Mathematics for elementary teachers

National Council of Teachers of MathematicsPrinciples and Standards for School Mathematics

Principles for School Mathematics

EQUITY. Excellence in mathematics education requires eq-uity high expectations and strong support for all students.

CURRICULUM. A curriculum is more than a collection of ac-tivities: it must be coherent, focused on important mathematics,and well articulated across the grades.

TEACHING. Effective mathematics teaching requires under-standing what students know and need to learn and then chal-lenging and supporting them to learn it well.

LEARNING. Students must learn mathematics with understand-ing, actively building new knowledge from experience and priorknowledge.

ASSESSMENT. Assessment should support the learning of im-portant mathematics and furnish useful information to bothteachers and students.

TECHNOLOGY. Technology is essential in teaching and learn-ing mathematics; it influences the mathematics that is taughtand enhances students learning.

Standards for School MathematicsNUMBER AND OPERATIONS

Instructional programs from prekindergarten through grade 12should enable all students to

understand numbers, ways of representing numbers, relation-ships among numbers, and number systems;

understand meanings of operations and how they relate to oneanother;

compute fluently and make reasonable estimates.

ALGEBRA


understand patterns, relations, and functions;

represent and analyze mathematical situations and structuresusing algebraic symbols;

use mathematical models to represent and understand quanti-tative relationships;

analyze change in various contexts.

GEOMETRY


analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical argu-ments about geometric relationships;

specify locations and describe spatial relationships using co-ordinate geometry and other representational systems;

apply transformations and use symmetry to analyze mathe-matical situations;

use visualization, spatial reasoning, and geometric modelingto solve problems.

MEASUREMENT


understand measurable attributes of objects and the units, sys-tems, and processes of measurement;

apply appropriate techniques, tools, and formulas to deter-mine measurements.

DATA ANALYSIS AND PROBABILITY


formulate questions that can be addressed with data and col-lect, organize, and display relevant data to answer them;

select and use appropriate statistical methods to analyze data;

develop and evaluate inferences and predictions that are basedon data;

understand and apply basic concepts of probability.

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 solveproblems;

monitor and reflect on the process of mathematical problemsolving.

REASONING AND PROOF


recognize reasoning and proof as fundamental aspects ofmathematics;

make and investigate mathematical conjectures;

develop and evaluate mathematical arguments and proofs;

select and use various types of reasoning and methods of proof.

COMMUNICATION


organize and consolidate their mathematical thinking throughcommunication;

communicate their mathematical thinking coherently andclearly to peers, teachers, and others;

analyze and evaluate the mathematical thinking and strategiesof others;

use the language of mathematics to express mathematicalideas precisely.

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CONNECTIONS


recognize and use connections among mathematical ideas;

understand how mathematical ideas interconnect and build onone another to produce a coherent whole;

recognize and apply mathematics in contexts outside of mathematics.

REPRESENTATION


create and use representations to organize, record, and com-municate mathematical ideas;

select, apply, and translate among mathematical representa-tions to solve problems;

use representations to model and interpret physical, social, andmathematical phenomena.

PREKINDERGARTENNumber and Operations: Developing an understanding ofwhole numbers, including concepts of correspondence, counting,cardinality, and comparison.Geometry: Identifying shapes and describing spatial relationships.Measurement: Identifying measurable attributes and comparingobjects by using these attributes.

KINDERGARTENNumber and Operations: Representing, comparing and orderingwhole numbers, and joining and separating sets.Geometry: Describing shapes and space.Measurement: Ordering objects by measurable attributes.

GRADE 1Number and Operations and Algebra: Developing under-standings of addition and subtraction and strategies for basicaddition facts and related subtraction facts.Number and Operations: Developing an understanding ofwhole number relationships, including grouping in tens and ones.Geometry: Composing and decomposing geometric shapes.

GRADE 2Number and Operations: Developing an understanding of thebase-ten numeration system and place-value concepts.Number and Operations and Algebra: Developing quick recall of addition facts and related subtraction facts and fluencywith multidigit addition and subtraction.Measurement: Developing an understanding of linear mea-surement and facility in measuring lengths.

GRADE 3Number and Operations and Algebra: Developing under-standings of multiplication and division and strategies for basicmultiplication facts and related division facts.Number and Operations: Developing an understanding offractions and fraction equivalence.Geometry: Describing and analyzing properties of two-dimensional shapes.

GRADE 4Number and Operations and Algebra: Developing quick recall of multiplication facts and related division facts and fluency with whole number multiplication.

Curriculum Focal Pointsfor Prekindergarten through Grade 8 Mathematics

Number and Operations: Developing an understanding of decimals, including the connections between fractions and decimals.Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes.

GRADE 5Number and Operations and Algebra: Developing an under-standing of and fluency with division of whole numbers.Number and Operations: Developing an understanding of and fluency with addition and subtraction of fractions and decimals.Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, includingvolume and surface area.

GRADE 6Number and Operations: Developing an understanding of andfluency with multiplication and division of fractions and decimals.Number and Operations: Connecting ratio and rate to multi-plication and division.Algebra: Writing, interpreting, and using mathematical expres-sions and equations.

GRADE 7Number and Operations and Algebra and Geometry: Devel-oping an understanding of and applying proportionality, includ-ing similarity.Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areasand volumes of three-dimensional shapes.Number and Operations and Algebra: Developing an under-standing of operations on all rational numbers and solving linear equations.

GRADE 8Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations.Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle.Data Analysis and Number and Operations and Algebra:Analyzing and summarizing data sets.

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athematicsF o r E l e m e n t a r y Te a c h e r s

A CONTEMPORARY APPROACH

Gary L. MusserOregon State University

William F. Burger

Blake E. PetersonBrigham Young Univeristy

MathematicsME I G H T H E D I T I O N

John Wiley & Sons, Inc.

ffirs.qxd 11/15/07 12:53 PM Page iii

To:

Irene, my supportive wife of over 45 years; Greg, my son, for his continuing progressin life; Maranda, my granddaughter, for her enthusiasm and appreciation of her love;my parents, who have both passed away, but are always on my mind and in my heart;and Mary Burger, Bill Burgers wonderful daughter.

G.L.M.

Shauna, my eternal companion and best friend, for making me smile along thiswonderful journey called life; Quinn, Joelle, Taren, and Riley, my four children, forchoosing the right; Mark, Kent, and Miles, my brothers, for their examples and support.

B.E.P.PUBLISHER Laurie RosatoneACQUISITIONS EDITOR Jessica JacobsASSISTANT EDITOR Michael ShroffSENIOR PRODUCTION EDITOR Valerie A. VargasMARKETING MANAGER Jaclyn ElkinsCREATIVE DIRECTOR Harry NolanSENIOR DESIGNER Kevin MurphyPRODUCTION MANAGEMENT SERVICES mb editorial servicesSENIOR PHOTO EDITOR Lisa GeeEDITORIAL ASSISTANT Jeffrey BensonMEDIA EDITOR Stefanie LiebmanCOVER & TEXT DESIGN Michael JungCOVER IMAGE BY Miao Jin, Junho Kim, and Xianfeng David GuBICENTENNIAL LOGO DESIGN Richard J. Pacifico

This book was set in 10/12 Times New Roman by GGS Book Services and printed and bound byRRDJefferson City. The cover was printed by RRDJefferson City.

This book is printed on acid-free paper.

Copyright 2008 John Wiley & Sons, Inc. All rights reserved. No part of this publication may bereproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical,photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976United States Copyright Act, without either the prior written permission of the Publisher, or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222 RosewoodDrive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission shouldbe addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ07030-5774, (201) 748-6011, fax (201) 748-6008, website http://www.wiley.com/go/permissions.

To order books or for customer service please, call 1-800-CALL WILEY (225-5945).

ISBN-13 978-0470-10583-2

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About the Authors

v

GARY L. MUSSER is Professor Emeritus from Oregon State University. He earned bothhis B.S. in Mathematics Education in 1961 and his M.S. in Mathematics in 1963 at theUniversity of Michigan and his Ph.D. in Mathematics (Radical Theory) in 1970 at theUniversity of Miami in Florida. He taught at the junior and senior high, junior college,college, and university levels for more than 30 years. He served his last 24 years teachingprospective teachers in the Department of Mathematics at Oregon State University.While at OSU, Dr. Musser developed the mathematics component of the elementaryteacher program. Soon after Professor William F. Burger joined the OSU Departmentof Mathematics in a similar capacity, the two of them began to write the first edition 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. ProfessorPeterson 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, CanadianJournal of Mathematics, The Mathematics Association of America Monthly, the NCTMs The Mathematics Teacher, theNCTMs The Arithmetic Teacher, School Science and Mathematics, The Oregon Mathematics Teacher, and The ComputingTeacher. In addition, he is a coauthor of two other college mathematics books: College GeometryA Problem-SolvingApproach with Applications (2008) and A Mathematical View of Our World (2007). He also coauthored the K8 seriesMathematics in Action. He has given more than 65 invited lectures/workshops at a variety of conferences, includingNCTM 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 universitys prestigious College of Science Carter Awardfor Teaching. He is currently living in sunny Las Vegas, where he continues to write, ponder the mysteries of the stockmarket, and entertain both his wife and his faithful yellow lab, Zoey.

