A collection of 108 coins containing only quarters and nickels is worth $21.

BMIndex.indd 29 7/31/2013 7:29:35 AM

EQUITY.

CURRICULUM. -

TEACHING. - -

LEARNING. -

ASSESSMENT.

TECHNOLOGY. -

National Council of Teachers of Mathematics Principles and Standards for School Mathematics

Principles for School Mathematics

Standards for School Mathematics

NUMBER AND OPERATIONS

-

ALGEBRA

-

GEOMETRY

-

MEASUREMENT

-

DATA ANALYSIS AND PROBABILITY

-

PROBLEM SOLVING

REASONING AND PROOF

COMMUNICATION

FMEndpaper.indd 15 7/31/2013 10:58:25 AM

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

CONNECTIONS

-

REPRESENTATION

-

-

PREKINDERGARTEN Number and Operations:

-

Geometry: -

Measurement:

KINDERGARTEN Number and Operations:

Geometry: Measurement:

GRADE 1 Number and Operations Algebra: -

-

Number and Operations:

Geometry:

GRADE 2 Number and Operations:

Number and Operations Algebra:

Measurement: -

GRADE 3 Number and Operations Algebra: -

-

Number and Operations: -

Geometry: -

GRADE 4 Number and Operations Algebra:

Number and Operations:

Measurement: -

GRADE 5 Number and Operations Algebra: -

Number and Operations:

Geometry Measurement Algebra:

GRADE 6 Number and Operations: -

Number and Operations: -

Algebra: -

GRADE 7 Number and Operations Algebra Geometry: -

Measurement Geometry Algebra: -

Number and Operations Algebra: -

GRADE 8 Algebra: -

Geometry Measurement:

Data Analysis Number and Operations Algebra: -

FMEndpaper.indd 16 7/31/2013 10:58:25 AM

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MathematicsFor 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 Blake E. Peterson 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 SENIOR CONTENT MANAGER Karoline Luciano SENIOR PRODUCTION EDITOR Kerry Weinstein MARKETING MANAGER Kimberly Kanakes SENIOR PRODUCT DESIGNER Tom Kulesa OPERATIONS MANAGER Melissa Edwards ASSISTANT CONTENT EDITOR Jacqueline Sinacori SENIOR PHOTO EDITOR Lisa Gee MEDIA SPECIALIST Laura Abrams 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 Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions.

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

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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

ABOUT THE AUTHORS

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vi

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.

ABOUT THE COVER

FMWileyPlus.indd 6 7/31/2013 2:14:12 PM

viivii

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

BRIEF CONTENTS

FMBriefContents.indd 7 7/31/2013 12:29:55 PM

viii

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

CONTENTS

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ix

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

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x

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

FMBriefContents.indd 10 7/31/2013 12:29:55 PM

PREFACE

W elcome to the study of the foundations of ele-mentary school mathematics. We hope you will find your studies enlightening, useful, and fun. We salute you for choosing teaching as a 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 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 understand- ing of mathematics.

We have also sought to present this material in a man- ner consistent with the recommendations in (1) The Mathematical Education of Teachers prepared by the Conference Board of the Mathematical Sciences, (2) the National Council of Teachers of Mathematics’ Standards Documents, and (3) The Common Core State Standards for Mathematics. In addition, we have received valuable 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 book has mathematical integrity and is accessible and helpful to the variety of students who will use it. As al- ways, we look forward to hearing from you about your experiences with our text.

GARY L. MUSSER, glmusser@cox.net BLAKE E. PETERSON, peterson@mathed.byu.edu

Unique Content Features Number Systems The order in which we present the number systems in this book is unique and most relevant to elementary school teachers. The topics are covered to parallel their evolution historically and their development in the elementary/middle school curriculum. Fractions and integers are treated separately as an extension of the whole numbers. Then rational numbers can be treated at a brisk pace as extensions of both fractions (by adjoining their opposites) and integers (by adjoining their appro- priate 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 book have commented to us that this whole numbers-fractions-integers-rationals-reals

approach is clearly superior to the seemingly more effi- cient sequence of whole numbers-integers-rationals-reals that is more appropriate to use when teaching high school mathematics.

Approach to Geometry Geometry is organized from the point of view of the five-level van Hiele model of a child’s development in geometry. After studying shapes and measurement, geometry is approached more formally through Euclidean congruence and similarity, coordinates, and transformations. The Epilogue provides an eclectic approach by solving geometry problems using a variety of techniques.

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 reciprocal and hence may be most instructive after Chapter 6, “Fractions,” and Chapter 8, “Integers,” have been completed. This section also contains an introduction to modular arithmetic.

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

Deductive Reasoning The use of deduction is pro- moted throughout the book The approach is gradual, with later chapters having more multistep problems. In particular, the last sections of Chapters 14, 15, and 16 and the Epilogue offer a rich source of interesting theo- rems and problems in geometry.

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

In addition, the book companion Web site has eMa- nipulatives, spreadsheets, and sketches from Geometer’s

xi

FMPreface.indd 11 8/1/2013 12:05:27 PM

mailto:glmusser@cox.net
mailto:peterson@mathed.byu.edu
xii Preface

Sketchpad®. The eManipulatives are electronic versions of the manipulatives commonly used in the elementary classroom, such as the geoboard, base ten blocks, black and red chips, and pattern blocks. The spreadsheets contain dynamic representations of functions, statistics, and probability simulations. The sketches in Geometer’s Sketchpad® are dynamic representations of geomet- ric relationships that allow exploration. Exercises and problems that involve eManipulatives, spreadsheets, and Geometer’s Sketchpad® sketches have been integrated into the problem sets throughout the text.

Course Options We recognize that the structure of the mathematics for elementary teachers course will vary depending upon the college or university. Thus, we have organized this text so that it may be adapted to accommodate these differences.

Basic course: Chapters 1-7 Basic course with logic: Topic 1, Chapters 1–7 Basic course with informal geometry: Chapters 1–7,

12 Basic course with introduction to geometry and mea-

surement: Chapters 1–7, 12, 13

Summary of Changes to the Tenth Edition

Mathematical Tasks have been added to sections throughout the book to allow instructors more flex- ibility in how they choose to organize their classroom 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.

Chapter 6 contains a new discussion of fractions on a number line to be consistent with the Common Core standards.

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 through the chapter.

