 1 cm, 2 cm, 3 cm,

Problem

. . . , 10 cm. Using all the cubes, can the child build two towers of the same height by

7. Draw a Diagram

stacking one cube upon another? Why or why not?

8. Use Direct

Reasoning

9. Use Indirect

Reasoning

10. Use Properties of

Numbers

11. Solve an

Equivalent

Problem

Clues

The Solve an Equivalent Problem strategy may be appropriate when

r You can find an equivalent problem that is easier to solve.

r A problem is related to another problem you have solved previously.

r A problem can be represented in a more familiar setting.

r A geometric problem can be represented algebraically, or vice versa.

r Physical problems can easily be represented with numbers or symbols.

A solution of this Initial Problem is on page 247.

207

c06.indd 207

8/2/2013 3:13:11 PM - AUTHOR

I N T R O D U C T I O N

Chapters 2 to 5 have been devoted to the study of the system of whole numbers. Understanding the

system of whole numbers is necessary to ensure success in mathematics later. This chapter is devoted

to the study of fractions. Fractions were invented because it was not convenient to describe many

problem situations using only whole numbers. As you study this chapter, note the importance that

WALK-THROUGH the whole numbers play in helping to make fraction concepts easy to understand.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-2–NUMBER AND OPERATIONS

Understand and represent commonly used fractions, such as 1/4, 1/3, and 1/2.

r GRADES 3-5–NUMBER AND OPERATIONS

Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and

as divisions of whole numbers.

Use models, benchmarks, and equivalent forms to judge the size of fractions.

r GRADES 6-8–NUMBER AND OPERATIONS

Understand the meaning and effects of arithmetic operations with fractions.

Develop and analyze algorithms for computing with fractions and develop fluency in their use.

Key Concepts from the NCTM Curriculum Focal Points

r GRADE 3: Developing an understanding of fractions and fraction equivalence.

r GRADE 5: Developing an understanding of and fluency with addition and subtraction of fractions and decimals.

r GRADE 6: Developing an understanding of and fluency with multiplication and division of fractions and decimals.

Key Concepts from the Common Core State Standards for Mathematics

r GRADE 3: Develop understanding of fractions as numbers. Specifically, explain equivalence of fractions and com-

pare fractions with the same numerator or same denominator.

r GRADE 4: Extend understanding of fraction equivalence and ordering by comparing two fractions with different

numerators and different denominators. Add and subtract fractions with like denominators. Multiply fractions

by whole numbers.

r GRADE 5: Use equivalent fractions as a strategy to add and subtract fractions with unlike denominators. Apply

and extend previous understandings of multiplication and division to multiply fractions by fractions and divide

whole numbers by a unit fraction or a unit fraction by a nonzero whole number.

r GRADE 6: Apply and extend previous understandings of multiplication and division to divide fractions by

fractions.

6.1

208

c06.indd 208

7/30/2013 2:58:38 PM - Section 6.1

The Set of Fractions 209

THE SET OF FRACTIONS

For each of the following visual representations of fractions, there is a corresponding

incorrect symbolic expression. Discuss what aspects of the visual representation might

lead a student to the incorrect expression.

Children’s Literature

The Concept of a Fraction

www.wiley.com/college/musser

See “Give me Half” by Stuart J.

There are many times when whole numbers do not fully describe a mathematical situ-

Murphy.

ation. For example, using whole numbers, try to answer the following questions that

refer to Figure 6.1: (1) How much pizza is left? (2) How much of the stick is shaded?

(3) How much paint is left in the can?

Algebraic Reasoning

While the equation x − 6 = 2 can

be solved using whole numbers,

an equation such as 3x = 2 has a

fraction as a solution.

Figure 6.1

Reflection from Research

Although it is not easy to provide whole-number answers to the preceding questions,

Students need experiences that

the situations in Figure 6.1 can be conveniently described using fractions. Reconsider

build on their informal frac-

the preceding questions in light of the subdivisions added in Figure 6.2. Typical answers

tion knowledge before they are

introduced to fraction symbols

to these questions are (1) “Three-fourths of the pizza is left,” (2) “Four-tenths of the

(Sowder & Schappelle, 1994).

stick is shaded,” (3) “The paint can is three-fifths full.”

Reflection from Research

When using area models to

represent fractions, it should

be noted that students often

confuse length and area (Kouba,

Brown, Carpenter, Lindquist,

Silver, & Swafford, 1988).

Figure 6.2

The term fraction is used in two distinct ways in elementary mathematics. Initially,

fractions are used as numerals to indicate the number of parts of a whole to be consid-

ered. In Figure 6.2, the pizza was cut into 4 equivalent pieces, and 3 remain. In this case

we use the fraction 34 to represent the 3 out of 4 equivalent pieces (i.e., equivalent in size).

The use of a fraction as a numeral in this way is commonly called the “part-to-whole”

a

model. Succinctly, if a and b are whole numbers, where b ≠ 0, then the fraction or a b

/ ,

b

represents a of b equivalent parts; a is called the numerator and b is called the denominator.

c06.indd 209

7/30/2013 2:58:39 PM - 210 Chapter 6 Fractions

The word denominator comes from the word nomenclature which means “to name.”

Thus, the denominator tells you the name of the parts: fourths, tenths, fifths, etc. The

word numerator means “to number” so the numerator is the number of parts.

Reflection from Research

The term equivalent parts means equivalent in some attribute, such as length, area,

Students often think of a fraction

volume, number, or weight, depending on the composition of the whole and appro-

as two numbers separated by a

priate parts. In Figure 6.2, since 4 of 10 equivalent parts of the stick are shaded, the

line rather than as representing

fraction 4 describes the shaded part when it is compared to the whole stick. Also,

a single number (Bana, Farrell, &

10

McIntosh, 1995).

the fraction 3 describes the filled portion of the paint can in Figure 6.2.

5

As with whole numbers, a fraction also has an abstract meaning as a number.

What do you think of when you look at the relative amounts represented by the

NCTM Standard

shaded regions in Figure 6.3?

All students should develop

understanding of fractions as

parts of unit wholes, as a part

of a collection, as locations on

number lines, and as divisions of

whole numbers.

Figure 6.3

Although the various diagrams are different in size and shape, they share a com-

mon attribute—namely, that 5 of 8 equivalent parts are shaded. That is, the same

relative amount is shaded. This attribute can be represented by the fraction. 5 . Thus,

8

in addition to representing parts of a whole, a fraction is viewed as a number repre-

senting a relative amount. Hence we make the following definition.

D E F I N I T I O N 6 . 1

Reflection from Research

Students need to develop not

Fractions

only a conceptual understanding

of fractions, but also an under-

A fraction is a number that can be represented by an ordered pair of whole num-

a

standing of the proper notation

bers (or a b

/ ), where b ≠ 0. In set notation, the set of fractions is

of fractions (Brizuela, 2006).

b

⎧a

⎫

F =

| a and b are whole numbers, b

⎨

≠ 0⎬.

⎩b

⎭

On a piece of paper explain, using pictures, why 3 -and 6 are equivalent. Compare and

4

8

contrast your explanation with that of your peers.

Common Core – Grade 3

1

Fractions can also be represented on a number line. The fraction is located on a

Understand a fraction 1/b as the

b

quantity formed by 1 part when a

number line by defining the interval from 0 to 1 as the whole and partitioning it into b

whole is partitioned into b equal

1

1

parts; understand a fraction a/b

equal parts. Each part is of length and the number is located at the right endpoint

as the quantity formed by a parts

b

b

of size 1/b.

of the part with its left endpoint at 0 as shown in Figure 6.4.

Figure 6.4

c06.indd 210

7/30/2013 2:58:41 PM - Section 6.1 The Set of Fractions 211

a

The fraction is located on the number line by starting at 0 and marking off a

b

1

copies of the length determined by the fraction end to end as shown in Figure 6.5.

b

a

The right endpoint of the ath copy is the location of the fraction .

b

Common Core – Grade 3

Understand a fraction as a

number on the number line;

represent fractions on a number

line diagram.

0

1

Figure 6.5

Before proceeding with the computational aspects of fractions as numbers, it is

instructive to comment further on the complexity of this topic—namely, viewing

fractions as numerals and as numbers. Recall that the whole number three was the

attribute common to all sets that match the set { ,

a ,

b }

c . Thus if a child who under-

stands the concept of a whole number is shown a set of three objects, the child will

answer the question “How many objects?” with the word “three” regardless of the

size, shape, color, and so on of the objects themselves. That is, it is the “numerous-

ness” of the set on which the child is focusing.

With fractions, there are two attributes that the child must observe. First, when consid-

ering a fraction as a number, the focus is on relative amount. For example, in Figure 6.6

the relative amount represented by the various shaded regions is described by the fraction

1 (which is considered as a number). Notice that 1 describes the relative amount shaded

4

4

without regard to size, shape, arrangement, orientation, number of equivalent parts, and

so on; thus it is the “numerousness” of a fraction on which we are focusing.

Reflection from Research

The complex nature of fractions

requires students to develop

Figure 6.6

more than a part-to-whole under-

standing. Students also need to

develop an understanding based

Second, when considering a fraction as a numeral representing a part-to-whole

on the quotient 3

( ÷ 4 3

= ) and

4

relationship, many numerals can be used for the relationship. For example, the three

measurement (location on a

diagrams in Figure 6.6 can be labeled differently (Figure 6.7).

number line) interpretations

of fractions (Flores, Samson, &

Yanik, 2006).

Figure 6.7

In Figures 6.7(b) and (c), the shaded regions have been renamed using the fractions

2 and 3 , respectively, to call attention to the different subdivisions. The notion of

8

12

c06.indd 211

7/30/2013 2:58:42 PM - 212 Chapter 6 Fractions

fraction as a numeral displaying a part-to-whole relationship can be merged with the

concept of fraction as a number. That is, the fraction (number) 1 can also be thought

4

NCTM Standard

of and represented by any of the fractions 2 , 3 , 4 , and so on. Figure 6.8 brings this

8

12

16

All students should understand

into sharper focus.

and represent commonly used

On a more general note, a model such as those in Figure 6.8, which uses geometric

fractions, such as 1 1

, , and 1 .

4

3

2

shapes or “regions” to represent fractions, is called a region model.

Figure 6.8

In each of the pairs of diagrams in Figure 6.8, the same relative amount is shaded,

although the subdivisions into equivalent parts and the sizes of the diagrams are dif-

ferent. As suggested by the shaded regions, the fractions 4 , 2 , and 3 all represent the

16

8

12

same relative amount as 1 .

4

Two fractions that represent the same relative amount are said to be equivalent

fractions.

The diagrams in Figure 6.8 were different shapes and sizes. Fraction strips, which

can be constructed out of paper or cardboard, can be used to visualize fractional

parts (Figure 6.9). The advantage of this model is that the unit strips are the same

size—only the shading and number of subdivisions vary.

Figure 6.9

Equivalent fractions are also located at the same point on a number line. The

equivalence of 3 and 6 can be seen by their common location on the number line in

4

8

Figure 6.10.

Figure 6.10

Reflection from Research

It is useful to have a simple test to determine whether fractions such as 2 , 3 , and 4

8

12

16

A child’s understanding of frac-

represent the same relative amount without having to draw a picture of each represen-

tions needs to coordinate the

knowledge of notation and the

tation. Two approaches can be taken. First, observe that 2

1¥ 2

3

1¥ 3

=

,

=

, and 4

1 4

= ¥

8

4 ¥ 2 12

4 ¥ 3

16

4 ¥ .

4

knowledge of some part-whole

These equations illustrate the fact that 2 can be obtained from 1 by equally subdivid-

relations. However, these two

8

4

types of knowledge develop

ing each portion of a representation of 1 by 2 [Figure 6.8(b)]. A similar argument can

4

independently (Saxe, Taylor,

be applied to the equations 3

1 3

McIntosh, & Gearhart, 2005).

= ¥

1 4

= ¥

12

4 ¥ [Figure 6.8(c)] and 4

3

16

4 ¥ [Figure 6.8(a)]. Thus it

4

1¥ n

appears that any fraction of the form

, where n is a counting number, represents

4 ¥ n

c06.indd 212

7/30/2013 2:58:45 PM - From Lesson 10-5 “Finding Equivalent Fractions” from Envision Math Common Core, by Randall I.Charles et al., Grade 3,

copyright © by Pearson Education.

213

c06.indd 213

7/30/2013 2:58:47 PM - 214 Chapter 6 Fractions

an

a

an

a

the same relative amount as 1 , or that

= , in general. When is replaced with ,

4

bn

b

bn

b

an

where n ≠ 1, we say that

has been simplified.

bn

To determine whether 3 and 4 are equal, we can simplify each of them: 3

1

=

12

16

12

4

and 4

1

= . Since they both equal 1 , we can write 3

4

= . Alternatively, we can view

16

4

4

12

16

3 and 4 as 3¥16

¥

¥

= and 4 ¥12 = 48 are

12

16

12 ¥

and 4 12

16

16 ¥

instead. Since the numerators 3 16

48

12

equal and the denominators are the same, namely 12 ¥16, the two fractions 3 and 4

12

16

must be equal. As the next diagram suggests, the numbers 3 ¥16 and 4 ¥12 are called

the cross-products of the fractions 3 and 4 .

12

16

The technique, which can be used for any pair of fractions, leads to the following

Common Core – Grade 3

definition of fraction equality.

Explain equivalence of fractions

in special cases, and understand

two fractions as equivalent

D E F I N I T I O N 6 . 2

(equal) if they are the same size,

or the same point on a number

Fraction Equality

line.

a

c

a

c

Let and be any fractions. Then = if and only if ad = bc.

b

d

b

d

Algebraic Reasoning

In words, two fractions are equal fractions if and only if their cross-products, that

Although established on the prin-

is, products ad and bc obtained by cross-multiplication, are equal. The first method

ciples of equivalent fractions, the

described for determining whether two fractions are equal is an immediate conse-

process of cross-multiplication

quence of this definition, since a(bn) = b(an) by associativity and commutativity. This

is commonly used when solving

proportions algebraically.

is summarized next.

T H E O R E M 6 . 1

Common Core – Grade 4

a

Explain why a fraction a/b

Let be any fraction and n a nonzero whole number. Then

is equivalent to a fraction

b

(n × a)/(n × b) by using visual

a

an

na

fraction models, with attention

=

=

.

b

bn

nb

to how the number and size of

the parts differ even though the

two fractions themselves are

the same size. Use this principle

to recognize and generate

It is important to note that this theorem can be used in two ways: (1) to replace the

equivalent fractions.

a

an

an

a

fraction with

and (2) to replace the fraction

with . Occasionally, the term

b

bn

bn

b

reducing is used to describe the process in (2). However, the term reducing can be mis-

leading, since fractions are not reduced in size (the relative amount they represent) but

only in complexity (the numerators and denominators are smaller).

Verify the following equations using the definition of fraction

equality or the preceding theorem.

5

25

27

54

16

1

a. =

b.

=

c.

=

6

30

36

72

48

3

S O L U T I O N

5

5 ¥ 5

25

a. =

=

by the preceding theorem.

6

6 ¥ 5

30

c06.indd 214

7/30/2013 2:58:52 PM - Section 6.1 The Set of Fractions 215

54

27 ¥ 2

27

27

3 ¥ 9

3

b.

=

=

by simplifying. Alternatively,

=

= and

72

36 ¥ 2

36

36

4 ¥ 9

4

54

3 ¥18

3

=

=

27

54

, so

=

.

72

4 ¥18

4

36

72

16

1

c.

= , since their cross-products, 16 ¥ 3 and 48 ¥1, are equal.

■

48

3

Fraction equality can readily be checked on a calculator using an alternative version

a

c

ad

of cross-multiplication—namely, = if and only if

= c. Thus the equality 20 30

b

d

b

36 54

can be checked by pressing 20 × 54 ÷ 36 = 30 . Since 30 obtained in this way is

equal to the numerator of 30 , the two fractions are equal.

54

a

an

Since =

for n = 1, 2, 3, . . . , every fraction has an infinite number of representa-

b

bn

tions (numerals). For example,

1

2

3

4

5

6

= = = =

=

= ⋅⋅⋅

2

4

6

8

10

12

are different numerals for the number 1. In fact, another way to view a fraction as

2

a number is to think of the idea that is common to all of its various representations.

That is, the fraction 2 , as a number, is the idea that one associates with the set of all

3

fractions, as numerals, that are equivalent to 2 , namely 2 , 4 , 6 , 8 , . . . . Notice that the

3

3

6

9

12

term equal refers to fractions as numbers, while the term equivalent refers to fractions

as numerals.

Every whole number is a fraction and hence has an infinite number of fraction

representations. For example,

1

2

3

2

4

6

0

0

1 = =

= = ⋅⋅⋅ , 2 = = = = ⋅⋅⋅ , 0 = = = ⋅⋅⋅ ,

1

2

3

1

2

3

1

2

and so on. Thus the set of fractions extends the set of whole numbers.

A fraction is written in its simplest form or lowest terms when its numerator and

denominator have no common prime factors.

Find the simplest form of the following fractions.

12

36

9

a.

b.

c.

d. (23 35 57 )/(26 34

¥ ¥

¥ ¥ 7)

18

56

31

S O L U T I O N

12

2 ¥ 6

2

a.

=

=

18

3 ¥ 6

3

36

18 ¥ 2

18

9 ¥ 2

9

36

2 ¥ 2 ¥ 3 ¥ 3

3 ¥ 3

9

b.

=

=

=

=

or

=

=

=

56

28 ¥ 2

28

14 ¥ 2

14

56

2 ¥ 2 ¥ 2 ¥ 7

2 ¥ 7

14

9

c.

is in simplest form, since 9 and 31 have only 1 as a common factor.

31

d. To write (23 35 57 )/(26 34

¥ ¥

¥ ¥ 7) in simplest form, fi rst fi nd the GCF of 23 35 57

¥ ¥ and

26 34

¥ ¥ 7:

GCF(23 ¥ 35 ¥ 57 26 ¥ 34 ¥ 7 = 23 ¥ 34

,

)

.

Then

23 ¥ 35 ¥ 57

(3 ¥ 57 )(23 ¥ 34 )

3 ¥ 57

=

=

.

■

26 ¥ 34 ¥ 7

(23 ¥ 7)(23 ¥ 34 )

23 ¥ 7

c06.indd 215

7/30/2013 2:58:56 PM - 216

Chapter 6 Fractions

Reflection from Research

Fractions with numerators greater than or equal to their denominators fall into

When dealing with improper frac-

two categories. The fractions 1 , 2 , 3 , 4 , . . . , in which the numerators and denomina-

1

2

3

4

tions, like 12 , students struggle to

11

tors are equal, represent the whole number 1. Fractions where the numerators are

focus on the whole, 11, and not the

11

greater than the denominators are called improper fractions. For example, 7 , 8 , and

number of parts, 12 (Tzur, 1999).

2

5

117 are improper fractions. The fraction 7 would mean that an object was divided into

35

2

2 equivalent parts and 7 such parts are designated. Figure 6.11 illustrates a model for

7 because seven halves are shaded where one shaded circle represents one unit. The

2

diagram in Figure 6.11 illustrates that 7 can also be viewed as 3 wholes plus 1 , or 3 1 .

2

2

2

A combination of a whole number with a fraction juxtaposed to its right is called a

mixed number. Mixed numbers will be studied in Section 6.2.

Figure 6.11

The TI-34 MultiView calculator can be used to convert improper fractions to mixed

numbers. For example, to convert 1234 to a mixed number in its simplest form press

378

1234 n 378 enter which yields 3 100 . Pressing csimp enter yields 3 50 , which is in

d

378

189

simplest form. For calculators without the fraction function, 1234 ÷ 378 = 3 2645502

.

shows the whole-number part, namely 3, together with a decimal fraction. The calcula-

tion 1234 − 3 × 378 = 100 gives the numerator of the fraction part; thus 1234

100

= 3 ,

378

378

which is 3 50 in simplest form.

189

Check for Understanding: Exercise/Problem Set A #1–11

✔

Order the following fractions from smallest to largest without converting them to deci-

mals. Justify your method of ordering.

12

8

10

,

,

22

17

21

Ordering Fractions

The concepts of less than and greater than in fractions are extensions of the respective

whole-number concepts. Fraction strips can be used to order fractions (Figure 6.12).

Next consider the three pairs of fractions on the fraction number line in Figure 6.13.

Figure 6.12

Figure 6.13

Reflection from Research

Young children have a very diffi-

As it was in the case of whole numbers, the smaller of two fractions is to the

cult time believing that a fraction

left of the larger fraction on the fraction number line. Also, the three examples

such as one-eighth is smaller than

in Figure 6.13 suggest the following definition, where fractions having common

one-fifth because the number

denominators can be compared simply by comparing their numerators (which are

eight is larger than five (Post,

Wachsmuth, Lesh, & Behr, 1985).

whole numbers).

c06.indd 216

7/30/2013 2:59:00 PM - Section 6.1

The Set of Fractions 217

Common Core – Grade 3

D E F I N I T I O N 6 . 3

Compare two fractions with the

same numerator or the same

Less Than for Fractions

denominator by reasoning about

a

b

a

b

their size. Recognize that com-

Let

and be any fractions. Then < if and only if a < b.

c

c

c

c

parisons are valid only when the

two fractions refer to the same

whole. Record the results of com-

parisons with the symbols >, =,

NOTE: Although the definition is stated for “less than,” a corresponding state-

or <, and justify the conclusions

ment holds for “greater than.” Similar statements hold for “less than or equal to” and

(e.g., by using a visual fraction

“greater than or equal to.”

model).

For example, 3

5

< , since 3 < 5; 4 10

< , since 4 < 10; and so on. The numbers 2 and

7

7

13

13

7

4 can be compared by getting a common denominator.

13

Children’s Literature

2

2 ¥13

26

4

4 ¥ 7

28

www.wiley.com/college/musser

=

=

and

=

=

7

7

See “Fraction Action” by

¥13 91

13

13 ¥ 7

91

Loreen Leedy.

Since 26

28

< , we conclude that 2

4

< .

91

91

7

13

This last example suggests a convenient shortcut for comparing any two frac-

tions. To compare 2 and 4 , we compared 26 and 28 and, eventually, 26 and 28. But

7

13

91

91

26 = 2 ¥13 and 28 = 7 ¥ 4. In general, this example suggests the following theorem.

T H E O R E M 6 . 2

Cross-Multiplication of Fraction Inequality

a

c

a

c

Let

and be any fractions. Then < if and only if ad < bc.

b

d

b

d

Notice that this theorem reduces the ordering of fractions to the ordering of whole

a

c

c

a

c

a

numbers. Also, since < if and only if >

, we can observe that

> if and only

b

d

d

b

d

b

if bc > ad .

Arrange in order.

a. 7 and 9 b. 17 and 19

8

11

32

40

S O L U T I O N

a. 7

9

< if and only if 7 ¥11< 8 ¥ 9. But 77 > 72; therefore, 7

9

> .

8

11

8

11

b. 17 ¥ 40 = 680 and 32 ¥19 = 608. Since 32 ¥19 < 17 ¥ 40, we have 19

17

< .

■

40

32

Children’s Literature

Often fractions can be ordered mentally using your “fraction sense.” For example,

www.wiley.com/college/musser

fractions like 4 , 7 , 11 , and so on are close to 1, fractions like 1 , 1 , 1 , and so on are

5

8

12

15

14

10

See “Inchworm and a Half” by

close to 0, and fractions like 6 , 8

12

<

1

≈

14

, 11 , and so on are close to 1 . Thus 4

, since 4

15

20

2

7

13

7

2

Elinor Pinczes.

and 12 ≈ 1. Also, 1

7

< , since 1 ≈ 0 and 7

1

≈ .

13

11

13

11

13

2

Keep in mind that this procedure is just a shortcut for finding common denomina-

tors and comparing the numerators.

Cross-multiplication of fraction inequality can also be adapted to a T1-34 MultiView cal-

a

c

ad

a

c

ad

culator as follows:

< if and only if

< c (or > if and only if

> c). To order

b

d

b

b

d

b

17 and 19 using a fraction calculator, press 17 n 32 c × 40 enter which yields 21 8 ,

32

40

d

32

or 21 1 . Since 21 1 > 19, we conclude that 17

19

> . On a decimal calculator, the follow-

4

4

32

40

ing sequence leads to a similar result: 17 × 40 ÷ 32 =

21 2

. 5 . Since 21.25 is greater

than 19, 17

19

> . Ordering fractions using decimal equivalents, which is shown next, will

32

40

be covered in Chapter 7. In this case, 17 ÷ 32 = 0 53125

.

and 19 ÷ 40 =

0 475

.

.

Therefore, 19

17

< , since 0.475 < 0.53125. Finally, one could have ordered these two frac-

40

32

tions mentally by observing that 19 is less than 1

20

(= ), whereas 17 is greater than 1 16

(= ).

40

2

40

32

2

32

c06.indd 217

7/30/2013 2:59:07 PM - 218 Chapter 6 Fractions

Reflection from Research

On the whole-number line, there are gaps between the whole numbers (Figure 6.14).

A variety of alternative approach-

es and forms of presenting frac-

tions will help students build a

foundation that will allow them

to later deal in more systematic

ways with equivalent fractions,

convert fractions from one repre-

sentation to another, and move

Figure 6.14

back and forth between division

of whole numbers and rational

However, when fractions are introduced to make up the fraction number line,

numbers (Flores & Klein, 2005).

many new fractions appear in these gaps (Figure 6.15).

Figure 6.15

Unlike the case with whole numbers, it can be shown that there is a fraction

between any two fractions. For example, consider 3 and 5. Since 3

18

= and 5 20

= ,

4

6

4

24

6

24

we have that 19

24 is between 3

4 and 5 . Now consider 18 and 19 . These equal 36 and 38 ,

6

24

24

48

48

respectively; thus 37 is between 3 and 5 also. Continuing in this manner, one can show

48

4

6

that there are infinitely many fractions between 3 and 5 . From this it follows that

4

6

there are infinitely many fractions between any two different fractions.

Find a fraction between these pairs of fractions.

a. 7 and 8 b. 9 and 12

11

11

13

17

S O L U T I O N

a. 7

14

= and 8

16

= . Hence 15 is between 7 and 8 .

11

22

11

22

22

11

11

b. 9

9 ¥17

153

=

=

12 ¥13

156

=

=

13

13 ¥17

13 ¥

and 12

17

17

17 ¥ 3

17 ¥

. Hence both 154

13

13 ¥

and 155

17

13 ¥

are between 9

17

13

and 12 .

■

17

Sometimes students incorrectly add numerators and denominators to find the sum

of two fractions. It is interesting, though, that this simple technique does provide an

easy way to find a fraction between two given fractions. For example, for fractions

2 and 3 , the number 5 satisfies 2

5

3

< < since 2 ¥ 7 < 3 ¥ 5 and 5 ¥ 4 < 7 ¥ 3. This idea is

3

4

7

3

7

4

generalized next.

Algebraic Reasoning

T H E O R E M 6 . 3

Commutativity and distributivity

are introduced when children real-

a

c

a

c

Let and be any fractions, where

ize expressions like 2 + 5 = 5 + 2

< . Then

b

d

b

d

or 2(4 + 6) = 2 ¥ 4 + 2 ¥ 6 will work

a

a

c

c

for any numbers. These properties

< + < .

are also used to simplify algebraic

b

b + d

d

expressions like those used in the

proof at the right.

P R O O F

a

c

Let < . Then we have ad < bc. From this inequality, it follows that ad + ab < bc + ab,

b

d

or a(b + d ) < b(a + c). By cross-multiplication of fraction inequality, this last inequal-

a

a

c

ity implies <

+ , which is “half” of what we are to prove. The other half can be

b

b + d

proved in a similar fashion.

■

c06.indd 218

7/30/2013 2:59:12 PM - Section 6.1 The Set of Fractions 219

To find a fraction between 9 and 12 using this theorem, add the numerators and

13

17

denominators to obtain 21

30 . This theorem shows that there is a fraction between any

two fractions. The fact that there is a fraction between any two fractions is called the

density property of fractions.

Check for Understanding: Exercise/Problem Set A #12–15

✔

Although decimals are prevalent in our monetary system, we

commonly use the fraction terms “a quarter” and a “half dol-

lar.” It has been suggested that this tradition was based on

the Spanish milled dollar coin and its fractional parts that were

used in our American colonies. Its fractional parts were called

“bits” and each bit had a value of 12 1 cents (thus a quarter

2

is known as “two bits”). A familiar old jingle (which seems

impossible today) is “Shave and a haircut, two-bits.” Some

also believe that this fraction system evolved from the British

shilling and pence. A shilling was one-fourth of a dollar and a

sixpence was one-eighth of a dollar, or 12 1 cents.

2

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. What fraction is represented by the shaded portion of each

following set of objects, four out of the total of five objects

diagram?

are triangles.

a.

b.

We could say that 4 of the objects are triangles. This inter-

c.

d.

5

pretation of fractions compares part of a set with all of the

set. Draw pictures to represent the following fractions using

sets of nonequivalent objects.

a. 3 b. 3 c. 1

5

7

3

2. In addition to fraction strips, a region model, and a set

model, fractions can also be represented on the number

4. Does the following picture represent 1 ? Explain.

3

line by selecting a unit length and subdividing the interval

into equal parts. For example, to locate the fraction 3 ,

4

subdivide the interval from 0 to 1 into four parts and mark

off three as shown.

0

1

5. Using the diagram below, represent each of the

following as a fraction of all shapes shown.

Represent the following fractions using the given models.

i. 4 ii. 3

5

8

a. Region model

b. Set model

c. Fraction strips

d. Number line

3. In this section fractions were represented using equivalent

parts. Another representation uses a set model. In the

c06.indd 219

7/30/2013 2:59:14 PM - 220 Chapter 6 Fractions

a. What fraction is made of circular shapes?

11. Rewrite as a mixed number in simplest form.

b. What fraction is made of square shapes?

a. 525

b. 1234

96

432

c. What fraction is not made of triangular shapes?

12. a. Arrange each of the following from smallest to largest.

6. Fill in the blank with the correct fraction.

i. 11 13 12

,

,

ii. 1 1 1

, ,

17

17

17

5

6

7

a. 10 cents is _____ of a dollar.

b. What patterns do you observe?

b. 15 minutes is _____ of an hour.

c. If you sleep eight hours each night, you spend _____ of a

13. Arrange each of the following from smallest to largest.

day sleeping.

a. 14 3 9

, ,

17

25

,

,

27

7

20

b. 6

13

39

51

d. Using the information in part c, what part of a year do

14. The Chapter 6 eManipulative Comparing Fractions on our

you sleep?