BLAKE E. PETERSON is currently a Professor in the Department of Mathematics Educationat Brigham Young University. He was born and raised in Logan, Utah, where he graduatedfrom Logan High School. Before completing his B.A. in secondary mathematics education at Utah State University, he spent two years in Japan as a missionary for TheChurch 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 Schoolfor two years. In 1988, he began graduate school at Washington State University,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 inthe Department of Mathematics at Oregon State University in Corvallis, Oregon,where he taught for three years. It was at OSU that he met Gary Musser. He has sincemoved his wife and four children to Provo, Utah, to assume his position at Brigham

Young University where he is currently a full professor. As a professor, his first love is teaching, for which he has received a College Teaching Award in the College of Science.

Dr. Peterson has published papers in Rocky Mountain Mathematics Journal, The American Mathematical Monthly, TheMathematical Gazette, Mathematics Magazine, The New England Mathematics Journal, and The Journal of MathematicsTeacher Education as well as NCTMs Mathematics Teacher, and Mathematics Teaching in the Middle School. Afterstudying mathematics student teachers at a Japanese junior high school, he implemented some elements he observedinto the student teaching structure at BYU. In addition to teaching, research, and writing, Dr. Peterson has done consultingfor the College Board, founded the Utah Association of Mathematics Teacher Educators, is on the editorial panel for theMathematics Teacher, and is the associate chair of the department of mathematics education at BYU.

Aside from his academic interests, Dr. Peterson enjoys spending time with his family, fulfilling his church responsibilities,playing basketball, mountain biking, water skiing, and working in the yard.

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About the Cover

vi

The checkered figure on the cover, which is called Costasminimal surface, was discovered in 1982 by Celso Costa. Itis studied in the field of mathematics called differentialgeometry. There are many different fields of mathematicsand in each field there are different tools used to solveproblems. Particularly difficult problems, however, may require reaching from one field of mathematics into an-other to find the tools to solve it. One such problem wasposed in 1904 by Henri Poincar and is in a branch ofmathematics called topology. The problem, stated as a con-jecture, is that the three-sphere is the only compact three-manifold which has the property that each simple closedcurve can be contracted. While this conjecture is likely notunderstandable to one not well versed in topology, thestory surrounding its eventual proof is quite interesting.

This conjecture, stated in three dimensions, had severalproof attempts in the early 1900s that were initiallythought to be true only to be proven false later. In 1960,Stephen Smale proved an equivalent conjecture for dimen-sions 5 and higher. For this he received the Fields medalfor it in 1966. The Fields medal is the equivalent of theNobel prize for mathematics and to receive it, the recipientmust be 40 years of age or younger. The medal is awardedevery 4 years to between two and four mathematicians. In1983, Michael Freedman proved the equivalent conjec-ture for the 4th dimension and he also received a FieldsMedal in 1986.

In 2003, Grigory Grisha Perelman of St. Petersburg,Russia, claimed to have proved the Poincar conjecture asstated for three-dimensions when he posted three short papers on the internet. These postings were followed by aseries of lectures in the United States discussing the papers. Typically, a proof like this would be carefully writtenand submitted to a prestigious journal for peer review. Thebrevity of these papers left the rest of the mathematicscommunity wondering if the proof was correct. As other

mathematicians have filled in the gaps, their resulting papers (3 in total) were about 1000 pages long of densemathematics. One of the creative aspects of Perelmansproof is the tools that he used. He reached beyond the fieldof topology into the field of differential geometry and useda tool called a Ricci flow.

In August of 2006, Dr. Perelman was awarded the Fieldsmedal along with three other mathematicians. However, hedid not attend the awards ceremony in Spain and declinedto accept the medal along with its $13,400 stipend. In the70-year history of the Fields medal, there have only been48 recipients of the award and none have refused it beforePerelman. His colleagues indicate that he is only interestedin knowledge and not in awards or money.

Such an attitude is even more remarkable when youconsider the $1,000,000 award that is also available forproving the Poincar conjecture. In 2000, the Clay Mathe-matics Institute in Cambridge, Massachusetts, identifiedseven historic, unsolved mathematics problems that theywould offer a $1 million prize for the proof of each. ThePoincar conjecture is one of those 7 historic unsolvedproblems. Each proof requires a verification period beforethe prize would be awarded. As of this writing, it is un-known if Dr. Perelman would accept the $1 million prize.

There is one other interesting twist to this story. Becauseof the brevity of Perelmans proofs, other mathematiciansfilled in some details. In particular, two Chinese mathemati-cians, Professors Cao and Zhu, wrote a paper on this subjectentitled A Complete Proof of the Poincar and Geometriza-tion ConjecturesApplication of the Hamilton-PerelmanTheory of Ricci Flow. This paper was 327 pages long!

But what about the $1,000,000? If you are interested,search the internet periodically to see if Perelman acceptsall or part of the $1,000,000.The image on the cover was created by Miao Jin, Junho Kim andXianfeng David Gu.

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Brief Contents

vii

1 Introduction to Problem Solving 12 Sets, Whole Numbers, and Numeration 433 Whole Numbers: Operations and Properties 1074 Whole Number Computation: Mental, Electronic, and Written 1555 Number Theory 2036 Fractions 2377 Decimals, Ratio, Proportion, and Percent 2858 Integers 3419 Rational Numbers, Real Numbers, and Algebra 379

10 Statistics 43911 Probability 51312 Geometric Shapes 58113 Measurement 66514 Geometry Using Triangle Congruence and Similarity 73915 Geometry Using Coordinates 80716 Geometry Using Transformations 849

Epilogue: An Eclectic Approach to Geometry 909Topic 1 Elementary Logic 912Topic 2 Clock Arithmetic: A Mathematical System 923Answers to Exercise/Problem Sets-Part A, Chapter Tests, and Topics A1Index I1Contents of Book Companion Web Site

Resources for Technology Problems

Technology Tutorials

Webmodules

Additional Resources

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Contents

viii

Preface xi

1 Introduction to Problem Solving 11.1 The Problem Solving Process and Strategies 31.2 Three Additional Strategies 20

2 Sets, Whole Numbers, and Numeration 432.1 Sets as a Basis for Whole Numbers 452.2 Whole Numbers and Numeration 592.3 The HinduArabic System 702.4 Relations and Functions 82

3 Whole Numbers: Operations and Properties 1073.1 Addition and Subtraction 1093.2 Multiplication and Division 1233.3 Ordering and Exponents 140

4 Whole Number ComputationMental, Electronic, and Written 1554.1 Mental Math, Estimation, and Calculators 1574.2 Written Algorithms for Whole-Number Operations 1714.3 Algorithms in Other Bases 192

5 Number Theory 2035.1 Primes, Composites, and Tests for Divisibility 2055.2 Counting Factors, Greatest Common Factor, and Least Common Multiple 219

6 Fractions 2376.1 The Sets of Fractions 2396.2 Fractions: Addition and Subtraction 2556.3 Fractions: Multiplication and Division 266

7 Decimals, Ratio, Proportion, and Percent 2857.1 Decimals 2877.2 Operations with Decimals 2977.3 Ratios and Proportion 3107.4 Percent 320

8 Integers 3418.1 Addition and Subtraction 3438.2 Multiplication, Division, and Order 357

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9 Rational Numbers, Real Numbers, and Algebra 3799.1 The Rational Numbers 3819.2 The Real Numbers 3999.3 Functions and Their Graphs 417

10 Statistics 43910.1 Organizing and Picturing Information 44110.2 Misleading Graphs and Statistics 46410.3 Analyzing Data 484

11 Probability 51311.1 Probability and Simple Experiments 51511.2 Probability and Complex Experiments 53211.3 Additional Counting Techniques 54911.4 Simulation, Expected Value, Odds, and Conditional Probability 560

12 Geometric Shapes 58112.1 Recognizing Geometric Shapes 58312.2 Analyzing Shapes 60012.3 Properties of Geometric Shapes: Lines and Angles 61512.4 Regular Polygons and Tessellations 62812.5 Describing Three-Dimensional Shapes 640

13 Measurement 66513.1 Measurement with Nonstandard and Standard Units 66713.2 Length and Area 68613.3 Surface Area 70713.4 Volume 717