Chapter 12 has been substantially revised. Sections 12.1, 12.2, and 12.3 have been organized to parallel the first three van Hiele levels. In this way, students will be able to pass through the levels in a more meaningful fashion so that they will get a strong feeling about how

their students will view geometry at various van Hiele levels.

Chapter 13 contains several new examples to give stu- dents the opportunity to see how the various equations for area and volume are applied in different contexts.

Children’s Videos are videos of children solving math- ematical problems linked to QR codes placed in the margin of the book in locations where the content being discussed is related to the content of the prob- lems being solved by the children. These videos will bring the mathematical content being studied to life.

Author Walk-Throughs are videos linked to the QR code on the third page of each chapter. These brief videos are of an author, Blake Peterson, describing and showing points of major emphasis in each chapter so students’ study can be more focused.

Children’s Literature and Reflections from Research margin notes have been revised/refreshed.

Common Core margin notes have been added through- out the text to highlight the correlation between the content of this text and the Common Core standards.

Professional recommendation statements from the Common Core State Standards for Mathematics, the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics, and the Curriculum Focal Points, have been compiled on the third page of each chapter.

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

The three dimensions of the cube—cognitive levels, representational levels, and mathematical content—are 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 problems (to support problem solv- ing and applications).

FMPreface.indd 12 8/1/2013 12:05:27 PM

Preface xiii

We have developed new pedagogical features to imple- ment and reinforce the goals discussed above and to address the many challenges in the course.

Summary of Pedagogical Changes to the Tenth Edition

Student Page Snapshots have been updated.

Reflection from Research margin notes have been edited and updated.

Mathematical Structure reveals the mathematical ideas of the book. Main Definitions, Theorems, and Properties in each section are highlighted in boxes for quick review.

Children’s Literature references have been edited and updated. Also, there is additional material offered on the Web site on this topic.

Check for Understanding have been updated to reflect the revision of the problem sets.

Mathematical Tasks have been integrated throughout.

Author Walk-Throughs videos have been made avail- able via QR codes on the third page of every chapter.

Children’s videos, produced by Blake Peterson and available via QR codes, have been integrated through- out.

Key Features Problem-Solving Strategies are integrated throughout the book. Six strategies are introduced in Chapter 1. The last strategy in the strategy box at the top of the second page of each chapter after Chapter l 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

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.

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.

B la

ke E

. P et

er so

n

See one Live!

Reflection from Research Extensive research has been done in the mathematics education community that

focuses on the teaching and learning of elemen- tary mathematics. Many important quotations from research are given in the margins to sup- port the content nearby.

Children’s Literature These margin inserts provide many examples of books that can be used to connect reading and mathematics. They should be invaluable to you when you begin teachig.

FMPreface.indd 15 8/1/2013 12:05:34 PM

xvi Preface

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:

Hands-On Activities: Activities that help develop initial understandings at the concrete level. Discussion Questions for the Classroom: Tasks designed to engage students with mathematical ideas by stimulating communication. Mental Math: Short activities to help develop mental math skills. Exercises: Additional practice for building skills in concepts. 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 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

Hints: Give 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 978-1-118-67925-8

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.

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.

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.

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 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.

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 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.

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.

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http://www.wiley.com/college/musser
xviii Preface

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:

Links to NCTM Standards

Links to Common Core 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 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:

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 of the strategies introduced to that point.

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:

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

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Preface xix

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.

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http://edugen.wiley.com
ACKNOWLEDGMENTS

During the development of Mathematics for Elementary Teach- ers, Eighth, Ninth, and Tenth Editions, we benefited from comments, suggestions, and evaluations from many of our col- leagues. We would like to acknowledge the contributions made by the following people:

Reviewers for the Tenth Edition

Meg Kiessling, University of Tennessee at Chattanooga Juli Ratheal, University of Texas Permian Basin Marie Franzosa, Oregon State University Mary Beth Rollick, Kent State University Linda Lefevre, SUNY Oswego

Reviewers for the Ninth Edition

Larry Feldman, Indiana University of Pennsylvania Sarah Greenwald, Appalachian State University Leah Gustin, Miami University of Ohio, Middleton Linda LeFevre, State University of New York, Oswego Bethany Noblitt, Northern Kentucky University Todd Cadwallader Olsker, California State University, Fullerton Cynthia Piez, University of Idaho Tammy Powell-Kopilak, Dutchess Community College Edel Reilly, Indiana University of Pennsylvania Sarah Reznikoff, Kansas State University Mary Beth Rollick, Kent State University

Ninth Edition Interviewees

John Baker, Indiana University of Pennsylvania Paulette Ebert, Northern Kentucky University Gina Foletta, Northern Kentucky University Leah Griffith, Rio Hondo College Jane Gringauz, Minneapolis Community College Alexander Kolesnick, Ventura College Gail Laurent, College of DuPage Linda LeFevre, State University of New York, Oswego Carol Lucas, University of Central Oklahoma Melanie Parker, Clarion University of Pennsylvania Shelle Patterson, Murray State University Cynthia Piez, University of Idaho Denise Reboli, King’s College Edel Reilly, Indiana University of Pennsylvania Sarah Reznikoff, Kansas State University Nazanin Tootoonchi, Frostburg State University

Ninth Edition Focus Group Participants

Kaddour Boukkabar, California University of Pennsylvania Melanie Branca, Southwestern College Tommy Bryan, Baylor University Jose Cruz, Palo Alto College Arlene Dowshen, Widener University Rita Eisele, Eastern Washington University Mario Flores, University of Texas at San Antonio Heather Foes, Rock Valley College

Mary Forintos, Ferris State University Marie Franzosa, Oregon State University Sonia Goerdt, St. Cloud State University Ralph Harris, Fresno Pacific University George Jennings, California State University, Dominguez Hills Andy Jones, Prince George’s Community College Karla Karstens, University of Vermont Margaret Kidd, California State University, Fullerton Rebecca Metcalf, Bridgewater State College Pamela Miller, Arizona State University, West Jessica Parsell, Delaware Technical Community College Tuyet Pham, Kent State University Mary Beth Rollick, Kent State University Keith Salyer, Central Washington University Sherry Schulz, College of the Canyons Carol Steiner, Kent State University Abolhassan Tagavy, City College of Chicago Rick Vaughan, Paradise Valley Community College Demetria White, Tougaloo College John Woods, Southwestern Oklahoma State University