Web site uses number lines and common denominators to

7. Use the area model illustrated on the Chapter 6 eManipu-

compare fractions. The eManipulative will have you plot

lative activity Equivalent Fractions on our Web site to

a

c

two fractions, and , on a number line. Do a few exam-

show that 5

15

= . Explain how the eManipulative was used

8

24

b

d

to do this.

a + c

ples and plot

on the number line as well. How does

8. Determine whether the following pairs are equal by writ-

b + d

ing each in simplest form.

a + c

a

c

the fraction

compare to and ? Does this relation-

a. 5 and 625 b. 11 and 275 c. 24 and 50 d. 14 and 8

b + d

b

d

8

1000

18

450

36

72

98

56

a

c

9. Determine which of the following pairs are equal.

ship appear to hold for all fractions and ?

b

d

a. 349 569

,

b. 734 468

,

c. 156 52

,

d. 882 147

,

568

928

957

614

558 186

552

91

15. Order the following sets of fractions from smaller to larger

10. Rewrite in simplest form.

and find a fraction between each pair

a. 21

b. 49

c. 108

d. 220

a. 17 51

,

b. 43

50

,

c. 214 597

,

d. 93

3

,

28

56

156

100

23

68

567

687

897

2511

2811 87

PROBLEMS

16. According to Bureau of the Census data, in 2005 in the

United States there were about

113,000,000 total households

58,000,000 married-couple households

5,000,000 family households with a male head of

household

14,000,000 family households with a female head

of household

30,000,000 households consisting of one person

Trace the outline of a figure like those shown and shade in

(NOTE: Figures have been rounded to simplify calculations.)

portions that represent the following fractions.

a. What fraction of U.S. households in 2005 were married-

a. 1 (different from the one shown)

couple households? To what fraction with a denomina-

2

b. 1 (different from the one shown)

tor of 100 is this fraction closest?

3

c. 1 d. 1 e. 1 f. 1

b. What fraction of U.S. households in 2005 consisted

4

6

12

24

of individuals living alone? To what fraction with a

20. A student simplifies 286 by “canceling” the 8s, obtaining

583

denominator of 100 is this closest?

26 , which equals 286 . He uses the same method with 28 886

,

,

c. What fraction of U.S. family households were headed

53

583

58 883

,

simplifying it to 26 also. Does this always work with simi-

by a woman in 2005? To what fraction with a denomina-

53

tor of 100 is this fraction closest?

lar fractions? Why or why not?

21. I am a proper fraction. The sum of my numerator and

17. What is mathematically inaccurate about the following

denominator is a one-digit square. Their product is a cube.

sales “pitches”?

What fraction am I?

a. “Save 1 , , ,

2 1 1 and even more!”

3

4

b. “You’ll pay only a fraction of the list price!”

22. True or false? Explain.

a. The greater the numerator, the greater the fraction.

18. In 2000, the United States generated 234,000,000 tons of

b. The greater the denominator, the smaller the fraction.

waste and recycled 68,000,000 tons. In 2003, 236,000,000

c. If the denominator is fixed, the greater the numerator,

tons of waste were generated and 72,000,000 tons were

the greater the fraction.

recycled. In which year was a greater fraction of the waste

d. If the numerator is fixed, the greater the denominator,

recycled?

the smaller the fraction.

19. The shaded regions in the figures represent the fractions 12

23. The fraction 12 is simplified on a fraction calculator

18

and 1 ,

3 respectively.

and the result is 2 . Explain how this result can be used

3

c06.indd 220

7/30/2013 2:59:19 PM - Section 6.1 The Set of Fractions 221

to find the GCF(12, 18). Use this method to find the

she take if she always moves closer to B? One route is

following.

shown.

a. GCF(72, 168)

b. GCF(234, 442)

24. Determine whether the following are correct or

incorrect. Explain.

ab + c

a + b

b

ab + ac

b

c

30. If you place a 1 in front of the number 5, the new

a.

= a + c b.

= c.

= +

b

a + c

c

ad

d

number is 3 times as large as the original number.

25. Three-fifths of a class of 25 students are girls. How many

a. Find another number (not necessarily a one-digit

are girls?

number) such that when you place a 1 in front

of it, the result is 3 times as large as the original

26. The Independent party received one-eleventh of the

number.

6,186,279 votes cast. How many votes did the party

b. Find a number such that when you place a 1 in front of

receive?

it, the result is 5 times as large as the original

27. Seven-eighths of the 328 adults attending a school bazaar

number. Is there more than one such number?

were relatives of the students. How many attendees were

c. Find a number such that, when you place a 2 in front of

not relatives?

it, the result is 6 times as large as the original

number. Can you find more than one such number?

28. The school library contains about 5280 books. If five-

d. Find a number such that, when you place a 3 in front

twelfths of the books are for the primary grades, how

of it, the result is 5 times as large as the original

many such books are there in the library?

number. Can you find more than one such number?

29. Talia walks to school at point B from her house at point

31. After using the Chapter 6 eManipulative activity

A, a distance of six blocks. For variety she likes to try

Comparing Fractions on our Web site, identify two

different routes each day. How many different paths can

fractions between 47 and 1 .

2

EXERCISE/PROBLEM SET B

EXERCISES

1. What fraction is represented by the shaded portion of each

4. Does the following picture represent 3 ? Explain.

4

diagram?

a.

5. Using the diagram, represent each of the following as a

b.

fraction of all the dots.

c.

d.

a. The part of the collection of dots inside the circle

2. Represent the following fractions using the given models.

b. The part of the collection of dots inside both the circle

i. 7 ii. 8

and the triangle

10

3

c. The part of the collection of dots inside the triangle but

a. Region model

b. Set model

outside the circle

c. Fraction strips d. Number line (see Set A, Exercise 2)

6. True or false? Explain.

3. Use the set model with non-equivalent parts as described in

a. 5 days is 16 of a month.

Set A, Exercise 3 to represent the following fractions.

b. 4 days is 47 of a week.

a. 5

b. 2

c. 3

6

9

4

c. 1 month is 1

12 of a year.

c06.indd 221

7/30/2013 2:59:22 PM - 222 Chapter 6 Fractions

7. Use the area model illustrated on the Chapter 6

13. Arrange each of the following from smallest to largest.

eManipulative activity Equivalent Fractions on our Web

a. 4 7 14

,

,

7

2

5

7

13

25 b. 3 ,

, ,

11

23

9

18

site to show that 3

9

= .

2

6 Explain how the eManipulative

was used to do this.

14. Use the Chapter 6 eManipulative Comparing Fractions

8. Decide which of the following are true. Do this mentally.

a

on our Web site to plot several pairs of fractions, and

a. 7

29

= b. 3

20

= c. 7 63

= d. 7

105

=

b

c

6

64

12

84

9

81

12

180

, with small denominators on a number line. For each

d

9. Determine which of the following pairs are equal.

ad + bc

example, plot

on the number line as well. How

a. 693 42

,

b. 873 184

,

c. 48 756

,

d. 468 156

,

2bd

858

52

954

212

84

1263

891

297

ad + bc

a

c

10. Rewrite in simplest form.

does the fraction

relate to and ? Does this

2bd

b

d

a. 189 b. 294 c. 480 d. 3335

a

c

153

63

672

230

relationship appear to hold for all fractions and ?

b

d

11. Rewrite as a mixed number in simplest form.

Explain.

a. 2232 b. 8976

444

144

15. Determine whether the following pairs are equal. If they

12. For each of the following, arrange from smallest to largest

are not, order them and find a fraction between them.

and explain how this ordering could be done mentally.

a. 231 and 308

b. 1516 and 2653 c. 516 and 1376

654

872

2312

2890

892

2376

a. 5 5 5

, ,

b. 7 6 8

, ,

11

9

13

8

7

9

PROBLEMS

16. According to the Bureau of the Census data on living

arrangements of Americans 15 years of age and older

in 2005 there were about

230,000,000 people over 15

20,000,000 people from 25–34 years old

4,000,000 people from 25–34 living alone

17,000,000 people over 75 years old

7,000,000 people over 75 living alone

3,000,000 people over 75 living with other persons

Trace outlines of figures like the ones shown and shade in

(NOTE: Figures have been rounded to simplify

portions that represent the following fractions.

calculations.)

a. 12 (different from the one shown)

a. In 2005, what fraction of people over 15 in the United

b. 1 c. 1 d. 1 e. 1

3

4

8

9

States were from 25 to 34 years old? To what fraction

20. A popular math trick shows that a fraction like 16 can be

with a denominator of 100 is this fraction closest?

64

simplified by “canceling” the 6s and obtaining 1 . There

4

b. In 2005, what fraction of 25 to 34 years olds were living

are many other fractions for which this technique yields a

alone? To what fraction with a denominator of 100 is

correct answer.

this fraction closest?

a. Apply this technique to each of the following fractions

c. In 2005, what fraction of people over 15 in the United

and verify that the results are correct.

States were over 75 years old? To what fraction with a

i. 16

64

ii. 19

95

iii. 26

iv. 199

65

995

v. 26666

66665

denominator of 100 is this fraction closest?

b. Using the pattern established in parts iv and v of part

d. In 2005, what fraction of people over 75 were living

a, write three more examples of fractions for which this

alone? To what fraction with a denominator of 100 is

method of simplification works.

this fraction closest?

21. Find a fraction less than 1 . Find another fraction less

12

than the fraction you found. Can you continue this pro-

17. Frank ate 12 pieces of pizza and Dave ate 15 pieces. “I ate

1

cess? Is there a “smallest” fraction greater than 0? Explain.

4 more,” said Dave. “I ate 1

5 less,” said Frank. Who was

right?

22. What is wrong with the following argument?

18. Mrs. Wills and Mr. Roberts gave the same test to their

fourth-grade classes. In Mrs. Wills’s class, 28 out of 36

students passed the test. In Mr. Roberts’s class, 26 out of

32 students passed the test. Which class had the higher

passing rate?

19. The shaded regions in the following figures represent the

Therefore, 1

1

> , since the area of the shaded square is

4

2

fractions 1

,

greater than the area of the shaded rectangle.

2 and 1

6 respectively.

c06.indd 222

7/30/2013 2:59:27 PM - Section 6.2 Fractions: Addition and Subtraction 223

23. Use a method like the one in Set A, Problem 23, find the

31. Pattern blocks

can be used to represent

following.

a. LCM(224, 336)

b. LCM(861, 1599)

fractions. For example, if

is the whole, then

24. If the same number is added to the numerator and denom-

inator of a proper fraction, is the new fraction greater

is 12 because it takes two trapezoids to cover a hexagon.

than, less than, or equal to the original fraction? Justify

Use the Chapter 6 eManipulative Pattern Blocks on our

your answer. (Be sure to look at a variety of fractions.)

Web site to determine representations of the following

25. Find 999 fractions between 13 and 1 such that the difference

2

fractions. In each case, you will need to identify what the

between pairs of numbers next to each other is the same.

whole and what the part is.

(Hint: Find convenient equivalent fractions for 1 and 1 .)

3

2

a. 1 in two different ways.

b. 1

c. 1

d. 1

3

6

4

12

26. About one-fifth of a federal budget goes for defense.

If the total budget is $400 billion, how much is spent on

32. Refer to the Chapter 6 eManipulative activities Parts

defense?

of a Whole and Visualizing Fractions on our Web site.

27. The U.S. Postal Service delivers about 170 billion pieces

Conceptually, what are some of the advantages of having

of mail each year. If approximately 90 billion of these are

young students use such activities?

first class, what fraction describes the other classes of mail

delivered?

Analyzing Student Thinking

28. Tuition in public universities is about two-ninths of tuition

33. David claims that an improper fraction is always greater

at private universities. If the average tuition at private uni-

than a proper fraction. Is his statement true or false?

versities is about $12,600 per year, what should you expect

Explain.

to pay at a public university?

34. How would you respond to a student who says that frac-

29. A hiker traveled at an average rate of 2 kilometers per

tions don’t change in value if you multiply the top and

hour (km/h) going up to a lookout and traveled at an

bottom by the same number or add the same number to

average rate of 5 km/h coming back down. If the entire

the top and bottom?

trip (not counting a lunch stop at the lookout) took

approximately 3 hours and 15 minutes, what is the total

35. Brandon claims that he cannot find a number between 3

4

distance the hiker walked? Round your answer to the

and 3 because they are “right next to each other.” How

5

nearest tenth of a kilometer.

should you respond?

30. Five women participated in a 10-kilometer (10 K)

36. Rafael asserts that 2

1

> since 2 > 1 and 8 > 4. How should

Volkswalk, but started at different times. At a certain time

8

4

you respond?

in the walk the following descriptions were true.

1. Rose was at the halfway point (5 K).

37. Regarding two fractions, Collin says that the one with the

2. Kelly was 2 K ahead of Cathy.

larger numerator is the larger. Is he correct? Explain.

3. Janet was 3 K ahead of Ann.

38. You have a student who says she knows how to divide a

4. Rose was 1 K behind Cathy.

circle into pieces to illustrate what 2 means but wonders

5. Ann was 3.5 K behind Kelly.

3

how she can divide a circle to show 3 . What could you say

a. Determine the order of the women at that point in time.

2

to help her understand the situation?

That is, who was nearest the finish line, who was second

a

c

closest, and so on?

39. Hailey says that if ab < cd, then <

. Is she correct?

b

d

b. How far from the finish line was Janet at that time?

Explain.

FRACTIONS: ADDITION AND SUBTRACTION

NCTM Standard

Addition and Its Properties

All students should use visual

models, benchmarks, and equiva-

Addition of fractions is an extension of whole-number addition and can be motivated

lent forms to add and subtract

using models. To find the sum of 1 and 3 , consider the following measurement mod-

5

5

commonly used fractions and

els: the region model and number-line model in Figure 6.16.

decimals.

Figure 6.16

c06.indd 223

7/30/2013 2:59:30 PM - 224

Chapter 6 Fractions

The idea illustrated in Figure 6.16 can be applied to any pair of fractions that have

the same denominator. Figure 6.17 shows how fraction strips, which are a blend of these

two models, can be used to find the sum of 1 and 3. That is, the sum of two fractions

5

5

with the same denominator can be found by adding the numerators, as stated next.

Figure 6.17

D E F I N I T I O N 6 . 4

Reflection from Research

Addition of Fractions with Common Denominators

A curriculum for teaching fractions

a

c

that focuses on the use of physical

Let

and be any fractions. Then

models, pictures, verbal symbols,

b

b

a

c

a

c

and written symbols allows

+ = + .

students to better conceptually

b

b

b

understand fraction ideas (Cramer,

Post, & delMas, 2002).

Some students initially view the addition of fractions as add-

ing the numerators and adding the denominators as follows:

3

1

4

+ = . Using this example, discuss why such a method for addition is unreasonable.

4

2

6

Figure 6.18 illustrates how to add fractions when the denominators are not the

same.

Similarly, to find the sum 2

3

+ ,

7

5 use the equality of fractions to express the

fractions with common denominators as follows:

2

3

2 ¥ 5

3 ¥ 7

+ =

+

7

5

7 ¥ 5

5 ¥ 7

10

21

=

+

35

35

31

=

.

35

Figure 6.18

This procedure can be generalized as follows.

T H E O R E M 6 . 4

Addition of Fractions with Unlike Denominators

a

c

Let

and be any fractions. Then

b

d

a

c

ad

bc

+

=

+

.

Common Core – Grade 5

b

d

bd

Add and subtract fractions with

unlike denominators (including

PROOF

mixed numbers) by replacing

given fractions with equivalent

a

c

ad

bc

fractions in such a way as to

+ =

+

Equality of fractions

produce an equivalent sum

b

d

bd

bd

or difference of fractions with

ad

bc

like denominators. (In general,

=

+

Addition with common denominators ■

bd

a/b + c /d = a

( d + bc)/bd.)

Fourth graders know what addition means and know how to represent fractions in several

different ways. Thinking like a fourth grader use pictures and words to find the following

sum: 1

2

+ .

4

3

NOTE: Since you are thinking like a fourth grader, you may not know the usual procedure. If you use a common denomi-

nator, you must explain why and how.

c06.indd 224

7/30/2013 2:59:32 PM - Section 6.2 Fractions: Addition and Subtraction 225

Reflection from Research

In words, to add fractions with unlike denominators, find equivalent fractions with

The most common error when

common denominators. Then the sum will be represented by the sum of the numera-

adding two fractions is to add

tors over the common denominator.

the denominators as well as the

numerators; for example, 1

1

+

2

4

becomes 2 (Bana, Farrell, &

6

Find the following sums and simplify.

McIntosh, 1995).

3

2

5

3

17

5

a.

+

b.

+ c.

+

7

7

9

4

15

12

S O L U T I O N

3

2

3

2

5

a.

+ = + =

7

7

7

7

5

3

5 ¥ 4

9 ¥ 3

20

27

47

b.

+ =

+

=

+

=

9

4

9 ¥ 4

9 ¥ 4

36

36

36

17

5

17 ¥12 15 ¥ 5

204

75

279

31

c.

+

=

+

=

+

=

=

■

15

12

15 ¥12

180

180

20

In Example 6.5(c), an alternative method can be used. Rather than using 15 ¥12 as

the common denominator, the least common multiple of 12 and 15 can be used. The

LCM(15,12) = 22 ¥ 3 ¥ 5 = 60. Therefore,

17

5

17 ¥ 4

5 ¥ 5

68

25

93

31

+

=

+

=

+

=

=

.

15

12

15 ¥ 4

12 ¥ 5

60

60

60

20

Although using the LCM of the denominators (called the least common denominator

and abbreviated LCD) simplifies paper-and-pencil calculations, using this method

does not necessarily result in an answer in simplest form as in the previous case. For

example, to find 3

8

+ , use 30 as the common denominator since LCM( ,

10 15) = 30.

10

15

Thus 3

8

9

16

25

+

=

+

= , which is not in simplest form.

10

15

30

30

30

Calculators and computers can also be used to calculate the sums of fractions. A

common four-function calculator can be used to find sums, as in the following example.

The T1-34 MultiView calculator can be used to find sums as follows: To calculate

23

38

+ , press 23 n 48 c + 38 n 51 enter which yields1183 . This mixed number can

48

51

d

d

816

be simplified by pressing csimp enter to yield 1 61 . The sum 237

384

+

may not

272

496

517

fit on a common fraction calculator display. In this case, the common denominator

a

c

ad

bc

approach to addition can be used instead, namely +

=

+

. Here

b

d

bd

ad = 237 × 517

= 122529

bc = 496 × 384

= 1990464

ad + bc = 122529 +190464 = 312993

bd = 496 × 517

= 256432.

Therefore, the sum is 312993 . Notice that the latter method may be done on any four-

256432

function calculator.

The following properties of fraction addition can be used to simplify computa-

tions. For simplicity, all properties are stated using common denominators, since any

two fractions can be expressed with the same denominator.

P R O P E R T Y

Closure Property for Fraction Addition

The sum of two fractions is a fraction.

c06.indd 225

7/30/2013 2:59:35 PM - 226 Chapter 6 Fractions

a

b

a

b

This follows from the equation +

= + , since a + b and c are both whole

c

c

c

numbers and c ≠ 0.

P R O P E R T Y

Commutative Property for Fraction Addition

a

c

Let b and b be any fractions. Then

a

c

c

a

+ = + .

Problem-Solving Strategy

b

b

b

b

Draw a Picture

Figure 6.19 provides a visual justification using fraction strips. The following is a

formal proof of this commutativity.

a

c

a

c

+ = +

Addition of fractions

b

b

b

c

a

= +

Commutative property of whole-number addition

b

c

a

= +

Addition of fractions

Figure 6.19

b

b

P R O P E R T Y

Associative Property for Fraction Addition

a c

e

Let , , and be any fractions. Then

b b

b

⎡a c ⎤ e a ⎡ c e

+

⎤

⎣⎢ b b ⎦⎥ + = +

+

b

b

⎣⎢b b ⎦⎥.

The associative property for fraction addition is easily justified using the associa-

tive property for whole-number addition.

P R O P E R T Y

Additive Identity Property for Fraction Addition

a

0

Let be any fraction. There is a unique fraction, , such that

b

b

a

0

a

0

a

+ = = + .

b

b

b

b

b

The following equations show how this additive identity property can be justified

using the corresponding property in whole numbers.

a

0

a

+ = + 0

Addition of fractions

b

b

b

a

=

Additive identity property of whole-number addition

b

0

0

The fraction is also written as or 0. It is shown in the problem set that this is

b

1

the only fraction that serves as an additive identity.

The preceding properties can be used to simplify computations.

c06.indd 226

7/30/2013 2:59:37 PM - Section 6.2 Fractions: Addition and Subtraction 227

3

⎡ 4 2 ⎤

Compute:

+

+

5

⎣⎢7 5 ⎦⎥.

S O L U T I O N

3

⎡ 4 2 ⎤ 3 ⎡ 2 4

+

+

⎤

Commutativity

5

⎣⎢7 5 ⎦⎥ = +

+

5

⎣⎢5 7 ⎦⎥

⎡3 2 ⎤ 4

=

+

Associativity

⎣⎢5 5 ⎦⎥ + 7

4

= 1 +

Addition

■

7

The number 1

4

+ can be expressed as the mixed number 1 4 . As in Example 6.6,

7

7

any mixed number can be expressed as a sum. For example, 3 2 = 3

2

+ . Also, any

5

5

mixed number can be changed to an improper fraction, and vice versa, as shown in

the next example.

a. Express 3 2 as an improper fraction.

5

b. Express 36 as a mixed number.

7

S O L U T I O N

2

2

15

2

17

a. 3

= 3 + =

+ =

5

5

5

5

5

⎡

2

5 3

2

17 ⎤

Shortcut : 3

= ⋅ + =

⎣⎢

5

5

5 ⎦⎥

36

35

1

1

1

b.

=

+ = 5 + = 5

■

7

7

7

7

7

Check for Understanding: Exercise/Problem Set A #1–8

✔

Subtraction

Problem-Solving Strategy

Subtraction of fractions can be viewed in two ways as we did with whole-number

Draw a Picture

subtraction—either as (1) take-away or (2) using the missing-addend approach.

Find 4

1

− .

7

7

S O L U T I O N From Figure 6.20, 4

1

3

− = .

■

7

7

7

This example suggests the following defi nition.

Figure 6.20

D E F I N I T I O N 6 . 5

Subtraction of Fractions with Common Denominators

a

c

Let

and be any fractions with a ≥ c. Then

b

b

a

c

a

c

− = − .

b

b

b

c06.indd 227

7/30/2013 2:59:39 PM - 228 Chapter 6 Fractions

Now consider the subtraction of fractions using the missing-addend approach. For

4

1

n

4

1

example, to find − , find a fraction such that =

+ n. The following argument

7

7

7

7

7

7

shows that the missing-addend approach leads to the take-away approach.

a

c

n

a

c

n

If

− = , and the missing-addend approach holds, then = + , or

b

b

b

b

b

b

a

c

n

= + . This implies that a = c + n or a − c = n.

b

b

a

c

a

c

That is, −

= − .

b

b

b

Also, it can be shown that the missing-addend approach is a consequence of the

take-away approach.

Let

be the standard pattern block shapes. Use these blocks to

model 3 1 − 15 and sketch the method used. Compare and contrast your method with that of a peer.

6

6

If fractions have different denominators, subtraction is done by first finding com-

mon denominators, then subtracting as before.

a

c

a

c

ad

bc

ad

bc

Suppose that

≥

Then

− =

−

=

−

.

.

b

d

b

d

bd

bd

bd

Therefore, fractions with unlike denominators may be subtracted as follows.

T H E O R E M 6 . 5

Subtraction of Fractions with Unlike Denominators

a

c

a

c

Let and be any fractions, where ≥

. Then

b

d

b

d

a

c

ad

bc

− =

−

.

b

d

bd

Find the following differences.

4

3

25

7

7

8

a. −

b.

−

c.

−

7

8

12

18

10

15

S O L U T I O N

4

3

4 ¥ 8

7 ¥ 3

32

21

11

a. −

=

−

=

−

=

7

8

7 ¥ 8

7 ¥ 8

56

56

25

7

b.

−

. NOTE: LCM( ,

12 18) = 36.

NCTM Standard

12

18

All students should develop and

25

7

25 ¥ 3

7 ¥ 2

75 14

61

−

=

−

=

−

=

use strategies to estimate com-

12

18

12 ¥ 3 18 ¥ 2

36

36

putations involving fractions and

7

8

21

16

5

1

decimals in situations relevant to

c.

−

=

−

=

=

■

students’ experience.

10

15

30

30

30

6

Check for Understanding: Exercise/Problem Set A #9–14

✔

c06.indd 228

7/30/2013 2:59:42 PM - Section 6.2 Fractions: Addition and Subtraction 229

Mental Math and Estimation for Addition

and Subtraction

Mental math and estimation techniques similar to those used with whole numbers can

be used with fractions.

Calculate mentally.

⎛ 1 3⎞

4

4

2

3

a.

+

+

−

⎝⎜

2 c. 40

8

5

4 ⎠⎟ +

b. 3

5

5

5

7

S O L U T I O N

⎛ 1 3⎞

4

4

⎛ 1 3⎞ ⎛ 4 1⎞ 3

3

a.

+

1

⎝⎜ 5 4⎠⎟ + = +

+

5

5

⎝⎜ 5 4⎠⎟ =

+

⎝⎜ 5 5⎠⎟ + =

4

4

Commutativity and associativity were used to be able to add 4 and 1 , since they

5

5

are compatible fractions (their sum is 1).

4

2

1

1

b. 3

+ 2 = 4 + 2 = 6

5

5

5

5

This is an example of additive compensation, where 3 4 was increased by 1 to 4 and

5

5

consequently 2 2 was decreased by 1 to 2 1 .

5

5

5

3

4

4

c. 40 − 8 = 40

− 9 = 31

7

7

7

Here 4 was added to both 40 and 8 3 (since 8 3

4

+ = 9), an example of the

7

7

7

7

equal-additions method of subtraction. This problem could also be viewed as

39 7 − 8 3 = 31 4 .

■

7

7

7

NCTM Standard

Estimate using the indicated techniques.

All students should develop and

use strategies to estimate the

3

2

results of rational-number compu-

a. 3

+ 6 using range estimation

tations and judge the reasonable-

7

5

ness of the results.

4

3

b. 15

− 7 using front-end with adjustment

5

8

5

1

3

1

c. 3

+ 5 + 8 rounding to the nearest or whole

6

8

8

2

S O L U T I O N

3

2

Algebraic Reasoning

a. 3

+ 6 is between 3 + 6 = 9 and 4 + 7 = 11.

7

5

When solving algebraic

expressions that involve

4

3

1

4

3

1

fractions, it is worthwhile

b. 15

− 7 ≈ 8 since 15 − 7 = 8 and − ≈ .

5

8

2

5

8

2

to be able to estimate the

result of computations so the

5

1

3

1

1

c. 3

+ 5 + 8 ≈ 4 + 5 + 8 = 17 .

reasonableness of the algebraic

6

8

8

2

2

■

solution can be evaluated.

Check for Understanding: Exercise/Problem Set A #15–19

✔

c06.indd 229

7/30/2013 2:59:45 PM - 230 Chapter 6 Fractions

Around 2900 B.C.E. the Great Pyramid of Giza was constructed.

It covered 13 acres and contained over 2,000,000 stone blocks

averaging 2.5 tons each. Some chamber roofs are made of 54-ton

granite blocks, 27 feet long and 4 feet thick, hauled from a quarry

600 miles away and set into place 200 feet above the ground.

The relative error in the lengths of the sides of the square base

was 1 and in the right angles was 1

14,000

. This construction was

27,000

estimated to have required 100,000 laborers working for about

30 years.

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. Illustrate the problem 2

1

+ using the following.

9. On a number line, demonstrate the following problems

5

3

a. A region model b. A number-line model

using the take-away approach.

3

1

1

c. Fraction strips

a. 7 −

b. 5 −

c. 2 −

10

10

12

12

3

4

2. Find 5

7

+ using four different denominators.

10. Perform the following subtractions.

9

12

a. 9

5

− b. 3 2

− c. 4 3

−

11

11

7

9

5

4

3. Using rectangles or circles as the whole, represent the

d. 13

8

− e. 21

7

− f. 11

99

−

following problems. The Chapter 6 eManipulative activity

18

27

51

39

100

1000

Adding Fractions on our Web site will help you

11. Find the difference for the following pairs of numbers.

understand these representations.

Answers should be written as mixed numbers.

a. 2

1

+ b. 1 3

+ c. 3 2

+

a.

5 1 6

, b. 2 2 ,1 1 c. 7 5 , 5 2 d. 22 1 ,15 11

3

6

4

8

4

3

3

7

3

4

7

3

6

12

4. Find the following sums and express your answer in

12. Find the simplest form of the following differences using

simplest form.

the method described in Exercise 7.

a. 1

5

+ b. 1 1

+ c. 3 1

+

a. 25

4

− b. 3 4

−

8

8

4

2

7

3

8

5

5

7

d. 8

1

3

+

+ e. 8

4

+

f. 9

89

+

9

12

16

13

51

22

121

13. Compute the following. Use a fraction calculator if available.

g. 61

7

+

h. 7

20

+

i. 143

759

+

a. 5

8

− b. 7 3

−

100

1000

10

100

1000

100,000

6

15

9

5

14. An alternative definition of “less than” for fractions is as

5. Change the following mixed numbers to improper

follows:

fractions.

a

c

a. 3 5 b. 2 7 c. 5 1 d. 7 1

< if and only if

6

8

5

9

b

d

6. Find the sum for the following pairs of numbers. Answers

a

m

c

+

= for a

should be written as mixed numbers.

b

n

d m

a. 3 3 4

, b. 2 2 ,1 1 c. 7 5 , 5 2 d. 22 1 ,15 11

nonzero

4

5

3

4

7

3

6

12

n

7. To find the sum 2

3

+ on a scientific calculator,

Use this definition to confirm the following statements.