14 Geometry Using Triangle Congruence and Similarity 73914.1 Congruence of Triangles 74114.2 Similarity of Triangles 75214.3 Basic Euclidean Constructions 76514.4 Additional Euclidean Constructions 77714.5 Geometric Problem Solving Using Triangle Congruence and Similarity 790

15 Geometry Using Coordinates 80715.1 Distance and Slope in the Coordinate Plane 80915.2 Equations and Coordinates 82215.3 Geometric Problem Solving Using Coordinates 834

16 Geometry Using Transformations 84916.1 Transformations 85116.2 Congruence and Similarity Using Transformations 87516.3 Geometric Problem Solving Using Transformations 893

Contents ix

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x Contents

Epilogue: An Eclectic Approach to Geometry 909

Topic 1. Elementary Logic 912

Topic 2. Clock Arithmetic: A Mathematical System 923

Answers to Exercise/Problem SetsPart A, Chapter Tests, and Topics A1

Photograph Credits P1

Index I1

Contents of Book Companion Web Site

Resources for Technology Problems eManipulatives Spreadsheets Geometers Sketchpad

Technology Tutorials Spreadsheets Geometers Sketchpad Programming in Logo Graphing Calculators

Webmodules Algebraic Reasoning Using Childrens Literature Introduction to Graph Theory Guide to Problem Solving

Additional Resources Research Articles Web Links

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Preface

xi

Welcome to the study of the foundations of elementary school mathematics. We hope youwill find your studies enlightening, useful, and fun. We salute you for choosing teaching asa profession and hope that your experiences with this book will help prepare you to be the

best possible teacher of mathematics that you can be. We have presented this elementary mathematicsmaterial from a variety of perspectives so that you will be better equipped to address the broad rangeof learning styles that you will encounter in your future students. This book also encourages prospec-tive 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 understandingof mathematics.

We have also sought to present this material in a manner consistent with the recommendations in(1) The Mathematical Education of Teachers prepared by the Conference Board of the MathematicalSciences; and (2) the National Council of Teachers of Mathematics Principles and Standards forSchool Mathematics, and Curriculum Focal Points. In addition, we have received valuable advicefrom many of our colleagues around the United States through questionnaires, reviews, focusgroups, and personal communications. We have taken great care to respect this advice and to ensurethat the content of the book has mathematical integrity and is accessible and helpful to the variety ofstudents who will use it. As always, we look forward to hearing from you about your experienceswith our text.

GARY L. MUSSER, [email protected] E. PETERSON, [email protected]

Unique Content FeaturesNumber Systems The order in which we present the number systems in this book is unique andmost relevant to elementary school teachers. The topics are covered to parallel their evolution histori-cally and their development in the elementary/middle school curriculum. Fractions and integers aretreated separately as an extension of the whole numbers. Then rational numbers can be treated at abrisk pace as extensions of both fractions (by adjoining their opposites) and integers (by adjoiningtheir appropriate quotients) since students have a mastery of the concepts of reciprocals from fractions(and quotients) and opposites from integers from preceding chapters. Longtime users of this bookhave commented to us that this whole numbers-fractions-integers-rationals-reals approach is clearlysuperior to the seemingly more efficient sequence of whole numbers-integers-rationals-reals that ismore appropriate to use when teaching high school mathematics.

Approach to Geometry Geometry is organized from the point of view of the five-level vanHiele model of a childs development in geometry. After studying shapes and measurement, geometryis approached more formally through Euclidean congruence and similarity, coordinates, and transfor-mations. The Epilogue provides an eclectic approach by solving geometry problems using a variety oftechniques.

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xii Preface

Additional Topics Topic 1, Elementary Logic, may be used anywhere in a course. Topic 2, Clock Arithmetic: A Mathematical System, uses the concepts of opposite and recip-

rocal and hence may be most instructive after Chapter 6, Fractions, and Chapter 8, Inte-gers, have been completed. This section also contains an introduction to modular arithmetic.

Underlying ThemesProblem Solving An extensive collection of problem-solving strategies is developed throughoutthe book; these strategies can be applied to a generous supply of problems in the exercise/problemsets. The depth of problem-solving coverage can be varied by the number of strategies selectedthroughout the book and by the problems assigned.

Deductive Reasoning The use of deduction is promoted throughout the book. The approach isgradual, with later chapters having more multistep problems. In particular, the last sections of Chapters14, 15, and 16 and the Epilogue offer a rich source of interesting theorems and problems in geometry.

Technology Various forms of technology are an integral part of society and can enrich the mathe-matical understanding of students when used appropriately. Thus, calculators and their capabilities(long division with remainders, fraction calculations, and more) are introduced throughout the bookwithin the body of the text.

In addition, the book companion Web site has eManipulatives, spreadsheets, and sketches fromGeometers Sketchpad. The eManipulatives are electronic versions of the manipulatives commonly used in the elementary classroom, such as the geoboard, base ten blocks, black and redchips, and pattern blocks. The spreadsheets contain dynamic representations of functions, statistics,and probability simulations. The sketches in Geometers Sketchpad are dynamic representationsof geometric relationships that allow exploration. Exercises and problems that involve eManipulatives,spreadsheets, and Geometers Sketchpad sketches have been integrated into the problem setsthroughout the text.

Course OptionsWe recognize that the structure of the mathematics for elementary teachers course will vary depend-ing upon the college or university. Thus, we have organized this text so that it may be adapted to ac-commodate these differences.

Basic course: Chapters 17Basic course with logic: Topic 1, Chapters 17Basic course with informal geometry: Chapters 17, 12.Basic course with introduction to geometry and measurement: Chapters 17, 12, 13

Summary of Changes to the Eighth Edition Exercise sets have been revised and enriched, where necessary, to assure that they are

closely aligned with and provide complete coverage of the section material. In addition, the exercises are in matched pairs between Part A and Part B.

New problems have been added and Problems for Writing/Discussion at the end of the sections in the Seventh Edition have been appended to the end of the problem sets.

All Spotlights in Technology that were in Seventh Edition, other than those involving calculators, have been converted to exercises or problems.

fpref.qxd 11/15/07 1:07 PM Page xii

Sections 10.2 and 10.3 have been interchanged. NCTMs Curriculum Focal Points are listed at the beginning of the book and cited in each

chapter introduction.

A set of problems based on the NCTM Standards and Focal Points has been added to the endof each section.

Several changes have been made in the body of the text throughout the book based on rec-ommendations of our reviewers.

New Mathematical Morsels have been added where appropriate. The Table of Contents now includes a listing of resources on the Web site. Topic 3, Introduction to Graph Theory, has been moved to our Web site. Complete reference lists for both Reflections from Research and Childrens Literature are lo-

cated in the Web site.

PedagogyThe general organization of the book was motivated by the following mathematics learning cube:

Preface xiii

Goal ofMathematicsInstruction

Measurement

Number systems Geometry

AbstractPictorial

Concrete

Knowledge

Skill

Understanding

Problem solving

Applications

START

The three dimensions of the cubecognitive levels, representational levels, and mathematical con-tentare integrated throughout the textual material as well as in the problem sets and chapter tests.Problem sets are organized into exercises (to support knowledge, skill, and understanding) and prob-lems (to support problems solving and applications).

We have developed new pedagogical features to implement and reinforce the goals discussed aboveand to address the many challenges in the course.

Summary of Pedagogical Changes to the Eighth Edition Student Page Snapshots have been updated. Reflections from Research have been edited and updated. Childrens Literature references have been edited and updated. Also, there is additional mate-

rial offered on the Web site on this topic.

fpref.qxd 11/15/07 1:07 PM Page xiii

Ones point of view or interpretation of a problem can often change a seem-ingly difficult problem into one that is easily solvable. One way to solve thenext problem is by drawing a picture or, perhaps, by actually finding somerepresentative blocks to try various combinations. On the other hand, an-other approach is to see whether the problem can be restated in an equivalentform, say, using numbers. Then if the equivalent problem can be solved, thesolution can be interpreted to yield an answer to the original problem.

STRATEGY 11Solve an Equivalent Problem

Problem-Solving Strategies1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler Problem

7. Draw a Diagram

8. Use Direct Reasoning

9. Use Indirect Reasoning

10. Use Properties ofNumbers

11. Solve an EquivalentProblem

INITIAL PROBLEM

CLUES

A child has a set of 10 cubical blocks. The lengths of the edges are 1 cm, 2cm, 3 cm, . . . , 10 cm. Using all the cubes, can the child build two towers ofthe same height by stacking one cube upon another? Why or why not?