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

Reviewers for the Eighth Edition

Seth Armstrong, Southern Utah University Elayne Bowman, University of Oklahoma Anne Brown, Indiana University, South Bend David C. Buck, Elizabethtown Alison Carter, Montgomery College Janet Cater, California State University, Bakersfield Darwyn Cook, Alfred University Christopher Danielson, Minnesota State University, Mankato Linda DeGuire, California State University, Long Beach Cristina Domokos, California State University, Sacramento Scott Fallstrom, University of Oregon Teresa Floyd, Mississippi College Rohitha Goonatilake, Texas A&M International University Margaret Gruenwald, University of Southern Indiana Joan Cohen Jones, Eastern Michigan University Joe Kemble, Lamar University Margaret Kinzel, Boise State University J. Lyn Miller, Slippery Rock University Girija Nair-Hart, Ohio State University, Newark Sandra Nite, Texas A&M University Sally Robinson, University of Arkansas, Little Rock Nancy Schoolcraft, Indiana University, Bloomington Karen E. Spike, University of North Carolina, Wilmington Brian Travers, Salem State Mary Wiest, Minnesota State University, Mankato Mark A. Zuiker, Minnesota State University, Mankato

Student Activity Manual Reviewers

Kathleen Almy, Rock Valley College Margaret Gruenwald, University of Southern Indiana

xx

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Acknowledgments xxi

Kate Riley, California Polytechnic State University Robyn Sibley, Montgomery County Public Schools

State Standards Reviewers

Joanne C. Basta, Niagara University Joyce Bishop, Eastern Illinois University Tom Fox, University of Houston, Clear Lake Joan C. Jones, Eastern Michigan University Kate Riley, California Polytechnic State University Janine Scott, Sam Houston State University Murray Siegel, Sam Houston State University Rebecca Wong, West Valley College

Reviewers

Paul Ache, Kutztown University Scott Barnett, Henry Ford Community College Chuck Beals, Hartnell College Peter Braunfeld, University of Illinois Tom Briske, Georgia State University Anne Brown, Indiana University, South Bend Christine Browning, Western Michigan University Tommy Bryan, Baylor University Lucille Bullock, University of Texas Thomas Butts, University of Texas, Dallas Dana S. Craig, University of Central Oklahoma Ann Dinkheller, Xavier University John Dossey, Illinois State University Carol Dyas, University of Texas, San Antonio Donna Erwin, Salt Lake Community College Sheryl Ettlich, Southern Oregon State College Ruhama Even, Michigan State University Iris B. Fetta, Clemson University Marjorie Fitting, San Jose State University Susan Friel, Math/Science Education Network, University of

North Carolina Gerald Gannon, California State University, Fullerton Joyce Rodgers Griffin, Auburn University Jerrold W. Grossman, Oakland University Virginia Ellen Hanks, Western Kentucky University John G. Harvey, University of Wisconsin, Madison Patricia L. Hayes, Utah State University, Uintah Basin Branch

Campus Alan Hoffer, University of California, Irvine Barnabas Hughes, California State University, Northridge Joan Cohen Jones, Eastern Michigan University Marilyn L. Keir, University of Utah Joe Kennedy, Miami University Dottie King, Indiana State University Richard Kinson, University of South Alabama Margaret Kinzel, Boise State University John Koker, University of Wisconsin David E. Koslakiewicz, University of Wisconsin, Milwaukee Raimundo M. Kovac, Rhode Island College Josephine Lane, Eastern Kentucky University Louise Lataille, Springfield College Roberts S. Matulis, Millersville University Mercedes McGowen, Harper College Flora Alice Metz, Jackson State Community College J. Lyn Miller, Slippery Rock University Barbara Moses, Bowling Green State University

Maura Murray, University of Massachusetts Kathy Nickell, College of DuPage Dennis Parker, The University of the Pacific William Regonini, California State University, Fresno James Riley, Western Michigan University Kate Riley, California Polytechnic State University Eric Rowley, Utah State University Peggy Sacher, University of Delaware Janine Scott, Sam Houston State University Lawrence Small, L.A. Pierce College Joe K. Smith, Northern Kentucky University J. Phillip Smith, Southern Connecticut State University Judy Sowder, San Diego State University Larry Sowder, San Diego State University Karen Spike, University of Northern Carolina, Wilmington Debra S. Stokes, East Carolina University Jo Temple, Texas Tech University Lynn Trimpe, Linn–Benton Community College Jeannine G. Vigerust, New Mexico State University Bruce Vogeli, Columbia University Kenneth C. Washinger, Shippensburg University Brad Whitaker, Point Loma Nazarene University John Wilkins, California State University, Dominguez Hills

Questionnaire Respondents

Mary Alter, University of Maryland Dr. J. Altinger, Youngstown State University Jamie Whitehead Ashby, Texarkana College Dr. Donald Balka, Saint Mary’s College Jim Ballard, Montana State University Jane Baldwin, Capital University Susan Baniak, Otterbein College James Barnard, Western Oregon State College Chuck Beals, Hartnell College Judy Bergman, University of Houston, Clearlake James Bierden, Rhode Island College Neil K. Bishop, The University of Southern Mississippi,

Gulf Coast Jonathan Bodrero, Snow College Dianne Bolen, Northeast Mississippi Community College Peter Braunfeld, University of Illinois Harold Brockman, Capital University Judith Brower, North Idaho College Anne E. Brown, Indiana University, South Bend Harmon Brown, Harding University Christine Browning, Western Michigan University Joyce W. Bryant, St. Martin’s College R. Elaine Carbone, Clarion University Randall Charles, San Jose State University Deann Christianson, University of the Pacific Lynn Cleary, University of Maryland Judith Colburn, Lindenwood College Sister Marie Condon, Xavier University Lynda Cones, Rend Lake College Sister Judith Costello, Regis College H. Coulson, California State University Dana S. Craig, University of Central Oklahoma Greg Crow, John Carroll University Henry A. Culbreth, Southern Arkansas University, El Dorado Carl Cuneo, Essex Community College 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 Jennifer Davis, Ulster County Community College Dennis De Jong, Dordt College Mary De Young, Hop College Louise Deaton, Johnson Community College Shobha Deshmukh, College of Saint Benedict/St.