5

4

a. 3

5

< b. 1 1

<

press 2 ÷ 5 + 3 ÷ 4 = 1 1

. 5 . The whole-number

7

7

3

2

part of the sum is 1. Subtract it: − 1 = 0 1

. 5 . This repre-

15. Use the properties of fraction addition to calculate each of

sents the fraction part of the answer in decimal form. Since

the following sums mentally.

the denominator of the sum should be 5 × 4 = 20,

a. 3

1

4

( + )+

7

9

7

multiply by 20: × 20 = 3 . This is the numerator of the

b. 1 9

5

4

+ +

13

6

13

fraction part of the sum. Thus 2

3

3

+ = 1 . Find the sim-

2

5

4

3

5

4

20

c. (2 + 3

1

(

2

)+ + )

5

8

5

8

plest form of the sums using this method.

16. Find each of these differences mentally using the equal-

a. 3

5

+ b. 3 4

+

7

8

8

5

additions method. Write out the steps that you thought

8. Compute the following. Use a fraction calculator if

through.

available.

a. 8 2 − 2 6 b. 9 1 − 2 5

7

7

8

8

a. 3

7

+ b. 5 5

+

3

5

−

1

5

−

4

10

6

8

c. 11

6 d. 8

3

7

7

6

6

c06.indd 230

7/30/2013 2:59:52 PM - Section 6.2 Fractions: Addition and Subtraction 231

17. Estimate each of the following using (i) range and

18. Estimate each of the following using “rounding to the

(ii) front-end with adjustment estimation.

nearest whole number or 1 .”

2

a. 6 7 + 7 3 b. 9 1 − 6 4

a. 9 7 + 3 6 b. 9 5 − 5 4 c. 7 2 + 5 3 + 2 7

11

9

8

7

9

13

8

9

11

13

12

c. 8 2 + 2 7 + 5 2

1

4

6

+

+

11

11

9

19. Estimate using cluster estimation: 5

4

5 .

3

5

7

PROBLEMS

20. Sally, her brother, and another partner own a pizza

29. In the first 10 games of the baseball season, Jim has 15

restaurant. If Sally owns 1 and her brother owns 1 of the

hits in 50 times at bat. The fraction of his times at bat that

3

4

restaurant, what part does the third partner own?

were hits is 15 . In the next game he is at bat 6 times and

50

gets 3 hits.

21. John spent a quarter of his life as a boy growing up,

a. What fraction of at-bats are hits in this game?

one-sixth of his life in college, and one-half of his life as a

b. How many hits does he now have this season?

teacher. He spent his last six years in retirement. How old

was he when he died?

c. How many at-bats does he now have this season?

d. What is his record of hits/at-bats this season?

22. Rafael ate one-fourth of a pizza and Rocco ate one-third

e. In this setting “baseball addition” can be defined as

of it. What fraction of the pizza did they eat?

a

c

a

c

⊕ = +

23. Greg plants two-fifths of his garden in potatoes and

b

d

b + d

one-sixth in carrots. What fraction of the garden remains

(Use ⊕ to distinguish from ordinary +.)

for his other crops?

Using this definition, do you get an equivalent answer

24. About eleven-twelfths of a golf course is in fairways,

when fractions are replaced by equivalent fractions?

one-eighteenth in greens, and the rest in tees. What part of

30. Fractions whose numerators are 1 are called unitary frac-

the golf course is in tees?

tions. Do you think that it is possible to add unitary frac-

25. David is having trouble when subtracting mixed numbers.

tions with different odd denominators to obtain 1? For

What might be causing his difficulty? How might you help

example, 1

1

1

+ + = 1, but 2 and 6 are even. How about

2

3

6

David?

the following sum?

3 2 = 2 12

1

1

1

1

1

1

1

1

1

1

1

5

5

+ + + +

+

+

+

+

+

+

3

3

− =

3

5

7

9

15

21

27

35

63

105

135

5

5

2 9 = 3 4

31. The Egyptians were said to use only unitary fractions with

5

5

the exception of 2 . It is known that every unitary frac-

3

26. a. The divisors (other than 1) of 6 are 2, 3, and 6.

tion can be expressed as the sum of two unitary fractions

Compute 1

1

1

+ + .

2

3

6

in more than one way. Represent the following fractions

b. The divisors (other than 1) of 28 are 2, 4, 7, 14, and 28.

as the sum of two different unitary fractions. (NOTE:

Compute 1

1

1

1

1

+ + +

+ .

1

1

1

1

1

2

4

7

14

28

= + , but 1 = + is requested.)

2

4

4

2

3

6

c. Will this result be true for 496? What other numbers will

a. 1 b. 1 c. 1

5

7

17

have this property?

32. At a round-robin tennis tournament, each of eight players

plays every other player once. How many matches are there?

27. Determine whether 1+ 3

1 + 3 + 5

=

+ + +

. Is 1 3 5 7

5 + 7

7 + 9 + 11

9 + 11 + 13 +

also the

15

33. New lockers are being installed in a school and will be

same fraction? Find two other such fractions. Prove why

numbered from 0001 to 1000. Stick-on digits will be used to

this works. (Hint: 1 3

22

+ = , 1 3 5 32

+ + = , etc.)

number the lockers. A custodian must calculate the number

1

1

1

1

28. Find this sum: +

+

+ ⋅⋅⋅ +

.

of packages of numbers to order. How many 0s will be

2

22

23

2100

needed to complete the task? How many 9s?

EXERCISE/PROBLEM SET B

EXERCISES

1. Illustrate 3

2

+ using the following.

Adding Fractions on our Web site will help you understand

4

3

a. A region model

this process.)

1

1

5

b. A number-line model

a. 3 − b. 3 − 1

4

3

3

6

c. Fraction strips

4. Find the following sums and express your answer in

2. Find 5

1

+ using four different denominators.

simplest form. (Leave your answers in prime factorization

8

6

form.)

3. Using rectangles as the whole, represent the following prob-

1

1

a.

+

lems. (The Chapter 6 eManipulative activity

22

32

2 × 33

×

c06.indd 231

7/30/2013 2:59:55 PM - 232 Chapter 6 Fractions

1

1

12. Find the simplest form of the following differences using

b.

+

32

73

53 × 72

×

× 29

the method described in Exercise/Problem of Set A.

4

19

1

1

a. 23 −

b. 37 −

14

9

52

78

c.

+

54

75

132

32 × 5 × 133

×

×

13. Compute the following. Use a fraction calculator if

1

1

d.

+

available.

173

535

6713

115 × 172 × 679

×

×

a. 7

3

−

b. 4

2

−

6

5

5

7

5. Change the following improper fractions to mixed numbers.

14. Prove that 2

5

< in two ways.

a. 35 b. 19 c. 49 d. 17

5

8

3

4

6

5

6. Calculate the following and express as mixed numbers in

15. Use properties of fraction addition to calculate each of the

simplest form.

following sums mentally.

a. 7 5 + 13 2

b. 11 3 + 9 8

5

3

2

5

8

3

5

9

a. 2

( + )+

b. 4 + ( + 2 )

5

8

5

9

15

9

7. Using a scientific calculator, find the simplest form of

c. 1 3

( + 3 5 1 8

(

2 1

)+ + )

4

11

11

4

each sum.

a. 19

51

+

b. 37

19

+

16. Find each of these differences mentally using the equal-

135

75

52

78

additions method. Write out the steps that you thought

8. Compute the following. Use a fraction calculator if available.

through.

a. 7

4

+

b. 4

5

+

6

5

9

12

a. 5 2 − 2 7 b. 9 1 − 2 5

9

9

6

6

9. On a number line, demonstrate the following problems

c. 21 3 − 8 5 d. 5 3 − 2 6

7

7

11

11

using the missing-addend approach.

a. 9

5

− b. 2 1

− c. 3 1

−

17. Estimate each of the following using (i) range and (ii)

12

12

3

5

4

3

front-end with adjustment estimation.

10. a. Compute the following.

a. 5 8 + 6 3 b. 7 4 − 5 6

c. 8 2 + 2 8 + 7 3

i. 7

2

1

− ( − ) ii. 7 2 1

( − )−

9

13

5

7

11

9

13

8

3

6

8

3

6

b. The results of i and ii illustrate that a property of frac-

18. Estimate each of the following using “rounding to the

tion addition does not hold for fraction subtraction.

nearest whole number or 1 ” estimation.

2

What property is it?

a. 5 8 + 6 4 b. 7 4 − 5 5 c. 8 2 + 2 7 + 7 3

9

7

5

9

11

12

13

11. Calculate the following and express as mixed numbers in

simplest form.

19. Estimate using cluster estimation:

a. 11 3 − 9 8

2

5

−

6 1 + 6 2 + 5 8 + 6 3

5

b. 13

7

9

3

8

8

11

9

13

PROBLEMS

20. Grandma was planning to make a red, white, and blue

Amy:

quilt. One-third was to be red and two-fifths was to be

6

6

4

6

white. If the area of the quilt was to be 30 square feet,

1

1

1

2

3

7

= 7

5

= 5

5

how many square feet would be blue?

6

6

3

6

8

2

4

1

3

1

21. A recipe for cookies will prepare enough for three-

− 5 = 5

− 2 = 2

− 2

3

6

2

6

2

sevenths of Ms. Jordan’s class of 28 students. If she makes

three batches of cookies, how many extra

2

1

3

1

1 = 1

2

= 2

students can she feed?

6

3

6

2

22. Karl wants to fertilize his 6 acres. If it takes 8 2 bags of

Robert:

3

fertilizer for each acre, how much fertilizer does Karl need

1

55

1

19

1

to buy?

9

= 9

6

= 6

5

6

48

3

3

3

23. During one evening Kathleen devoted 2 of her study

7

23

2

5

4

5

− 2 = 2

− 1

= 1 − 1

time to mathematics, 3 of her time to Spanish, 1 of

20

3

8

48

3

3

5

her time to biology, and the remaining 35 minutes to

32

14

English. How much time did she spend studying her

7

5

48

3

Spanish?

What concept of fractions might you use to help these

24. A man measures a room for a wallpaper border and finds

students?

he needs lengths of 10 ft. 6 3 in., 14 ft. 9 3 in., 6 ft. 5 1 in.,

8

4

2

and 3 ft. 2 7 in. What total length of wallpaper border

26. Consider the sum of fractions shown next.

8

does he need to purchase? (Ignore amount needed for

1

1

8

matching and overlap.)

+ +

3

5

15

25. Following are some problems worked by students.

The denominators of the first two fractions differ by two.

Identify their errors and determine how they would

a. Verify that 8 and 15 are two parts of a Pythagorean

answer the final question.

triple. What is the third number of the triple?

c06.indd 232

7/30/2013 3:00:00 PM - Section 6.3 Fractions: Multiplication and Division 233

b. Verify that the same result holds true for the following sums:

equations. For example, in the row 1 2 1 11 2 , a correct

4 3 7 21 15

i. 1

1

+ ii. 1

1

+ iii. 1

1

+

equation is 2

1

11

− − .

7

9

11

13

19

21

3

7

21

c. How is the third number in the Pythagorean triple

a. 11 7 3 5 4 5 1 1

b. 2 3 3 6 1 1 1

21 12 4 9 5 12 3 12

3 5 7 12 6 3 9

related to the other two numbers?

32. If one of your students wrote 1

2

3

+ = , how would you

d. Use a variable to explain why this result holds. For example,

4

3

7

convince him or her that this is incorrect?

you might represent the two denominators by n and n + 2.

33. A classroom of 25 students was arranged in a square with

27. Find this sum:

five rows and five columns. The teacher told the students

1

1

1

1

+

+

+ ⋅⋅⋅ +

that they could rearrange themselves by having each student

1 × 3

3 × 5

5 × 7

21 × 23

move to a new seat directly in front or back, or directly to the

28. The unending sum 1

1

1

1

+ + +

+ ⋅⋅⋅, where each term is

right or left—no diagonal moves were permitted. Determine

2

4

8

16

a fixed multiple (here 1 ) of the preceding term, is called an

how this could be accomplished (if at all). (Hint: Solve an

2

infinite geometric series. The sum of the first two terms is 3 .

equivalent problem—consider a 5 × 5 checkerboard.)

4

a. Find the sum of the first three terms, first four terms,

Analyzing Student Thinking

first five terms.

b. How many terms must be added in order for the sum to

34. Cristobal says that you don’t necessarily have to find the

exceed 99 ?

100

LCD to add two fractions and that any common denomi-

c. Guess the sum of the geometric series.

nator will do. Is he correct? Explain.

29. By giving a counterexample, show for fractions that

35. Whitley claims that she can add two fractions with-

a. subtraction is not closed.

out finding a common denominator. How should you

b. subtraction is not commutative.

respond?

c. subtraction is not associative.

36. To add 3 1 and 5 2 , a student claims that you have to

4

7

30. The triangle shown is called the harmonic triangle. Observe

change these two mixed numbers into improper fractions

the pattern in the rows of the triangle and then answer the

first. Is the student correct? Explain.

questions that follow.

37. Instead of adding 3 and 7 first as shown in the problem

4

8

3

7

1

( + )+ , Kalin adds 3 and 1 first. Is this okay? Explain.

4

8

4

4

4

38. You asked Marcos to determine whether certain fractions

were closer to 0, 1 , or to 1. He answered that since frac-

2

tions were always small, they would all be close to 0. How

would you respond to him?

a. Write down the next two rows of the triangle.

39. Brenda asks why she needs to find a common denomina-

b. Describe the pattern in the numbers that are first in each

tor when adding or subtracting fractions. How should you

row.

respond?

c. Describe how each fraction in the triangle is related to

40. Marilyn said her family made two square pizzas at home,

the two fractions directly below it.

one ′′

8 on a side and the other 12′′ on a side. She ate 1 of

4

31. There is at least one correct subtraction equation of

the small pizza and 1 of the larger pizza. So Marilyn says

6

the form a − b = c in each of the following. Find all such

she ate 5 of the pizza. Do you agree? Explain.

12

FRACTIONS: MULTIPLICATION AND DIVISION

Let

be the standard pattern block shapes. Use pattern blocks to

illustrate how 4

1

× is conceptually different than 1 × 4 and yet has the same resulting product.

2

2

Multiplication and Its Properties

Extending the repeated-addition approach of whole-number multiplication to frac-

Reflection from Research

tion multiplication is an interesting challenge. Consider the following cases.

In the teaching of fractions, even

low performing students seem to

benefit from task-based partici-

Case 1: A Whole Number Times a Fraction

pant instruction where students

1

1

1

1

3

are given tasks or problems to

3 ×

= + + =

solve on their own or in groups.

4

4

4

4

4

Such instruction allows students

1

1

1

1

1

1

1

to be engaged in the mathemati-

6 ×

= + + + + + = 3

2

2

2

2

2

2

2

cal practices of problem solving

(Empson, 2003).

Here repeated addition works well since the fi rst factor is a whole number.

c06.indd 233

7/30/2013 3:00:04 PM - 234

Chapter 6 Fractions

Case 2: A Fraction Times a Whole Number

1 × 6

2

Here, we cannot apply the repeated-addition approach directly, since that would

literally say to add 6 one-half times. But if multiplication is to be commutative,

then 1 × 6 would have to be equal to 6

1

× , or 3, as in case 1. Thus a way of inter-

2

2

preting 1 × 6 would be to view the 1 as taking “one-half of 6” or “one of two equal

2

2

parts of 6,” namely 3. Similarly, 1 × 3 could be modeled by fi nding “one-fourth of

4

3” on the fraction number line to obtain 3 (Figure 6.21).

4

Figure 6.21

In the following fi nal case, it is impossible to use the repeated-addition

approach, so we apply the new technique of case 2.

Case 3: A Fraction of a Fraction

1

5

1

5

× means

of

3

7

3

7

First picture 5 . Then take one of the three equivalent parts of 5 (Figure 6.22).

7

7

After subdividing the region in Figure 6.22 horizontally into seven equivalent

parts and vertically into three equal parts, the shaded region consists of 5 of the 21

smallest rectangles (each small rectangle represents 1 ). Therefore,

21

1

5

5

× =

3

7

21

Similarly, 2

5

× would comprise 10 of the smallest rectangles, so

3

7

2

5

10

× =

.

3

7

21

Figure 6.22

Fourth graders know what multiplication means and know the meaning of mixed

numbers. Thinking like a fourth grader, use pictures and words to explain 1 1 × 12.

4

5

NOTE: Since you are thinking like a fourth grader, you may not know the procedure of converting to improper fractions.

If you use improper fractions, you must explain why and how.

The discussion above Mathematical Task 6.3B should make the following

defi nition of fraction multiplication seem reasonable.

D E F I N I T I O N 6 . 6

Reflection from Research

Multiplication of Fractions

Students who only view fractions

a

c

like 3 as “three out of 4 parts”

4

Let

and be any fractions. Then

struggle to handle fraction mul-

b

d

tiplication problems such as 2 of

3

a c

ac

9 . Students who can more flex-

¥ =

.

10

b d

bd

ibly view 3 as “three-fourths of

4

one whole or three units of one

fourth” can better solve multi-

plication of two proper fractions

Compute the following products and express the answers in

(Mack, 2001).

simplest form.

2

5

3 28

1

2

a.

¥

b.

¥

c. 2 ¥ 7

3 13

4 15

3

5

c06.indd 234

7/30/2013 3:00:08 PM - Section 6.3 Fractions: Multiplication and Division 235

S O L U T I O N

2

5

2 5

10

a.

¥

¥

=

=

3 13

3 ¥13

39

3 28

3 28

84

21¥ 4

21

7 ¥ 3

7

b.

¥

¥

=

=

=

=

=

=

4 15

4 ¥15

60

15 ¥ 4

15

5 ¥ 3

5

1

2

7 37

259

4

c. 2 ¥ 7 =

¥

=

, or 17

3

5

3

5

15

15

■

Two mixed numbers were multiplied in Example 6.12(c). Children may incorrectly

multiply mixed numbers as follows:

2

1

2 1 ⎞

2

¥ 3 = (2 ¥ 3) + ⎛ ¥

3

2

⎝⎜ 3 2⎠⎟

1

= 6 .

3

However, Figure 6.23 shows that multiplying mixed numbers is more complex.

In Example 6.12(b), the product was easy to find but the process of simplification

Figure 6.23

required several steps.

When using a fraction calculator, the answers may not be given in simplest form. For

example, on the TI-34 MultiView, Example 6.12(b) is obtained as follows:

3 n 4 c × 28 n 15 enter which yields 1 24

d

d

60

This mixed number is further simplified one step at a time by pressing the simplify

key repeatedly as follows:

csimp enter 112

6

2 .

30 csimp

enter 115 csimp enter 1 5

To convert this mixed number into an improper fraction press 2nd n

n

U

bc

,

d

d

yielding 7.

5

The next example shows how one can simplify first (a good habit to cultivate) and

then multiply.

3 28

Compute and simplify: ¥

.

4 15

S O L U T I O N Instead, simplify, and then compute.

3 28

3 28

3 ¥ 28

3

28

1 7

7

¥

¥

=

=

=

¥

= ¥ = .

4 15

4 ¥15

15 ¥ 4

15

4

5 1

5

Another way to calculate this product is

3 28

3 2 2 7

7

¥

¥ ¥ ¥

=

= .

■

4 15

2 ¥ 2 ¥ 3 ¥ 5

5

a c

a c

The equation ¥

= ¥ is a simplification of the essence of the procedure in

b d

d b

Example 6.13. That is, we simply interchange the two denominators to expedite the

simplification process. This can be justified as follows:

a c

ac

¥ =

Multiplication of fractions

b d

bd

ac

=

Commutativity for whole-number multiplication

bd

a c

= ⋅ .

Multiplication of fractions

d b

You may have seen the following even shorter method.

c06.indd 235

7/30/2013 3:00:11 PM - 236 Chapter 6 Fractions

Compute and simplify.

18 39

50 39

a.

¥

b.

¥

13 72

15 55

S O L U T I O N

3

1

3

18 39

18

39

18

39

3

a.

¥

=

¥

=

¥

=

13 72

13

72

13

72

4

1

1

4

10

2

13

50 39

50

39

10

39

2

39

26

b.

¥

=

¥

=

¥

=

¥

=

■

15 55

15

55

3

55

3

11

11

3

11

1

The definition of fraction multiplication together with the corresponding prop-

erties for whole-number multiplication can be used to verify properties of fraction

multiplication. A verification of the multiplicative identity property for fraction mul-

tiplication is shown next.

a

a

¥

1

1 =

¥

1

2

3

Recall that 1= =

= = ⋅⋅⋅.

b

b 1

1

2

3

a ¥1

=

Fraction multiplication

b ¥1

a

=

Identity for whole-number multiplication

b

It is shown in the problem set that 1 is the only multiplicative identity.

The properties of fraction multiplication are summarized next. Notice that frac-

Reflection from Research

tion multiplication has an additional property, different from any whole-number

Although there are inherent dan-

properties—namely, the multiplicative inverse property.

gers in using rules without mean-

ing, teachers can introduce rules

that are both useful and math-

ematically meaningful to enhance

P R O P E R T Y

children’s understanding of frac-

tions (Ploger & Rooney, 2005).

Properties of Fraction Multiplication

a c

e

Let , , and be any fractions.

b d

f

Closure Property for Fraction Multiplication

The product of two fractions is a fraction.

Commutative Property for Fraction Multiplication

a c

c a

¥ = ¥

b d

d b

Associative Property for Fraction Multiplication

⎛ a c ⎞ e

a ⎛ c

e ⎞

¥

¥

¥

⎝⎜ b d ⎠⎟

=

f

b ⎝⎜ d

f ⎠⎟

Multiplicative Identity Property for Fraction Multiplication

a

a

a

⎛

m

¥

⎞

1 =

= ¥

1

1 =

, m ≠ 0

b

b

b

⎝⎜

m

⎠⎟

Multiplicative Inverse Property for Fraction Multiplication

a

b

For every nonzero fraction , there is a unique fraction such that

b

a

a b

¥ = 1.

b a

c06.indd 236

7/30/2013 3:00:14 PM - Section 6.3 Fractions: Multiplication and Division 237

Algebraic Reasoning

a

b

a

When

≠ 0,

is called the multiplicative inverse or reciprocal of . The

Being comfortable with the mul-

b

a

b

tiplicative inverse property is very

multiplicative inverse property is useful for solving equations involving fractions.

valuable when solving equations

like 2 x = 5 . This property allows

3

7

one to isolate x by multiplying

3

5

both sides of this equation by 3

Solve: x = .

2

7

8

to get 3 2

3 5

¥ x = ¥ or x = 15 .

2

3

2 7

14

S O L U T I O N

3

5

x =

7

8

7 ⎛ 3 ⎞

7 5

x

3 ⎝⎜ 7 ⎠⎟ =

¥

Multiplication

3 8

⎛ 7 3⎞

7 5

¥

x

¥

⎝⎜

Associative property

3 7 ⎠⎟

= 3 8

35

1¥ x =

Multiplicative inverse property

24

35

x =

Multiplicate identity property ■

24

Finally, as with whole numbers, distributivity holds for fractions. This property

can be verified using distributivity in the whole numbers.

Algebraic Reasoning

P R O P E R T Y

This property can be verified (a)

Distributive Property for Fraction Multiplication over Addition

by adding the variable expression

on the left side of the equation

a c

e

and then multiplying, (b) by multi-

Let , , and be any fractions. Then

plying the variable expression on

b d

f

the right side of the equation and

a ⎛ c

e ⎞

a

c

a

e

then adding, and (c) by observing

+

the resulting expressions in (a)

b ⎝⎜ d

f ⎠⎟ =

× + × .

b

d

b

f

and (b) are the same.

Distributivity of multiplication over subtraction also holds; that is,

a ⎛ c

e ⎞

a

c

a

e

−

b ⎝⎜ d

f ⎠⎟ =

×

− × .

b

d

b

f

Check for Understanding: Exercise/Problem Set A #1–10

✔

Write a word problem for each of the following expressions. Each word problem should

have the corresponding expression as part of its solution.

1

1

5

1

(a) 3 ÷ (b) ÷ 4 (c) ÷

2

3

8

4

An example of a word problem for the expression 4

2

× is the following:

3

Darius had 4 pies for his birthday party. By the end of his party, two-thirds of each pie had been eaten. How much pie

had been eaten altogether?

Division

Division of fractions is a difficult concept for many children (and adults), in part

because of the lack of simple concrete models. We will view division of fractions as

an extension of whole-number division. Several other approaches will be used in this

section and in the problem set. These approaches provide a meaningful way of learn-

ing fraction division. Such approaches are a departure from simply memorizing the

rote procedure of “invert and multiply,” which offers no insight into fraction division.

c06.indd 237

7/30/2013 3:00:19 PM - 238

Chapter 6 Fractions

By using common denominators, division of fractions can be viewed as an exten-

sion of whole-number division. For example, 6

2

÷ is just a measurement division

7

7

problem where we ask the question, “How many groups of size 2 are in 6

7

?” The

7

answer to this question is the equivalent measurement division problem of 6 ÷ 2,

where the question is asked, “How many groups of size 2 are in 6?” Since there are

three 2s in 6, there are three 2 s in 6 . Figure 6.24 illustrates this visually. In general,

Figure 6.24

7

7

the division of fractions in which the divisor is not a whole number can be viewed

as a measurement division problem. On the other hand, if the division problem

has a divisor that is a whole number, then it should be viewed as a sharing division

problem.

Find the following quotients.

a. 12

4

÷

b. 6

3

÷ c. 16

2

÷

13

13

17

17

19

19

Reflection from Research

S O L U T I O N

Until students understand opera-

4

÷

=

tions with fractions, they often

a. 12

3, since there are three 4 in 12 .

13

13

13

13

have misconceptions that multi-

b. 6

3

÷

= 2, since there are two 3 in 6 .

17

17

17

17

plication always results in a larger

2

answer and division in a smaller

c. 16 ÷

= 8, since there are eight 2 in 16 .

19

19

19

19

answer (Greer, 1988).

Notice that the answers to all three of these problems can be found simply by dividing

the numerators in the correct order.

■

In the case of 12

5

÷ , we ask ourselves, “How many 5 make 12 ?” But this is the

13

13

13

13

same as asking, “How many 5s (including fractional parts) are in 12?” The answer is

2 and 2 fives or 12 fives. Thus 12

5

12

÷

= . Generalizing this idea, fraction division is

5

5

13

13

5

defined as follows.

D E F I N I T I O N 6 . 7

Division of Fractions with Common Denominators

a

c

Let

and be any fractions with c ≠ 0. Then

b

b

a

c

a

÷ = .

b

b

c

To divide fractions with different denominators, we can rewrite the fractions so

that they have the same denominator. Thus we see that

a

c

ad

bc

ad ⎛ a

d

÷ =

÷

=

= × ⎞

b

d

bd

bd

bc ⎝⎜

b

c ⎠⎟

using division with common denominators. For example,

3

5

27

35

27

÷ =

÷

=

.

7

9

63

63

35

a

c

a

d

Notice that the quotient ÷

is equal to the product ×

, since they are both

b

d

b

c

ab

equal to

. Thus a procedure for dividing fractions is to invert the divisor and multiply.

bc

Another interpretation of division of fractions using the missing-factor approach

refers directly to multiplication of fractions.

c06.indd 238

7/30/2013 3:00:26 PM - Section 6.3 Fractions: Multiplication and Division 239

21

7

Find:

÷ .

40

8

S O L U T I O N

21

7

21

7

Let

÷ = e . If the missing-factor approach holds, then

= × e . Then

40

8

f

40

8

f

7 × e

21

=

e

21

7

3

and we can take e = 3 and f = 5. Therefore,

= 3 , or

÷ = .

■

8 × f

40

f

5

40

8

5

In Example 6.17 we have the convenient situation where one set of numerators

and denominators divides evenly into the other set. Thus a short way of doing this

problem is

21

7

21 7

3

÷ =

÷ = ,

40

8

40 ÷ 8

5

since 21 ÷ 7 = 3 and 40 ÷ 8 = 5. This “divide-the-numerators-and-denominators

approach” can be adapted to a more general case, as the following example shows.

21

6

Find:

÷

.

40

11

S O L U T I O N

21

6

21 6 11

6

÷

=

× ×

÷

40

11

40 × 6 × 11 11

(21 6 11)

6

=

× ×

÷

(40 × 6 × 11) ÷ 11

21 11

=

×

40 × 6

231

21

11

=

=

×

■

240

40

6

Notice that this approach leads us to conclude that 21

6

21

11

÷ =

× . Generalizing

40

11

40

6

from these examples and results using the common-denominator approach, we are

led to the following familiar “invert-the-divisor-and-multiply” procedure.

T H E O R E M 6 . 6

Common Core – Grade 6

Division of Fractions with Unlike Denominators—

Interpret and compute quotients

Invert the Divisor and Multiply

of fractions, and solve word prob-

lems involving division of frac-

a

c

tions by fractions (e.g., by using

Let and be any fractions with c ≠ 0. Then

b

d

visual fraction models and equa-

tions to represent the problem).

a

c

a

d

In general, (a/b) ÷ (c /d) = ad/bc.

÷

= × .

b

d

b

c

A more visual way to understand why you “invert and multiply” in order to divide

fractions is described next. Consider the problem 3

1

÷ . Viewing it as a measurement

2

division problem, we ask “How many groups of size 1 are in 3?” Let each rectangle

2

in Figure 6.25 represent one whole. Since each of the rectangles can be broken into

two halves and there are three rectangles, there are 3 × 2 = 6 one-halfs in 3. Thus we

see that 3

1

÷ = 3 × 2 = 6.

2

c06.indd 239

7/30/2013 3:00:35 PM - 240 Chapter 6 Fractions

Figure 6.25

Reflection from Research

The problem 4

2

÷ should also be approached as a measurement division problem

3

As children are given opportuni-

for which the question is asked, “How many groups of size 2 are in 4?” In order to

3

ties to create fractional amounts

answer this question, we must first determine how many groups of size 2 are in one

3

from wholes (partitioning) and

whole. Let each rectangle in Figure 6.26 represent one whole and divide each of them

wholes from fractional amounts

(iterating), they will naturally

into three equal parts.

develop images that can be used

as tools for working with fractions

and fraction operations (Siebert

& Gaskin, 2006).

Figure 6.26

As shown in Figure 6.26, there is one group of size 2 in the rectangle and one-half

3

of a group of size 2 in the rectangle. Therefore, each rectangle has 1 1 groups of size

3

2

2 in it. In this case there are 4 rectangles, so there are 4 × 1 1 = 6 groups of size 2 in

3

2

3

4. Since 1 1 can be rewritten as 3 , the expression 4 × 1 1 is equivalent to 4

3

× . Thus,

2

2

2

2

4

2

÷ = 4 3

× = 6.