The Solve an Equivalent Problem strategy may be appropriate when

You can find an equivalent problem that is easier to solve. A problem is related to another problem you have solved previously. A problem can be represented in a more familiar setting. A geometric problem can be represented algebraically, or vice versa. Physical problems can easily be represented with numbers or symbols.A solution of this Initial Problem is on page 281.

xiv Preface

Key FeaturesProblem-Solving Strategies are integrated through-out the book. Six strategies are introduced in Chapter 1. Thelast strategy in the strategy box at the top of the second pageof each chapter after Chapter One contains a new strategy.

Addition of Fractions with Common Denominators

Let and be any fractions. Then

a

b+

c

b=

a + cb

.

c

b

a

b

D E F I N I T I O ND E F I N I T I O N

Addition of Fractions with Unlike Denominators


a

b+

c

d=

ad + bcbd

.

c

d

a

b

T H E O R E MT H E O R E M

Commutative Property for Fraction Addition


a

b+

c

b=

c

b+

a

b.

c

b

a

b

P R O P E R T YP R O P E R T Y

Following recess, the 1000 students of Wilson School lined up for the followingactivity: The first student opened all of the 1000 lockers in the school. The sec-ond student closed all lockers with even numbers. The third student changedall lockers that were numbered with multiples of 3 by closing those that wereopen and opening those that were closed. The fourth student changed eachlocker whose number was a multiple of 4, and so on. After all 1000 studentshad entered completed the activity, which lockers were open? Why?

STARTING POINT

Mathematical Structure reveals the mathematicalideas of the book. Main Definitions, Theorems, and Properties in each section are highlighted in boxes forquick review.

Starting Points are located at the beginning of each section. These Starting Points can be used in a variety ofways. First, they can be used by an instructor at the beginningof class to have students engage in some novel thinkingand/or discussion about forthcoming material. Second, theycan be used in small groups where students discuss thequery presented. Third, they can be used as an advanced organizer homework piece where a class begins with a discussion of what individual students have discovered.

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Technology Problems appear in the Exercise/Problem sets through the book.These problems rely on and are enriched by the use of technology. The technologyused includes activities from the eManipulatives (virtual manipulatives), spread-sheets, Geometers Sketchpad, and the TI-34 II calculator. Most of these technologicalresources can be accessed through the accompanying book companion Web site.

Preface xv

S T U D E N T PA G E S N A P S H O T

From Harcourt Mathematics, Level 5, p. 500. Copyright 2004 by Harcourt.

Key Concepts from NCTM Curriculum Focal Points

GRADE 1: Developing an understanding of whole number relationships, includ-ing grouping in tens and ones.

GRADE 2: Developing quick recall of addition facts and related subtraction factsand fluency with multidigit addition and subtraction.

8. Using the Chapter 6 eManipulative activity DividingFractions on our Web site, construct representations of thefollowing division problems. Sketch each representation.a. b. c. 214 ,

58

34 ,

23

74 ,

12

Student Page Snapshots have been updated. Eachchapter has a page from an elementary school textbookrelevant 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 areonly provided in the Instructors Resource Manual). In addition, exercises and problems are distinguished so thatstudents can learn how they differ.

Problems for Writing/Discussion have been integrated into the problem sets throughout the book andare designated by a writing icon. They are also includedas part of the chapter review.

NCTM Standards and Curriculum FocalPoints In previous editions the NCTM Standards thathave been listed at the beginning of the book and thenhighlighted in margin notes throughout the book. Theeighth edition also lists the Curriculum Focal Points fromNCTM at the beginning of the book. At the beginning ofeach chapter, the Curriculum Focal Points that are rele-vant to that particular chapter are listed again.

8. The Chapter 11 dynamic spreadsheet Roll the Dice on our Website simulates the rolling of 2 dice and computing the sum. Usethis spreadsheet to simulate rolling a pair of dice 100 times andfind the experimental probability for the events in parts ae.a. The sum is even.b. The sum is not 10.c. The sum is a prime.d. The sum is less than 9.e. The sum is not less than 9.f. Repeat parts ae with 500 rolls.

17. The Chapter 12 Geometers Sketchpad activity Name That Quadrilateral on our Web site displays seven different quadrilaterals in the shape of a square.However, each quadrilateral is constructed with different properties. Some have right angles, some have congruent sides, and some have parallel sides. By dragging each of the points on each of thequadrilaterals, you can determine the most general name of each quadrilateral. Name all seven of thequadrilaterals.

22. The fraction is simplified on a fraction calculator and the

result is Explain how this result can be used to find the

GCF(12, 18). Use this method to find the following.a. GCF(72, 168)b. GCF(234, 442)

23.

1218

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Problems from the NCTM Standards and Curriculum Focal Points To further help studentsunderstand and be aware of these documents from theNational Council of Teachers of Mathematics, new prob-lems have been added at the end of every section. Theseproblems ask students to connect the mathematics beinglearned from the book with the K8 mathematics outlinedby NCTM.

xvi Preface

Problems Relating to the NCTM Standards and Curriculum Focal Points

1. The Focal Points for Grade 3 state Developing anunderstanding of and fluency with addition and subtraction offractions and decimals. Based on the discussions in thissection, explain at least one main concept essential tounderstanding addition and subtraction of fractions.

2. The NCTM Standards state All students use visual models,benchmarks, and equivalent forms to add and subtractcommonly used fractions and decimals. Explain what is

meant by visual models when adding and subtractingfractions.

3. The NCTM Standards state All students should develop anduse strategies to estimate computations involving fractionsand decimals in situations relevant to students experience.List and explain some examples of strategies to estimatefraction computations.

Famous Unsolved Problems

Number theory provides a rich source of intriguingproblems. Interestingly, many problems in numbertheory are easily understood, but still have never

been solved. Most of these problems are statements orconjectures that have never been proven right or wrong.The most famous unsolved problem, known as FermatsLast Theorem, is named after Pierre de Fermat who is pic-tured below. It states There are no nonzero whole num-bers a, b, c, where an bn cn, for n a whole numbergreater than two.

The following list contains several such problems thatare still unsolved. If you can solve any of them, you willsurely become famous, at least among mathematicians.

1. Goldbachs conjecture. Every even number greaterthan 4 can be expressed as the sum of two odd primes.For example, 6 3 3, 8 3 5, 10 5 5, 12 5 7, and so on. It is interesting to note that if Gold-bachs conjecture is true, then every odd number greaterthan 7 can be written as the sum of three odd primes.

2. Twin prime conjecture. There is an infinite number ofpairs of primes whose difference is two. For example,(3, 5), (5, 7), and (11, 13) are such prime pairs. Noticethat 3, 5, and 7 are three prime numbers where 5 32 and 7 5 2. It can easily be shown that this is theonly such triple of primes.

3. Odd perfect number conjecture. There is no odd per-fect number; that is, there is no odd number that is thesum of its proper factors. For example, 6 1 2 3;hence 6 is a perfect number. It has been shown that theeven perfect numbers are all of the form 2 p 1 (2p 1),where 2 p 1 is a prime.

4. Ulams conjecture. If a nonzero whole number is even,divide it by 2. If a nonzero whole number is odd, multi-ply it by 3 and add 1. If this process is applied repeat-edly to each answer, eventually you will arrive at 1. Forexample, the number 7 yields this sequence of numbers:7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.Interestingly, there is a whole number less than 30 thatrequires at least 100 steps before it arrives at 1. It can beseen that 2n requires n steps to arrive at 1. Hence onecan find numbers with as many steps (finitely many) asone wishes.

FOCUS ON

Fermat left a note in the margin of a book saying that he didnot have room to write up a proof of what is now called Fer-mats Last Theorem. However, it remained an unsolvedproblem for over 350 years because mathematicians wereunable to prove it. In 1993, Andrew Wiles, an English math-ematician on the Princeton faculty, presented a proof at aconference at Cambridge University. However, there was ahole in his proof. Happily, Wiles and Richard Taylor pro-duced a valid proof in 1995, which followed from workdone by Serre, Mazur, and Ribet beginning in 1985.

The University of Oregon football team has developed quite awardrobe. Most football teams have two different uniforms: onefor home games and one for away games. The University of Oregonteam will have as many as 384 different uniform combinationsfrom which to choose. Rather than the usual light and dark jerseys, they have 4 different colored jerseys: white, yellow,green, and black. Beyond that, however, they have 4 different colored pants, 4 different colored pairs of socks, 2 different colored pairs of shoes and 2 different colored helmets with a 3rdone on the way. If all color combinations are allowed, the fundamental counting principle would suggest that they have 4 4 4 2 3 384 possible uniform combinations. Whethera uniform consisting of a green helmet, black jersey, yellow pants,white socks and black shoes would look stylish is debatable.

MATHEMATICALMORSELMATHEMATICALMORSEL

Reflection from ResearchGiven the proper experiences,children as young as eight andnine years of age can learn tocomfortably use letters torepresent unknown values andcan operate on representationsinvolving letters and numberswhile fully realizing that theydid not know the values of the unknowns (Carraher,Schliemann, Brizuela, &Earnest, 2006).

