Mathematics.pdf

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

r EQUITY. &YDFMMFODF JO NBUIFNBUJDT FEVDBUJPO SFRVJSFTFRVJUZIJHIFYQFDUBUJPOTBOETUSPOHTVQQPSUGPSBMMTUVEFOUTr CURRICULUM. " DVSSJDVMVN JT NPSF UIBO B DPMMFDUJPO PGBDUJWJUJFTJUNVTUCFDPIFSFOUGPDVTFEPOJNQPSUBOUNBUIFNBU-JDTBOEXFMMBSUJDVMBUFEBDSPTTUIFHSBEFTr TEACHING. &GGFDUJWF NBUIFNBUJDT UFBDIJOH SFRVJSFT VOEFS-TUBOEJOHXIBUTUVEFOUTLOPXBOEOFFEUPMFBSOBOEUIFODIBM-MFOHJOHBOETVQQPSUJOHUIFNUPMFBSOJUXFMM

r LEARNING. 4UVEFOUT NVTU MFBSO NBUIFNBUJDT XJUI VOEFS-TUBOEJOHBDUJWFMZCVJMEJOHOFXLOPXMFEHFGSPNFYQFSJFODFBOEQSJPSLOPXMFEHFr ASSESSMENT. "TTFTTNFOU TIPVME TVQQPSU UIF MFBSOJOH PGJNQPSUBOUNBUIFNBUJDTBOEGVSOJTIVTFGVMJOGPSNBUJPOUPCPUIUFBDIFSTBOETUVEFOUTr TECHNOLOGY. 5FDIOPMPHZJTFTTFOUJBMJOUFBDIJOHBOEMFBSO-JOHNBUIFNBUJDT JU JOGMVFODFT UIFNBUIFNBUJDT UIBU JT UBVHIUBOEFOIBODFTTUVEFOUTMFBSOJOH

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

Principles for School Mathematics

Standards for School Mathematics

NUMBER AND OPERATIONS

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr VOEFSTUBOEOVNCFSTXBZTPGSFQSFTFOUJOHOVNCFSTSFMBUJPO-TIJQTBNPOHOVNCFSTBOEOVNCFSTZTUFNTr VOEFSTUBOENFBOJOHT PG PQFSBUJPOT BOEIPX UIFZ SFMBUF UPPOFBOPUIFSr DPNQVUFVFOUMZBOENBLFSFBTPOBCMFFTUJNBUFT

ALGEBRA

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr VOEFSTUBOEQBUUFSOTSFMBUJPOTBOEGVODUJPOTr SFQSFTFOUBOEBOBMZ[FNBUIFNBUJDBMTJUVBUJPOTBOETUSVDUVSFTVTJOHBMHFCSBJDTZNCPMTr VTFNBUIFNBUJDBMNPEFMTUPSFQSFTFOUBOEVOEFSTUBOERVBO-UJUBUJWFSFMBUJPOTIJQTr BOBMZ[FDIBOHFJOWBSJPVTDPOUFYUT

GEOMETRY

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr BOBMZ[F DIBSBDUFSJTUJDT BOE QSPQFSUJFT PG UXP BOE UISFFEJNFOTJPOBM HFPNFUSJD TIBQFT BOE EFWFMPQ NBUIFNBUJDBMBSHVNFOUTBCPVUHFPNFUSJDSFMBUJPOTIJQTr TQFDJGZ MPDBUJPOT BOE EFTDSJCF TQBUJBM SFMBUJPOTIJQT VTJOHDPPSEJOBUFHFPNFUSZBOEPUIFSSFQSFTFOUBUJPOBMTZTUFNTr BQQMZ USBOTGPSNBUJPOT BOE VTF TZNNFUSZ UP BOBMZ[FNBUI-FNBUJDBMTJUVBUJPOTr VTFWJTVBMJ[BUJPOTQBUJBMSFBTPOJOHBOEHFPNFUSJDNPEFMJOHUPTPMWFQSPCMFNT

MEASUREMENT

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr VOEFSTUBOE NFBTVSBCMF BUUSJCVUFT PG PCKFDUT BOE UIF VOJUTTZTUFNTBOEQSPDFTTFTPGNFBTVSFNFOUr BQQMZ BQQSPQSJBUF UFDIOJRVFT UPPMT BOE GPSNVMBT UP EFUFS-NJOFNFBTVSFNFOUT

DATA ANALYSIS AND PROBABILITY

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr GPSNVMBUFRVFTUJPOTUIBUDBOCFBEESFTTFEXJUIEBUBBOEDPM-MFDUPSHBOJ[FBOEEJTQMBZSFMFWBOUEBUBUPBOTXFSUIFNr TFMFDUBOEVTFBQQSPQSJBUFTUBUJTUJDBMNFUIPETUPBOBMZ[FEBUBr EFWFMPQ BOE FWBMVBUF JOGFSFODFT BOE QSFEJDUJPOT UIBU BSFCBTFEPOEBUBr VOEFSTUBOEBOEBQQMZCBTJDDPODFQUTPGQSPCBCJMJUZ