3

2

In a similar way, this visual approach can be extended to handle problems with a

dividend that is not a whole number.

Find the following quotients using the most convenient divi-

sion method.

17

4

3

5

6

2

5

13

a.

÷

b. ÷ c.

÷ d. ÷

11

11

4

7

25

5

19

11

S O L U T I O N

17

4

17

a.

÷

=

, using the common-denominator approach.

11

11

4

3

5

3

7

21

b. ÷

= × =

, using the invert-the-divisor-and-multiply approach.

4

7

4

5

20

6

2

6

2

3

c.

÷ =

÷

= , using the divide-numerators-and-denominators approach.

25

5

25 ÷ 5

5

5

13

5

11

55

d.

÷

=

×

=

, using the invert-the-divisor-and-multiply approach.

■

19

11

19

13

247

In summary, there are three equivalent ways to view the division of fractions:

1. The common-denominator approach

2. The divide-the-numerators-and-denominators approach

3. The invert-the-divisor-and-multiply approach

Now, through the division of fractions, we can perform the division of any whole

numbers without having to use remainders. (Of course, we still cannot divide by

a

b

a

1

a

zero.) That is, if a and b are whole numbers and b ≠ 0, then a ÷ b =

÷ = × = .

1

1

1

b

b

This approach is summarized next.

c06.indd 240

7/30/2013 3:00:37 PM - Section 6.3 Fractions: Multiplication and Division 241

Common Core – Grade 5

For all whole numbers a and

Interpret a fraction as division of

b, b ñ 0,

the numerator by the denomi-

a

nator (a/b = a ÷ b). Solve word

a ÷ b = .

b

problems involving division

of whole numbers leading to

answers in the form of fractions

or mixed numbers (e.g., by using

Find 17

visual fraction models or equa-

÷ 6 using fractions.

tions to represent the problem).

S O L U T I O N

17

5

17 ÷ 6 =

= 2

■

6

6

There are many situations in which the answer 2 5 is more useful than 2 with a

6

remainder of 5. For example, suppose that 17 acres of land were to be divided among

6 families. Each family would receive 2 5 acres, rather than each receiving 2 acres with

6

5 acres remaining unassigned.

Expressing a division problem as a fraction is a useful idea. For example, complex

1

fractions such as 2 may be written in place of 1

3

÷ . Although fractions are comprised

3

2

5

5

of whole numbers in elementary school mathematics, numbers other than whole

numbers are used in numerators and denominators of “fractions” later. For example,

2

the “fraction”

is simply a symbolic way of writing the quotient 2 ÷ p (numbers

p

such as 2 and p are discussed in Chapter 9).

Complex fractions are used to divide fractions, as shown next.

Find 1

3

÷ using a complex fraction.

2

5

S O L U T I O N

1

3

1

5

1

5

¥

5

2

3

2

3

÷ = ¥ =

=

2

5

3

5

1

6

5

3

Notice that the multiplicative inverse of the denominator 3 was used to form a

5

complex fraction form of one.

■

Check for Understanding: Exercise/Problem Set A #11–20

✔

Mental Math and Estimation for Multiplication

and Division

Mental math and estimation techniques similar to those used with whole numbers can

be used with fractions.

Calculate mentally.

1

4

4

a. (25 × 16) ×

b. 24

÷ 4 c. × 15

4

7

5

S O L U T I O N

1

⎛

1 ⎞

a. (25 × 16) ×

= 25 × 16 ×

25

4 100

4

⎝⎜

4 ⎠⎟ =

× =

Associativity was used to group 16 and 1 together, since they are compatible num-

4

bers. Also, 25 and 4 are compatible with respect to multiplication.

c06.indd 241

7/30/2013 3:00:43 PM - 242 Chapter 6 Fractions

4

⎛

4 ⎞

⎛ 4

⎞

1

1

b. 24

÷ 4 = 24 +

4

(24 4) + ÷ 4 6

6

7

⎝⎜

7 ⎠⎟ ÷

=

÷

⎝⎜ 7

⎠⎟ = + =

7

7

Right distributivity can often be used when dividing mixed numbers, as illustrated

here.

4

⎛

1⎞

⎛ 1

⎞

c.

× 15 = 4 ×

15

4

15

4

3 12

5

⎝⎜

5⎠⎟ ×

= ×

×

⎝⎜ 5

⎠⎟ = × =

The calculation also can be written as 4

15

× 15 = 4 ×

= 4 × 3 = 12. This product also

5

5

can be found as follows: 4

4

× 15 = ( × 5 3 4 3 12

)× = × = .

■

5

5

Estimate using the indicated techniques.

a. 5 1 × 7 5 using range estimation b. 4 3 × 9 1 rounding to the nearest 1 or whole

8

6

8

16

2

c. 14 8 ÷ 2 3 rounding to the nearest 1 or whole

9

8

2

S O L U T I O N

a. 5 1 × 7 5 is between 5 × 7 = 35 and 6 × 8 = 48.

8

6

3

1

1

1

1

b. 4

× 9

≈ 4 × 9 = 36 + 4 = 40

8

16

2

2

2

8

3

1

c. 14

÷ 2 ≈ 15 ÷ 2 = 6

■

9

8

2

Check for Understanding: Exercise/Problem Set A #21–24

✔

The Hindu mathematician Bhaskara (1119–1185) wrote an arithmetic text called the

Lilavat (named after his wife). The following is one of the problems contained in this

text. “A necklace was broken during an amorous struggle. One-third of the pearls fell

to the ground, one-fifth stayed on the couch, one-sixth were found by the girl, and

one-tenth were recovered by her lover; six pearls remained on the string. Say of how

many pearls the necklace was composed.”

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. Use a number line to illustrate how 1 × 5 is different

a.

b.

3

from 5

1

× .

3

2. Use the Chapter 6 eManipulative activity Multiplying Fractions

on our Web site or the rectangular area model to sketch repre-

sentations of the follow ing multiplication problems.

a. 1

2

×

×

7

×

3

5

b. 3

5

c. 2

8

6

3

10

3. What multiplication problems are represented by each of

the following area models? What are the products?

c06.indd 242

7/30/2013 3:00:49 PM - Section 6.3 Fractions: Multiplication and Division 243

4. Find reciprocals for the following numbers.

14. Use the fact that the numerators and denominators divide

a. 11 b. 9 c. 13 4 d. 108

evenly to simplify the following quotients.

21

3

9

a. 15

3

÷ b. 21 7

÷ c. 39 3

÷ d. 17 17

÷

5. a. Insert the appropriate equality or inequality symbol in

16

4

27

9

56

8

24

12

the following statement:

15. The missing-factor approach can be applied to fraction

3 3

division, as illustrated.

4

2

4

2

2

4

b. Find the reciprocals of 3 and 3 and complete the following

÷ = so × =

4

2

statement, inserting either < or > in the center blank.

7

5

5

7

Since we want 4 to be the result, we insert that in the box.

reciprocal of 3 reciprocal of 3

7

4

2

Then if we put in the reciprocal of 2 , we have

5

c. What do you notice about ordering reciprocals?

2

5

4

4

×

×

=

so

6. Identify which of the properties of fractions could be

5

2

7

7

applied to simplify each of the following computations.

4

2

5

4

20

10

a. 5

2

7

× × b. 3

2

2

2

( × )+ ( × ) c. 8 3 13

( × )×

÷ =

×

=

=

7

9

5

5

11

5

11

5

13

3

7

5

2

7

14

7

7. Perform the following operations and express your answer

Use this approach to do the following division problems.

in simplest form.

a. 3

2

÷ b. 13 3

÷ c. 12 6

÷

5

7

6

7

13

5

a. 4

3

×

b. 6

5

3

× ×

7

8

25

9

2

16. Find the following quotients using the most convenient

c. 2

9

2

4

× + ×

d. 11

5

5

7

× +

×

5

13

5

13

12

13

13

12

of the three methods for division. Express your answer in

e. 4

3

×

f. 2

4

5

× ×

simplest form.

7

14

9

3

7

g. 7

1

3

× 3 ×

h. 1

5

1

5

× ( − ) ×

a. 5

4

÷ b. 33 11

÷ c. 5

3

÷ d. 3

8

÷

15

2

5

3

4

4

6

7

9

14

7

13

13

11

22

8. Calculate using a fraction calculator if available.

17. Perform the following operations and express your answer

in simplest form.

a. 7

4

× b. 3 × 5

6

5

7

a. 7

11

÷ b. 1 5 1 5

+ ( ÷ ) c. 7 21

÷

11

7

3

4

4

6

8

4

9. Find the following products.

18. Calculate using a fraction calculator if available.

a. 2

1

× 4

b. 5 1 × 2 1

3

2

3

6

5

2

c. 3 7

2 3

×

d. 3 3 × 2 2

a. 3 ÷ b. 4 ÷ c. 8

1

÷

8

6

5

7

2

8

4

4

5

19. Find the following quotients.

10. We usually think of the distributive property for fractions

a. 8 1 ÷ 2 1 b. 6 1 ÷ 1 2 c. 16 2 ÷ 2 7

as “multiplication of fractions distributes over addition of

3

10

4

3

3

9

fractions.” Which of the following variations of the distrib-

20. Change each of the following complex fractions into

utive property for fractions holds for arbitrary fractions?

ordinary fractions.

a. Addition over subtraction

7

2

a. 9 b. 3

b. Division over multiplication

13

3

14

2

11. Suppose that the following unit square represents the

21. Calculate mentally using properties.

whole number 1. □ We can use squares like this one to

a. 15

3

× + 6 3

× b. 35 6

× − 35 3

×

7

7

7

7

represent division problems like 3

1

÷ , by asking how

2

c. ( 2

3 )

5

×

× d. 3 5 × 54

many 1 s are in 3. □ □ □ 3

1

÷ = 6, since there are six

5

8

2

9

2

2

one-half squares in the three squares. Draw similar fig-

22. Estimate using compatible numbers.

ures and calculate the quotients for the following division

a. 29 1 × 4 2 b. 57 1 ÷ 7 4

problems.

3

3

5

5

c. 70 3 ÷ 8 5 d. 31 1 × 5 3

a. 4

1

÷ b. 2 1 1

÷ c. 3 3

÷

5

8

4

4

3

2

4

4

23. Estimate using cluster estimation.

12. Using the Chapter 6 eManipulative activity Dividing

Fractions on our Web site, construct representations of the

a. 12 1 × 11 5 b. 5 1 × 4 8 × 5 4

4

6

10

9

11

following division problems. Sketch each representation.

24. Here is a shortcut for multiplying by 25:

a. 7

1

÷ b. 3 2

÷ c. 2 1 5

÷

4

2

4

3

4

8

100

36

25 × 36 =

× 36 = 100 ×

= 900.

13. Use the common-denominator method to divide the

4

4

following fractions.

Use this idea to find the following products mentally.

a. 15

3

÷ b. 4 3

÷ c. 33 39

÷

17

17

7

7

51

51

a. 25 × 44 b. 25 × 120 c. 25 × 488 d. 1248 × 25

PROBLEMS

25. The introduction of fractions allows us to solve equations

26. Another way to find a fraction between two given fractions

of the form ax = b by dividing whole numbers. For exam-

a

c

and is to find the average of the two fractions. For

ple, 5x = 16 has as its solution x = 16 (which is 16 divided

b

d

5

1

2

7

by 5). Solve each of the following equations and check

example, the average of 1 and 2 is 1 ( + ) =

. Use this

2

3

2

2

3

12

your results.

method to find a fraction between each of the given pairs.

8

11

a. 31x = 15 b. 67x = 56 c. 102x = 231

a. 7 , b. 7 ,

8

9

12

16

c06.indd 243

7/30/2013 3:00:59 PM - 244 Chapter 6 Fractions

27. You buy a family-size box of laundry detergent that con-

35. A piece of office equipment purchased for $60,000 depre-

tains 40 cups. If your washing machine calls for 1 1 cups

ciates in value each year. Suppose that each year the value

4

per wash load, how many loads of wash can you do?

of the equipment is 1 less than its value the preceding

20

year.

28. In December of 2012, about 481/999 of the oil refined in

a. Calculate the value of the equipment after 2 years.

the United States was produced in the United States. If

the United States produced 7,030,000 barrels per day in

b. When will the piece of equipment first have a value less

December of 2012, how much oil was being refined at that

than $40,000?

time? (Source: U.S. Energy Information Administration)

36. If a nonzero number is divided by one more than itself,

29. All but 1 of the students enrolled at a particular elemen-

the result is one-fifth. If a second nonzero number is divid-

16

tary school participated in “Family Fun Night” activities.

ed by one more than itself, the answer is one-fifth of the

If a total of 405 students were involved in the evening’s

number itself. What is the product of the two numbers?

activities, how many students attend the school?

37. Carpenters divide fractions by 2 in the following way:

11

11

11

÷ = × =

30. The directions for Weed-Do-In weed killer recommend mix-

2

(doubling the denominator)

16

16 2

32

ing 2 1 ounces of the concentrate with 1 gallon of water. The

a. How would they find 11 ÷ 5?

2

16

bottle of Weed-Do-In contains 32 ounces of concentrate.

a

a

b. Does ÷ n =

always?

a. How many gallons of mixture can be made from the

b

b × n

3 ÷

bottle of concentrate?

c. Find a quick mental method for finding 5

2. Do the

8

same for 10 9 ÷ 2.

b. Since the weed killer is rather expensive, one gardener

16

decided to stretch his dollar by mixing only 1 3 ounces

4

38. a. Following are examples of student work in multiplying

of concentrate with a gallon of water. How many more

fractions. In each case, identify the error and answer the

gallons of mixture can be made this way?

given problem as the student would.

31. A number of employees of a company enrolled in a fit-

Sam: 1

2

3

4

12

× = × =

= 2

2

3

6

6

6

ness program on January 2. By March 2, 4 of them were

5

3

6

6

3

still participating. Of those, 5 were still participating on

1

1

× = × = =

3

1

× = ?

6

4

8

8

8

8

4

4

6

May 2 and of those, 9 were still participating on July

10

Sandy: 3

5

3

6

18

9

× = × =

=

2. Determine the number of employees who originally

8

6

8

5

40

20

enrolled in the program if 36 of the original participants

2

2

2

3

6

3

× = × =

=

5

3

× = ?

5

3

5

2

10

5

6

8

were still active on July 2.

b.

Each student is confusing the multiplication algorithm

32. Each morning Tammy walks to school. At one-third of

with another algorithm. Which one?

the way she passes a grocery store, and halfway to school

she passes a bicycle shop. At the grocery store, her watch

39. Mr. Chen wanted to buy all the grocer’s apples for a

says 7:40 and at the bicycle shop it says 7:45. When does

church picnic. When he asked how many apples the store

Tammy reach her school?

had, the grocer replied, “If you added 1 , 1 , and 1 of them,

4

5

6

it would make 37.” How many apples were in the store?

33. A recipe that makes 3 dozen peanut butter cookies calls

for 1 1 cups of flour.

40. Seven years ago my son was one-third my age at that time.

4

a. How much flour would you need if you doubled the recipe?

Seven years from now he will be one-half my age at that

time. How old is my son?

b. How much flour would you need for half the recipe?

c. How much flour would you need to make 5 dozen cookies?

41. Try a few examples on the Chapter 6 eManipulative

Dividing Fractions on our Web site. Based on these exam-

34. A softball team had three pitchers: Gale, Ruth, and

ples, answer the following question: “When dividing 1

Sandy. Gale started in 3 of the games played in one sea-

8

whole by 3 , it can be seen that there is 1 group of 3 and a

son. Sandy started in one more game than Gale, and Ruth

5

5

part of a group of 3 . Why is the part of a group described

started in half as many games as Sandy. In how many of

5

as 2 and not 2 ?”

the season’s games did each pitcher start?

3

5

EXERCISE/PROBLEM SET B

EXERCISES

1. Use a number line to illustrate how 3 × 2 is different

a.

b.

5

from 2

3

× .

5

2. Use the Chapter 6 eManipulative activity Multiplying Fractions

on our Web site or the rectangular area model to sketch repre-

sentations of the following multiplication problems.

a. 1

3

× b. 4 2

× c. 3 5

×

4

5

7

3

4

6

3. What multiplication problems are represented by each of

the following area models? What are the products?

c06.indd 244

7/30/2013 3:01:04 PM - Section 6.3 Fractions: Multiplication and Division 245

4. a. What is the reciprocal of the reciprocal of 4 ?

13. Use the common-denominator method to divide the

13

b. What is the reciprocal of the multiplicative inverse of 4 ?

following fractions.

13

a. 5

3

÷

b. 12

4

÷ c. 13 28

÷

5. a. Order the following numbers from smallest to largest.

8

8

13

13

15

30

5

3

7

9

14. Use the fact that the numerators and denominators

divide evenly to simplify the following quotients.

8

16

5

10

a. 12

4

÷ b. 18 9

÷ c. 30

6

÷ d. 28 14

÷

b. Find the reciprocals of the given numbers and order

15

5

24

6

39

13

33

11

them from smallest to largest.

15. Use the method described in Set A, Exercise 15 to find the

c. What do you observe about these two orders?

following quotients.

a. 3

6

÷ b. 10

8

÷ c. 5 2

÷

6. Identify which of the properties of fractions could be

4

9

7

11

6

3

applied to simplify each of the following computations.

16. Find the following quotients using the most convenient of

a. 3

7

3

5

( × )+( × )

the three methods for division. Express your answer in sim-

8

6

8

6

plest form.

b. 6

11

2

× ( × )

11

3

7

a. 48

12

÷ b. 8

4

÷ c. 9 3

÷ d. 3 5

÷

63

21

11

11

3

5

7

8

c. 6

2

13

15

× × ×

13

5

6

2

17. Perform the following operations and express your answer

7. Perform the following operations and express your answer

in simplest form.

in simplest form.

a. 9

2

÷ b. 17

9

÷

c. 6

4

÷

11

3

100

10 000

,

35

21

a. 3

4

×

b. 2

21

×

5

9

7

10

18. Calculate using a fraction calculator if available.

c. 7

11

×

d. 4

8

7

8

× + ×

100

10 000

,

9

11

9

11

a. 7 ÷ 14 b. 12

2

÷ c. 3

5

÷

9

3

7

14

e. 3

2

4

× +

f. 3

2

4

+ ×

5

3

7

5

3

7

19. Calculate the following and express as mixed numbers in

g. 3

2

4

× ( + )

h. 7 2 × 5 4

5

3

7

5

7

simplest form.

a. 11 3 ÷ 9 8 b. 7 5 ÷ 13 2 c. 4 3 ÷ 3 8

8. Calculate using a fraction calculator if available.

5

9

8

3

4

11

a. 2

3

× b. 3 4

×

8

8

5

20. Change each of the following complex fractions into

ordinary fractions.

9. Calculate the following and express as mixed numbers

2

1 4

in simplest form.

a. 3 b. 7

2

3 7

a. 6

3

× 2 b. 8 1 × 3 4

3

8

7

4

4

5

21. Calculate mentally using properties.

c. 7 5 × 13 2

d. 11 3 × 9 8

8

3

5

9

a. 52 7

¥ − 52 3

¥

b. ( 2

5

+ ) 3

+

8

8

5

8

5

10. Which of the following variations of the distributive prop-

c. ( 3

1

× ) 7

×

d. 23 3

¥ + 7 + 23 4

¥

7

9

3

7

7

erty for fractions holds for arbitrary fractions?

a. Multiplication over subtraction

22. Estimate using compatible numbers.

b. Subtraction over addition

a. 19 1 × 5 3 b. 77 1 × 23 4

3

5

5

5

c. 54 3 ÷ 7 5 d. 25 2 × 3 3

11. Draw squares similar to those show in Set A, Exercise 11

5

8

3

4

to illustrate the following division problems and calculate

23. Estimate using cluster estimation.

the quotients.

a. 5 2 × 6 1 b. 3 1 × 2 8 × 3 2

3

8

10

9

11

a. 2

2

÷

b. 4

2

÷ c. 1 3 1

÷

3

5

4

2

24. Make up your own shortcuts for multiplying by 50 and 75

12. Using the Chapter 6 eManipulative activity Dividing

(see Set A, Exercise 24) and use them to compute the

Fractions on our Web site, construct representations

following products mentally. Explain your shortcut for

of the following division problems. Sketch each

each part.

representation.

a. 50 × 246

b. 84,602 × 50

a. 5

1

÷ b. 4 2

÷ c. 2 1 5

÷

6

3

3

5

3

6

c. 75 × 848

d. 420 × 75

PROBLEMS

25. Solve the following equations involving fractions.

cans sold in the United States. How many aluminum cans

a. 2

3

x = b. 1

5

x =

c. 2

7

x = d. 5

1

x =

were sold in the United States in 2010?

5

7

6

12

9

9

3

10

26. Find a fraction between 2 and 3 in two different ways.

28. Kids belonging to a Boys and Girls Club collected cans

7

8

and bottles to raise money by returning them for the

27. According to a report, approximately 57 billion alumi-

deposit. If 54 more cans than bottles were collected and

num cans were recycled in the United States in 2010. That

the number of bottles was 5 of the total number of bever-

11

amount was about 5 of the total number of aluminum

age containers collected, how many bottles were collected?

11

c06.indd 245

7/30/2013 3:01:13 PM - 246 Chapter 6 Fractions

29. Mrs. Martin bought 20 1 yards of material to make 4

39. Observe the following pattern:

4

bridesmaid dresses and 1 dress for the flower girl.

1

1

The flower girl’s dress needs only half as much

3 + 1 = 3 × 1

2

2

material as a bridesmaid dress. How much material

4 + 1 1 = 4 × 1 1

3

3

is needed for a bridesmaid dress? For the flower girl’s

5 + 1 1 = 5 × 1 1

dress?

4

4

30. In a cost-saving measure, Chuck’s company reduced all

a. Write the next two equations in the list.

salaries by 1 of their present amount. If Chuck’s monthly

b. Determine whether this pattern will always hold true. If

8

salary was $2400, what will he now receive? If his new sal-

so, explain why.

ary is $2800, what was his old salary?

40. Using the alternative definition of “less than,” prove the

31. If you place one full container of flour on one pan of a

following statements. Assume that the product of two

balance scale and a similar container 3 full and a

4

fractions is a fraction in part c.

1 -pound weight on the other pan, the pans balance. How

3

much does the full container of flour weigh?

a

c

c

e

a

e

a. If <

and < , then < .

b

d

d

f

b

f

32. A young man spent 1 of his allowance on a movie.

4

a

c

a

e

c

e

He spent 11 of the remainder on after-school snacks.

b. If < , then +

< + .

18

Then from the money remaining, he spent $3.00 on

b

d

b

f

d

f

a magazine, which left him 1 of his original allowance

a

c

a

e

c

e

e

24

c. If < , then ×

< × for any nonzero .

to put into savings. How much of his allowance did

b

d

b

f

d

f

f

he save?

41. How many guests were present at a dinner if

33. An airline passenger fell asleep halfway to her destination.

every two guests shared a bowl of rice, every

When she awoke, the distance remaining was half the dis-

three guests shared a bowl of broth, every four guests

tance traveled while she slept. For how much of the entire

shared a bowl of fowl, and 65 bowls were used

trip was she asleep?

altogether?

34. A recipe calls for 2 of a cup of sugar. You find that you

3

7

7

3

+

=

+

3

42. a. Does 2

5

2

5 ? Explain.

4

8

8

4

only have 1 a cup of sugar left. What fraction of the recipe

3

7

7

3

2

b. Does 2 × 5 = 2 × 5 ? Explain.

4

8

8

4

can you make?

35. The following students are having difficulty with division

Analyzing Student Thinking

of fractions. Determine what procedure they are using,

43. Devonnie asserts that she needs to find common denomi-

and answer their final question as they would.

nators to multiply fractions. Is she correct? Explain.

Abigail: 4

2

2

÷ = Harold: 2 3 3 3

9

÷ = × =

6

6

6

3

8

2

8

16

44. When asked to simplify 12

9

× , Cameron did the

6

2

3

÷

=

3

5

4

5

20

÷ = × =

27

24

10

10

10

4

6

3

6

18

following:

8

2

÷

=

5

3

÷ =

12

12

8

4

9

12

1

1

1

36. A chicken and a half lays an egg and a half in a day and a

×

= × = .

27

24

3

2

6

half. How many eggs do 12 chickens lay in 12 days?

a. How long will it take 3 chickens to lay 2 dozen eggs?

Is his method okay? Explain.

b. How many chickens will it take to lay 36 eggs in 6 days?

45. Damon asks you if you can draw a picture to explain what

37. Fill in the empty squares with different fractions to

3 of 5 means. What would you draw?

4

7

produce equations.

46. Hans asks if you can illustrate what 8

3

÷ means. What

4

would you draw?

47. Robbyn noticed that 9 ÷ 6

9

= whereas 6 ÷ 9 6

= . She

6

9

wonders if turning a division problem around will always

give answers that are reciprocals. How would

you respond?

48. Katrina said that dividing always makes numbers

smaller, for example, 10 ÷ 5 = 2 and 2 is smaller than 10.

She wonders how 6

1

÷ could give a result that is bigger

2

38. If the sum of two numbers is 18 and their product is 40,

than 6. How could you help Katrina make sense of this

find the following without finding the two numbers.

situation?

a. The sum of the reciprocals of the two numbers

49. To estimate 6 1 × 4 5 Blair uses range estimation, but

b. The sum of the squares of the two numbers

9

8

Paulo uses rounding. Should one method be preferred

[Hint: What is (x + y)2?]

over the other? Explain.

c06.indd 246

7/30/2013 3:01:17 PM - Chapter Review 247

END OF CHAPTER MATERIAL

equal? Answer—No! If the sums are equal in each set and if

these two sums are added together, the resulting sum would be

A child has a set of 10 cubical blocks. The lengths of the edges

even. However, the sum of 1 through 10 is 55, an odd number!

are 1 cm, 2 cm, 3 cm, . . . , 10 cm. Using all the cubes, can the

Additional Problems Where the Strategy “Solve an Equivalent

child build two towers of the same height by stacking one cube

Problem” Is Useful

upon another? Why or why not?

1. How many numbers are in the set { ,

11

,

18

,

25 . . . ,

}

396 ?

Strategy: Solve an Equivalent Problem

2. Which is larger: 230 or 320?

This problem can be restated as an equivalent problem: Can

3. Find eight fractions equally spaced between 0 and 1 on the

3

the numbers 1 through 10 be put into two sets whose sums are

number line.

Evelyn Boyd Granville (1924–)

Paul Erdos

Evelyn Boyd Granville was a

(1913–1996)

mathematician in the Mercury

Paul Erdos was one of

and Apollo space programs,

the most prolifi c mathe-

specializing in orbit and tra-

maticians of the modern

jectory computations. She says

, The University of

era. Erdos (pronounced

that if she had foreseen the

“air-dish”) authored or

Courtesy of Evelyn Granville

space program and her role in

Paul Halmos Collection,

e_ph_0080_01, The Dolph

Briscoe Center for American

History

Texas at Austin

coauthored approxi-

it, she would have been an astronomer. Granville grew up

mately 900 research papers. He was called an “itinerant

in Washington, D.C., at a time when the public schools

mathematician” because of his penchant for traveling

were racially segregated. She was fortunate to attend an

to mathematical conferences around the world. His

African-American high school with high standards and

achievements in number theory are legendary. At one

was encouraged to apply to the best colleges. In 1949, she

mathematical conference, he was dozing during a lec-

graduated from Yale with a Ph.D. in mathematics, one of

ture of no particular interest to him. When the speaker

two African-American women to receive doctorates in

mentioned a problem in number theory, Erdos perked

mathematics that year and the fi rst ever to do so. After

up and asked him to explain the problem again. The

the space program, she joined the mathematics faculty at

lecture then proceeded, and a few minutes later Erdos

California State University. She has written (with Jason

interrupted to announce that he had the solution! Erdos

Frand) the text Theory and Application of Mathematics

was also known for posing problems and offering mon-

for Teachers. “I never encountered any problems in com-

etary awards for their solution, from $25 to $10,000. He

bining career and private life. Black women have always

also was known for the many mathematical prodigies he

had to work.”

discovered and “fed” problems to.

CHAPTER REVIEW

Review the following terms and exercises to determine which require learning or relearning—page numbers are provided for easy

reference.

THE SET OF FRACTIONS

Vocabulary/Notation

Numerator 209

Set of fractions (F ) 210

Fraction strips 212

Denominator 209

Region model 212

Simplified 214

Fraction (a/b) 210

Equivalent fractions 212

Equal fractions 214

c06.indd 247

7/30/2013 3:01:19 PM - 248 Chapter 6 Fractions

Cross-product 214

Improper fraction 216

Greater than (>) 217

Cross-multiplication 214

Mixed number 216

Less than or equal to (≤) 217

Simplest form 215

Fraction number line 216

Greater than or equal to (≥) 217

Lowest terms 215

Less than (<) 217

Density property 219

Exercises

1. Explain why a child might think that 1 is greater than 1 .

4. Determine whether the following are equal. If not, deter-

4

2

mine the smaller of the two.

2. Draw a sketch to show why 3

6

= .

a. 24 8

,

b. 12 15

,

4

8

56

19

28

35

5. Express each fraction in Exercise 4 in simplest form.

3. Explain the difference between an improper fraction and a

mixed number.

6. Illustrate the density property using 2 and 5 .

5

12

FRACTIONS: ADDITION AND SUBTRACTION

Vocabulary/Notation

Least common denominator (LCD) 225

Exercises

1. Use fraction strips to find the following.

4. Which of the following properties hold for fraction

a. 1

5

+ b. 7 3

−

subtraction?

6

12

8

4

a. Closure

b. Commutative

2. Find the following sum/difference, and express your

c. Associative

d. Identity

answers in simplest form.

a. 12

13

+ b. 17

7

−

5. Calculate mentally, and state your method.

27

15

25

15

a. 5 3 + 3 7

b. 31 − 4 7 c. 2

3

5

( + )+

3. Name the property of addition that is used to justify each

8

8

8

7

5

7

of the following equations.

6. Estimate using the techniques given.

a. 3

2

2

3

+ = + b. 4

0

4

+

=

7

7

7

7

15

15

15

a. Range: 5 2 + 7 1

3

6

c. 2

3

4

2

3

4

+ ( + ) = ( + ) +

1

2

5

5

7

5

5

7

b. Rounding to the nearest 1 :17 + 24

2

8

5

d. 2

3

+ is a fraction

3

2

1

+

+

5

7

c. Front-end with adjustment: 9

7

5

4

3

6

FRACTIONS: MULTIPLICATION AND DIVISION

Vocabulary/Notation

Multiplicative inverse 237

Reciprocal 237

Complex fraction 241

Exercises

1. Use a model to find 2

4

× .