Reflections from Research Extensive research has been done in the mathematicseducation community that focuses on the teaching and learning of elementary mathe-matics. Many important quotations from research are given in the margins to supportthe content nearby.

Historical vignettes open each chapter and introduceideas and concepts central to each chapter.

Mathematical Morsels end every section with an interesting historical tidbit. One of our students referred tothese as a reward for completing the section.

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People in Mathematics, a feature near the end ofeach chapter, highlights many of the giants in mathemat-ics throughout history.

Preface xvii

John Von Neumann(19031957)John von Neumann was one ofthe most remarkable mathemati-cians of the twentieth century.His logical power was legendary.It is said that during and afterWorld War II the U.S. govern-

ment reached many scientific decisions simply by askingvon Neumann for his opinion. Paul Halmos, his one-timeassistant, said, The most spectacular thing about Johnnywas not his power as a mathematician, which was great,but his rapidity; he was very, very fast. And like themodern computer, which doesnt memorize logarithms,but computes them, Johnny didnt bother to memorizethings. He computed them. Appropriately, von Neu-

Julia Bowman Robinson(19191985)Julia Bowman Robinson spenther early years in Arizona, nearPhoenix. She said that one of herearliest memories was of arrang-ing pebbles in the shadow of agiant saguaroIve always had

a basic liking for the natural numbers. In 1948, Robin-son earned her doctorate in mathematics at Berkeley; shewent on to contribute to the solution of Hilberts tenthproblem. In 1975 she became the first woman mathe-matician elected to the prestigious National Academy ofSciences. Robinson also served as president of theAmerican Mathematical Society, the main professionalorganization for research mathematicians. Rather than

People in Mathematics

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 StudentsStudent Activity Manual This activity manual is designed to enhance student learning as well asto model effective classroom practices. Since many instructors are working with students to create apersonalized journal, this edition of the manual is shrink-wrapped and three-hole punched for easycustomization. This supplement is an extensive revision of the Student Resource Handbook that wasauthored by Karen Swenson and Marcia Swanson for the first six editions of this book.

ISBN 978-0470-10584-9

FEATURES INCLUDE Hands-On Activities: Activities that help develop initial understandings at the concrete level. Exercises: Additional practice for building skills in concepts. Connections to the Classroom: Classroom-like questions to provoke original thought. Mental Math: Short activities to help develop mental math skills. 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.

Solutions: Solutions to all items in the handbook to enhance self-study. Two-Dimensional Manipulatives: Cutouts are provided on cardstock

Prepared by Lyn Riverstone of Oregon State University

The ETA Cuisenaire 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 theStudent Activity Manual. It is available to be packaged with the text. Please contact your local Wileyrepresentative for ordering information.

ISBN 978-0470-13552-5

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State Correlation Guidebooks In an attempt to help preservice teachers prepare for state licensing exams and to inform their future teaching, Wiley has updated seven completely uniquestate-specific correlation guidebooks. These 35-page pamphlets provide a detailed correlation between the textbook and key supplements with state standards for the following states: CA, FL, IL,MI, NY, TX, VA. Each guidebook may be packaged with the text. Please contact your local Wileyrepresentative for further information.

Prepared by Chris Awalt of the Princeton Review[CA: 9-780-470-23172-2; FL: 9-780-470-23173-9; IL: 9-780-470-23171-5; MI: 9-780-470-23174-6;

NY: 9-780-470-23175-3; TX: 9-780-470-23176-0; VA: 9-780-470-23177-7]

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

Hints: Gives students a start on all Part A problems in the text. Additional Hints: A second hint is provided for more challenging problems. 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 978-0470-10585-6

Companion Web site http://www.wiley.com/college/musserThe companion Web site provides a wealth of resources for students.

Resources for Technology ProblemsThese problems are integrated into the problem sets throughout the book and are denoted by a mouseicon.

eManipulatives mirror physical manipulatives as well as provide dynamic representations ofother mathematical situations. The goal of using the eManipulatives is to engage learners in away that will lead to a more in-depth understanding of the concepts and to give them experi-ence 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.ISBN 978-0470-13551-8

The Geometers Sketchpad activities allow students to use the dynamic capabilities of thissoftware to investigate geometric properties and relationships. They are accessible through aWeb browser so having the software is not necessary.

The Spreadsheet activities utilize the iterative properties of spreadsheets and the user-friendly interface to investigate problems ranging from graphs of functions to standard devi-ation to simulations of rolling dice.

Tutorials The Geometers 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 tutorialgives basic instruction on how to use the software and includes some sample problems that

xviii Preface

fpref.qxd 11/15/07 1:07 PM Page xviii

will help the students gain a better understanding of the software and the geometry that couldbe learned by using it.

Prepared by Armando Martinez-Cruz, California State University, Fullerton.

The Spreadsheet Tutorial is written for students who are interested in learning how to usespreadsheets to investigate mathematical problems. The tutorial describes some of the func-tions of the software and provides exercises for students to investigate mathematics using thesoftware.

Prepared by Keith Leatham, Brigham Young University.

Webmodules

The Algebraic Reasoning Webmodule helps students understand the critical transition fromarithmetic to algebra. It also highlights situations when algebra is, or can be, used. Marginalnotes are placed in the text at the appropriate locations to direct students to the webmodule.

Prepared by Keith Leatham, Brigham Young University.

The Childrens Literature Webmodule provides references to many mathematically relatedexamples of childrens books for each chapter. These references are noted in the marginsnear the mathematics that corresponds to the content of the book. The webmodule also con-tains ideas about using childrens literature in the classroom.

Prepared by Joan Cohen Jones, Eastern Michigan University.

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 todownload it from the Web if they choose to use it.

The companion Web site also includes:

Links to NCTM Standards A Logo and TI-83 graphing calculator tutorial Four cumulative tests covering material up to the end of Chapters 4, 9, 12, and 16. Research Article References: A complete list of references for the research articles that are

mentioned in the Reflections from Research margin notes throughout the book.

Guide to Problem Solving This valuable resource, available as a webmodule on the companionWeb site, contains more than 200 creative problems keyed to the problem solving strategies in thetextbook and includes:

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

Practice Problems: A second problem that uses the same strategy together with a worked-out solution and two practice problems.

Mixed Strategy Practice: Four practice problems that can be solved using one or more ofthe strategies introduced to that point.

Preface xix

fpref.qxd 11/15/07 1:07 PM Page xix

Additional Practice Problems and Additional Mixed Strategy Problems: Sections thatprovide more practice for particular strategies as well as many problems for which studentsneed to identify appropriate strategies.

Prepared by Don Miller, who retired as a professor of mathematics at St. Cloud State University.

The Geometers Sketchpad Developed by Key Curriculum Press, this dynamic geometryconstruction and exploration tool allows users to create and manipulate precise figures while preserv-ing geometric relationships. This software is only available when packaged with the text. Please con-tact your local Wiley representative for further details.

WileyPLUS WileyPLUS is a powerful online tool that will help you study more effectively, get im-mediate feedback when you practice on your own, complete assignments and get help with problemsolving, and keep track of how youre doingall at one easy-to-use Web site.

Resources for the InstructorCompanion Web SiteThe companion Web site is available to text adopters and provides a wealth of resources including:

PowerPoint Slides of more than 190 images that include figures from the text and severalgeneric masters for dot paper, grids, and other formats.

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 textmaterial, student expectations (objectives) for each chapter, answers for all Part B exercises andproblems, and answers for all of the even-numbered problems in the Guide to Problem-Solving.