John’s University Sheila Doran, Xavier University Randall L. Drum, Texas A&M University P. R. Dwarka, Howard University Doris Edwards, Northern State College Roger Engle, Clarion University Kathy Ernie, University of Wisconsin Ron Falkenstein, Mott Community College Ann Farrell, Wright State University Francis Fennell, Western Maryland College Joseph Ferrar, Ohio State University Chris Ferris, University of Akron Fay Fester, The Pennsylvania State University Marie Franzosa, Oregon State University Margaret Friar, Grand Valley State College Cathey Funk, Valencia Community College Dr. Amy Gaskins, Northwest Missouri State University Judy Gibbs, West Virginia University Daniel Green, Olivet Nazarene University Anna Mae Greiner, Eisenhower Middle School Julie Guelich, Normandale Community College Ginny Hamilton, Shawnee State University Virginia Hanks, Western Kentucky University Dave Hansmire, College of the Mainland Brother Joseph Harris, C.S.C., St. Edward’s University John Harvey, University of Wisconsin Kathy E. Hays, Anne Arundel Community College Patricia Henry, Weber State College Dr. Noal Herbertson, California State University Ina Lee Herer, Tri-State University Linda Hill, Idaho State University Scott H. Hochwald, University of North Florida Susan S. Hollar, Kalamazoo Valley Community College Holly M. Hoover, Montana State University, Billings Wei-Shen Hsia, University of Alabama Sandra Hsieh, Pasadena City College Jo Johnson, Southwestern College Patricia Johnson, Ohio State University Pat Jones, Methodist College Judy Kasabian, El Camino College Vincent Kayes, Mt. St. Mary College Julie Keener, Central Oregon Community College Joe Kennedy, Miami University Susan Key, Meridien Community College Mary Kilbridge, Augustana College Mike Kilgallen, Lincoln Christian College Judith Koenig, California State University, Dominguez Hills Josephine Lane, Eastern Kentucky University Don Larsen, Buena Vista College Louise Lataille, Westfield State College Vernon Leitch, St. Cloud State University Steven C. Leth, University of Northern Colorado Lawrence Levy, University of Wisconsin Robert Lewis, Linn-Benton Community College Lois Linnan, Clarion University

Jack Lombard, Harold Washington College Betty Long, Appalachian State University Ann Louis, College of the Canyons C. A. Lubinski, Illinois State University Pamela Lundin, Lakeland College Charles R. Luttrell, Frederick Community College Carl Maneri, Wright State University Nancy Maushak, William Penn College Edith Maxwell, West Georgia College Jeffery T. McLean, University of St. Thomas George F. Mead, McNeese State University Wilbur Mellema, San Jose City College Clarence E. Miller, Jr. Johns Hopkins University Diane Miller, Middle Tennessee State University Ken Monks, University of Scranton Bill Moody, University of Delaware Kent Morris, Cameron University Lisa Morrison, Western Michigan University Barbara Moses, Bowling Green State University Fran Moss, Nicholls State University Mike Mourer, Johnston Community College Katherine Muhs, St. Norbert College Gale Nash, Western State College of Colorado T. Neelor, California State University Jerry Neft, University of Dayton Gary Nelson, Central Community College, Columbus Campus James A. Nickel, University of Texas, Permian Basin Kathy Nickell, College of DuPage Susan Novelli, Kellogg Community College Jon O’Dell, Richland Community College Jane Odell, Richland College Bill W. Oldham, Harding University Jim Paige, Wayne State College Wing Park, College of Lake County Susan Patterson, Erskine College (retired) Shahla Peterman, University of Missouri Gary D. Peterson, Pacific Lutheran University Debra Pharo, Northwestern Michigan College Tammy Powell-Kopilak, Dutchess Community College Christy Preis, Arkansas State University, Mountain Home Robert Preller, Illinois Central College Dr. William Price, Niagara University Kim Prichard, University of North Carolina Stephen Prothero, Williamette University Janice Rech, University of Nebraska Tom Richard, Bemidji State University Jan Rizzuti, Central Washington University Anne D. Roberts, University of Utah David Roland, University of Mary Hardin–Baylor Frances Rosamond, National University Richard Ross, Southeast Community College Albert Roy, Bristol Community College Bill Rudolph, Iowa State University Bernadette Russell, Plymouth State College Lee K. Sanders, Miami University, Hamilton Ann Savonen, Monroe County Community College Rebecca Seaberg, Bethel College Karen Sharp, Mott Community College Marie Sheckels, Mary Washington College Melissa Shepard Loe, University of St. Thomas Joseph Shields, St. Mary’s College, MN

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Acknowledgments xxiii

Lawrence Shirley, Towson State University Keith Shuert, Oakland Community College B. Signer, St. John’s University Rick Simon, Idaho State University James Smart, San Jose State University Ron Smit, University of Portland Gayle Smith, Lane Community College Larry Sowder, San Diego State University Raymond E. Spaulding, Radford University William Speer, University of Nevada, Las Vegas Sister Carol Speigel, BVM, Clarke College Karen E. Spike, University of North Carolina, Wilmington Ruth Ann Stefanussen, University of Utah Carol Steiner, Kent State University Debbie Stokes, East Carolina University Ruthi Sturdevant, Lincoln University, MO Viji Sundar, California State University, Stanislaus Ann Sweeney, College of St. Catherine, MN Karen Swenson, George Fox College Carla Tayeh, Eastern Michigan University Janet Thomas, Garrett Community College S. Thomas, University of Oregon Mary Beth Ulrich, Pikeville College Martha Van Cleave, Linfield College Dr. Howard Wachtel, Bowie State University Dr. Mary Wagner-Krankel, St. Mary’s University Barbara Walters, Ashland Community College Bill Weber, Eastern Arizona College Joyce Wellington, Southeastern Community College Paula White, Marshall University Heide G. Wiegel, University of Georgia Jane Wilburne, West Chester University Jerry Wilkerson, Missouri Western State College Jack D. Wilkinson, University of Northern Iowa Carole Williams, Seminole Community College Delbert Williams, University of Mary Hardin–Baylor Chris Wise, University of Southwestern Louisiana John L. Wisthoff, Anne Arundel Community College (retired) Lohra Wolden, Southern Utah University Mary Wolfe, University of Rio Grande Vernon E. Wolff, Moorhead State University

Maria Zack, Point Loma Nazarene College Stanley L. Zehm, Heritage College Makia Zimmer, Bethany College