5. Find 12

1

÷ in two ways.

3

5

25

5

2. Find the following product/quotient, and express your

6. Which of the following properties hold for fraction

answers in simplest form.

w division?

a. 16

15

× b. 17 34

÷

25

36

19

57

a. Closure b. Commutative

3. Name the property of multiplication that is used to justify

c. Associative d. Identity

each of the following equations.

7. Calculate mentally and state your method.

a. 6

7

× = 1

b. 7 3

5

7

3 5

( × )=( × )

7

6

5

4

7

5

4 7

a. 2 × (5 × 9)

b. 25 × 2 2

c. 5

6

5

× =

d. 2

3

× is a fraction

3

5

9

6

9

5

7

e. 3 8

4

3

8 4

( × )=( × )

8. Estimate using the techniques given.

8 3

7

8

3 7

2

1

×

4. State the distributive property of fraction multiplication over

a. Range: 5

7

3

6

addition and give an example to illustrate its usefulness.

b. Rounding to the nearest 1

3

1

: 3 × 4

2

5

7

c06.indd 248

7/30/2013 3:01:24 PM - Chapter Test 249

CHAPTER TEST

Knowledge

Understanding

1. True or false?

10. Using a carton of 12 eggs as a model, explain how the

a. Every whole number is a fraction.

fractions 6 and 12 are distinguishable.

12

24

b. The fraction 17 is in simplest form.

51

a

b

11. Show how the statement “ < if and only if a < b”

c. The fractions 2 and 15 are equivalent.

12

20

b

c

a

c

d. Improper fractions are always greater than 1.

can be used to verify the statement “ <

if and only if

b

d

e. There is a fraction less than

2

and greater

1 000

,

000

,

ad < bc,” where b and d are nonzero.

than

1

.

1 000

,

000

,

f. The sum of 5 and 3 is 8 .

12. Verify the distributive property of fraction multiplication

7

8

15

g. The difference 4

5

− does not exist in the set of fractions.

over subtraction using the distributive property of whole-

7

6

number multiplication over subtraction.

h. The quotient 6

7

÷ is the same as the product 11 7

¥ .

11

13

6

13

13. Make a drawing that would show why 2

3

> .

2. Select two possible meanings of the fraction 3 and explain

3

5

4

each.

14. Use rectangles to explain the process of adding 1

2

+ .

4

3

3. Identify a property of an operation that holds in the set

15. Use the area model to illustrate 3

3

× .

5

4

of fractions but does not hold for the same operation on

16. Write a word problem for each of the following.

whole numbers.

a. 2

3

× b. 2 1

÷ c. 2 ÷ 3

4

3

5

Skill

17. If

is one whole, then shade the following regions:

4. Write the following fractions in simplest form.

a. 12 b. 34 c. 34 d. 123 123

,

18

36

85

567 567

,

a. 1 b. 2 of 1

4

3

4

5. Write the following mixed numbers as improper fractions,

and vice versa.

a. 3 5 b. 91 c. 5 2 d. 123

Problem Solving/Application

11

16

7

11

6. Determine the smaller of each of the following pairs of frac-

18. Notice that 2

3

4

< < . Show that this sequence continues

3

4

5

tions.

n

n 1

indefinitely—namely, that

< + when n ≥ 0.

a. 3 10

,

b. 7 7

, c. 16 18

,

4

13

2

3

92

94

n + 1

n + 2

19. An auditorium contains 315 occupied seats and was 7

9

7. Perform the following operations and write your answer in

filled. How many empty seats were there?

simplest form.

a. 4

5

+ b. 7

8

− c. 4 ¥ 15 d. 8 7

÷

20. Upon his death, Mr. Freespender left 1 of his estate to his

9

12

15

25

5

16

7

8

2

wife, 1 to each of his two children, 1 to each of his three

8

16

8. Use properties of fractions to perform the following

grandchildren, and the remaining $15,000 to his favorite

computations in the easiest way. Write answers in simplest

university. What was the value of his entire estate?

form.

21. Find three fractions that are greater than 2 and less

a. 5

3

¥ ( ¥ 2 )

b. 4 3

4

3

¥ +

5

5

¥

2

4

5

7

5

5

than 3 .

c. ( 13

5

+ ) 4

+

d. 3 5

4

¥

3

−

7

7

¥

17

11

17

8

9

8

22. Inga was making a cake that called for 4 cups of flour.

9. Estimate the following and describe your method of

However, she could only find a two-thirds measuring cup.

estimation.

How many two-thirds measuring cups of flour will she

a. 35 4 ÷ 9 2 b. 3 5 × 14 2 c. 3 4 + 13 1

3

+

5

7

8

3

9

5

13

need to make her cake?

c06.indd 249

7/30/2013 3:01:30 PM - C H A P T E R

7 DECIMALS, RATIO,

PROPORTION, AND PERCENT

The Golden Ratio

The golden ratio, also called the divine proportion, right, a surprising result regarding the areas is obtained.

was known to the Pythagoreans in 500 B.C.E. and

(Check this!)

has many interesting applications in geometry.

The golden ratio may be found using the Fibonacci

sequence, 1, 1

, 2, 3, 5, 8, . . . , an, . . . , where an is obtained

by adding the previous two numbers. That is,

1 + 1 = 2, 1

+ 2 = 3, 2 + 3 = 5, and so on. If the quotient of

a

each consecutive pair of numbers, n , is formed, the

an−1

numbers produce a new sequence. The first several terms

Notice that the numbers 5, 8, 13, and 21 occur. If these

of this new sequence are 1, 2, 1

5

. , 1

66

.

. . . , 1

6

. , 1

625

.

,

numbers from the Fibonacci sequence are replaced by

1 61538

.

. . . , 1 61904

.

. . . , . . . . These numbers approach a

8, 13, 21, and 34, respectively, an even more surprising

decimal 1 61803

.

. . . , which is the golden ratio, technically

result occurs. These surprises continue when using the

1

5

f = +

Fibonacci sequence. However, if the four numbers are

. (Square roots are discussed in Chapter 9.)

2

replaced with 1, f, f + 1, and 2f + 1, respectively, all is

Following are a few of the remarkable properties asso-

in harmony.

ciated with the golden ratio.

3. Surprising places. Part of Pascal’s triangle is shown.

1. Aesthetics. In a golden rectangle, the ratio of the length

to the width is the golden ratio, f. Golden rectangles

were deemed by the Greeks to be especially pleasing to

the eye. The Parthenon at Athens can be surrounded by

such a rectangle.

However, if carefully rearranged, the Fibonacci

sequence reappears.

geyAK/

Ser

iStockphoto

Along these lines, notice how index cards are usually

dimensioned 3 × 5 and 5 × 8, two pairs of numbers in

the Fibonacci sequence whose quotients approximate f.

These are but a few of the many interesting relation-

2. Geometric fallacy. If one cuts out the square shown next

ships that arise from the golden ratio and its counterpart,

on the left and rearranges it into the rectangle shown at

the Fibonacci sequence.

250

c07.indd 250

7/31/2013 11:54:04 AM - Problem-Solving

Work Backward

Strategies

Normally, when you begin to solve a problem, you probably start at the beginning

1. Guess and Test

of the problem and proceed “forward” until you arrive at an answer by applying

2. Draw a Picture

appropriate strategies. At times, though, rather than start at the beginning of a prob-

lem statement, it is more productive to begin at the end of the problem statement and

3. Use a Variable

work backward. The following problem can be solved quite easily by this strategy.

4. Look for a

Pattern

Initial Problem

5. Make a List

A street vendor had a basket of apples. Feeling generous one day, he gave away

6. Solve a Simpler

one-half of his apples plus one to the first stranger he met, one-half of his remaining

Problem

apples plus one to the next stranger he met, and one-half of his remaining apples plus

one to the third stranger he met. If the vendor had one left for himself, with how

7. Draw a Diagram

many apples did he start?

8. Use Direct

Reasoning

9. Use Indirect

Reasoning

10. Use Properties

of Numbers

11. Solve an

Equivalent

Problem

12. Work Backward

Clues

The Work Backward strategy may be appropriate when

r The final result is clear and the initial portion of a problem is obscure.

r A problem proceeds from being complex initially to being simple at the end.

r A direct approach involves a complicated equation.

r A problem involves a sequence of reversible actions.

A solution of this Initial Problem is on page 298.

251

c07.indd 251

7/31/2013 11:54:04 AM - AUTHOR

I N T R O D U C T I O N

In Chapter 6, the set of fractions was introduced to permit us to deal with parts of a whole. In this

chapter we introduce decimals, which are a convenient numeration system to represent fractions, and

percents. Percents are representations of fractions in equivalent forms with powers of 10 as denomi-

nators and are convenient for commerce. The concepts of ratio and proportion are also developed

WALK-THROUGH because of their importance in applications throughout mathematics.

Key Concepts from the NCTM Principles and Standards for School Mathematics

r GRADES 3-5–NUMBER AND OPERATIONS

Understand the place-value structure of the base ten number system and be able to represent and compare whole num-

bers and decimals.

Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.

Develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students’

experience.

Use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.

r GRADES 6-8–NUMBER AND OPERATIONS

Work flexibly with fractions, decimals, and percents to solve problems.

Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.

Develop meaning for percents greater than 100 and less than 1.

Understand and use ratios and proportions to represent quantitative relationships.

Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.

Use the associative and commutative properties of addition and multiplication and the distributive property of multi-

plication over addition to simplify computations with integers, fractions, and decimals.

Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equiva-

lent ratios.

Key Concepts from the NCTM Curriculum Focal Points

r GRADE 2: Developing an understanding of the base ten numeration system and place-value concepts.

r GRADE 4: Developing an understanding of decimals, including the connections between fractions and decimals.

r GRADE 5: Developing an understanding of and fluency with addition and subtraction of fractions and decimals.

r GRADE 6: Developing an understanding of and fluency with multiplication and division of fractions and decimals.

r GRADE 7: Developing an understanding of and applying proportionality, including similarity.

Key Concepts from the Common Core State Standards for Mathematics

r GRADE 4: Understand decimal notation for fractions, and compare decimal fractions.

r GRADE 5: Understand the place value system for decimals to the thousandths place. Perform operations with

decimals to hundredths.

r GRADE 6: Compute fluently with multi-digit numbers (including decimals). Understand ratio concepts and use

ratio reasoning to solve problems.

r GRADE 7: Analyze proportional relationships and use them to solve real-world and mathematical problems.

252

c07.indd 252

7/31/2013 11:54:04 AM - Section 7.1 Decimals 253

DECIMALS

The numbers 0.1, 0.10, and 0.100 are all equal but can be represented differently. Use base

ten blocks to represent 0.1, 0.10, and 0.100 and demonstrate that they are, in fact, equal.

NCTM Standard

Decimals

All students should understand

the place-value structure of the

Decimals are used to represent fractions in our usual base ten place-value notation.

base ten number system and be

The method used to express decimals is shown in Figure 7.1.

able to represent and compare

whole numbers and decimals.

Children’s Literature

www.wiley.com/college/musser

See “If America Were a Village”

by David J. Smith.

Figure 7.1

Common Core – Grade 5

In the figure the number 3457.968 shows that the decimal point is placed between the

Read and write decimals to

ones column and the tenths column to show where the whole-number portion ends

thousandths using base ten

and where the decimal (or fractional) portion begins. Decimals are read as if they

numerals, number names,

were written as fractions and the decimal point is read “and.” The number 3457.968

and expanded form [e.g.,

is written in its expanded form as

347.392 = 3 × 100 + 4 × 10 +

7 × 1+ 3 × 1

( /10) + 9 × 1

( /100) +

1 ⎞

1 ⎞

1 ⎞

2 × 1

( /1000)].

3 1000

(

) + 4 100

(

) + 5 1

( 0) + 7 1

( ) + ⎛

9

6

8

⎝⎜10⎠⎟ + ⎛⎝⎜100⎠⎟ + ⎛⎝⎜1000⎠⎟

From this form one can see that 3457 968

.

= 3457 968 and so is read “three thousand

1000

Reflection from Research

four hundred fifty-seven and nine hundred sixty-eight thousandths.” Note that the

Students often have misconcep-

word and should only be used to indicate where the decimal point is located.

tions regarding decimals. Some

Figure 7.2 shows how a hundreds square can be used to represent tenths and hun-

students see the decimal point

as something that separates two

dredths. Notice that the large square represents 1, one vertical strip represents 0.1,

whole numbers (Greer, 1987).

and each one of the smallest squares represents 0.01.

Figure 7.2

c07.indd 253

7/31/2013 11:54:06 AM - 254 Chapter 7 Decimals, Ratio, Proportion, and Percent

A number line can also be used to picture decimals. The number line in Figure 7.3

shows the location of various decimals between 0 and 1.

Figure 7.3

Rewrite each of these numbers in decimal form, and state the

decimal name.

a. 7

b. 123 c. 17

100

10 000

,

8

S O L U T I O N

a. 7 = 0 0

. 7, read “seven hundredths”

100

b. 123

100

20

3

1

2

3

=

+

+

=

+

+

= 0.00123, read “one hundred twenty-

10 000

,

10 000

,

10 000

,

10 000

,

100

1000

10 000

,

three ten thousandths”

7

7

7 ¥ 5 ¥ 5 ¥ 5

875

800

70

5

8

7

c. 1

= 1 + = 1 +

= 1 +

= 1 +

+

+

= 1 +

+

+

8

8

2 ¥ 2 ¥ 2 ¥ 5 ¥ 5 ¥ 5

1000

1000

1000

1000

10

100

5

= 1 875

.

, read “one and eight hundred seventy-fi ve thousandths”

■

1000

All of the fractions in Example 7.1 have denominators whose only prime factors

are 2 or 5. Such fractions can always be expressed in decimal form, since they have

equivalent fractional forms whose denominators are powers of 10. This idea is illus-

trated in Example 7.2.

Reflection from Research

Express as decimals.

Students should be encouraged

to express decimal fractions with

3

7

43

meaningful language (rather than

a.

b.

c.

24

23

using “point”). It is sometimes

¥ 5

1250

helpful to have students break

S O L U T I O N

fractions down into composi-

tions of tenths; for instance, 0.35

3

3 ¥ 54

1875

7

7 ¥ 52

175

would be read three tenths plus

a.

=

=

= 0 1875

.

b.

=

=

= 0 175

.

24

24 ¥ 54

10,000

23 5

23 ¥ 53

¥

1000

five hundredths rather than 35

hundredths (Resnick, Nesher,

43

43

43 ¥ 23

344

Leonard, Magone, Omanson, &

c.

=

=

=

= 0 0344

.

■

1250

2 ¥ 54

24 ¥ 54

10,000

Peled, 1989).

The decimals we have been studying thus far are called terminating decimals,

since they can be represented using a finite number of nonzero digits to the right of

the decimal point. We will study nonterminating decimals later in this chapter. The

following result should be clear, based on the work we have done in Example 7.2.

Use whatever method you would like to find the decimal representations of the following

numbers.

1 , 3 , 2 , 4 , 9 , 25 , 7 , 27

3

8

7

5

36

30

15

40

Determine a criterion that, by looking at a fraction, you will know whether the decimal representation of the fraction will

terminate or not.

c07.indd 254

7/31/2013 11:54:32 AM - Section 7.1 Decimals 255

T H E O R E M 7 . 1

Fractions with Terminating Decimal Representations

a

a

Let be a fraction in simplest form. Then has a terminating decimal representa-

b

b

tion if and only if b contains only 2s and/or 5s in its prime factorization (since b

can be expanded to a power of 10).

Check for Understanding: Exercise/Problem Set A #1–7

✔

Reflection from Research

Ordering Decimals

Students mistakenly identify a

number such as 0.1814 as being

Terminating decimals can be compared using a hundreds square, using a number line,

larger than 0.3 because 0.1814

by comparing them in their fraction form, or by comparing place values one at a time

has more digits (Hiebert &

Wearne, 1986).

from left to right just as we compare whole numbers.

Determine the larger of each of the following pairs of numbers in

the four ways mentioned in the preceding paragraph.

a. 0.7, 0.23 b. 0.135, 0.14

S O L U T I O N

Common Core – Grade 5

a. Hundreds Square: See Figure 7.4. Since more is shaded in the 0.7 square, we con-

Compare two decimals to hun-

clude that 0 7

. > 0 23

.

.

dredths by reasoning about their

size. Recognize that comparisons

are valid only when the two

decimals refer to the same whole.

Record the results of comparisons

with the symbols >, =, or <, and

justify the conclusions, e.g., by

using a visual model.

Figure 7.4

Number Line: See Figure 7.5. Since 0.7 is to the right of 0.23, we have 0 7

. > 0 23

.

.

Fraction Method: First, 0 7

7

. =

, 0 2

. 3

23

=

. Now 7

70

=

and 70

23

>

since 70 > 23.

10

100

10

100

100

100

Therefore, 0 7

. > 0 23

.

.

Figure 7.5

Place-Value Method: 0 7

. > 0 2

. 3, since 7 > 2. The reasoning behind this method is

that since 7 > 2, we have 0 7

. > 0 2

. . Furthermore, in a terminating decimal, the

digits that appear after the 2 cannot contribute enough to make a decimal as

large as 0.3 yet have 2 in its tenths place. This technique holds for all terminating

decimals.

b. Hundreds Square: The number 0.135 is 1 tenth plus 3 hundredths plus 5 thou-

sandths. Since 5

1

1

1

1

=

= ¥

,13 squares on a hundreds square must be

1000

200

2

100

2

shaded to represent 0.135. The number 0.14 is represented by 14 squares on a hun-

dreds square. See Figure 7.6. Since an extra half of a square is shaded in 0.14, we

have 0 14

.

> 0 135

.

.

Number Line: See Figure 7.7. Since 0.14 is to the right of 0.135 on the number line,

0 14

.

> 0 135

.

.

c07.indd 255

7/31/2013 11:54:36 AM - 256

Chapter 7 Decimals, Ratio, Proportion, and Percent

Common Core – Grade 5

Compare two decimals to thou-

sandths based on meanings of

the digits in each place, using

>, =, and < symbols to record

the results of comparisons.

Figure 7.6

Figure 7.7

Fraction Method: 0 135

135

.

=

and 0 1

. 4

14

140

=

=

. Since 140 > 135, we have

1000

100

1000

0 14

.

> 0 135

.

. Many times children will write 0 135

.

> 0 14

.

because they know

135 > 14 and believe that this situation is the same. It is not! Here we are comparing

decimals, not whole numbers. A decimal comparison can be turned into a whole-

number comparison by getting common denominators or, equivalently, by having

the same number of decimal places. For example, 0 14

.

> 0 135

.

since 140

135

>

, or

1000

1000

0 140

.

> 0 135

.

.

Place-Value Method: 0 14

.

> 0 135

.

, since (1) the tenths are equal (both are 1), but

(2) the hundredths place in 0.14, namely 4, is greater than the hundredths place in

0.135, namely 3.

■

Check for Understanding: Exercise/Problem Set A #8–11

✔

NCTM Standard

Mental Math and Estimation

All students should use models,

benchmarks, and equivalent

The operations of addition, subtraction, multiplication, and division involving deci-

forms to judge the size of

mals are similar to the corresponding operations with whole numbers. In particular,

fractions.

place value plays a key role. For example, to find the sum 3 2

. + 5 7

. mentally, one may

add the whole-number parts, 3 + 5 = 8, and then the tenths, 0 2

. + 0 7

. = 0 9

. , to obtain

Children’s Literature

8.9. Observe that the whole-number parts were added first, then the tenths—that is,

www.wiley.com/college/musser

the addition took place from left to right. In the case of finding the sum 7 6

. + 2 5

. ,

See “Night Noises’’ by Mem Fox.

one could add the tenths first, 0 6

. + 0 5

. = 1 1

. , then combine this sum with 7 + 2 = 9 to

obtain the sum 9 + 1 1

. = 10 1

. . Thus, as with whole numbers, decimals may be added

from left to right or right to left.

Before developing algorithms for operations involving decimals, some mental

math and estimation techniques similar to those that were used with whole numbers

and fractions will be extended to decimal calculations.

Use compatible (decimal) numbers, properties, and/or compen-

sation to calculate the following mentally.

a. 1 7

. + (3 2

. + 4 3

. )

b. ( .

0 5 × .

6 7) × 4

c. 6 × 8 5

.

d. 3 76

.

+ 1 98

.

e. 7 32

.

− 4 94

.

f. 17 × 0 25

.

+ 0 25

.

× 23

S O L U T I O N

a. 1 7

. + (3 2

. + 4 3

. ) = 1

( 7

. + 4 3

. ) + 3 2

. = 6 + 3 2

. = 9 2

. . Here 1.7 and 4.3 are compatible

numbers with respect to addition, since their sum is 6.

Reflection from Research

Students often have difficulty

b. (0.5 × 6.7) × 4 = .

6 7 × ( .

0 5 × 4) = .

6 7 × 2 = 1 .

3 .

4 Since 0 5

. × 4 = 2, it is more con-

understanding the equivalence

venient to use commutativity and associativity to fi nd 0 5

. × 4 rather than to fi nd

between a decimal fraction and

0 5

. × 6 7

. fi rst.

a common fraction (for instance,

c. Using distributivity, 6

that 0.4 is equal to 2/5). Research

× 8 5

. = 6(8 + 0 5

. ) = 6 × 8 + 6 × 0 5

. = 48 + 3 = 51.

has found that this understanding

d. 3 76

.

+ 1 98

.

= 3 74

.

+ 2 = 5 74

.

using additive compensation.

can be enhanced by teaching the

e. 7 32

.

− 4 94

.

= 7 38

.

− 5 = 2 38

.

by equal additions.

two concurrently by using both a

f. 17 × 0 25

.

+ 0 25

.

× 23 = 17 × 0 25

.

+ 23 × 0 25

.

= 17

(

+ 23) × 0 25

.

= 40 × 0 25

.

= 10 using

decimal fraction and a common

fraction to describe the same

distributivity and the fact that 40 and 0.25 are compatible numbers with respect to

situation (Owens, 1990).

multiplication.

■

c07.indd 256

7/31/2013 11:54:43 AM - Section 7.1 Decimals 257

TABLE 7.1

Since common decimals have fraction representations, the fraction equivalents

DECIMAL

FRACTION

shown in Table 7.1 can often be used to simplify decimal calculations.

0.05

1

20

0.1

1

10

Find these products using fraction equivalents.

0.125

1

8

a. 68 × 0 5

.

b. 0 2

. 5 × 48

c. 0 2

. × 375

0.2

1

5

d. 0 05

.

× 280

e. 56 × 0 125

.

f. 0 7

. 5 × 72

0.25

1

4

0.375

3

S O L U T I O N

8

0.4

2

5

a. 68 × 0 5

. = 68

1

× = 34

2

0.5

1

1

2

b. 0 2

. 5 × 48 = × 48 = 12

4

0.6

3

1

5

c. 0 2

. × 375 = × 375 = 75

5

0.625

5

d. 0 05

.

× 280

1

=

× 280 1

= × 28 = 14

8

20

2

0.75

3

e. 56 × 0 125

.

= 56 1

× = 7

4

8

0.8

4

f. 0 75

.

× 72 3

= × 72 = 3 1

× × 72 = 3 × 18 = 54

■

5

4

4

0.875

7

8

Multiplying and dividing decimals by powers of 10 can be performed mentally in a

fashion similar to the way we multiplied and divided whole numbers by powers of 10.

Find the following products and quotients by converting to

fractions.

a. 3 75 104

.

×

b. 62 013 105

.

×

c. 127 9

. ÷ 10 d. 0 53 104

.

÷

S O L U T I O N

a. 3 75

.

104

375

10 000

,

×

=

×

= 37,500

100

1

b. 62 013

.

105

62 013

,

100 000

,

×

=

×

= 6,201,300

1000

1

c. 127 9

. ÷ 10

1279

=

÷ 10 1279

1

=

×

= 12 79

.

10

10

10

d. 0 53

.

÷ 104

53

=

÷ 104

53

1

=

×

= 0 000053

.

■

100

100

10 000

,

Notice that in Example 7.6(a), multiplying by 104 was equivalent to moving the

decimal point of 3.75 four places to the right to obtain 37,500. Similarly, in part (b),

because of the 5 in 105, moving the decimal point five places to the right in 62.013

results in the correct answer, 6,201,300. When dividing by a power of 10, the decimal

point is moved to the left an appropriate number of places. These ideas are summa-

rized next.

T H E O R E M 7 . 2

Multiplying/Dividing Decimals by Powers of 10

Let n be any decimal number and m represent any nonzero whole number.

Multiplying a number n by 10m is equivalent to forming a new number by moving

the decimal point of n to the right m places. Dividing a number n by 10m is equiva-

lent to forming a new number by moving the decimal point of n to the left m places.

Multiplying/dividing by powers of 10 can be used with multiplicative compensation

to multiply some decimals mentally. For example, to find the product 0 003

.

× 41,000,

one can multiply 0.003 by 1000 (yielding 3) and then divide 41,000 by 1000 (yielding

41) to obtain the product 3 × 41 = 123.

Previous work with whole-number and fraction computational estimation can also

be applied to estimate the results of decimal operations.

c07.indd 257

7/31/2013 11:54:51 AM - 258 Chapter 7 Decimals, Ratio, Proportion, and Percent

Estimate each of the following using the indicated estimation

techniques.

a. $ .

1 57 + $ .

4 36 + $ .

8 78 using (i) range, (ii) front-end with adjustment, and (iii) round-

ing techniques

b. 39 3

. 7 × 5 5

. using (i) range and (ii) rounding techniques

S O L U T I O N

a. Range: A low estimate for the range is $1 + $4 + $8 = $1 ,

3 and a high estimate is

$2 + $5 + $9 = $1 .

6 Thus a range estimate of the sum is $13 to $16.

Front-end: The one-column front-end estimate is simply the low estimate of the

range, namely $13. The sum of 0.57, 0.36, and 0.78 is about $1.50, so a good esti-

mate is $14.50.

Rounding: Rounding to the nearest whole or half yields an estimate of $1.50 +

$ .

4 50 + $ .

9 00 = $ .

15

.

00

b. Range: A low estimate is 30 × 5 = 150, and a high estimate is 40 × 6 = 240. Hence a

range estimate is 150 to 240.

Rounding: One choice for estimating this product is to round 39 3

. 7 × 5 5

. to 40 × 6 to

obtain 240. A better estimate would be to round to 40 × 5 5

. = 220.

■

Decimals can be rounded to any specified place as was done with whole numbers.

Round 56.94352 to the nearest

a. tenth

b. hundredth

c. thousandth

d. ten thousandth

Common Core – Grade 5

S O L U T I O N

Use place value understanding

to round decimals to any place.

a. First, 56 9

. < 56 94352

.

< 57 0

. . Since 56.94352 is closer to 56.9 than to 57.0, we round

to 56.9 (Figure 7.8).

Figure 7.8

b. 56 9

. 4 < 56 94352

.

< 56 95

. and 56.94352 is closer to 56.94 (since 352 < 500), so we

round to 56.94.

c. 56 943

.

< 56 94352

.

< 56 944

.

and 56.94352 is closer to 56.944, since 52 > 50. Thus we

round up to 56.944.

d. 56 9435

.

< 56 94352

.

< 56 9436

.

. Since 56 94352

.

< 56 94355

.

, and 56.94355 is the half-

way point between 56.94350 and 56.94360, we round down to 56.9435.

■

For decimals ending in a 5, we can use the “round a 5 up” method, as is usually done

in elementary school. For example, 1.835, rounded to hundredths, would round to 1.84.

Perhaps the most useful estimation technique for decimals is rounding to numbers

that will, in turn, yield compatible whole numbers or fractions.

Estimate.

a. 203 4

. × 47 8

.

b. 31 ÷ 1 9

. 3

c. 75 × 0 2

. 4

d. 124 ÷ 0 7

. 4

e. 0 0021

.

× 44 123

,

f. 3847 6

. ÷ 51 3

.

c07.indd 258

7/31/2013 11:54:59 AM - Section 7.1 Decimals 259

S O L U T I O N

a. 203 4

. × 47 8

. ≈ 200 × 50 = 10,000

b. 31 ÷ 1 93

.

≈ 30 ÷ 2 = 15

c. 75 × 0 2

. 4 ≈ 75

1

× ≈ 76 1

× = 19. (Note that 76 and 1 are compatible, since 76 has a

4

4

4

factor of 4.)

d. 124 ÷ 0 7

. 4 ≈ 124

3

÷ = 124 4

× ≈ 123 4

× = 164. (123 and 3 , hence 4 , are compatible,

4

3

3

4

3

since 123 has a factor of 3.)

e. 0 0021

.

× 44 123

,

= 0 2

. 1 × 441 23

1

.

≈ × 450 = 90. (Here multiplicative compensation

5

was used by multiplying 0.0021 by 100 and dividing 44,123 by 100.)

f. 3847 6

. ÷ 51 3

. = 38 476

.

÷ 0 513

.

≈ 38 1

÷ = 76;

÷

≈

2

alternatively, 3847 6

.

51 3

.

3500 ÷ 50 = 70

■

Check for Understanding: Exercise/Problem Set A #12–17

✔

Decimal notation has evolved over the years without universal

agreement. Consider the following list of decimal expressions for

the fraction 3142

1000 .

NOTATION

DATE INTRODUCED

3 142

1522, Adam Riese (German)

3 | 142⎫

1579, François Vieta (French)

⎬

,

3 142 ⎭

0

1

2

3

1585, Simon Stevin (Dutch)

3

1

4

2

3 ¥ 142

1614, John Napier (Scottish)

Today, Americans use a version of Napier’s “decimal point” notation

(3.142, where the point is on the line), the English retain the original

©Ron Bagwell

version (3 ¥142, where the point is in the middle of the line), and

the French and Germans retain Vieta’s “decimal comma” notation

(3,142). Hence the issue of establishing a universal decimal notation

remains unresolved to this day.