Prepared by Lyn Riverstone, Oregon State UniversityISBN 978-0470-23302-3

NEW! Computerized/Print Test Bank The Computerized/Printed Test Bank includes a collec-tion 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 CollegeComputerized Test Bank ISBN 978-0470-29296-9

Printed Test Bank ISBN 978-0470-29295-2

WileyPLUS WileyPLUS is a powerful online tool that provides instructors with an integrated suiteof resources, including an online version of the text, in one easy-to-use Web site. Organized aroundthe 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 visithttp://edugen.wiley.com or contact your local Wiley representative for a demonstration and further details.

xx Preface

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Acknowledgments

xxi

During the development of Mathematics for ElementaryTeachers, Eighth Edition, we benefited from comments, suggestions, and evaluations from many of our colleagues. Wewould like to acknowledge the contributions made by the following people:

Reviewers for the Eighth Edition

Seth Armstrong, Southern Utah UniversityElayne Bowman, University of OklahomaAnne Brown, Indiana University, South BendDavid C. Buck, ElizabethtownAlison Carter, Montgomery CollegeJanet Cater, California State University, BakersfieldDarwyn Cook, Alfred University Christopher Danielson, Minnesota State University

MankatoLinda DeGuire, California State University, Long BeachCristina Domokos, California State University,

SacramentoScott Fallstrom, University of OregonTeresa Floyd, Mississippi CollegeRohitha Goonatilake, Texas A&M International UniversityMargaret Gruenwald, University of Southern IndianaJoan Cohen Jones, Eastern Michigan University Joe Kemble, Lamar UniversityMargaret Kinzel, Boise State UniversityJ. Lyn Miller, Slippery Rock UniversityGirija Nair-Hart, Ohio State University, NewarkSandra Nite, Texas A&M UniversitySally Robinson, University of Arkansas, Little RockNancy Schoolcraft, Indiana University, BloomingtonKaren E. Spike, University of North Carolina,

WilmingtonBrian Travers, Salem StateMary Wiest, Minnesota State University, MankatoMark A. Zuiker, Minnesota State University,

Mankato

Student Activity Manual Reviewers

Kathleen Almy, Rock Valley CollegeMargaret Gruenwald, University of Southern IndianaKate Riley, California Polytechnic State UniversityRobyn Sibley, Montgomery County Public Schools

State Standards Reviewers

Joanne C. Basta, Niagara UniversityJoyce Bishop, Eastern Illinois UniversityTom Fox, University of Houston, Clear LakeJoan C. Jones, Eastern Michigan UniversityKate Riley, California Polytechnic State UniversityJanine Scott, Sam Houston State UniversityMurray Siegel, Sam Houston State UniversityRebecca Wong, West Valley College

In addition, we would like to acknowledge the contributionsmade by colleagues from earlier editions.

Reviewers

Paul Ache, Kutztown UniversityScott Barnett, Henry Ford Community CollegeChuck Beals, Hartnell CollegePeter Braunfeld, University of IllinoisTom Briske, Georgia State UniversityAnne Brown, Indiana University, South BendChristine Browning, Western Michigan UniversityTommy Bryan, Baylor UniversityLucille Bullock, University of TexasThomas Butts, University of Texas, DallasDana S. Craig, University of Central OklahomaAnn Dinkheller, Xavier UniversityJohn Dossey, Illinois State UniversityCarol Dyas, University of Texas, San AntonioDonna Erwin, Salt Lake Community CollegeSheryl Ettlich, Southern Oregon State CollegeRuhama Even, Michigan State UniversityIris B. Fetta, Clemson UniversityMajorie Fitting, San Jose State UniversitySusan Friel, Math/Science Education Network, University of

North CarolinaGerald Gannon, California State University, FullertonJoyce Rodgers Griffin, Auburn UniversityJerrold W. Grossman, Oakland UniversityVirginia Ellen Hanks, Western Kentucky UniversityJohn G. Harvey, University of Wisconsin, MadisonPatricia L. Hayes, Utah State University, Uintah Basin Branch

CampusAlan Hoffer, University of California, IrvineBarnabas Hughes, California State University, NorthridgeJoan Cohen Jones, Eastern Michigan UniversityMarilyn L. Keir, University of UtahJoe Kennedy, Miami UniversityDottie King, Indiana State UniversityRichard Kinson, University of South AlabamaMargaret Kinzel, Boise State UniversityJohn Koker, University of WisconsinDavid E. Koslakiewicz, University of Wisconsin, MilwaukeeRaimundo M. Kovac, Rhode Island CollegeJosephine Lane, Eastern Kentucky UniversityLouise Lataille, Springfield CollegeRoberts S. Matulis, Millersville UniversityMercedes McGowen, Harper CollegeFlora Alice Metz, Jackson State Community CollegeJ. Lyn Miller, Slippery Rock UniversityBarbara Moses, Bowling Green State UniversityMaura Murray, University of MassachusettsKathy Nickell, College of DuPageDennis Parker, The University of the PacificWilliam Regonini, California State University, Fresno

flast.qxd 11/15/07 1:11 PM Page xxi

James Riley, Western Michigan UniversityKate Riley, California Polytechnic State UniversityEric Rowley, Utah State UniversityPeggy Sacher, University of DelawareJanine Scott, Sam Houston State UniversityLawrence Small, L.A. Pierce CollegeJoe K. Smith, Northern Kentucky UniversityJ. Phillip Smith, Southern Connecticut State UniversityJudy Sowder, San Diego State UniversityLarry Sowder, San Diego State UniversityKaren Spike, University of Northern Carolina, WilmingtonDebra S. Stokes, East Carolina UniversityJo Temple, Texas Tech UniversityLynn Trimpe, LinnBenton Community CollegeJeannine G. Vigerust, New Mexico State UniversityBruce Vogeli, Columbia UniversityKenneth C. Washinger, Shippensburg UniversityBrad Whitaker, Point Loma Nazarene UniversityJohn Wilkins, California State University, Dominguez Hills

Questionnaire Respondents

Mary Alter, University of MarylandDr. J. Altinger, Youngstown State UniversityJamie Whitehead Ashby, Texarkana CollegeDr. Donald Balka, Saint Marys CollegeJim Ballard, Montana State UniversityJane Baldwin, Capital UniversitySusan Baniak, Otterbein CollegeJames Barnard, Western Oregon State CollegeChuck Beals, Hartnell CollegeJudy Bergman, University of Houston, ClearlakeJames Bierden, Rhode Island CollegeNeil K. Bishop, The University of Southern Mississippi Gulf

CoastJonathan Bodrero, Snow CollegeDianne Bolen, Northeast Mississippi Community CollegePeter Braunfeld, University of IllinoisHarold Brockman, Capital UniversityJudith Brower, North Idaho CollegeAnne E. Brown, Indiana University, South BendHarmon Brown, Harding UniversityChristine Browning, Western Michigan UniversityJoyce W. Bryant, St. Martins CollegeR. Elaine Carbone, Clarion UniversityRandall Charles, San Jose State UniversityDeann Christianson, University of the PacificLynn Cleary, University of MarylandJudith Colburn, Lindenwood CollegeSister Marie Condon, Xavier UniversityLynda Cones, Rend Lake CollegeSister Judith Costello, Regis CollegeH. Coulson, California State UniversityDana S. Craig, University of Central OklahomaGreg Crow, John Carroll UniversityHenry A. Culbreth, Southern Arkansas University, El Dorado

Carl Cuneo, Essex Community CollegeCynthia Davis, Truckee Meadows Community CollegeGregory Davis, University of Wisconsin, Green BayJennifer Davis, Ulster County Community CollegeDennis De Jong, Dordt CollegeMary De Young, Hop CollegeLouise Deaton, Johnson Community CollegeShobha Deshmukh, College of Saint Benedict/St. Johns

UniversitySheila Doran, Xavier UniversityRandall L. Drum, Texas A&M UniversityP. R. Dwarka, Howard UniversityDoris Edwards, Northern State CollegeRoger Engle, Clarion UniversityKathy Ernie, University of WisconsinRon Falkenstein, Mott Community CollegeAnn Farrell, Wright State UniversityFrancis Fennell, Western Maryland CollegeJoseph Ferrar, Ohio State UniversityChris Ferris, University of AkronFay Fester, The Pennsylvania State UniversityMarie Franzosa, Oregon State UniversityMargaret Friar, Grand Valley State CollegeCathey Funk, Valencia Community CollegeDr. Amy Gaskins, Northwest Missouri State UniversityJudy Gibbs, West Virginia UniversityDaniel Green, Olivet Nazarene UniversityAnna Mae Greiner, Eisenhower Middle SchoolJulie Guelich, Normandale Community CollegeGinny Hamilton, Shawnee State UniversityVirginia Hanks, Western Kentucky UniversityDave Hansmire, College of the MainlandBrother Joseph Harris, C.S.C., St. Edwards UniversityJohn Harvey, University of WisconsinKathy E. Hays, Anne Arundel Community CollegePatricia Henry, Weber State CollegeDr. Noal Herbertson, California State UniversityIna Lee Herer, Tri-State UniversityLinda Hill, Idaho State UniversityScott H. Hochwald, University of North FloridaSusan S. Hollar, Kalamazoo Valley Community CollegeHolly M. Hoover, Montana State University, BillingsWei-Shen Hsia, University of AlabamaSandra Hsieh, Pasadena City CollegeJo Johnson, Southwestern CollegePatricia Johnson, Ohio State UniversityPat Jones, Methodist CollegeJudy Kasabian, El Camino CollegeVincent Kayes, Mt. St. Mary CollegeJulie Keener, Central Oregon Community CollegeJoe Kennedy, Miami UniversitySusan Key, Meridien Community CollegeMary Kilbridge, Augustana CollegeMike Kilgallen, Lincoln Christian CollegeJudith Koenig, California State University, Dominguez HillsJosephine Lane, Eastern Kentucky University