Focus Group Participants

Mara Alagic, Wichita State University Robin L. Ayers, Western Kentucky University Elaine Carbone, Clarion University of Pennsylvania Janis Cimperman, St. Cloud State University Richard DeCesare, Southern Connecticut State University Maria Diamantis, Southern Connecticut State University Jerrold W. Grossman, Oakland University Richard H. Hudson, University of South Carolina, Columbia Carol Kahle, Shippensburg University Jane Keiser, Miami University Catherine Carroll Kiaie, Cardinal Stritch University Armando M. Martinez-Cruz, California State University, Fuller- ton Cynthia Y. Naples, St. Edward’s University David L. Pagni, Fullerton University Melanie Parker, Clarion University of Pennsylvania Carol Phillips-Bey, Cleveland State University

Content Connections Survey Respondents

Marc Campbell, Daytona Beach Community College Porter Coggins, University of Wisconsin–Stevens Point Don Collins, Western Kentucky University Allan Danuff, Central Florida Community College Birdeena Dapples, Rocky Mountain College Nancy Drickey, Linfield College Thea Dunn, University of Wisconsin–River Falls Mark Freitag, East Stroudsberg University Paula Gregg, University of South Carolina, Aiken Brian Karasek, Arizona Western College Chris Kolaczewski, Ferris University of Akron R. Michael Krach, Towson University Randa Lee Kress, Idaho State University Marshall Lassak, Eastern Illinois University Katherine Muhs, St. Norbert College 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.

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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.

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mailto:glmusser@cox.net
mailto:peterson@mathed.byu.edu
1

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.

A NOTE TO OUR STUDENTS

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2

G eorge Pólya was born in Hungary in 1887. He received his Ph.D. at the University of Budapest. In 1940 he came to Brown University and then joined the faculty at Stanford University in 1942.

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In his studies, he became interested in the process of discovery, which led to his famous four-step process for solving problems:

1. Understand the problem.

2. Devise a plan.

3. Carry out the plan.

4. Look back.

Pólya wrote over 250 mathematical papers and three books that promote problem solving. His most famous book, How to Solve It, which has been translated

into 15 languages, introduced his four-step approach together with heuristics, or strategies, which are helpful in solving problems. Other important works by Pólya 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.

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 know-how, mental attitudes, the habit of methodical work.

6. Let them learn guessing.

7. Let them learn proving.

8. Look out for such features of the problem at hand as may be useful in solving the problems to come—try to disclose the general pattern that lies behind the present concrete situation.

9. Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as is feasible.

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

C H A P T E R

1 INTRODUCTION TO PROBLEM SOLVING George Pólya—The Father of Modern Problem Solving

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3

Problem-Solving Strategies 1. 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

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 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 should your ability to solve problems.

Initial 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.

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AUTHOR

WALK-THROUGH

4

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, 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

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

KINDERGARTEN: Choose, combine, and apply effective strategies for answering quantitative questions. 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. GRADE 3: Apply increasingly sophisticated strategies … to solve multiplication and 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.

Key Concepts from the Common Core State Standards for Mathematics

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.

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Section 1.1 The Problem-Solving Process and Strategies 5

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.

Doing exercises is a very valuable aid in learning mathematics. Exercises help you to learn concepts, properties, procedures, and so on, which you can then apply when solving 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 problem solver and should show you how to help others develop their problem- solving skills.

A famous mathematician, George Pólya, devoted much of his teaching to helping students become better problem solvers. His major contribution is what has become known as Pólya’s four-step process for solving problems.

Step 1 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?

Step 2 Devise a Plan

Can one of the following strategies (heuristics) be used? (A strategy is defi ned as an artful means to an end.)

Reflection from Research Many children believe that the answer to a word problem can always be found by adding, sub- tracting, multiplying, or dividing two numbers. Little thought is given to understanding the con- text of the problem (Verschaffel, De Corte, & Vierstraete, 1999).

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents start by explaining to them- selves the meaning of a problem and looking for entry points to its solution.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solu- tion attempt.

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?

THE PROBLEM-SOLVING PROCESS AND STRATEGIES

1. 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 of numbers.

11. Solve an equivalent problem.

12. Work backward.

13. Use cases.

14. Solve an equation.

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6 Chapter 1 Introduction to Problem Solving

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

Step 3 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 which to solve the problem. If you are not successful, seek hints from others or put the problem aside for a while. (You may have a flash of insight when you least expect it!)

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

Step 4 Look Back

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

Can you see an easier solution?

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

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

Learning to utilize Pólya’s four steps and the diagram in Figure 1.1 are first steps in becoming a good problem solver. In particular, the “Devise a Plan” step is very important. In this chapter and throughout the book, you will learn the strategies listed under the “Devise a Plan” step, which in turn help you decide how to proceed to solve problems. However, selecting an appropriate strategy is critical! As we worked with students who were successful problem solvers, we asked them to share “clues” that they observed in statements of problems that helped them select appropriate 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 in several ways using different strategies.

In summary, this initial material on problem solving is a foundation for your success in problem solving. Review this material on Pólya’s four steps as well as the strategies and clues as you continue to develop your expertise in solving problems.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents consider analogous prob- lems and try special cases and simpler forms of the original problem in order to gain insight into its solution.

Common Core – Grades K-12 (Mathematical Practice 1) Mathematically proficient stu- dents monitor and evaluate their progress and change course if necessary.

Reflection from Research Researchers suggest that teach- ers think aloud when solving problems for the first time in front of the class. In so doing, teachers will be modeling suc- cessful problem-solving behaviors for their students (Schoenfeld, 1985).

NCTM Standard Instructional programs should enable all students to apply and adapt a variety of appropriate strategies to solve problems.

15. Look for a formula.

16. Do a simulation.

17. Use a model.

18. Use dimensional analysis.

19. Identify subgoals.

20. Use coordinates.

21. Use symmetry.

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7

From Chapter 6, Lesson “Problem Solving” from My Math, Volume 1 Common Core State Standards, Grade 2, copyright © 2013 by McGraw-Hill Education.

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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 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

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

Figure 1.2

Figure 1.3

Figure 1.4

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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.

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.