EXERCISE/PROBLEM SET A

EXERCISES

1. Write each of the following sums in decimal form.

4. Write the following numbers in words.

a. 7 10

(

) + 5 + 6 1

( ) + 3 1

(

)

a. 0.013

b. 68,485.532

10

1000

b. 6 1 2 + 3 1 3

( )

( ) c. 3 10 2 + 6 + 4 1 2 + 2 1 3

(

)

( )

( )

c. 0.0082

d. 859.080509

10

10

10

10

5. A student reads the number 3147 as “three thousand

2. Write each of the following decimals (i) in its expanded

one hundred and forty-seven.” What is wrong with this

form and (ii) as a fraction.

reading?

a. 0.45

b. 3.183

c. 24.2005

6. Determine, without converting to decimals, which of

3. Write the following expressions as decimal numbers.

the following fractions has a terminating decimal

a. Seven hundred forty-six thousand

representation.

b. Seven hundred forty-six thousandths

3

c. Seven hundred and forty-six thousandths

a. 21 b. 62 c. 6 d. 326 e. 39 f. 54

45

125

90

400

60

60

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7/31/2013 11:55:06 AM - 260 Chapter 7 Decimals, Ratio, Proportion, and Percent

7. Decide whether the following fractions terminate in their

e. 51 24 103

.

÷

f. 21 28

.

+ 17 79

.

decimal form. If a fraction terminates, tell in how many

g. 8(9.5)

h. 0 15 105

.

×

places and explain how you can tell from the fraction form.

a. 4 b. 7 c. 1 d. 3

13. Calculate mentally by using fraction equivalents.

3

8

15

16

a. 0 2

. 5 × 44

b. 0 7

. 5 × 80

8. Arrange the following numbers in order from smallest to

c. 35 × 0 4

.

d. 0 2

. × 65

largest.

e. 65 × 0 8

.

f. 380 × 0 0

. 5

a. 0.58, 0.085, 0.85

b. 781.345, 781.354, 780.9999

14. Find each of the following products and quotients.

c. 4.9, 4.09, 4.99, 4.099

a. (6.75)(1,000,000)

b. 19 514

.

÷ 100,000

9. One method of comparing two fractions is to find their

2 9

. 6 × 1016

c. (2.96 1016 )(1012

×

)

d.

decimal representations by calculator and compare them.

1012

For example, divide the numerator by the denominator.

15. Estimate, using the indicated techniques.

7 =

9

0 58333333

.

= 0 56250000

.

a. 4 7

. 5 + 5 9

. 1 + 7 3

. 6; range and rounding to the nearest

12

16

whole number

Thus 9

7

< . Use this method to compare the following

b. 74 5

. × 6 1

. ; range and rounding

16

12

fractions.

c. 3 18

.

+ 4 3

. 9 + 2 73

. ; front-end with adjustment

a. 5

d. 4 3

. × 9 7

. ; rounding to the nearest whole number

9 and 19 b. 38 and 18

34

52

25

10. Order each of the following from smallest to largest as

16. Estimate by rounding to compatible numbers and fraction

simply as possible by using any combination of these three

equivalents.

methods: (i) common denominator, (ii) cross-multiplication,

a. 47 1

. ÷ 2 9

.

b. 0 2

. 3 × 88

and (iii) converting to decimal.

c. 126 × 0 2

. 1

d. 56,324 × 0 2

. 5

a. 7 4 9

, ,

b. 27 43 539

,

,

c. 3 5 7

, ,

e. 14,897 ÷ 750

f. 0 5

. 9 × 474

8 5 10

25 40 500

5 8 9

11. The legal limit of blood alcohol content to drive a car is

17. Round the following.

0.08. Three drivers are tested at a police checkpoint. Juan

a. 97.26 to the nearest tenth

had a level of 0.061, Lucas had a level of 0.1, and Amy had

b. 345.51 to the nearest one

a level of 0.12. Who was arrested and who was let go?

c. 345.51 to the nearest ten

d. 0.01826 to the nearest thousandth

12. Calculate mentally. Describe your method.

e. 0.01826 to the nearest ten thousandth

a. 18 4

. 3 − 9 9

. 6

b. 1 3

. × 5 9

. + 64 1

. × 1 3

.

f. 0.498 to the nearest tenth

c. 4 6

. + (5 8

. + 2 4

. )

d. ( .

0 25 × 17) × 8

g. 0.498 to the nearest hundredth

PROBLEMS

18. The numbers shown next can be used to form an additive

19. Suppose that classified employees went on strike for 22

magic square:

working days. One of the employees, Kathy, made $9.74

per hour before the strike. Under the old contract, she

10.48, 15.72, 20.96, 26.2, 31.44,

worked 240 six-hour days per year. If the new contract is

36.68, 41.92, 47.16, 52.4.

for the same number of days per year, what increase in

her hourly wage must Kathy receive to make up for the

Use your calculator to determine where to place the num-

wages she lost during the strike in one year?

bers in the nine cells of the magic square.

20. Decimals are just fractions whose denominators are powers

of 10. Change the three decimals in the following sum to

fractions and add them by finding a common denominator.

0 6

. + 0 7

. 83 + 0 2

. 9

In what way(s) is this easier than adding fractions such as

2 5

, , and 3 ?

7 6

4

EXERCISE/PROBLEM SET B

EXERCISES

1. Write each of the following sums in decimal form.

2. Write each of the following decimals (i) in its expanded

a. 5 1 2 + 7 10 + 3 1 5

( )

(

)

( )

form and (ii) as a fraction.

10

10

a. 0.525

b. 8 1 + 3 10 3 + 9 1 2

( )

(

)

( )

10

10

b. 34.007

c. 5 1 3 + 2 1 2

1 6

( )

( ) + ( )

10

10

10

c. 5.0102

c07.indd 260

7/31/2013 11:55:12 AM - Section 7.1 Decimals 261

3. Write the following expressions as decimal numerals.

11. According to state law, the amount of radon released from

a. Seven hundred forty-six millionths

wastes cannot exceed a 0.033 working level. A study of

b. Seven hundred forty-six thousand and seven hundred

two locations reported a 0.0095 working level at one loca-

forty-six millionths

tion and 0.0039 at a second location. Does either of these

c. Seven hundred forty-six million and seven hundred

locations fail to meet state standards?

forty-six thousandths

12. Calculate mentally. Describe your method.

4. Write the following numbers in words.

a. 7 × 3 4

. + 6 6

. × 7

b. 26 5

. 3 − 8 9

. 5

c. 0 491 102

.

÷

a. 0.000000078

b. 7,589.12345

d. 5 8

. 9 + 6 2

. 7

e. ( .

5 7 + .

4 8) + 3.2

f. 67 32 103

.

×

c. 187,213.02003

d. 2.00031

g. 0 5

. × (639 × 2)

h. 6 5

. × 12

5. A student reads 0.059 as “point zero five nine thou-

13. Calculate mentally using fraction equivalents.

sandths.” What is wrong with this reading?

a. 230 × 0 1

.

b. 36 × 0 2

. 5

c. 82 × 0 5

.

×

×

6. Determine which of the following fractions have

d. 125

0 8

.

e. 175

0 2

.

f. 0 6

. × 35

terminating decimal representations.

14. Find each of the following products and quotients.

24 ¥1116 ¥1719

23 ¥ 311 ¥ 79 ¥1116

23 ¥ 39 ¥1117

Express your answers in scientific notation.

a.

b.

c.

512

713 ¥119 ¥ 57

28 ¥ 34 ¥ 57

7 8752

.

a. 12 6416

.

× 100

b. 10,000,000

7. Decide whether the following fractions terminate in their

decimal form. If a fraction terminates, tell in how many

8 2

. 5 × 1020

c. ( .

8 25 1020 )(107

×

)

d.

places and explain how you can tell from the fraction

107

form.

15. Estimate, using the indicated techniques.

1

17

3

17

a.

b.

c.

d.

a. 34 7

. × 3 9

. ; range and rounding to the nearest whole

11

625

12,800

219

523

×

number

8. Arrange the following from smallest to largest.

b. 15 7

. 1 + 3 23

.

+ 21 95

. ; two-column front-end

a. 3.08, 3.078, 3.087, 3.80

c. 13 7

. × 6 1

. ; one-column front-end and range

b. 8.01002, 8.010019, 8.0019929

d. 3 61

.

+ 4 9

. 1 + 1 3

. ; front-end with adjustment

c. 0.5, 0.505, 0.5005, 0.55

16. Estimate by rounding to compatible numbers and fraction

9. Order each of the following from smallest to largest by

equivalents.

changing each fraction to a decimal.

a. 123 9

. ÷ 5 3

.

b. 87 4

. × 7 9

.

c. 402 ÷ 1 2

. 5

a. 5 4 10

, ,

b. 4 3 2

, , c. 5 7 11

,

,

d. 3 11 17

,

,

d.

34,546 × 0 004

.

e. 0 0024

.

× 470,000

f. 3591 ÷ 0 6

. 1

7

5

13

11

7

5

9

13 18

5

18

29

10. Order each of the following from smallest to largest as

17. Round the following as specified.

simply as possible by using any combinations of the three

a. 321.0864 to the nearest hundredth

following methods: (i) common denominators, (ii) cross-

b. 12.16231 to the nearest thousandth

multiplication, and (iii) converting to a decimal.

c. 4.009055 to the nearest ten thousandth

d. 1.9984 to the nearest tenth

a. 5 1 17

, ,

b. 13 2 3

, , c. 8 26 50

,

,

8

2

23

16

3

4

5

15

31

e. 1.9984 to the nearest hundredth

PROBLEMS

18. Determine whether each of the following is an additive

21. Brigham says the fraction 42 should be a repeating deci-

150

magic square. If not, change one entry so that your result-

mal because the factors of the denominator include a 3 as

ing square is magic.

well as 2s and 5s. But on his calculator 42 ÷ 150 seems to

a.

b.

terminate. What could you say to clarify this confusion?

2 4

.

5 4

.

1 2

.

0 438

.

0 073

.

0 584

.

22. Mary Lou said she knows that when fractions are written

1 8

.

3

4 2

.

0 511

.

0 365

.

0 219

.

as decimals they either repeat or terminate. So 12 must not

17

be a fraction because when she divided 12 by 17 on her

4 8

.

1 4

.

3 6

.

0 146

.

0 657

.

0 292

.

calculator, she got a decimal that did not repeat or termi-

nate. How would you react to this?

19. Solve the following cryptarithm where D = 5.

23. Camille tells you that 6.45 is greater than 6.5 because 45 is

great than 5. How would you respond to this?

DONALD

+ GERALD

24. Caroline is rounding decimals to the nearest hundredth.

ROBERT

she rounds 19.67472 to 19.675, then rounds 19.675 to

19.68. Is her method correct? Explain.

25. Looking at the same problem that Caroline had in 24,

Analyzing Student Thinking

Amir comes up with the answer 19.66. When his teacher

20. Joseph read 357.8 as “three hundred and fifty seven and

asks him how he got the answer, he says, “Because of the

eight tenths.” Did he read it correctly? Explain.

4, I had to round down.” What is Amir’s misconception?

c07.indd 261

7/31/2013 11:55:24 AM - 262

Chapter 7 Decimals, Ratio, Proportion, and Percent

26. Merilee said her calculator changed 2 to 0.6666667, so

27. When asked to find a decimal number between 0.19 and

3

obviously it does not repeat. Therefore it must be a termi-

0.20, Alondra said there is none since 19 and 20 are con-

nating decimal. How would you respond to Merilee?

secutive whole numbers. How should you respond?

OPERATIONS WITH DECIMALS

Common Core – Grade 6

Algorithms for Operations with Decimals

Fluently add, subtract, multiply,

and divide multi-digit decimals

Algorithms for adding, subtracting, multiplying, and dividing decimals are simple

using the standard algorithm for

extensions of the corresponding whole-number algorithms.

each operation.

Perform the following computations

5 8

. + 2 3

. 7

5 8

. × 2 3

. 7

For each of these computations discuss with your peers:

a. How the placement of the decimal point is handled in the set up of each computation and in the solution;

b. Why the method for handling the decimal point described in part a works.

Addition

Add:

a. 3 56

.

+ 7 95

.

b. 0 0094

.

+ 80 183

.

S O L U T I O N We will find these sums in two ways: using fractions and using a decimal

algorithm.

Fraction Approach

356

795

94

80 183

,

a. 3 56

.

+ 7 95

.

=

+

b. 0 0094

.

+ 80 183

.

=

+

100

100

10,000

1000

= 1151

= 94 + 801,830

100

10,000

10,000

=

,

11 51

.

= 801 924

10,000

= 80 1924

.

Decimal Approach As with whole-number addition, arrange the digits in columns

according to their corresponding place values and add the numbers in each column,

regrouping when necessary (Figure 7.9).

This decimal algorithm can be stated more simply as “align the decimal points, add

the numbers in columns as if they were whole numbers, and insert a decimal point in

the answer immediately beneath the decimal points in the numbers being added.” This

algorithm can easily be justifi ed by writing the two summands in their expanded form

Figure 7.9

and applying the various properties for fraction addition and/or multiplication.

■

Subtraction

Subtract:

a. 14 793

.

− 8 95

.

b. 7 56

.

− 0 0008

.

S O L U T I O N Here we could again use the fraction approach as we did with addition.

However, the usual subtraction algorithm is more efficient.

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Pearson Education.

263

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Chapter 7 Decimals, Ratio, Proportion, and Percent

Reflection from Research

a.

Step 1:

Step 2:

Step 3:

Students tend to assume that

Align Decimal

Subtract as if

Insert Decimal

adding a zero to the end of a

Points

Whole Numbers

Point in Answer

decimal fraction is the same as

adding a zero to the end of a

14 793

.

14793

14 793

.

whole number. The most com-

− 8 9

. 5

− 8950

− 8 9

. 5

mon error on a test item for

which students were to write the

5843

5 843

.

number ten times bigger than

437.56 was 437.560 (Hiebert &

(Note: Step 2 is performed mentally—there is no need to rewrite the numbers with-

Wearne, 1986).

out the decimal points.)

b. Rewrite 7.56 as 7.5600.

7 5600

.

− 0 0008

.

7 5592

.

■

Now let’s consider how to multiply decimals.

Researchers have coined the phrase “multiplication makes bigger” to describe the

student misconception that in any multiplication problem, the product is always larger

than either of the factors. What is a problem or situation where “multiplication makes bigger” does not hold?

Similarly, “division makes smaller” is used to describe the student misconception that in a division problem, the quo-

tient is always smaller than the dividend. What is a problem or situation where this does not hold? Discuss where the

misconception “multiplication makes bigger, division makes smaller” might come from.

Multiplication

Multiply 437 09

.

× 3 8

. .

S O L U T I O N Refer to fraction multiplication.

43,709

38

43,709

38

437 09

.

× 3 8

. =

×

=

×

100

10

100 × 10

1,660,942

=

= 16660 942

.

■

1000

Observe that when multiplying the two fractions in Example 7.12, we multiplied

43,709 and 38 (the original numbers “without the decimal points”). Thus the proce-

dure illustrated in Example 7.12 suggests the following algorithm for multiplication.

Multiply the numbers “without the decimal points”:

43,709

×

38

1,660,942

Insert a decimal point in the answer as follows: The number of digits to the right of

the decimal point in the answer is the sum of the number of digits to the right of the

decimal points in the numbers being multiplied.

437 09

.

( 2 digits to the right of the decimal point)

×

3 8

.

(1digit to the right of the decimal point)

1660 942

.

( 2 + 1digits to the right of the decimal point)

Notice that there are three decimal places in the answer, since the product of the two

denominators (100 and 10) is 103. This procedure can be justified by writing the deci-

mals in expanded form and applying appropriate properties.

c07.indd 264

7/31/2013 11:55:34 AM - Section 7.2 Operations with Decimals 265

An alternative way to place the decimal point in the answer of a decimal mul-

tiplication problem is to do an approximate calculation. For example, 437 09

.

× 3 8

.

is approximately 400 × 4 or 1600. Hence the answer should be in the thousands—

namely, 1660.942, not 16,609.42 or 16.60942, and so on.

Compute: 57 98

.

× 1 371

.

using a calculator.

S O L U T I O N First, the answer should be a little less than 60 × 1 4

. , or 84. Using a

calculator we find 57.98 × 1.371 = 79.49058. Notice that the answer is close to the

estimate of 84.

■

Division

Divide: 154 63

.

÷ 4 7

. .

S O L U T I O N First let’s estimate the answer: 155 ÷ 5 = 31, so the answer should be

approximately 31. Next, we divide using fractions.

15, 463

47

15, 463

470

154 63

.

÷ 4 7

. =

÷

=

÷

100

10

100

100

15, 463

=

= 32 9

.

■

470

Reflection from Research

Notice that in the fraction method, we replaced our original problem in decimals

Division of decimals tends to be

with an equivalent problem involving whole numbers:

quite difficult for students, espe-

cially when the division problem

154 63

.

÷ 4 7

. ⎯ →

⎯ 15,463 ÷ 470.

requires students to add zeros as

place holders in either the divi-

Similarly, the problem 1546 3

. ÷ 47 also has the answer 32.9 by the missing-factor

dend or the quotient (Trafton &

approach. Thus, as this example suggests, any decimal division problem can be replaced

Zawojewski, 1984).

with an equivalent one having a whole-number divisor. This technique is usually used

when performing the long-division algorithm with decimals, as illustrated next.

Algebraic Reasoning

The problem 154.63 ÷ 4.7

Compute: 4 7

. )154 63

.

.

is equivalent to 1546.3 ÷ 47

because they both have the same

S O L U T I O N Replace with an equivalent problem where the divisor is a whole number.

solution. The latter problem is

47)1546 3

.

preferred because it is easier to

compute. This is similar to the

NOTE: Both the divisor and dividend have been multiplied by 10. Now divide as if it

algebraic process of converting

is whole-number division. The decimal point in the dividend is temporarily omitted.

3x + 7 = 19 into the equivalent

equation 3x = 12. Both equations

329

have the same solution but the

47)15463

solution is easier to see in the lat-

ter equation.

− 141

136

− 94

423

− 423

0

Replace the decimal point in the dividend, and place a decimal point in the quotient

directly above the decimal point in the dividend. This can be justifi ed using division

of fractions.

Check: 4 7

. × 32 9

. = 154 63

.

.

■

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7/31/2013 11:55:38 AM - 266 Chapter 7 Decimals, Ratio, Proportion, and Percent

The “moving the decimal points” step to obtain a whole-number divisor in

Example 7.15 can be justified as follows:

Let a and b be decimals.

If a ÷ b = c, then a = bc.

Then a

n

10 = bc

n

10 = (b

n

¥

¥

¥10 ) c

for any n.

Thus (a

n

10 ) ÷ (b

n

¥

¥10 ) = .

c

This last equation shows we can multiply both a and b (the dividend and divisor) by

the same power of 10 to make the divisor a whole number. This technique is similar to

equal-additions subtraction except that division and multiplication are involved here.

Check for Understanding: Exercise/Problem Set A #1–4

✔

Scientific Notation

Multiplying and dividing large numbers can sometimes be assisted by first express-

ing them in scientific notation. In Section 4.1, scientific notation was discussed in the

context of using a scientific calculator. Numbers are said to be in scientific notation

when expressed in the form a

n

× 10 , where 1 ≤ a < 10 and n is any whole number (the

case when n can be negative will be discussed in Chapter 8). The number a is called

the mantissa and n the characteristic of a

n

× 10 . The following table provides some

examples of numbers written in scientific notation.

SCIENTIFIC NOTATION

STANDARD NOTATION

Diameter of Jupiter

1 438 108

.

×

meters

143,800,000 meters

Total amount of gold in Earth's crust

1 2 1016

. ×

kilograms

12,000,000,000,000,000 kilograms

Distance from Earth to Jupiter

5 88 1011

.

×

meters

588,000,000,000 meters

Once large numbers are expressed in scientific notation they can be multiplied and

divided more easily as shown in the following example.

Compute the following using scientific notation.

a. 54 500

,

,000,000 × 346,000,000

b. 1,200,000,000,000 ÷ 62 500

,

,000

S O L U T I O N

a. 54 500 000 000 × 346 000 000 = 5 45 × 1010 × 3 46 × 108

,

,

,

,

,

( .

)

( .

)

= (5 4

. 5 × 3.446) × 1010

(

× 108)

= 18 857

.

× 1018

= 1 8857

.

× 101 × 1018

= 1 8857

.

× 1019

12

b. 1,200,000,000,000

1 2

.

10

=

×

62 500

,

,000

6 2

. 5 × 107

1 2

.

1012

=

×

6 2

. 5

107

= 0 192

.

× 105

= 0 192

.

× 10 × 104

= 1 9

. 2 × 104

■

c07.indd 266

7/31/2013 11:55:41 AM - Section 7.2 Operations with Decimals 267

Check for Understanding: Exercise/Problem Set A #5–11

✔

Classifying Repeating Decimals

In Example 7.2 we observed that fractions in simplest form whose denominators are

of the form 2m 5n

¥ have terminating decimal representations. Fractions of this type

can also be converted into decimals using a calculator or the long-division algorithm

for decimals.

Express 7 in decimal form (a) using a calculator and (b) using

40

the long-division algorithm.

S O L U T I O N

a. 7 ÷ 40 = 0 175

.

b.

0 175

.

40)7 000

.

− 40

300

− 280

200

− 200

0

Therefore, 7 = 0 175

.

.

■

40

Now, let’s express 1 as a decimal. Using a calculator, we obtain

3

1 ÷ 3 = 0 333333333

.

This display shows 1 as a terminating decimal, since the calculator can display only

3

finitely many decimal places. However, the long-division method adds some addi-

tional insight to this situation.

…

)0 333

.

3 1 000

.

− 9

10

− 9

10

− 9

1

Using long division, we see that the decimal in the quotient will never terminate,

since every remainder is 1. Similarly, the decimal for 1 is 0 0909

.

. . . . Instead of writing

11

dots, a horizontal bar may be placed above the repetend, the first string of repeating

digits. Thus

1

1

= 0.3,

= 0 0

. 9,

3

11

2

2

= 0 285714

.

,

= 0.2,

7

9

40

and

= 0 4

. 0.

99

(Use your calculator or the long-division algorithm to check that these are correct.)

Decimals having a repetend are called repeating decimals. (NOTE: Terminating

c07.indd 267

7/31/2013 11:55:46 AM - 268 Chapter 7 Decimals, Ratio, Proportion, and Percent

decimals are those repeating decimals whose repetend is zero.) The number of dig-

its in the repetend is called the period of the decimal. For example, the period of 1

11

is 2. To gain additional insight into why certain decimals repeat, consider the next

example.

Express 6 as a decimal.

7

S O L U T I O N

0 857142

.

Problem-Solving Strategy

7

Look for a Pattern

)6 000000

.

− 56

40

− 35

50

− 49

10

− 7

30

− 28

20

− 14

6

When dividing by 7, there are seven possible remainders—0, 1, 2, 3, 4, 5, 6. Thus,

when dividing by 7, either a 0 will appear as a remainder (and the decimal terminates)

or one of the other nonzero remainders must eventually reappear as a remainder. At

that point, the decimal will begin to repeat. Notice that the remainder 6 appears for a

second time, so the decimal will begin to repeat at that point. Therefore, 6 = 0 857142

.

.

7

Similarly, 1 will begin repeating no later than the 13th remainder, 7 will begin

13

23

repeating by the 23rd remainder, and so on.

■

By considering several examples where the denominator has factors other than 2

or 5, the following statement will be apparent.

T H E O R E M 7 . 3

Fractions with Repeating, Nonterminating Decimal Representations

a

a

Let be a fraction written in simplest form. Then has a repeating decimal

b

b

representation that does not terminate if and only if b has a prime factor other

than 2 or 5.

Find ten different fractions between 0.77 and 0.78 where five of these fractions have

terminating decimal representations and the other five have repeating, nonterminating

decimal representations. Share the methods used to find these fractions with a peer.

Earlier we saw that it was easy to express any terminating decimal as a fraction.

But suppose that a number has a repeating, nonterminating decimal representation.

Can we find a fractional representation for that number?

c07.indd 268

7/31/2013 11:55:47 AM - Section 7.2 Operations with Decimals 269

Algebraic Reasoning

Express 0 3

. 4 in its fractional form.

In the solution of Example 7.19,

a letter is used to represent a

specific number, namely n = 0.34,

S O L U T I O N Let n = 0 3

. 4. Thus 100n = 34 34

. .

rather than to represent many

Then 100n = 34 343434

.

⋅⋅⋅

possible numbers as a variable

does. Using a letter in this way

− n = 343434

.

. . .

allows one to manipulate the

so

99n

number in order to find its frac-

= 34

tional representation.

or

n = 34 .

■

99

This procedure can be applied to any repeating decimal that does not terminate, except

that instead of multiplying n by 100 each time, you must multiply n by 10m, where m is the

number of digits in the repetend. For example, to express 17 531

.

in its fractional form, let

n = 17 53

. 1 and multiply both n and 17 531

.

by 103, since the repetend .531 has three digits.

Then 103n − n = 17,531 531

.

− 17 531

.

= 17,514. From this we find that n = 17 514

,

.

999

Finally, we can state the following important result that links fractions and repeat-

ing decimals.

T H E O R E M 7 . 4

Every fraction has a repeating decimal representation, and every repeating decimal

has a fractional representation.

The following diagram provides a visual summary of this theorem.

Fraction

Repeating

(simplest form)

Decimal

Denominator

Terminating

with factors

Decimal

of only 2 and 5

(repetend is zero)

Denominator

Nonterminating

with at least

Decimal

one factor not

(repetend is not zero)

2 or 5

Check for Understanding: Exercise/Problem Set A #12–14

✔

Debugging is a term used to describe the process of checking a com-

puter program for errors and then correcting the errors. According

to legend, the process of debugging was adopted by Grace Hopper,

who designed the computer language COBOL. When one of her

programs was not running as it should, it was found that one of the

computer components had malfunctioned and that a real bug found

among the components was the culprit. Since then, if a program did

not run as it was designed to, it was said to have a “bug” in it. Thus it

had to be “debugged.”

©Ron Bagwell

c07.indd 269

7/31/2013 11:55:51 AM - 270 Chapter 7 Decimals, Ratio, Proportion, and Percent

EXERCISE/PROBLEM SET A

EXERCISES

1. a. Perform the following operations using the decimal algo-

8. Find the following quotients, and express the answers in

rithms of this section.

scientific notation.

i. 38 52

.

+ 9 251

.

ii. 534 51

.

− 48 67

.

1 612

.

× 105

8 019

.

× 109

9 0

. 2 × 105

a.

b.

c.

b. Change the decimals in part a to fractions, perform the

3 1

. × 102

9 9

. × 105

2 2

. × 103

computations, and express the answers as decimals.

9. A scientific calculator can be used to perform calculations

2. a. Perform the following operations using the decimal algo-

with numbers written in scientific notation. The SCI key

rithms of this section.

or EE key is used as shown in the following multiplica-

i. 5 23

.

× 0 034

.

ii. 8 272

.

÷ 1 7

. 6

tion example:

b. Change the decimals in part a to fractions, perform the

(3.41 1012 )( .

4 95 108

×

×

).

computations, and express the answers as decimals.

3 4

. 1 SCI 1

2 × 4

95

.

SCI 8

= 1 68795

.

21

3. Find answers on your calculator without using the deci-

So the product is 1 68795 × 1021

.

.

mal-point key. (Hint: Locate the decimal point by doing

approximate calculations.) Check using a written algorithm.

Try this example on your calculator. The sequence of steps

a. 48 62

.

× 52 7

. b. 123,658 57

.

÷ 17 9

.

and the appearance of the result may differ slightly from

what was shown. Consult your manual if necessary. Use

4. By thinking about the process of the division algorithm

your calculator to find the following products and quo-

for decimals, determine which of the following division

tients. Express your answers in scientific notation.

problems have the same quotient. NOTE: You don’t need to

24

×

6

8

6 4

.

10

complete the algorithm in order to answer this question.

a . ( .

7 19 × 10 )( .

1 4 × 10 ) b. 5 0. ×1010

a. 56)1680 b. 0 056

.

)0 168

.

c. 0 5

. 6)0 168

.

10. The Earth’s oceans have a total volume of approximately

5. It is possible to write any decimal as a number between 1

1,286,000,000 cubic kilometers. The volume of fresh water

and 10 (including 1) times a power of 10. This scientific

on the Earth is approximately 35,000,000 cubic kilometers.

notation is particularly useful in expressing large numbers.

a. Express each of these volumes in scientific notation.

For example,

b. The volume of salt water in the oceans is about how

many times greater than the volume of fresh water on

6321 = 6 321

.

× 103 and

the Earth?

760,000,000 = 7 6

. × 108.

11. The distance from Earth to Mars is 399,000,000 kilo-

Write each of the following in scientific notation.

meters. Use this information and scientific notation to

a.

59

b. 4326

answer the following questions.

c. 97,000

d. 1,000,000

a. If you traveled at 88 kilometers per hour (55 miles per

e. 64,020,000 f. 71,000,000,000

hour), how many hours would it take to travel from

Earth to Mars?

6. The nearest star (other than the sun) is Alpha Centauri,

b. How many years would it take to travel from Earth to

which is the brightest star in the constellation Centaurus.

Mars?

a. Alpha Centauri is 41,600,000,000,000,000 meters from

c. In order to travel from Earth to Mars in a year, how fast

our sun. Express this distance in scientific notation.

would you have to travel in kilometers per hour?

b. Although it appears to the naked eye to be one star,

12. Write each of the following using a bar over the repetend.

Alpha Centauri is actually a double star. The two stars

a. 0 7777

.

. . . b. 0 47121212

.

. . . c. 0 181818

.

. . .

that comprise it are about 3,500,000,000 meters apart.

Express this distance in scientific notation.

13. Write out the first 12 decimal places of each of the following.

a. 0 3

. 174 b. 0 3

. 174 c. 0 317

.

4

7. Find the following products, and express answers in scien-

tific notation.

14. Express each of the following repeating decimals as a frac-

a. (6.2 101)

(5.9 104

×

×

×

)

tion in simplest form.

b. ( .

7 1 102 )

( .

8 3 106

×

×

×

)

a. 0 1

. 6 b.

0 387

.

c.

0 7

. 25

PROBLEMS

15. The star Deneb is approximately 1 5 1019

. ×

meters from

16. Is the decimal expansion of 151/7,018,923,456,413 termi-

Earth. A light year, the distance that light travels in one

nating or nonterminating? How can you tell without com-

year, is about 9 46 1015

.