xxii Acknowledgments

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Acknowledgments xxiii

Don Larsen, Buena Vista CollegeLouise Lataille, Westfield State CollegeVernon Leitch, St. Cloud State UniversitySteven C. Leth, University of Northern ColoradoLawrence Levy, University of WisconsinRobert Lewis, Linn-Benton Community CollegeLois Linnan, Clarion UniversityJack Lombard, Harold Washington CollegeBetty Long, Appalachian State UniversityAnn Louis, College of the CanyonsC. A. Lubinski, Illinois State UniversityPamela Lundin, Lakeland CollegeCharles R. Luttrell, Frederick Community CollegeCarl Maneri, Wright State UniversityNancy Maushak, William Penn CollegeEdith Maxwell, West Georgia CollegeJeffery T. McLean, University of St. ThomasGeorge F. Mead, McNeese State UniversityWilbur Mellema, San Jose City CollegeDiane Miller, Middle Tennessee State UniversityClarence E. Miller, Jr. Johns Hopkins UniversityKen Monks, University of ScrantonBill Moody, University of DelawareKent Morris, Cameron UniversityLisa Morrison, Western Michigan UniversityBarbara Moses, Bowling Green State UniversityFran Moss, Nicholls State UniversityMike Mourer, Johnston Community CollegeKatherine Muhs, St. Norbert CollegeGale Nash, Western State College of ColoradoT. Neelor, California State UniversityJerry Neft, University of DaytonGary Nelson, Central Community College, Columbus CampusJames A. Nickel, University of Texas, Permian BasinKathy Nickell, College of DuPageSusan Novelli, Kellogg Community CollegeJon ODell, Richland Community CollegeJane Odell, Richland CollegeBill W. Oldham, Harding UniversityJim Paige, Wayne State CollegeWing Park, College of Lake CountySusan Patterson, Erskine College (retired)Shahla Peterman, University of MissouriGary D. Peterson, Pacific Lutheran UniversityDebra Pharo, Northwestern Michigan CollegeTammy Powell-Kopilak, Dutchess Community CollegeChristy Preis, Arkansas State University, Mountain HomeRobert Preller, Illinois Central CollegeDr. William Price, Niagara UniversityKim Prichard, University of North CarolinaStephen Prothero, Williamette UniversityJanice Rech, University of NebraskaTom Richard, Bemidji State UniversityJan Rizzuti, Central Washington UniversityAnne D. Roberts, University of UtahDavid Roland, University of Mary HardinBaylor

Frances Rosamond, National UniversityRichard Ross, Southeast Community CollegeAlbert Roy, Bristol Community CollegeBill Rudolph, Iowa State UniversityBernadette Russell, Plymouth State CollegeLee K. Sanders, Miami University, HamiltonAnn Savonen, Monroe County Community CollegeRebecca Seaberg, Bethel CollegeKaren Sharp, Mott Community CollegeMarie Sheckels, Mary Washington CollegeMelissa Shepard Loe, University of St. ThomasJoseph Shields, St. Marys College, MNLawrence Shirley, Towson State UniversityKeith Shuert, Oakland Community CollegeB. Signer, St. Johns UniversityRick Simon, Idaho State UniversityJames Smart, San Jose State UniversityRon Smit, University of PortlandGayle Smith, Lane Community CollegeLarry Sowder, San Diego State UniversityRaymond E. Spaulding, Radford UniversityWilliam Speer, University of Nevada, Las VegasSister Carol Speigel, BVM, Clarke CollegeKaren E. Spike, University of North Carolina, WilmingtonRuth Ann Stefanussen, University of UtahCarol Steiner, Kent State UniversityDebbie Stokes, East Carolina UniversityRuthi Sturdevant, Lincoln University, MOViji Sundar, California State University, StanislausAnn Sweeney, College of St. Catherine, MNKaren Swenson, George Fox CollegeCarla Tayeh, Eastern Michigan UniversityJanet Thomas, Garrett Community CollegeS. Thomas, University of OregonMary Beth Ulrich, Pikeville CollegeMartha Van Cleave, Linfield CollegeDr. Howard Wachtel, Bowie State UniversityDr. Mary Wagner-Krankel, St. Marys UniversityBarbara Walters, Ashland Community CollegeBill Weber, Eastern Arizona CollegeJoyce Wellington, Southeastern Community CollegePaula White, Marshall UniversityHeide G. Wiegel, University of GeorgiaJane Wilburne, West Chester UniversityJerry Wilkerson, Missouri Western State CollegeJack D. Wilkinson, University of Northern IowaCarole Williams, Seminole Community CollegeDelbert Williams, University of Mary HardinBaylorChris Wise, University of Southwestern LouisianaJohn L. Wisthoff, Anne Arundel Community College (retired)Lohra Wolden, Southern Utah UniversityMary Wolfe, University of Rio GrandeVernon E. Wolff, Moorhead State UniversityMaria Zack, Point Loma Nazarene CollegeStanley 1. Zehm, Heritage CollegeMakia Zimmer, Bethany College

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Focus Group Participants

Mara Alagic, Wichita State UniversityRobin L. Ayers, Western Kentucky UniversityElaine Carbone, Clarion University of PennsylvaniaJanis Cimperman, St. Cloud State UniversityRichard DeCesare, Southern Connecticut State UniversityMaria Diamantis, Southern Connecticut State UniversityJerrold W. Grossman, Oakland UniversityRichard H. Hudson, University of South Carolina, ColumbiaCarol Kahle, Shippensburg UniversityJane Keiser, Miami UniversityCatherine Carroll Kiaie, Cardinal Stritch UniversityCynthia Y. Naples, St. Edwards UniversityArmando M. Martinez-Cruz, California State University, FullertonDavid L. Pagni, Fullerton UniversityMelanie Parker, Clarion University of PennsylvaniaCarol Phillips-Bey, Cleveland State University

Content Connections Survey Respondents

Marc Campbell, Daytona Beach Community CollegePorter Coggins, University of WisconsinStevens PointDon Collins, Western Kentucky UniversityAllan Danuff, Central Florida Community CollegeBirdeena Dapples, Rocky Mountain CollegeNancy Drickey, Linfield CollegeThea Dunn, University of WisconsinRiver FallsMark Freitag, East Stroudsberg UniversityPaula Gregg, University of South Carolina AikenBrian Karasek, Arizona Western CollegeChris Kolaczewski, Ferris University of AkronR. Michael Krach, Towson UniversityRanda Lee Kress, Idaho State UniversityMarshall Lassak, Eastern Illinois UniversityKatherine Muhs, St. Norbert CollegeBethany Noblitt, Northern Kentucky University

xxiv Acknowledgments

We would like to acknowledge the following people for their assistance in the preparation of the firstseven editions of this book: Ron Bagwell, Jerry Becker, Julie Borden, Sue Borden, Tommy Bryan, JuliDixon, Christie Gilliland, Dale Green, Kathleen Seagraves Higdon, 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 and Becky Gwilliam for her research contributions toChapter 10 and the Reflections from Research. We also thank Lyn Riverstone and Vikki Maurer fortheir careful checking of the accuracy of the answers.

We also want to acknowledge Marcia Swanson and Karen Swenson for their creation of and contri-bution to our Student Resource Handbook during the first seven editions with a special thanks to LynRiverstone for her expert revision of the new Student Activity Manual for the seventh edition. Thanksare also due to Don Miller for his Guide to Problem Solving, to Lyn Trimpe, Roger Maurer, and VikkiMaurer, for their longtime authorship of our Student Hints and Solutions Manual, to Keith Leathem forthe Spreadsheet Tutorial and Algebraic Reasoning Web Module, Armando Martinez-Cruz for TheGeometers Sketchpad Tutorial, to Joan Cohen Jones for the Childrens Literature Webmodule, and toLawrence O. Cannon, E. Robert Heal, Joel Duffin, Richard Wellman, and Ethalinda K. S. Cannon forthe eManipulatives activities.

We are very grateful to our publisher, Laurie Rosatone, and our acquisitions editor, Jessica Jacobs, fortheir commitment and super teamwork, to our senior production editor, Valerie A. Vargas, for attendingto the details we missed, to Martha Beyerlein, our full-service representative and copyeditor, for lightingthe path as we went from manuscript to the final book, and to Melody Englund for creating the index.Other Wiley staff who helped bring this book and its print and media supplements to fruition are:Christopher Ruel, Executive Marketing Manager; Stefanie Liebman, media editor; Ann Berlin, VicePresident, Production and Manufacturing; Dorothy Sinclair, Production Services Manager; Kevin Murphy,Senior Designer; Lisa Gee, Photo Researcher; Michael Shroff, Assistant Editor; Jeffrey Benson, EditorialAssistant; and Matt Winslow, Production Assistant. They have been uniformly wonderful to work withJohn Wiley would have been proud of them.