Additional Problems Where the Strategy “Guess and Test” Is Useful

1. In the following cryptarithm—that is, a collection of words where the letters represent numbers—sun and fun represent two three-digit numbers, and swim is their four-digit sum. Using all of the digits 0, 1, 2, 3, 6, 7, and 9 in place of the letters where no letter represents two different digits, determine the value of 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) b

1 3

9 3

1 6

1 6

9 6

1 2

u

u

wi

u

u

wi

+ + ( )

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.

Figure 1.6

Figure 1.5

NCTM Standard Instructional programs should enable all students to monitor and reflect on the process of mathematical problem solving.

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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 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.)

Clues The Guess and Test strategy may be appropriate when

There is a limited number of possible answers to test.

You want to gain a better understanding of the problem.

You have a good idea of what the answer is.

You can systematically try possible answers.

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

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

Often problems involve physical situations. In these situations, drawing a picture can help you better understand the problem so that you can formulate a plan to solve the problem. As you proceed to solve the following “pizza” problem, see whether you can visualize the solution without looking at any pictures first. Then work through the given solution using pictures to see how helpful they can be.

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.

Figure 1.8

Figure 1.7

Children’s Literature www.wiley.com/college/musser See “Counting on Frank” by Rod Clement.

Reflection from Research Training children in the process of using pictures to solve problems results in more improved prob- lem-solving performance than training students in any other strategy (Yancey, Thompson, & Yancey, 1989).

NCTM Standard All students should describe, extend, and make generalizations about geometric and numeric patterns.

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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 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

Figure 1.10

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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

A physical situation is involved.

Geometric fi gures or measurements are involved.

You want to gain a better understanding of the problem.

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.

Use a Variable

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

By trying several examples, you might guess that 3 is the greatest such number. However, it is necessary to use a variable to account for all possible instances of three consecutive numbers.

Step 1 Understand the Problem

The whole numbers are 0 1 2 3, , , , . . . , so that consecutive whole numbers differ by 1. Thus an example of three consecutive whole numbers is the triple 3, 4, and 5. The sum of three consecutive whole numbers has a factor of 3 if 3 multiplied by another whole number produces the given sum. In the example of 3, 4, and 5, the 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 x, , . + +1 2 Now we can discover whether the sum has a factor of 3.

Step 3 Carry Out the Plan

The sum of x x, ,+ 1 and x + 2 is

x x x x x+ +( ) + +( ) = + = +( )1 2 3 3 3 1 .

NCTM Standard All students should represent the idea of a variable as an unknown quantity using a letter or a symbol.

Reflection from Research Given the proper experiences, children as young as eight and nine years of age can learn to comfortably use letters to represent unknown values and can operate on representations involving letters and numbers while fully realizing that they did not know the values of the unknowns (Carraher, Schliemann, Brizuela, & Earnest, 2006).

Algebraic Reasoning In algebra, the letter “x” is most commonly used for a variable. However, any letter (even Greek letters, for example) can be used as a variable.

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Section 1.1 The Problem-Solving Process and Strategies 13

Thus x x x+ + + +( ) ( )1 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 9 99 100+ + + + + +. . . 8 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 2 1+ + + + − + − +. . . ( ) ( ) .n n 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

1

3

2

2

3

1

2 1

1

n n n

n n n

n n

( ) ( )

( ) ( )

( ) ( 11 1 1 1 1) ( ) ( ) ( ) ( )+ + + ⋅ ⋅ ⋅ + + + + + +n n n n

= +n n• ( )1

Since each number was added twice, the desired sum is obtained by dividing n n¥ ( )+ 1 by 2 which yields

1 2 3 2 1 1

2 + + + ⋅ ⋅ ⋅ + −( ) + −( ) + = +( )n n n n n¥

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

1 2 3 8 9 10 10 10 1

2 55

1 2 3 98 99 100 100 101

+ + + ⋅ ⋅ ⋅ + + + = +( ) =

+ + + ⋅ ⋅ ⋅ + + + = (

¥

¥ )) =

+ + + ⋅ ⋅ ⋅ + + + = =

2 5050

1 2 3 498 499 500 500 501

2 125 250

¥ ,

Reflection from Research When asked to create their own problems, good problem solvers generated problems that were more mathematically complex than those of less successful problem solvers (Silver & Cai, 1996).

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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 3 6 3 3 3+ + + ⋅ ⋅ ⋅ + − + − +( ) ( )n n n 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

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

A problem suggests an equation.

A proof or a general solution is required.

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

There is a large number of cases.

There is an unknown quantity related to known quantities.

There is an infi nite number of numbers involved.

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.

A common way to introduce the use of variables is to find a general formula for a pattern of numbers such as 3 6 9 3, , , . . . .n One of the challenges for students is to see the role that each number plays in the expression. For example, the pattern 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 3 2n + . 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.

NCTM Standard All students should develop an initial conceptual understanding of different uses of variables.

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. Another student’s method is illustrated in Figure 1.12.Figure 1.11

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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

In the two previous methods, either corner dots are not counted (so they must be added on) or they are counted twice (so they must be subtracted to avoid double counting). The following fourth method assigns each corner to only one side (Figure 1.14).

4 4 16

4 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). ■

Reflection from Research Sixth-grade students with no formal instruction in algebra are “generally able to solve prob- 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 problems” (Swafford & Langrall, 2000).

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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16 Chapter 1 Introduction to Problem Solving

Since each expression on the right represents the total number of dots in Figure 1.15, they are all equal to each other. Using properties of numbers and equations, each equation can be rewritten as the same expression. Learning to simplify expressions and equations with variables is one of the most important processes in mathematics. Traditionally, this topic has represented a substantial portion of an entire course in introductory algebra. An equation is a sentence involving numbers, or symbols repre- senting numbers, where the verb is equals ( ).= There are various types of equations:

3 4 7+ = True equation 3 4 9+ = False equation

2 5 7x x x+ = 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+ = .

Reflection from Research The more an equation varies from the standard a b c+ = format, the more difficult it is for students to work with. The most difficult problems are those with opera- tions on both sides of the equal sign (Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012).

Algebraic Reasoning Variable is a central concept in algebra. Students may struggle with the idea that the letter x represents many numbers in y x= +3 4 but only one or two numbers in other situations. For example, the solution of the equation x2 9= consists of the numbers 3 and −3 since each of these numbers squared is 9. This means, that the equation is true whenever x = 3 or x = −3.

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 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 that represent the total number of dots on a square fi gure with n dots on a side are generalized as shown next.