PROBLEM SOLVING

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr CVJMEOFXNBUIFNBUJDBMLOPXMFEHFUISPVHIQSPCMFNTPMWJOHr TPMWFQSPCMFNTUIBUBSJTFJONBUIFNBUJDTBOEJOPUIFSDPOUFYUTr BQQMZBOEBEBQUBWBSJFUZPGBQQSPQSJBUFTUSBUFHJFT UPTPMWFQSPCMFNTr NPOJUPSBOESFFDUPOUIFQSPDFTTPGNBUIFNBUJDBMQSPCMFNTPMWJOH

REASONING AND PROOF

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr SFDPHOJ[F SFBTPOJOH BOE QSPPG BT GVOEBNFOUBM BTQFDUT PGNBUIFNBUJDTr NBLFBOEJOWFTUJHBUFNBUIFNBUJDBMDPOKFDUVSFTr EFWFMPQBOEFWBMVBUFNBUIFNBUJDBMBSHVNFOUTBOEQSPPGTr TFMFDUBOEVTFWBSJPVTUZQFTPGSFBTPOJOHBOENFUIPETPGQSPPG

COMMUNICATION

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr PSHBOJ[FBOEDPOTPMJEBUFUIFJSNBUIFNBUJDBMUIJOLJOHUISPVHIDPNNVOJDBUJPOr DPNNVOJDBUF UIFJS NBUIFNBUJDBM UIJOLJOH DPIFSFOUMZ BOEDMFBSMZUPQFFSTUFBDIFSTBOEPUIFSTr BOBMZ[FBOEFWBMVBUFUIFNBUIFNBUJDBMUIJOLJOHBOETUSBUFHJFTPGPUIFSTr VTF UIF MBOHVBHF PG NBUIFNBUJDT UP FYQSFTT NBUIFNBUJDBMJEFBTQSFDJTFMZ

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

Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics

CONNECTIONS

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr SFDPHOJ[FBOEVTFDPOOFDUJPOTBNPOHNBUIFNBUJDBMJEFBTr VOEFSTUBOEIPXNBUIFNBUJDBM JEFBT JOUFSDPOOFDU BOECVJMEPOPOFBOPUIFSUPQSPEVDFBDPIFSFOUXIPMFr SFDPHOJ[FBOEBQQMZNBUIFNBUJDTJODPOUFYUTPVUTJEFPGNBUI-FNBUJDT

REPRESENTATION

*OTUSVDUJPOBMQSPHSBNTGSPNQSFLJOEFSHBSUFOUISPVHIHSBEFTIPVMEFOBCMFBMMTUVEFOUTUPr DSFBUFBOEVTFSFQSFTFOUBUJPOTUPPSHBOJ[FSFDPSEBOEDPN-NVOJDBUFNBUIFNBUJDBMJEFBTr TFMFDUBQQMZBOEUSBOTMBUFBNPOHNBUIFNBUJDBMSFQSFTFOUB-UJPOTUPTPMWFQSPCMFNTr VTF SFQSFTFOUBUJPOT UPNPEFM BOE JOUFSQSFU QIZTJDBM TPDJBMBOENBUIFNBUJDBMQIFOPNFOB

PREKINDERGARTENNumber and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGXIPMFOVNCFSTJODMVEJOHDPODFQUTPGDPSSFTQPOEFODFDPVOUJOHDBS-EJOBMJUZBOEDPNQBSJTPOGeometry: *EFOUJGZJOH TIBQFT BOE EFTDSJCJOH TQBUJBM SFMBUJPO-TIJQTMeasurement:*EFOUJGZJOHNFBTVSBCMFBUUSJCVUFTBOEDPNQBSJOHPCKFDUTCZVTJOHUIFTFBUUSJCVUFT

KINDERGARTENNumber and Operations:3FQSFTFOUJOHDPNQBSJOHBOEPSEFSJOHXIPMFOVNCFSTBOEKPJOJOHBOETFQBSBUJOHTFUTGeometry:%FTDSJCJOHTIBQFTBOETQBDFMeasurement:0SEFSJOHPCKFDUTCZNFBTVSBCMFBUUSJCVUFT

GRADE 1Number and OperationsBOEAlgebra:%FWFMPQJOHVOEFSTUBOE-JOHTPGBEEJUJPOBOETVCUSBDUJPOBOETUSBUFHJFTGPSCBTJDBEEJ-UJPOGBDUTBOESFMBUFETVCUSBDUJPOGBDUTNumber and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGXIPMFOVNCFSSFMBUJPOTIJQTJODMVEJOHHSPVQJOHJOUFOTBOEPOFTGeometry:$PNQPTJOHBOEEFDPNQPTJOHHFPNFUSJDTIBQFT