Finally, we welcome comments from colleagues and students. Please feel free to send suggestionsto Gary at [email protected] and Blake at [email protected]. Please include both of us inany communications.

G.L.M.B.E.P.

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Introduction to Problem Solving

George PlyaThe Father of Modern Problem Solving

CHAPTER

1

George Plya was born in Hungary in 1887. He received his Ph.D. at the University of Budapest. In1940 he came to Brown University and then joined

the faculty at Stanford University in 1942.

book, How to Solve It, which has been translated into 15languages, introduced his four-step approach together withheuristics, or strategies, which are helpful in solving prob-lems. Other important works by Plya are MathematicalDiscovery, Volumes 1 and 2, and Mathematics and Plausi-ble Reasoning, Volumes 1 and 2.

He died in 1985, leaving mathematics with the impor-tant legacy of teaching problem solving. His Ten Com-mandments 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 theirexpectations and difficulties; put yourself in their place.

4. Realize that the best way to learn anything is to dis-cover it by yourself.

5. Give your students not only information, but alsoknow-how, mental attitudes, the habit of methodicalwork.

6. Let them learn guessing.

7. Let them learn proving.

8. Look out for such features of the problem at hand asmay be useful in solving the problems to cometry todisclose the general pattern that lies behind the presentconcrete situation.

9. Do not give away your whole secret at oncelet thestudents guess before you tell itlet them find out bythemselves as much as is feasible.

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

1

FOCUS ON

In his studies, he became interested in the process ofdiscovery, which led to his famous four-step process forsolving problems:

1. Understand the problem.

2. Devise a plan.

3. Carry out the plan.

4. Look back.

Plya wrote over 250 mathematical papers and threebooks that promote problem solving. His most famous

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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.

2

Because problem solving is the main goal of mathematics, this chapter introduces the six strategies listed in the Problem-Solving Strategies boxthat are helpful in solving problems. Then, at the beginning of each chapter,an initial problem is posed that can 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, as shouldyour ability to solve problems.

Problem-Solving Strategies1. Guess and Test

2. Draw a Picture

3. Use a Variable

4. Look for a Pattern

5. Make a List

6. Solve a Simpler ProblemINITIAL PROBLEM

A solution to this Initial Problem is on page 38.

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Section 1.1 The Problem-Solving Process and Strategies 3

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

Key Concepts from NCTM Curriculum Focal Points

KINDERGARTEN: Choose, combine, and apply effective strategies for answer-ing quantitative questions.

GRADE 1: Develop an understanding of the meanings of addition and subtractionand strategies to solve such arithmetic problems. Solve problems involving the rel-ative sizes of whole numbers.

GRADE 3: Apply increasingly sophisticated strategies . . . to solve multiplicationand division problems.

GRADE 4 AND 5: Select appropriate units, strategies, and tools for solving problems. GRADE 6: Solve a wide variety of problems involving ratios and rates. GRADE 7: Use ratio and proportionality to solve a wide variety of percent problems.

nce, at an informal meeting, a social scientist asked of a mathematics professor,Whats the main goal of teaching mathematics? The reply was, Problemsolving. 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,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 structurethat can help to facilitate problem solving. Because of the universal importance of prob-lem solving, the main professional group in mathematics education, the National Councilof Teachers of Mathematics (NCTM), recommended in its 1980 An Agenda for Actionthat problem solving be the focus of school mathematics in the 1980s. The NationalCouncil of Teachers of Mathematics 1989 Curriculum and Evaluation Standards forSchool Mathematics called for increased attention to the teaching of problem solving inK8 mathematics. Areas of emphasis include word problems, applications, patterns andrelationships, open-ended problems, and problem situations represented verbally, numerically, graphically, geometrically, or symbolically. The NCTMs 2000 Principlesand Standards for School Mathematics identified problem solving as one of theprocesses by which all mathematics should be taught.

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

O

1.1 THE PROBLEM-SOLVING PROCESS AND STRATEGIES

Use any strategy you know to solve the problem below. As you solve the prob-lem below, pay close attention to the thought processes and steps that you use.Write down these strategies and compare them to a classmates. Are there anysimilarities in your approaches to solving the problem below?

Problem: Lins garden has an area of 78 square yards. The length of the gardenis 5 less than three times its width. What are the dimensions of Lins garden?

STARTING POINT

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Reflection from ResearchMany children believe that theanswer to a word problem canalways be found by adding,subtracting, multiplying, ordividing two numbers. Littlethought is given tounderstanding the context of theproblem (Verschaffel, De Corte,& Vierstraete, 1999).

Developing AlgebraicReasoningwww.wiley.com/college/musserSee Mathematizing.

4 Chapter 1 Introduction to Problem Solving

Childrens Literaturewww.wiley.com/college/musserSee The Math Curse by JonSciezke.

Step 1

Step 2

Plyas Four StepsIn this book we often distinguish between exercises and problems. Unfortunately, thedistinction cannot be made precise. To solve an exercise, one applies a routine procedureto arrive at an answer. To solve a problem, one has to pause, reflect, and perhaps takesome original step never taken before to arrive at a solution. This need for some sort ofcreative step on the solvers part, however minor, is what distinguishes a problem from anexercise. 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 equallyamong 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 aproblem can vary, since it depends on the state of mind of the person who is to solve it.

Doing exercises is a very valuable aid in learning mathematics. Exercises help you tolearn concepts, properties, procedures, and so on, which you can then apply when solv-ing problems. This chapter provides an introduction to the process of problem solving.The techniques that you learn in this chapter should help you to become a better problemsolver and should show you how to help others develop their problem-solving skills.

A famous mathematician, George Plya, devoted much of his teaching to helpingstudents become better problem solvers. His major contribution is what has becomeknown as Plyas four-step process for solving problems.

Understand the Problem Do you understand all the words? Can you restate the problem in your own words? Do you know what is given? Do you know what the goal is? Is there enough information? Is there extraneous information? Is this problem similar to another problem you have solved?

Devise a PlanCan one of the following strategies (heuristics) be used? (A strategy is defined asan artful means to an end.)

1. Guess and test. 12. Work backward.

2. Draw a picture. 13. Use cases.

3. Use a variable. 14. Solve an equation.

4. Look for a pattern. 15. Look for a formula.

5. Make a list. 16. Do a simulation.

6. Solve a simpler problem. 17. Use a model.

7. Draw a diagram. 18. Use dimensional analysis.

8. Use direct reasoning. 19. Identify subgoals.

9. Use indirect reasoning. 20. Use coordinates.

10. Use properties of numbers. 21. Use symmetry.

11. Solve an equivalent problem.

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

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S T U D E N T PA G E S N A P S H O T

From Mathematics, Grade 2 Pupil Edition, p. 233. Copyright 2005 by Scott Foresman-Addison Wesley.

5

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6 Chapter 1 Introduction to Problem Solving

Reflection from ResearchResearchers suggest thatteachers think aloud whensolving problems for the firsttime in front of the class. In sodoing, teachers will bemodeling successful problem-solving behaviors for theirstudents (Schoenfeld, 1985).

NCTM StandardInstructional programs shouldenable all students to apply andadapt a variety of appropriatestrategies to solve problems.

Originalproblem

Translate

Answer tooriginal problem

Interpret

Check Solve

Mathematicalversion

of the problem

Solution to themathematical

version

Figure 1.1

Learning to utilize Plyas four steps and the diagram in Figure 1.1 are first stepsin becoming a good problem solver. In particular, the Devise a Plan step is very im-portant. In this chapter and throughout the book, you will learn the strategies listedunder the Devise a Plan step, which in turn help you decide how to proceed to solveproblems. However, selecting an appropriate strategy is critical! As we worked withstudents who were successful problem solvers, we asked them to share clues thatthey observed in statements of problems that helped them select appropriate strate-gies. Their clues are listed after each corresponding strategy. Thus, in addition tolearning how to use the various strategies herein, these clues can help you decidewhen to select an appropriate strategy or combination of strategies. Problem solvingis as much an art as it is a science. Therefore, you will find that with experience youwill develop a feeling for when to use one strategy over another by recognizing cer-tain clues, perhaps subconsciously. Also, you will find that some problems may besolved in several ways using different strategies.

In summary, this initial material on problem solving is a foundation for your suc-cess in problem solving. Review this material on Plyas four steps as well as thestrategies and clues as you continue to develop your expertise in solving problems.

Step 3

Step 4

Carry Out the Plan Implement the strategy or strategies that you have chosen until the problem is

solved or until a new course of action is suggested.

Give yourself a reasonable amount of time in w

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