×

meters. What is the distance

puting the decimal expansion?

from Earth to Deneb measured in light years?

c07.indd 270

7/31/2013 11:55:56 AM - Section 7.2 Operations with Decimals 271

17. Give an example of a fraction whose decimal expansion

26. Gary cashed a check from Joan for $29.35. Then he

terminates in the following numbers of places.

bought two magazines for $1.95 each, a book for $5.95,

a. 3 b. 4 c. 8 d. 17

and a CD for $5.98 in a used book store. He had $21.45

left. How much money did he have before cashing the

18. From the fact that 0. 1

1

= , mentally convert the following

9

check?

decimals into fractions.

a. 0.3 b. 0.5 c. 0.7 d. 2.8 e. 5.9

27. Each year a car depreciates to about 0.8 of its value the

year before. What was the original value of a car that is

19. From the fact that 0 0

. 1

1

= , mentally convert the follow-

99

worth $16,000 at the end of 3 years?

ing decimals into fractions.

a. 0 0

. 3

b. 0 0

. 5

c. 0 0

. 7

28. A regional telephone company advertises calls for $0.11 a

d. 0 3

. 7

e. 0 6

. 4

f. 5 97

.

minute. How much will an hour and 21 minute call cost?

20. From the fact that

29. Juanita’s family’s car odometer read 32,576.7 at the begin-

0 001

1

.

=

, mentally convert the

999

following decimals into fractions.

ning of the trip and 35,701.2 at the end. If $282.18 worth

of gasoline at $2.89 per gallon was purchased during the

a. 0 003

.

b. 0 005

.

c. 0 007

.

trip, how many miles per gallon (to the nearest mile) did

d. 0 019

.

e. 0 827

.

f. 3 217

.

they average?

21. a. Use the pattern you have discovered in Problems 18

30. In 2007, the exchange rate for the Japanese yen was 121 yen

to 20 to convert the following decimals into fractions

per U.S. dollar. How many dollars should one receive in

mentally.

exchange for 10,000 yen (round to the nearest hundredth)?

i. 0 2

. 3

ii. 0 010

.

iii. 0 769

.

iv. 0.9

v. 0 5

. 7

vi. 0 1827

.

31. A typical textbook measures 8 inches by 10 inches. There

b. Verify your answers by using the method taught in the

are exactly 2.54 centimeters per inch. What are the dimen-

text for converting repeating decimals into fractions us-

sions of a textbook in centimeters?

ing a calculator.

32. Inflation causes prices to increase about 0.03 per year. If

a textbook costs $115 in 2010, what would you expect the

22. a. Give an example of a fraction whose decimal represen-

book to cost in 2014 (round to the nearest dollar)?

tation has a repetend containing exactly five digits.

b. Characterize all fractions whose decimal representa-

33. Sport utility vehicles advertise the following engine capaci-

tions are of the form 0.

,

abcde where a, b, c, d, and e are

ties: a 2.4-liter 4-cylinder, a 3.5-liter V-6, a 4.9-liter V-8,

arbitrary digits 0 through 9 and not all five digits are the

and a 6.8-liter V-10. Compare the capacities of these

same.

engines in terms of liters per cylinder.

23. a. What is the 11th digit to the right of the decimal in the

34. Three nickels, one penny, and one dime are placed as

decimal expansion of 1 ?

shown. You may move only one coin at a time, to an

13

b. What is the 33rd digit of 1 ?

adjacent empty square. Move the coins so that the penny

13

c. What is the 2731st digit of 1 ?

and the dime have exchanged places and the lower middle

13

d. What is the 11,000,000th digit of 1 ?

square is empty. Try to find the minimum number of

13

such moves.

24. From the observation that 100 × 1 = 100 = 29

1 , what con-

71

71

71

clusion can you draw about the relationship between the

decimal expansions of 1 and 29 ?

71

71

25. It may require some ingenuity to calculate the following

number on an inexpensive four-function calculator.

Explain why, and show how one can, in fact, calculate it.

364 × 363 × 362 × 361 × 360 × 359

365 × 365 × 365 × 365 × 365 × 365

EXERCISE/PROBLEM SET B

EXERCISES

1. Perform the following operations using the decimal algo-

2. Perform the following operations using the decimal algo-

rithms of this section.

rithms of this section.

a. 7 482

.

+ 94 3

.

a. 16 4

. × 2 8

.

b. 100 63

.

− 72 495

.

b. 0 065

.

× 1 9

. 2

c. 0 0

. 8 + 0 1234

.

c. 44 4

. ÷ 0 3

.

d. 24 − 2 099

.

d. 129 168

.

÷ 4 1

. 4

c07.indd 271

7/31/2013 11:56:01 AM - 272 Chapter 7 Decimals, Ratio, Proportion, and Percent

3. Find answers on your calculator without using the deci-

9. Use your calculator to find the following products and

mal-point key. (Hint: Locate the decimal point by doing

quotients. Express your answers in scientific notation.

approximate calculations.) Check using a written algorithm.

a. ( .

1 2 1010 )( .

3 4 1012 )( .

8 5 1017

×

×

×

)

a. 473 92

.

× 49 12

.

b. 537,978 4146

.

÷ 1379 4

.

( .

4 56 × 109 )(7.0 × 1021)

b.

4. By thinking about the process of the division algorithm

(1.2 × 106 )(2.8 × 1010 )

for decimals, determine which of the following division

c. (3.6 1018 )3

×

problems have the same quotient. NOTE: You don’t need to

complete the algorithm in order to answer this question.

10. At a height of 8.488 kilometers, the highest mountain in

the world is Mount Everest in the Himalayas. The deepest

a. 5 6

. )16 8

. b. 0 056

.

)1 6.8 c. 0 5.6)16 8.

part of the oceans is the Marianas Trench in the Pacific

5. Write each of the following numbers in scientific notation.

Ocean, with a depth of 11.034 kilometers. What is the ver-

a. 860

b. 4520

tical distance from the top of the highest mountain in the

world to the deepest part of the oceans?

c. 26,000,000

d. 315,000

e. 1,084,000,000

f. 54,000,000,000,000

11. The amount of gold in the Earth’s crust is about

120,000,000,000,000 kilograms.

6. a. The longest human life on record was more than 122

a. Express this amount of gold in scientific notation.

years, or about 3,850,000,000 seconds. Express this num-

b. The market value of gold in March 2007 was about

ber of seconds in scientific notation.

$20,700 per kilogram. What was the total market

b. Some tortoises have been known to live more than 150

value of all the gold in the Earth’s crust at that time?

years, or about 4,730,000,000 seconds. Express this num-

c. The total U.S. national debt in March 2007 was about

ber of seconds in scientific notation.

$ .

8 6 10 .

12

×

How many times would the value of the gold

c. The oldest living plant is probably a bristlecone pine

pay off the national debt?

tree in Nevada; it is about 4900 years old. Its age in

d. If there were about 300,000,000 people in the United

seconds would be about 1 545 1011

.

×

seconds. Express

States in March 2007, how much do each of us owe on

this number of seconds in standard form and write a

the national debt?

name for it.

12. Write each of the following using a bar over the repetend.

7. Perform the following operations, and express answers in

a. 0.35 b. 0 141414

.

. . . c. 0 45315961596

.

. . .

scientific notation.

a. (2.3 102 )

(3.5 104

×

×

×

) b. ( .

7 3 103 )

( .

8 6 106

×

×

×

)

13. Write out the first 12 decimal places of each of the following.

a. 0 3174

.

b. 0 3

. 174 c. 0 1159123

.

8. Find the following quotients, and express the answers in

scientific notation.

14. Express each of the following decimals as fractions.

1 357

.

× 1027

4 894689

.

× 1023

5 561

.

× 107

a. 0.5

b. 0 7

. 8

c. 0 123

.

a.

b.

c.

2 3

. × 103

5 1

. 9 × 1018

6 7

. × 102

d. 0 1

. 24

e. 0 0

. 178

f. 0 123456

.

PROBLEMS

15. Determine whether the following are equal. If not, which

of reversing the digits of any number and adding until a

is smaller, and why?

palindrome is obtained was described. The same technique

works for decimal numbers, as shown next.

0 2

. 525

0 2

. 525

7 9

. 5

16. Without doing any written work or using a calculator,

order the following numbers from largest to smallest.

+59 7

.

Step 1

x =

67 65

.

0 00000456789

.

÷ 0 00000987654

.

+

Step 2

y =

56 76

.

0 00000456789

.

× 0 00000987654

.

124 41

.

z = 0 00000456789

.

+ 0 00000987654

.

14

+ 421

.

Step 3

17. Look for a pattern in each of the following sequences of

138 831

.

A palindrome

decimal numbers. For each one, write what you think the

next two terms would be.

a. Determine the number of steps required to obtain a

a. 11 5

. , 14

7

. , 1

7 9

. , 21 1

. , . . .

palindrome from each of the following numbers.

b. 24, 33 6

. , 47 04

.

, 65 856

.

, . . .

i. 16.58 ii. 217.8

c. 0 5

. , 0 0

. 5, 0 055

.

, 0 0055

.

, 0 00555

.

, . . .

iii. 1.0097 iv. 9.63

d. 0 5

. , 0 6

. , 1

0

. , 1

9

. , 3 5

. , 6 0

. , . . .

b. Find a decimal number that requires exactly four steps

e. 1 0

. , 0 5

. , 0.6, 0 7

. 5, 0 8

. , . . .

to give a palindrome.

18. In Chapter 3 a palindrome was defined to be a number such

19. a. Express each of the following as fractions.

as 343 that reads the same forward and backward. A process

i. 0. 1 ii. 0 0

. 1 iii. 0 001

.

iv. 0 0001

.

c07.indd 272

7/31/2013 11:56:12 AM - Section 7.2 Operations with Decimals 273

b. What fraction would you expect to be given by

b. What is the maximum amount? (Write out your answer

0 000000001

.

?

to the ten-thousandths place.)

c. What would you expect the decimal expansion of 1 to be?

90

28. A map shows a scale of 1 in. = 73 6

. mi.

20. Change 0.9 to a fraction. Can you explain your result?

a. How many miles would 3.5 in. represent?

b. How many inches would represent 576 miles (round to

21. Consider the decimals: a = . ,

= . , =

1

0 9 a2

0 99 a3 0 9

. 99,

the nearest tenth)?

a = .

, . . . ,

=

4

0 9999

an

0 9999

.

. . . 9(with n digits of 9).

a. Give an argument that 0 < a

29. If you invest $7500 in a mutual fund at $34.53 per share,

n < an+ <

1

1 for each n.

b. Show that there is a term a

how much profit would you make if the price per share

n in the sequence such that

increases to $46.17?

1

1 − an <

.

10100

30. A casino promises a payoff on its slot machines of 93

cents on the dollar. If you insert 188 quarters one at a

(Find a value of n that works.)

time, how much would you expect to win?

c. Give an argument that the sequence of terms gets arbi-

trarily close to 1. That is, for any distance d, no matter

31. Your family takes a five-day trip logging the following

how small, there is a term an in the sequence such that

miles and times: (503 mi, 9 hr), (480 mi, 8.5 hr), (465 mi,

1 − d < an < 1.

7.75 hr), (450 mi, 8.75 hr), and (490 mi, 9.75 hr). What

was the average speed (to the nearest mile per hour) of

the trip?

32. The total value of any sum of money that earns inter-

est at 9% per year doubles about every 8 years. What

amount of money invested now at 9% per year will

d. Use parts a to c to explain why 0.9 = 1.

accumulate to about $120,000 in about 40 years (assum-

ing no taxes are paid on the earnings and no money is

22. a. Write 1 2 3 4 5

, , , , , and 6 in their decimal expansion

7

7

7

7

7

7

withdrawn)?

form. What do the repetends for each expansion have

in common?

33. An absentminded bank teller switched the dollars and

b. Write 1

2

3

11

,

,

, . . . ,

, and 12 in decimal expansion

cents when he cashed a check for Mr. Spencer, giving him

13 13 13

13

13

form. What observations can you make about the repe-

dollars instead of cents, and cents instead of dollars. After

tends in these expansions?

buying a 5-cent newspaper, Mr. Spencer discovered that

he had left exactly twice as much as his original check.

23. Characterize all fractions a/b, a

< b, whose decimal expan-

What was the amount of the check?

sions consist of n random digits to the right of the decimal

followed by a five-digit repetend. For example, suppose

Analyzing Student Thinking

that n = 7; then 0 2135674

.

51139 would be the decimal

34. Bhumi adds 6.45 and 2.3 and states that the answer is

expansion of such a fraction.

6.68. What mistake did the student make and how would

24. If 9 =

you help her understand her misconception?

0 3913043478260869565217

.

, what is the 999th digit

3

2

to the right of the decimal?

35. Kaisa multiplies 7.2 and 3.5 and gets 2.52, which is clearly

wrong. What mistake did she make?

25. Name the digit in the 4321st place of each of the following

decimals.

36. Tyler says that 50 times 4.68 is the same as 0.5 times 468.

a. 0 142857

.

b. 0 1234567891011121314

.

. . .

Thus, he simply takes half of 468 to get the answer. Is his

method acceptable? Explain.

26. What happened to the other 1 ?

4

37. To find 0.33 times 24, a student takes one-third of 24 and

16 1

says the answer is 8. Is she correct? Explain.

2

16 5

.

× 12 1

38. When changing 7 452

.

to a fraction, Henry set n = 7 452

.

×

2

12 5

.

32

and multiplied both sides by 100. However, when he

8 2

. 5

subtracted, he got another repeating decimal. How

160

33

could you help?

8 1

1

(one half of 16 )

165

4

2

39. Lauren asserts that 3 1211211121111

.

. . . is a repeating

6 1

(one halff of 12 1 )

206 25

.

4

2

decimal. How should you respond?

206 12

40. A student says that the sum of two repeating decimals

must be a repeating decimal. How should you respond?

27. The weight in grams, to the nearest hundredth, of a

particular sample of toxic waste was 28.67 grams.

41. Barry says he can’t find the product of

a. What is the minimum amount the sample could

12,750,000,000,000 × 3,987,000,000 on a standard

have weighed? (Write out your answer to the ten-

calculator that has a ten digit display. How should you

thousandths place.)

respond?

c07.indd 273

7/31/2013 11:56:18 AM - 274 Chapter 7 Decimals, Ratio, Proportion, and Percent

RATIO AND PROPORTION

Children’s Literature

Ratio

www.wiley.com/college/musser

See “Beanstalk: The Measure of a

The concept of ratio occurs in many places in mathematics and in everyday life, as

Giant’’ by Ann McCallum.

the next example illustrates.

a. In Washington School, the ratio of students to teachers is 17 :1, read “17 to 1.”

b. In Smithville, the ratio of girls to boys is 3 : 2.

c. A paint mixture calls for a 5 : 3 ratio of blue paint to red paint.

d. The ratio of centimeters to inches is 2 54

.

:1.

■

Common Core – Grade 6

In this chapter the numbers used in ratios will be whole numbers, fractions, or dec-

Understand the concept of a

imals representing fractions. Ratios involving real numbers are studied in Chapter 9.

ratio and use ratio language

In English, the word per means “for every” and indicates a ratio. For example,

to describe a ratio relationship

rates such as miles per gallon (gasoline mileage), kilometers per hour (speed), dollars

between two quantities.

per hour (wages), cents per ounce (unit price), people per square mile (population

density), and percent are all ratios.

D E F I N I T I O N 7 . 1

Ratio

A ratio is an ordered pair of numbers, written a : b, with b ≠ 0.

For his construction project, José needed to cut some boards into two pieces A and B so

that piece A is 1 as big as piece B. For this situation, discuss the following questions:

3

1. Piece A is how much of the board?

2. Piece B is ____ times as big as piece A?

3. What is the ratio of the length of piece A to the length of piece B?

Repeat for the following two situations:

Piece A is 3 as big as piece B

4

Piece A is 2 as big as piece B

5

Unlike fractions, there are instances of ratios in which b could be zero. For example,

the ratio of men to women on a starting major league baseball team could be reported

as 9 : 0. However, since such applications are rare, the definition of the ratio a : b

excludes cases in which b = 0.

Ratios allow us to compare the relative sizes of two quantities. This comparison

a

can be represented by the ratio symbol a : b or as the quotient . Quotients occur

b

quite naturally when we interpret ratios. In Example 7.20(a), there are 1 as many

17

teachers as students in Washington School. In part (b) there are 3 as many girls as

2

boys in Smithville. We could also say that there are 2 as many boys as girls, or that

3

the ratio of boys to girls is 2 : 3. This is illustrated in Figure 7.10.

Notice that there are several ratios that we can form when comparing the popula-

Figure 7.10

tion of boys and girls in Smithville, namely 2 : 3 (boys to girls), 3 : 2 (girls to boys), 2 : 5

c07.indd 274

7/31/2013 11:56:23 AM - Section 7.3 Ratio and Proportion 275

Algebraic Reasoning

(boys to children), 5 : 3 (children to girls), and so on. Some ratios give a part-to-part

A common ratio used in algebra

comparison, as in Example 7.20(c). In mixing the paint, we would use 5 units of blue

is that of slope. The slope is the

paint and 3 units of red paint. (A unit could be any size—milliliter, teaspoon, cup, and

ratio of the “change in y ” com-

so on.) Ratios can also represent the comparison of part-to-whole or whole-to-part.

pared to the “change in x .” More

generally, the slope is the ratio of

In Example 7.20(b) the ratio of boys (part) to children (whole) is 2 : 5. Notice that the

the amount one variable changes

part-to-whole ratio, 2 : 5, is the same concept as the fraction of the children that are

with respect to the amount of

boys, namely 2 . The comparison of all the children to the boys can be expressed in a

5

change in another variable.

whole-to-part ratio as 5 : 2, or as the fraction 52 .

In Example 7.20(b), the ratio of girls to boys indicates only the relative sizes of

Reflection from Research

the populations of girls and boys in Smithville. There could be 30 girls and 20 boys,

Children understand and can

300 girls and 200 boys, or some other pair of numbers whose ratio is equivalent.

work with part-whole relation-

It is important to note that ratios always represent relative, rather than absolute,

ships of quantities even before

amounts. In many applications, it is useful to know which ratios represent the same

they start school. However, this

concept is often not introduced

relative amounts. Consider the following example.

in schools for at least the first 2

years (Irwin, 1996).

In class 1 the ratio of girls to boys is 8 : 6. In class 2 the ratio is

4 : 3. Suppose that each class has 28 students. Do these ratios

represent the same relative amounts?

S O L U T I O N Notice that the classes can be grouped in different ways (Figure 7.11).

Figure 7.11

The subdivisions shown in Figure 7.11 do not change the relative number of girls

to boys in the groups. We see that in both classes there are 4 girls for every 3 boys.

Hence we say that, as ordered pairs, the ratios 4 : 3 and 8 : 6 are equivalent, since they

represent the same relative amount. They are equivalent to the ratio 16 :12.

■

From Example 7.21 it should be clear that the ratios a : b and ar : br, where r ≠ 0,

represent the same relative amounts. Using an argument similar to the one used

with fractions, we can show that the ratios a : b and c : d represent the same relative

amounts if and only if ad = bc. Thus we have the following definition.

D E F I N I T I O N 7 . 2

Equality of Ratios

a

c

a

c

Let and be any two ratios. Then =

if and only if ad = bc.

b

d

b

d

Just as with fractions, this definition can be used to show that if n is a nonzero

an

a

a

c

number, then

= , or an : bn = a : b. In the equation = , a and d are called the

bn

b

b

d

extremes, since a and d are at the “extremes” of the equation a : b = c : d, while b and

c are called the means. Thus the equality of ratios states that two ratios are equal if

and only if the product of the means equals the product of the extremes.

Check for Understanding: Exercise/Problem Set A #1–5

✔

c07.indd 275

7/31/2013 11:56:28 AM - 276 Chapter 7 Decimals, Ratio, Proportion, and Percent

When making orange juice, a common ratio of orange juice concentrate to water is 1: 3.

Some children believe that an orange juice concentrate to water ratio of 2 : 4 would form

the same concentration as the 1: 3 ratio. Why might children think this? That is, would an orange juice mixture with a

concentrate to water ratio of 2 : 4 be proportional to a mixture with a 1: 3 ratio?

Proportion

Reflection from Research

The concept of proportion is useful in solving problems involving ratios.

Sixth-grade students “seem able

to generalize the arithmetic that

they know well, but they have dif-

D E F I N I T I O N 7 . 3

ficulty generalizing the arithmetic

with which they are less familiar.

Proportion

In particular, middle school stu-

dents would benefit from more

A proportion is a statement that two given ratios are equal.

experiences with a rich variety of

multiplicative situations, including

proportionality, inverse variation

and exponentiation” (Swafford &

The equation 10

5

= is a proportion since 10 5¥2 5

=

22

¥ =

. Also, the equation 14 =

12

6

12

6 2

6

21

33

Langrall, 2000).

a

c

is an example of a proportion, since 14 ¥ 33 = 21¥ 22. In general, =

is a proportion

b

d

if and only if ad = bc. The next example shows how proportions are used to solve

Children’s Literature

everyday problems.

www.wiley.com/college/musser

See “What’s Faster than a

Speeding Cheetah?” by Robert

Adams School orders 3 cartons of chocolate milk for every

E. Wells.

7 students. If there are 581 students in the school, how many

cartons of chocolate milk should be ordered?

S O L U T I O N Set up a proportion using the ratio of cartons to students. Let n be the

unknown number of cartons. Then

3 (cartons)

(cartons)

= n

.

7 (students)

581 (students)

NCTM Standard

Using the cross-multiplication property of ratios, we have that

The so-called cross-multiplication

3

method can be developed

× 581 = 7 × n,

meaningfully if it arises naturally

so

in students’ work, but it can also

3

581

n

have unfortunate side effects

= ×

= 249.

7

when students do not adequately

understand when the method is

The school should order 249 cartons of chocolate milk.

■

appropriate to use.

In Example 7.22, the number of cartons of milk was compared with the number of

students. Ratios involving different units (here cartons to students) are called rates.

Commonly used rates include miles per gallon, cents per ounce, and so on.

When solving proportions like the one in Example 7.22, it is important to set up

the ratios in a consistent way according to the units associated with the numbers. In

our solution, the ratios 3 : 7 and n : 581 represented ratios of cartons of chocolate milk

to students in the school. The following proportion could also have been used.

⎛cartons of chocolate⎞

⎛ students ⎞

3

7

⎝⎜ milk in the ratio ⎠⎟

⎝⎜ in the ratio⎠⎟

=

⎛cartons of chocolate⎞

⎛ students ⎞

n

581

⎝⎜

milk in school

⎠⎟

⎝⎜ in school⎠⎟

3

581

Here the numerators show the original ratio. (Notice that the proportion =

n

7

would not correctly represent the problem, since the units in the numerators and

denominators would not correspond.)

c07.indd 276

7/31/2013 11:56:30 AM - Section 7.3 Ratio and Proportion 277

Solve the following problem in as many different ways as possible. Compare and contrast

your methods with those of your peers.

“Dan uses a 117 ounce bottle of liquid detergent in 6.5 weeks. At this rate, how many ounces of detergent is a

year's supply?”

In general, the following proportions are equivalent (i.e., have the same solutions).

This can be justified by cross-multiplication.

a

c

a

b

b

d

c

d

=

=

=

=

b

d

c

d

a

c

a

b

Thus there are several possible correct proportions that can be established when

equating ratios.

A recipe calls for 1 cup of mix, 1 cup of milk, the whites from 4

Algebraic Reasoning

The proportion in Example 7.23

eggs, and 3 teaspoons of oil. If this recipe serves 6 people, how

can be solved by reasoning that

many eggs are needed to make enough for 15 people?

since there are 2.5 times as many

people, there will need to be 2.5

S O L U T I O N When solving proportions, it is useful to list the various pieces of infor-

times as many eggs. Such reason-

mation as follows:

ing is algebraic even if the typical

cross-multiplication techniques

are not used.

ORIGINAL RECIPE

NEW RECIPE

Number of eggs

4

x

Number of people

6

15

4

Thus = x . This proportion can be solved in two ways.

6

15

CROSS-MULTIPLICATION

EQUIVALENT RATIOS

4 = x

4 = x

6

15

6

15

4

2 ¥ 2

2

2 ¥ 5

10

x

4 ¥ 15 = 6x

=

= =

=

=

6

2 ¥ 3

3

3 ¥ 5

15

15

60 = 6x

Thus x = 10.

10 = x

Notice that the table in Example 7.23 showing the number of eggs and people can

NCTM Standard

All students should solve simple

be used to set up three other equivalent proportions:

problems involving rates and

derived measurements for such

4

6

x

15

6

15

=

=

=

.

■

attributes as velocity and density.

x

15

4

6

4

x

If your car averages 29 miles per gallon, how many gallons

should you expect to buy for a 609-mile trip?

Reflection from Research

S O L U T I O N

Students often see no difference

in meaning between expressions

AVERAGE

TRIP

such as 5 km per hour and 5

Miles

29

609

hours per km. The meanings of

the numerator and denominator

Gallons

1

x

with respect to rate are not cor-

rectly understood (Bell, 1986).

29

609

x

609

Therefore,

=

, or

=

. Thus x = 21.

■

1

x

1

29

c07.indd 277

7/31/2013 11:56:33 AM - 278 Chapter 7 Decimals, Ratio, Proportion, and Percent

In a scale drawing, 0.5 centimeter represents 35 miles.

a. How many miles will 4 centimeters represent?

b. How many centimeters will represent 420 miles?

S O L U T I O N

a.

SCALE

ACTUAL

Centimeters

0.5

4

Miles

35

x

0 5

.

4

¥

Thus,

= . Solving, we obtain x = 35 4 , or x = 280.

35

x

0 5

.

b.

SCALE

ACTUAL

Centimeters

0.5

y

Miles

35

420

0 5

.

0 5

. × 420

210

Thus,

= y , or

= y. Therefore, y =

= 6 centimeters.

■

35

420

35

35

NCTM Standard

Example 7.25 could have been solved mentally by using the following mental tech-

All students should develop,

nique called scaling up/scaling down, that is, by multiplying/dividing each number in

analyze, and explain methods

a ratio by the same number. In Example 7.25(a) we can scale up as follows:

for solving problems involving

proportions such as scaling and

0 5

. centimeter : 35 miles = 1 centimeter : 70 miles

finding equivalent ratios.

= 2 centimeters :140 miles

= 4 centimeters : 280 miles.

Similarly, the number of centimeters representing 420 miles in Example 7.25(b) could

have been found mentally by scaling up as follows:

35 miles : 0 5

. centimeter = 70 miles :1 centimeter

= 6 × 70 miles : 6 × 1 centimeters.

Thus, 420 miles is represented by 6 centimeters.

In Example 7.23, to solve the proportion 4 : 6 = x :15, the ratio 4 : 6 was scaled

down to 2 : 3, then 2 : 3 was scaled up to 10 :15. Thus x = 10.

Two neighbors were trying to decide whether their property

taxes were fair. The assessed value of one house was $175,800

and its tax bill was $2777.64. The other house had a tax bill of $3502.85 and was

assessed at $189,300. Were the two houses taxed at the same rate?

S O L U T I O N Since the ratio of property taxes to assessed values should be the same,

the following equation should be a proportion:

2777 64

.

3502 85

.

=

175,800

189,300

Equivalently, we should have 2777 64

.

× 189,300 = 175,800 × 3502 85

.

. Using a calcu-

lator, 2777 64

.

× 189,300 = 525,807,252 and 3502 85

.

× 175,800 = 615,801,030. Thus,

the two houses are not taxed the same, since 525,807,252 ≠ 615,801,030.

c07.indd 278

7/31/2013 11:56:36 AM - Section 7.3 Ratio and Proportion 279

An alternative solution to this problem would be to determine the tax rate per

$1000 for each house.

.

2777 64

r

First house:

=

yields r = $ .

15 80 per $1000.

,

175 800

1000

3502 85

.

r

Second hous :

e

=

yields r = 18

$

50

.

per $1000.

189,300

1000

Thus, it is likely that two digits of one of the tax rates were accidentally interchanged

when calculating one of the bills.

■

Check for Understanding: Exercise/Problem Set A #6–11

✔

The famous mathematician Pythagoras founded a school that bore

his name. As lore has it, to lure a young student to study at this

school, he agreed to pay the student a penny for every theorem the

student mastered. The student, motivated by the penny-a-theorem

offer, studied intently and accumulated a huge sum of pennies.

However, he so enjoyed the geometry that he begged Pythagoras

for more theorems to prove. Pythagoras agreed to provide him with

more theorems, but for a price—namely, a penny a theorem. Soon,

Pythagoras had all his pennies back, in addition to a sterling student.

©Ron Bagwell

EXERCISE/PROBLEM SET A

EXERCISES

1. Write a ratio based on each of the following.

4. Determine whether the given ratios are equal.

a. Two-fifths of Ted’s garden is planted in tomatoes.

a. 3 : 4 and 15 : 22

b. The certificate of deposit you purchased earns $6.18

b. 11: 6 and 66 : 36

interest on every $100 you deposit.

5. When blood cholesterol levels are tested, sometimes a car-

c. Three out of every four voters surveyed favor ballot mea-

diac risk ratio is calculated.

sure 5.

d. There are five times as many boys as girls in Mr.

total cholesterol level

Cardiac risk ratio =

Wright’s physics class.

high-density lipo

oprotein level (HDL)

e. There are half as many sixth graders in Fremont School

For women, a ratio between 3.0 and 4.5 is desirable.

as eighth graders.

A woman’s blood test yields an HDL cholesterol level of

f. Nine of every 16 students in the hot-lunch line are girls.

60 mg/dL and a total cholesterol level of 225 mg/dL. What

is her cardiac risk ratio, expressed as a one-place decimal?

2. Explain how each of the following rates satisfies the

Is her ratio in the normal range?

definition of ratio. Give an example of how each is used.

a. 250 miles/11.6 gallons

6. Solve each proportion for n.

b. 25 dollars/3.5 hours

n

6

n

3

7

42

12

18

a.

=

b.

=

c.

=

d.

=

c. 1 dollar (American)/0.65 dollar (Canadian)

70

21

84

14

n

48

n

45

d. 2.5 dollars/0.96 British pound

7. Solve each proportion for x. Round each answer to two

3. Write a fraction in the simplest form that is equivalent to

decimal places.

each ratio.

9

10 8

35 2

.

5

a. 16 :125 = x : 5

b. =

.

c.

=

a. 16 to 64

7

x

19 6

.

3x

b. 30 to 75

x

3

3 1

2x

0 04

d.