4 3 4

4 5 2 4 4 2 4

4 5 4 4 4

4 4

4

× + × −( ) +

⎫ ⎬ ⎭

⎯ →⎯ −( ) +

× − ⎯ →⎯ − ×

n

n

×× −( ) ⎫ ⎬ ⎭

⎯ →⎯ −( ) 5 1

4 1 n ■

Figure 1.15

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

Figure 1.16

S O L U T I O N From Example 1.2, a square with n dots on a side has 4 4n − total dots. Thus, we have the equation 4 4 84n − = . Three elementary methods that can be used to solve equations such as 4 4 84n − = are Guess and Test, Cover Up, and Work Backward.

Reflection from Research Even 6-year-olds can solve alge- braic equations when they are rewritten as a story problem, logic puzzle, or some other problem with meaning (Femiano, 2003).

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

Try n = − = ≠10 4 10 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 h− =4 84 88 4. , . To make a true equation the must be Thus n == 88 Since we have 4 22 88 22• = =, .n

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:

4 4 84

4 84 4

4 88

88 4

22

n

n

n

n

n

− = = + = = ÷ =

Because so much algebra can be done with intuitive reasoning, it is important to help students realize when they are reasoning algebraically. One way to better understand the underlying principles of algebraic reasoning is to look at the Algebraic Reasoning Web Module on our Web site: www.wiley.com/college/musser/

Some researchers say that arithmetic is the foundation of learning algebra. Arithmetic typically means computation with different kinds of numbers and the underlying properties that make the computation work. Much of what is discussed 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.

Reflection from Research We are proposing that the teaching and learning of arithmetic be conceived as part of the foundation of learning algebra, not that algebra be conceived only as an extension of arithmetic procedures (Carpenter, Levi, Berman, & Pligge, 2005).

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.

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18 Chapter 1 Introduction to Problem Solving

1. a. If the diagonals of a square are drawn in, how many triangles of all sizes are formed?

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

2. Scott and Greg were asked to add two whole numbers. Instead, Scott subtracted the two numbers and got 10, and Greg multiplied them and got 651. What was the correct 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?

5. Show how 9 can be expressed as the sum of two consecutive numbers. Then decide whether every odd number can be expressed as the sum of two consecutive counting numbers. Explain your reasoning.

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 =

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.

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

10. Arrange the numbers 2 3 10, , ,. . . in the accompanying tri- angle so that each side sums to 21.

11. Find a set of consecutive counting numbers whose sum is each of the following. Each set may consist of 2, 3, 4, 5, or 6 consecutive integers. Use the spreadsheet activity Consecutive Integer Sum on our Web site to assist you.

a. 84 b. 213 c. 154

12. Place the digits 1 through 9 so that you can count from 1 to 9 by following the arrows in the diagram.

13. Using a 5-minute and an 8-minute hourglass timer, how can you measure 1 minute?

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

a. The largest possible sum:

b. The smallest possible (positive) difference:

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).

16. Solve this cryptarithm, where each letter represents a digit and no digit represents two different letters:

USSR

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.

PROBLEM SET A

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Section 1.1 The Problem-Solving Process and Strategies 19

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 sum of cubes in two different ways. Find a b c, , , and d such that a b3 3 1729+ = and c d3 3 1729+ = .

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.

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.

24. The hexagon below has a total of 126 dots and an equal number of dots on each side. How many dots are on each side?

1. Find the largest eight-digit number made up of the digits 1, 1, 2, 2, 3, 3, 4, and 4 such that the 1s are separated by one digit, the 2s by two digits, the 3s by three digits, and the 4s 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- picks, and five squares with 12 toothpicks.

5. A textbook is opened and the product of the page numbers of the two facing pages is 6162. What are the numbers of the pages?

6. Place numbers 1 through 19 into the 19 circles below so that any three numbers in a line through the center will give the same sum.

7. Using three of the symbols +, , ,− × and ÷ once each, fill in the following three blanks to make a true equation. (Parentheses are allowed.)

6 6 6 6 66 =

PROBLEM SET B

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20 Chapter 1 Introduction to Problem Solving

8. A water main for a street is being laid using a particu- lar kind of pipe that comes in either 18-foot sections or 20-foot sections. The designer has determined that the water main would require 14 fewer sections of 20-foot pipe than if 18-foot sections were used. Find the total length of the water main.

9. Mike said that when he opened his book, the product of the page numbers of the two facing pages was 7007. Without performing any calculations, prove that he was wrong.

10. The Smiths were about to start on an 18,000-mile automo- bile trip. They had their tires checked and found that each 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 make the trip without having to buy a new tire?

11. What is the maximum number of pieces of pizza that can result from 4 straight cuts?

12. Given: Six arrows arranged as follows:

↑ ↑ ↑ ↓ ↓ ↓

Goal: By inverting two adjacent arrows at a time, rear- range to the following:

↑ ↓ ↑ ↓ ↑ ↓

Can you find a minimum number of moves?

13. Two friends are shopping together when they encounter a special “3 for 2” shoe sale. If they purchase two pairs of shoes at the regular price, a third pair (of lower or equal value) will be free. Neither friend wants three pairs 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

DCBA

× 4

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 13

2 1

19 8 14

3

10 15 11

16

18. Together, a baseball and a football weigh 1.25 pounds, the baseball and a soccer ball weigh 1.35 pounds, and the football and the soccer ball weigh 1.9 pounds. How much does each of the balls weigh?

19. Pick any two consecutive numbers. Add them. Then add 9 to the sum. Divide by 2. Subtract the smaller of the original numbers from the answer. What did you get? Repeat this process with two other consecutive numbers. Make a con- jecture (educated guess) about the answer, and prove it.

20. An additive magic square has the same sum in each row, column, and diagonal. Find the error in this magic square and correct it.

47 56 34 22 83 7 24 67 44 26 13 75 29 52 3 99 18 48 17 49 89 4 53 37 97 6 3 11 74 28 35 19 46 87 8 54

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. How many triangles of all sizes are formed?

22. Using the triangle in Problem 10 in Set A, determine whether you can make similar triangles using the digits 1 2 9, , ,. . . , where the side sums are 18, 19, 20, 21, and 22.

23. Using the Chapter 1 eManipulative activity, Number Puzzles, Exercise 4 on our Web site, arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in the circles below so the sum of the numbers along each line of four is 20.

24. Using the Chapter 1 eManipulative activity Circle 99 on our Web site, find an arrangement of the numbers provided in the 7 circles below so that the sum of the three numbers in each circle is 99.

25. An arrangement of dots forms the perimeter of an equi- lateral triangle. There are 87 evenly spaced dots on each side including the dots at the vertices. How many dots are there altogether?

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Section 1.2 Three Additional Strategies 21

26. The equation y 5

12 23+ = can be solved by subtracting

12 from both sides of the equation to yield y 5

12 12+ − =

23 12− . Similarly, the resulting equation y 5

11= can be

solved by multiplying both sides of the equation by 5 to obtain y = 55. Explain how this process is related to the Work Backward method described in Example 1.3.

Analyzing Student Thinking

27. When the class was asked to solve the equation 3 8 27x − = , Wesley asked if he could use guess and test. How would you respond?

28. Rosemary said that she felt the ‘guess and test’ method was a waste of time; she just wanted to get an answer. What could you tell her about the value of using guess and test?

29. Even though you have taught your students how to ‘draw a picture’ to solve a problem, Cecelia asks if she has to draw a picture because she can solve the problems in class without it. How would you respond?

30. When Damian, a second grader, was asked to solve problems like + =3 5, he said that he had seen his older sister working on problems like x + =3 5 and wondered if these equations were different. How would you respond?

31. After the class had found three consecutive odd numbers whose sum is 99, Byron tried to find three consecutive odd numbers that would add to 96. He said he was struggling to find a solution. How could you help him understand the solution to this problem?

32. Consider the following problem:

The amount of fencing needed to enclose a rectangu- lar field was 92 yards and the length of the field was 3 times as long as the width. What were the dimensions of the field?

Vance solved this problem by drawing a picture and using guess and test. Jolie set up an equation with x being the width of the field and solved it. They got the same answer but asked you which method was better. How would you respond?

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?

THREE ADDITIONAL STRATEGIES

Look for a Pattern

When using the Look for a Pattern strategy, one usually lists several specific instances of a problem and then looks to see whether a pattern emerges that suggests a solu- 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+ + = = ×( ), 1 3 5 7 16 4 4+ + + = = ×( ), 1 3 5 7 9 25 5 5+ + + + = = ×( ), and so on. Based on the pat- tern generated by these five examples, one might expect that such a sum will always be a perfect square.

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

Each consecutive odd number of dots can be added to the previous square arrange- ment to form another square. Thus, the sum of the first n odd numbers is n2.

NCTM Standard All students should represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, sym- bolic rules.

Common Core – Grades K-12 (Mathematical Practice 7) Mathematically proficient stu- dents look closely to discern a 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 the most general case is a natural place to discuss variables.

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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

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 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- 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 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, and the individual numbers in the sequence are called terms of the sequence. The counting numbers, 1 2 3 4, , , , ,. . . give rise to many sequences. (An ellipsis, the three periods after the 4, means “and so on.”) Several sequences of counting numbers follow.

Figure 1.18

Figure 1.19

Figure 1.20

Figure 1.21

Children’s Literature www.wiley.com/college/musser See “There Was an Old Lady Who Swallowed a Fly” by Simms Taback.

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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, 3 3 ,2 3, . . . 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)

Inductive reasoning is used to draw conclusions or make predictions about a large collection of objects or numbers, based on a small representative subcollection. For example, inductive reasoning can be used to find the ones digit of the 400th term of 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.

Additional Problems Where the Strategy “Look for a Pattern” Is Useful

1. Find the ones digit in 399.

Step 1 Understand the Problem

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

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 3 3 3 3 3 3 3 31 2 3 4 5 6 7 8, , , , , , , . 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

3 3 3 3 3 24 3 72 3 218 3 6561 2 3 4 5 6 7 8= = = = = = = =3 9 7 1 3 9 7 1, , , , , , ,2 8 .. 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.

NCTM Standard All students should analyze how both repeating and growing pat- terns are generated.

Reflection from Research In classrooms where problem solving is valued and teachers have knowledge of children’s mathematical thinking, children see mathematics as a problem- solving endeavor in which com- municating mathematical thinking is important (Franke & Carey, 1997).

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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 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

Clues The Look for a Pattern strategy may be appropriate when

A list of data is given.

A sequence of numbers is involved.

Listing special cases helps you deal with complex problems.

You are asked to make a prediction or generalization.

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 6 5 6 9 4 1 0

1 4 9 6 5 6 9

, , , , , , , , , ,

, , , , , , ,

1 2 3 4 6 8 10

12 14 16 19 22 25 28 3244 1 0, ,36 40

The pattern in this list can be used to see that the ones digits of squares must be one of 0, 1, 4, 5, 6, or 9. This list suggests that a perfect square can never end in a 2, 3, 7, or 8.

Problem The number 10 can be expressed as the sum of four odd numbers in three ways: (i) 10 7 1 1 1= + + + , (ii) 10 5 3 1 1= + + + , and (iii) 10 3 3 3 1= + + + . In how many ways can 20 be expressed as the sum of eight odd numbers?

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.

Children’s Literature www.wiley.com/college/musser See “Math Appeal” by Greg Tang.

Reflection from Research Problem-solving abililty develops with age, but the relative dif- ficulty inherent in each problem is grade independent (Christou & Philippou, 1998).

NCTM Standard Instructional programs should enable all students to build new mathematical knowledge through problem solving.

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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 33 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

Step 4 Look Back

Could you have used the three sums to 10 to help find these 11 sums to 20? Can you think of similar problems to solve? For example, an easier one would be to express 8 as the sum of four odd numbers, and a more difficult one would be to express 40 as the sum of 16 odd numbers. We could also consider sums of even numbers, expressing 20 as the sum of six even numbers.

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- 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

+ + = + + = + + = + + =

, ,

, 00 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?

Reflection from Research Correct answers are not a safe indicator of good thinking. Teachers must examine more than answers and must demand from students more than answers (Sowder, Threadgill-Sowder, Moyer, & Moyer, 1983).

Figure 1.22

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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 The Make a List strategy may be appropriate when

Information can easily be organized and presented.

Data can easily be generated.

Listing the results obtained by using Guess and Test.

Asked “in how many ways” something can be done.

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

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 at most two balancings to fi nd the heavier coin.

Figure 1.23

Figure 1.24

Figure 1.25

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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 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.