=

e. 4 = x

f.

= .

c. 82.5 to 16.5

4 − x

4

2 1

11

x + 10

1 8

. 5

4

2

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7/31/2013 11:56:39 AM - 280 Chapter 7 Decimals, Ratio, Proportion, and Percent

8. Write three other proportions for the given proportion.

d. 20 inches in 15 hours is equal to 16 inches in _____ hours.

e. 32 cents for 8 ounces is equal to _____ cents for 12 ounces.

36 cents

42 cents

=

18 ounces

21 ounces

10. If you are traveling 100 kilometers per hour, how

fast are you traveling in mph? For this exercise, use

9. Solve these proportions mentally by scaling up or

50 mph = 80 kph (kilometers per hour). (The exact metric

scaling down.

equivalent is 80.4672 kph.)

a. 24 miles for 2 gallons is equal to _____ miles for

16 gallons.

11. Which of the following is the better buy?

b. $13.50 for 1 day is equal to _____ for 6 days.

a. 67 cents for 58 ounces or 17 cents for 15 ounces

c. 300 miles in 12 hours is equal to _____ miles in

b. 29 ounces for 13 cents or 56 ounces for 27 cents

8 hours. (Hint: Scale down to 4 hours, then scale up

c. 17 ounces for 23 cents, 25 ounces for 34 cents, or

to 8 hours.)

73 ounces for 96 cents

PROBLEMS

12. Grape juice concentrate is mixed with water in a ratio

b. If the teacher-pupil ratio remains at 1: 35 and if the cost

of 1 part concentrate to 3 parts water. How much grape

to the district for one teacher is $33,000 per year, how

juice can be made from a 10-ounce can of concentrate?

much will be spent per pupil per year?

c. Answer part b for a ratio of 1: 20.

13. A crew clears brush from 1 acre of land in 3 days. How

2

long will it take the same crew to clear the entire plot of

22. An astronomical unit (AU) is a measure of distance used

2 3 acres?

by astronomers. In October 1985 the relative distance

4

from Earth to Mars in comparison with the distance from

14. A recipe for peach cobbler calls for 6 small peaches for 4

Earth to Pluto was 1:12 37

.

.

servings. If a large quantity is to be prepared to serve 10

a. If Pluto was 30.67 AU from Earth in October 1985, how

people, about how many peaches would be needed?

many astronomical units from Earth was Mars?

15. Elise and her family went for a 7.5 mile mountain bike

b. Earth is always about 1 AU from the sun (in fact, this is

ride. It took them 65 minutes to ride 3.25 miles. If they

the basis of this unit of measure). In October 1985, Pluto

continue at the same pace, how much longer will it take

was about 2 85231 109

.

×

miles from Earth. About how

them to complete their entire bike ride?

many miles is Earth from the sun?

c. In October 1985 about how many miles was Mars from

16. If a 92-year-old man has averaged 8 hours per 24-hour

Earth?

day sleeping, how many years of his life has he been

asleep?

23. According to the “big-bang” hypothesis, the universe

was formed approximately 1010 years ago. The analogy

17. A man who weighs 175 pounds on Earth would weigh 28

of a 24-hour day is often used to put the passage of this

pounds on the moon. How much would his 30-pound dog

amount of time into perspective. Imagine that the universe

weigh on the moon?

was formed at midnight 24 hours ago and answer the fol-

18. Suppose that you drive an average of 4460 miles every

lowing questions.

half-year in your car. At the end of 2 3 years, how far will

a. To how many years of actual time does 1 hour correspond?

4

your car have gone?

b. To how many years of actual time does 1 minute

correspond?

19. Becky is climbing a hill that has a 17° slope. For every 5

c. To how many years of actual time does 1 second

feet she gains in altitude, she travels about 16.37 horizon-

correspond?

tal feet. If at the end of her uphill climb she has traveled 1

d. The Earth was formed, according to the hypothesis,

mile horizontally, how much altitude has she gained?

approximately 5 billion years ago. To what time in the

24-hour day does this correspond?

e. Earliest known humanlike remains have been determined

by radioactive dating to be approximately 2.6 million

years old. At what time of the 24-hour day did the

creatures who left these remains die?

f. Intensive agriculture and the growth of modern civiliza-

20. The Spruce Goose, a wooden flying boat built for Howard

tion may have begun as early as 10,000 years ago. To

Hughes, had the world’s largest wingspan, 319 ft 11 in.

what time of the 24-hour day does this correspond?

according to the Guinness Book of World Records. It flew

only once in 1947, for a distance of about 1000 yards.

24. Cary was going to meet Jane at the airport. If he traveled

Shelly wants to build a scale model of the 218 ft 8 in.–long

60 mph, he would arrive 1 hour early, and if he traveled

Spruce Goose. If her model will be 20 inches long, what

30 mph, he would arrive 1 hour late. How far was it to the

will its wingspan be (to the nearest inch)?

airport? (Recall: Distance = rate ¥ time.)

21. Jefferson School has 1400 students. The teacher–pupil

25. Seven children each had a different number of pennies.

ratio is 1: 35.

The ratio of each child’s total to the next poorer was a

a. How many additional teachers will have to be hired to

whole number. Altogether they had $28.79. How much

reduce the ratio to 1: 20?

did each have?

c07.indd 280

7/31/2013 11:56:41 AM - Section 7.3 Ratio and Proportion 281

26. Two baseball batters, Eric and Morgan, each get 31 hits in

30. Beginning with 100, each of two persons, in turn, subtracts

69 at-bats. In the next week, Eric slumps to 1 hit in 27 at-

a single-digit number. The player who ends at zero is the

bats and Morgan bats 4 for 36 (1 out of 9). Without doing

loser. Can you explain how to play so that one player

any calculations, which batter do you think has the higher

always wins?

average? Check your answer by calculating the two averages

31. How can you cook something for exactly 15 minutes

(the number of hits divided by the number of times at bat).

if all you have are a 7-minute and an 11-minute

27. A woman has equal numbers of pennies, nickels, and

egg timer?

dimes. If the total value of the coins is $12.96, how many

32. Twelve posts stand equidistant along a race track.

dimes does she have?

Starting at the first post, a runner reaches the eighth

28. A man walked into a store to buy a hat. The hat he

post in 8 seconds. If she runs at a constant velocity, how

selected cost $20. He said to his father, “If you will lend

many seconds are needed to reach the twelfth post?

me as much money as I have in my pocket, I will buy that

33. Melvina was planning a long trip by car. She knew she

$20 hat.” The father agreed. Then they did it again with

could average about 180 miles in 4 hours, but she was

a $20 pair of slacks and again with a $20 pair of shoes.

trying to figure out how much farther she could get each

The man was finally out of money. How much did he

day if she and her friend (who drives about the same

have when he walked into the store?

speed) shared the driving and they drove for 10 hours per

29. What is the largest sum of money in U.S. coins that you

day. She figured they could travel an extra 450 miles, so

could have without being able to give change for a nickel,

altogether they could do 630 miles a day. Is she on track?

dime, quarter, half-dollar, or dollar?

How would you explain this?

EXERCISE/PROBLEM SET B

EXERCISES

1. The ratio of girls to boys in a particular classroom is 6 : 5.

6. Solve for the unknown in each of the following proportions.

a. What is the ratio of boys to girls?

3

B

2 1

X

5 = D

4

=

4 8

= .

b. What fraction of the total number of students are boys?

a.

b.

c.

6

25

8

18

100

1 5

.

c. How many boys are in the class?

57 4

.

7 4

.

d. How many boys are in the class if there are 33 students?

d.

=

(to one decimal place)

39 6

.

P

2. Explain how each of the following rates satisfies the defini-

7. Solve each proportion for x. Round your answers to two

tion of ratio. Give an example of how each is used.

decimal places where decimal answers do not terminate.

a. 1580 people/square mile

b. 450 people/year

7

12

40

c. 360 kilowatt-hours/4 months

d. 355 calories/6 ounces

a. = x

b.

=

c. 2 : 9 = x : 3

5

40

35

x

3. Write a fraction in the simplest form that is equivalent to

3

15

x

3x

12

x

d. : 8 = 9 : x

e.

=

f.

=

−

each ratio.

4

32

x + 2

4

6

a. 17 to 119 b. 26 to 91 c. 97.5 to 66.3

8. Write three other proportions for each given proportion.

4. Determine whether the given ratios are equal.

35 miles

87 5 miles

a. 5 : 8 and 15 : 25

b. 7 :12 and 36 : 60

=

.

2 hours

5 hours

5. In one analysis of people of the world, it was reported that

9. Solve these proportions mentally by scaling up or scaling

of every 1000 people the following numbers speak the indi-

down.

cated language as their native tongue.

a. 26 miles for 6 hours is equal to _____ miles for 24 hours.

165 speak Mandarin

b. 84 ounces for each 6 square inches is equal to _____

86 speak English

ounces for each 15 square inches.

83 speak Hindi/Urdu

c. 40 inches in 12 hours is equal to _____ inches in 9 hours.

64 speak Spanish

d. $27.50 for 1.5 days is equal to _____ for 6 days.

58 speak Russian

e. 750 people for each 12 square miles is equal to _____

37 speak Arabic

people for each 16 square miles.

a. Find the ratio of Spanish speakers to Russian speakers.

10. If you are traveling 55 mph, how fast are you traveling

b. Find the ratio of Arabic speakers to English speakers.

in kph?

c. The ratio of which two groups is nearly 2 :1?

d. Find the ratio of persons who speak Mandarin, English,

11. Determine which of the following is the better buy.

or Hindi/Urdu to the total group of 1000 people.

a. 60 ounces for 29 cents or 84 ounces for 47 cents

e. What fraction of persons in the group of 1000 world

b. $45 for 10 yards of material or $79 for 15 yards

citizens are not accounted for in this list? These persons

c. 18 ounces for 40 cents, 20 ounces for 50 cents, or

speak one of the more than 200 other languages spoken

30 ounces for 75 cents (Hint: How much does $1

in the world today.

purchase in each case?)

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7/31/2013 11:56:44 AM - 282 Chapter 7 Decimals, Ratio, Proportion, and Percent

PROBLEMS

12. Three car batteries are advertised with warranties as

b. In 1996, 163 returns per 10,000 were audited. How many

follows.

more of the 12,500 returns would be expected to be

Model XA: 40-month warranty, $34.95

audited for 1996 than for 1994?

Model XL: 50-month warranty, $39.95

23. a. A baseball pitcher has pitched a total of 25 innings so

Model XT: 60-month warranty, $49.95

far during the season and has allowed 18 runs. At this

Considering only the warranties and the prices, which

rate, how many runs, to the nearest hundredth, would

model of car battery is the best buy?

he allow in nine innings? This number is called the

pitcher’s earned run average, or ERA.

13. Cari walked 3.4 kilometers in 45 minutes. At that rate,

b. Randy Johnson of the Arizona Diamondbacks had

how long will it take her to walk 11.2 kilometers? Round

an ERA of 2.64 in 2000. At that rate, how many runs

to the nearest minute.

would he be expected to allow in 100 innings pitched?

14. A family uses 5 gallons of milk every 3 weeks. At that

Round your answer to the nearest whole number.

rate, how many gallons of milk will they need to purchase

24. Many tires come with 13 inch of tread on them. The first

in a year’s time?

32

2 inch wears off quickly (say, during the first 1000 miles).

32

15. A couple was assessed property taxes of $1938.90 on a

From then on the tire wears uniformly (and more slowly).

home valued at $168,600. What might Frank expect to

A tire is considered “worn out” when only 2 inch of tread

2

3

pay in property taxes on a home he hopes to purchase

is left.

in the same neighborhood if it has a value of $181,300?

a. How many 32nds of an inch of usable tread does a tire

Round to the nearest dollar.

have after 1000 miles?

16. By reading just a few pages at night before falling asleep,

b. A tire has traveled 20,000 miles and has 5 inch of tread

32

Randy finished a 248-page book in 4 1 weeks. He just

remaining. At this rate, how many total miles should the

2

started a new book of 676 pages. About how long should

tire last before it is considered worn out?

it take him to finish the new book if he reads at the same

25. In classroom A, there are 12 boys and 15 girls. In class-

rate?

room B, there are 8 boys and 6 girls. In classroom C, there

17. a. If 1 inch on a map represents 35 miles, how many miles

are 4 boys and 5 girls.

are represented by 3 inches? 10 inches? n inches?

a. Which two classrooms have the same boys-to-girls ratio?

b. Los Angeles is about 1000 miles from Portland. About

b. On one occasion classroom A joined classroom B. What

how many inches apart would Portland and Los Angeles

was the resulting boys-to-girls ratio?

be on this map?

c. On another occasion classroom C joined classroom B.

What was the resulting ratio of boys to girls?

18. A farmer calculates that out of every 100 seeds of corn he

d. Are your answers to parts b and c equivalent? What

plants, he harvests 84 ears of corn. If he wants to harvest

does this tell you about adding ratios?

7200 ears of corn, how many seeds must he plant?

26. An old picture frame has dimensions 33 inches by 24

19. A map is drawn to scale such that 1 inch represents

8

inches. What one length must be cut from each dimension

65 feet. If the shortest route from your house to the

so that the ratio of the shorter side to the longer side is 2 ?

grocery store measures 23 7 inches, how many miles is it

3

6

1

to the grocery store?

27. The Greek musical scale, which very closely resembles the

12-note tempered scale used today, is based on ratios of

20. a. If 13 cups of flour are required to make 28 cookies,

4

frequencies. To hear the first and fifth tones of the scale

how many cups are required for 88 cookies?

is equivalent to hearing the ratio 3 , which is the ratio of

b. If your car gets 32 miles per gallon, how many gallons

2

their frequencies.

do you use on a 160-mile trip?

c. If your mechanic suggests 3 parts antifreeze to 4 parts

water, and if your radiator is 14 liters, how many liters

of antifreeze should you use?

d. If 11 ounces of roast beef cost $1.86, how much does

roast beef cost per pound?

a. If the frequency of middle C is 256 vibrations per

21. Two professional drag racers are speeding down a 1 -mile

4

second, find the frequencies of each of the other notes

track. If the lead driver is traveling 1.738 feet for every

given. For example, since G is a fifth above middle C, it

1.670 feet that the trailing car travels, and if the trailing

follows that G : 256 = 3 : 2 or G = 384 vibrations/second.

car is going 198 miles per hour, how fast in miles per hour

(NOTE: Proceeding beyond B would give sharps, below

is the lead car traveling?

F, flats.)

22. In 1994, the Internal Revenue Service audited 107 of every

b. Two notes are an octave apart if the frequency of one

10,000 individual returns.

is double the frequency of the other. For example, the

a. In a community in which 12,500 people filed returns,

frequency of C above middle C is 512 vibrations per

how many returns might be expected to be audited?

second. Using the values found in part a, find the

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7/31/2013 11:56:47 AM - Section 7.4 Percent 283

frequencies of the corresponding notes in the octave

33. Ms. Price has three times as many girls as boys in her

above middle C (in the following range).

class. Ms. Lippy has twice as many girls as boys. Ms.

256512

Price has 60 students in her class and Ms. Lippy has

135 students. If the classes were combined into one,

C

DEFGABC

what would be the ratio of girls to boys?

c. The aesthetic effect of a chord depends on the ratio of

its frequencies. Find the following ratios of seconds.

Analyzing Student Thinking

D : C

E : D

A : G

34. Carlos says that if the ratio of oil to vinegar in a salad

What simple ratio are these equivalent to?

dressing is 3 : 4, that means that 75% of the dressing is oil.

d. Find the following ratio of fourths.

How should you respond?

F : C G : D A : E

35. Scott heard that the ratio of boys to girls in his class next

What simple ratio are these equivalent to?

year was going to be 5 : 4. He asks, “Does that mean that

there are going to be 5 boys in my class?” How should

28. Ferne, Donna, and Susan have just finished playing three

you respond?

games. There was only one loser in each game. Ferne lost

the first game, Donna lost the second game, and Susan lost

36. Ashlee writes that for ratios a/b and c/d, if a/b = c/d, then

the third game. After each game, the loser was required to

a/c = b/d. Is she correct? Explain.

double the money of the other two. After three rounds, each

37. Caleb used scaling up to find how many miles would be

woman had $24. How much did each have at the start?

represented by 8 meters if 0.25 centimeters represents 12

29. A ball, when dropped from any height, bounces 1 of the

miles as follows: 0 2

. 5 cm :12 miles = 100 cm : 48 miles =

3

original height. If the ball is dropped, bounces back up,

800 cm : 384 miles. Is this correct? Explain.

and continues to bounce up and down so that it has trav-

38. When mixing orange juice concentrate, the ratio of

eled 106 feet when it strikes the ground for the fourth time,

juice to water is 1: 3. Micala thought that using 2 cans

what is the original height from which it was dropped?

of juice would require using 4 cans of water or a ratio of

30. Mary had a basket of hard-boiled eggs to sell. She first

2 : 4 to have the same flavor. What is Micala’s misunder-

sold half her eggs plus half an egg. Next she sold half her

standing?

eggs and half an egg. The same thing occurred on her third,

39. Sabrina noticed that 60% of her English class was girls

fourth, and fifth times. When she finished, she had no eggs

and so she concluded that the ratio of girls to boys was

in her basket. How many did she have when she started?

3 : 5. Is she correct? Explain.

31. Joleen had a higher batting average than Maureen for the

40. Marvin is trying to find the height of a tree in the school

first half of the season, and Joleen also had a higher batting

yard. He is using the proportion

average than Maureen for the second half of the season.

Does it follow that Joleen had a better batting average

Marvin’s height

tree’s height

=

than Maureen for the entire season? Why or why not?

Marvin’s shadow length

tree’s shadow length

32. A box contains three different varieties of apples. What

His height is 4 feet. His shadow length is 15 inches. The

is the smallest number of apples that must be taken to be

length of the tree’s shadow is 12 feet. Marvin used the

sure of getting at least 2 of one kind? How about at least 3

4

proportion

= tree’s height . But that gave the tree’s

of one kind? How about at least 10 of a kind? How about

15

12

at least n of a kind?

height as being shorter than Marvin’s! What went wrong?

PERCENT

Converting Percents

Like ratios, percents are used and seen commonly in everyday life.

Solve the following problem and explain

your solution method with a peer.

In the rectangle at the right, shade six small squares.

Use the diagram to justify what percent of the rectangle is shaded.

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7/31/2013 11:56:49 AM - 284 Chapter 7 Decimals, Ratio, Proportion, and Percent

a. The Dow Jones stock index declined by 1.93%.

b. The BYU basketball team made 41.3% of the three-point shots attempted.

c. A spring clearance sale advertised jeans at 30% off the retail price.

d. The land prices in Mapleton today are up 150% from 5 years ago.

In each case, the percent represents a ratio, a fraction, or a decimal. The percent in

part (b) represents the fact that a basketball team made 247 out of 598 three-point

shots, a ratio of 247 or about 0.413. Thus, it was reported that they made 41.3%.

598

The jeans sale described in part (c) offered buyers a discount of $18 off of $60. This

fraction 18 is equal to 0.3 or 30%.

■

60

Common Core – Grade 6

The word percent has a Latin origin that means “per hundred.” Thus 25 percent

Find a percent of a quantity as

means 25 per hundred, 25 , or 0.25. The symbol “%” is used to represent percent. So

100

a rate per 100 (e.g., 30% of a

n

420% means 420 , 4.20, or 420 per hundred. In general, n% represents the ratio

.

quantity means 30/100 times the

100

100

quantity); solve problems involv-

Since percents are alternative representations of fractions and decimals, it is

ing finding the whole, given a

important to be able to convert among all three forms, as suggested in Figure 7.12.

part and the percent.

Figure 7.12

Since we have studied converting fractions to decimals, and vice versa, there are

only four cases of conversion left to consider in Figure 7.12.

Case 1: Percents to Fractions

Use the defi nition of percent. For example, 63

63

% =

by the meaning of percent.

100

Case 2: Percents to Decimals

Since we know how to convert fractions to decimals, we can use this skill to con-

vert percents to fractions and then to decimals. For example, 63

63

% =

= 0 6

. 3 and

100

27

27

% =

= 0 2

. 7. These two examples suggest the following shortcut, which elimi-

100

nates the conversion to a fraction step. Namely, to convert a percent directly to a

decimal, “drop the % symbol and move the number’s decimal point two places to

the left.” Thus 31% = 0 31

. , 213

% = 2 1

. 3, 0

5

. % = 005

.

, and so on. These examples

can also be seen visually in Figure 7.13 on a 10 by 10 grid, where 31% is rep-

resented by shading 31 out of 100 squares, 213% is represented by shading 213

squares (2 full grids and 13 squares on a third grid), and 0.5% is represented by

shading half of 1 square on the grid.

Figure 7.13

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7/31/2013 11:56:52 AM - Section 7.4 Percent 285

NCTM Standard

Case 3: Decimals to Percents

All students should recognize

and generate equivalent forms of

Here we merely reverse the shortcut in case 2. For example, 0 8

. 3 = 83%, 5 1

. = 510%,

commonly used fractions, deci-

and 0 0001

.

= 0 0

. 1% where the percents are obtained from the decimals by

mals, and percents.

“moving the decimal point two places to the right and writing the % symbol on

the right side.”

Case 4: Fractions to Percents

Some fractions that have terminating decimals can be converted to percents by

expressing the fraction with a denominator of 100. For example, 17

2

4

= 17%, =

=

100

5

10

40

3

12

= 40%,

=

= 12%, and so on. Also, fractions can be converted to decimals

100

25

100

(using a calculator or long division), and then case 3 can be applied.

A calculator is useful when converting fractions to percents. For example,

3 ÷ 13 = 0 23076923

.

shows that 3 ≈ 0 2

. 3 or 23%. Also,

13

5 ÷ 9 = 0 555555556

.

shows that 5 ≈ 56%.

9

NCTM Standard

Write each of the following in all three forms: decimal, per-

All students should work flexibly

cent, fraction (in simplest form).

with fractions, decimals, and per-

1

cents to solve problems.

a. 32% b. 0.24 c. 450% d. 16

S O L U T I O N

32

8

a. 32% = 0 3

. 2 =

=

100

25

24

6

b. 0 2

. 4 = 24% =

=

100

25

450

1

c. 450% =

= 4 5

. = 4

100

2

1

1

1¥ 54

625

6 2

. 5

d.

=

=

=

= 0 0625

.

=

= 6 2

. 5%

■

16

24

24 ¥ 54

10,000

100

Check for Understanding: Exercise/Problem Set A #1–2

✔

Mental Math and Estimation Using Fraction Equivalents

TABLE 7.2

Since many commonly used percents have convenient fraction equivalents, it is often

PERCENT

FRACTION

easier to find the percent of a number mentally, using fractions (Table 7.2). Also, as

was the case with proportions, percentages of numbers can be estimated by choosing

5%

1

20

compatible fractions.

10%

1

10

20%

1

5

Find the following percents mentally, using fraction equiva-

25%

1

4

lents. Notice that a helpful interchanging technique is used in

33 1 %

1

3

3

parts e and f.

50%

1

2

a. 25% × 44

b. 75% × 24

66 2 %

2

3

3

c. 50% × 76

d. 33 1 % × 93

3

75%

3

4

e. 38% × 50

f. 84% × 25

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7/31/2013 11:56:55 AM - 286 Chapter 7 Decimals, Ratio, Proportion, and Percent

S O L U T I O N

a. 25% × 44

1

= × 44 = 11

b. 75% × 24

3

= × 24 = 18

4

4

c. 50% × 76

1

= × 76 = 38

d. 33 1 % × 93

1

= × 93 = 31

2

3

3

e. 38% × 50 = 38 × 50% = 38

1

× = 19

2

f. 84% × 25 = 84 × 25% = 84

1

× = 21

■

4

Estimate the following percents mentally, using fraction

equivalents.

a. 48% × 73

b. 32% × 95

c. 24% × 71

d. 123% × 54

e. 0 4

. 5% × 57

f. 59% × 81

S O L U T I O N

a. 48% × 73 ≈ 50% × 72 = 36. (Since 50% > 48%, 73 was rounded down to 72 to

compensate.)

b. 32% × 95 ≈ 331 % × 93

1

= × 93 = 31. (Since 331 % > 32%, 95 was rounded down to

3

3

3

93, which is a multiple of 3.)

c. 24% × 71 ≈ 1 × 72 = 1 × 8 × 9 = 18

4

4

d. 123% × 54 ≈ 125% × 54

5

≈ × 56 = 5 × 14 = 70; alternatively,

4

123% × 54 = 123 × 54% ≈ 130 × 50% = 130

1

× = 65

2

e. 0 4

. 5% × 57 ≈ 0 5

. % × 50 = 0 5

. × 50% = 0 2

. 5

f. 59% × 81 ≈ 60% × 81 3

≈ × 80 = 3 × 16 = 48

■

5

Check for Understanding: Exercise/Problem Set A #3–8

✔

If the wholesale price of a jacket is marked up 40% to obtain the retail price and the retail

price is then marked down 40% to a sale price, are the wholesale price and the sale price

the same? If not, explain why not and determine which is larger.

Reflection from Research

Solving Percent Problems

Students who work on decimal

problems that are given in famil-

Since percents can be expressed as fractions using a denominator of 100, percent

iar, everyday contexts increase

problems involve three pieces of information: a percent, p, and two numbers, a and

their knowledge much more

n, as the numerator and denominator of a fraction. The relationship between these

significantly than those who work

numbers is shown in the proportion

on noncontextualized problems

(Irwin, 2001).

p

a

= .

100

n

p

This proportion can be rewritten as the equation

¥ n = a. Related to these three

100

quantities, there are three common types of problems involving percents and each

type is determined by what piece of information is unknown: p, a,

or

n.

The following questions illustrate three common types of problems involving

percents.

a. A car was purchased for $13,000 with a 20% down payment. How much was the

down payment?

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7/31/2013 11:56:59 AM - Section 7.4 Percent 287

b. One hundred sixty-two seniors, 90% of the senior class, are going on the class trip.

How many seniors are there?

c. Susan scored 48 points on a 60-point test. What percent did she get correct?

There are three approaches to solving percent problems such as the preceding three

problems. The first of the three approaches is the grid approach and relies on the 10

by 10 grids introduced earlier in this section. This approach is more concrete and aids

in understanding the underlying concept of percents. The more common approaches,

proportions and equations, are more powerful and can be used to solve a broader

range of problems.

Grid Approach Since percent means “per hundred,” solving problems to find a

missing percent can be visualized by using the 10 by 10 grids introduced earlier in

the section.

Answer the preceding three problems using the grid approach.

S O L U T I O N

a. A car was purchased for $13,000 with a 20% down payment. How much was the

down payment?

Figure 7.14

Let the grid in Figure 7.14 represent the total cost of the car, or $13,000. Since

the down payment was 20%, shade 20 out of 100 squares. The solution can be

found by reasoning that since 100 squares represent $13,000, then 1 square rep-

resents 13 000

,

= 130

$

and therefore 20 squares represent the down pay ment of

100

20 × $130 = $2600.

b. One hundred sixty-two seniors, 90% of the senior class, are going on the class trip.

How many seniors are there?

Let the grid in Figure 7.15 represent the total class size. Since 90% of the stu-

Figure 7.15

dents will go on the class trip, shade 90 of the 100 squares. The reasoning used to

solve this problem is that since 90 squares represent 162 students, then 1 square

represents 162 = 1 8

. students. Thus 100 squares, the whole class, is 100 × 1 8

. = 180

90

students.

c. Susan scored 48 points on a 60-point test. What percent did she get correct?

Let the grid in Figure 7.16 represent all 60 points on the test. In this case, the per-

cent is not given, so determining how many squares should be shaded to represent

Susan’s score of 48 points becomes the focus of the problem. Reasoning with the

grid, it can be seen that since 100 squares represent 60 points, then 1 square rep-

Figure 7.16

resents 0.6 points. Thus 10 squares is 6 points and 80 squares is Susan’s 6 × 8 = 48

point score. Thus she got 80% correct.

■

Proportion Approach Since percents can be written as a ratio, solving percent

problems may be done using proportions. For problems involving percents between

0 and 100, it may be helpful to think of a fuel gauge that varies from empty (0%) to

full (100%) (Figure 7.17). The next example shows how this visual device leads to

solving a proportion.

Figure 7.17

Answer the preceding three problems using the proportion

approach.

Common Core – Grade 7

S O L U T I O N

Use proportional relationships to

solve multistep ratio and percent

a. A car was purchased for $13,000 with a 20% down payment. How much was the

problems.

down payment (Figure 7.18)?

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7/31/2013 11:57:00 AM - 288 Chapter 7 Decimals, Ratio, Proportion, and Percent

DOLLARS

PERCENT

Down payment

x

20

Purchase price

13,000

100

x

20

13,000

Thus

=

, or x =

= 2600

$

.

13,000

100

5

b. One hundred sixty-two seniors, 90% of the senior class, are going on the class trip.

Figure 7.18

How many seniors are there (Figure 7.19)?

SENIORS

PERCENT

Class trip

162

90

Class total

x

100

162

90

10 ⎞

Thus

=

, or x =

⎛

162

180.

x

100

⎝⎜ 9 ⎠⎟ =

c. Susan scored 48 points on a 60-point test. What percent did she get correct

(Figure 7.20)?

Figure 7.19

TEST

PERCENT

Score

48

x

Total

60

100

48

4

Thus

= x , or x = 100 ¥ = 80.

■

60

100

5

Notice how (a), (b), and (c) lead to the following generalization:

Figure 7.20

Part

percent

=

.

Whole

100

In other examples, if the “part” is larger than the “whole,” the percent is larger than 100%.

Algebraic Reasoning

Equation Approach An equation can be used to represent each of the problems

As can be seen here, equations

in Example 7.32 as follows:

and variables can be used to

solve percent problems that are

(a) 20% ¥ ,

13 000 = x

commonly found in daily life.

(b)

%

90

¥ x = 162

(c) x% ¥ 60 = 48.

The following equations illustrate these three forms, where x represents an unknown

and takes the place of p, n

, or a depending on what is given and what is unknown in

the problem.

TRANSLATION OF PROBLEM

EQUATION

⎛ p ⎞

(a)

p% of n is x

n

x

⎝⎜100⎠⎟ =

⎛ p ⎞

(b)

p% of x is a

x

a

⎝⎜100⎠⎟ =

⎛ x ⎞

(c)

x% of n is a

n

a

⎝⎜100⎠⎟ =

Once we have obtained on