GRADE 2Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPG UIFCBTFUFOOVNFSBUJPOTZTUFNBOEQMBDFWBMVFDPODFQUTNumber and OperationsBOEAlgebra:%FWFMPQJOHRVJDLSFDBMMPGBEEJUJPOGBDUTBOESFMBUFETVCUSBDUJPOGBDUTBOEVFODZXJUINVMUJEJHJUBEEJUJPOBOETVCUSBDUJPOMeasurement:%FWFMPQJOHBOVOEFSTUBOEJOHPGMJOFBSNFBTVSF-NFOUBOEGBDJMJUZJONFBTVSJOHMFOHUIT

GRADE 3Number and OperationsBOEAlgebra:%FWFMPQJOHVOEFSTUBOE-JOHTPGNVMUJQMJDBUJPOBOEEJWJTJPOBOETUSBUFHJFTGPSCBTJDNVM-UJQMJDBUJPOGBDUTBOESFMBUFEEJWJTJPOGBDUTNumber and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGGSBD-UJPOTBOEGSBDUJPOFRVJWBMFODFGeometry:%FTDSJCJOHBOEBOBMZ[JOHQSPQFSUJFTPGUXPEJNFO-TJPOBMTIBQFT

GRADE 4Number and OperationsBOEAlgebra:%FWFMPQJOHRVJDLSFDBMMPGNVMUJQMJDBUJPO GBDUT BOE SFMBUFE EJWJTJPO GBDUT BOE VFODZXJUIXIPMFOVNCFSNVMUJQMJDBUJPO

Number and Operations: %FWFMPQJOH BO VOEFSTUBOEJOH PGEFDJNBMT JODMVEJOH UIF DPOOFDUJPOT CFUXFFO GSBDUJPOT BOEEFDJNBMTMeasurement:%FWFMPQJOHBOVOEFSTUBOEJOHPGBSFBBOEEFUFS-NJOJOHUIFBSFBTPGUXPEJNFOTJPOBMTIBQFT

GRADE 5Number and Operations BOE Algebra: %FWFMPQJOH BO VOEFS-TUBOEJOHPGBOEVFODZXJUIEJWJTJPOPGXIPMFOVNCFSTNumber and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGBOEVFODZXJUIBEEJUJPOBOETVCUSBDUJPOPGGSBDUJPOTBOEEFDJNBMTGeometry BOEMeasurement BOE Algebra: %FTDSJCJOH UISFFEJNFOTJPOBM TIBQFT BOE BOBMZ[JOH UIFJS QSPQFSUJFT JODMVEJOHWPMVNFBOETVSGBDFBSFB

GRADE 6Number and Operations:%FWFMPQJOHBOVOEFSTUBOEJOHPGBOEV-FODZXJUINVMUJQMJDBUJPOBOEEJWJTJPOPGGSBDUJPOTBOEEFDJNBMTNumber and Operations:$POOFDUJOHSBUJPBOESBUFUPNVMUJQMJ-DBUJPOBOEEJWJTJPOAlgebra:8SJUJOHJOUFSQSFUJOHBOEVTJOHNBUIFNBUJDBMFYQSFT-TJPOTBOEFRVBUJPOT

GRADE 7Number and Operations BOEAlgebra BOEGeometry:%FWFMPQ-JOHBOVOEFSTUBOEJOHPGBOEBQQMZJOHQSPQPSUJPOBMJUZJODMVEJOHTJNJMBSJUZMeasurementBOEGeometryBOEAlgebra:%FWFMPQJOHBOVOEFS-TUBOEJOHPGBOEVTJOHGPSNVMBTUPEFUFSNJOFTVSGBDFBSFBTBOEWPMVNFTPGUISFFEJNFOTJPOBMTIBQFTNumber and Operations BOE Algebra: %FWFMPQJOH BO VOEFS-TUBOEJOH PG PQFSBUJPOT PO BMM SBUJPOBM OVNCFST BOE TPMWJOHMJOFBSFRVBUJPOT

GRADE 8Algebra:"OBMZ[JOHBOESFQSFTFOUJOHMJOFBSGVODUJPOTBOETPMW-JOHMJOFBSFRVBUJPOTBOETZTUFNTPGMJOFBSFRVBUJPOTGeometry BOE Measurement: "OBMZ[JOH UXP BOE UISFFEJNFOTJPOBMTQBDFBOEHVSFTCZVTJOHEJTUBODFBOEBOHMFData AnalysisBOE Number and OperationsBOEAlgebra:"OBMZ[-JOHBOETVNNBSJ[JOHEBUBTFUT

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

For more information, visit www.wileyplus.com

WileyPLUS builds students confidence because it takes the guesswork out of studying by providing students with a clear roadmap:

what to do how to do it if they did it right

It offers interactive resources along with a complete digital textbook that help students learn more. With WileyPLUS, students take more initiative so youll have

greater impact on their achievement in the classroom and beyond.

WileyPLUS is a research-based online environment for effective teaching and learning.

Now available for

ALL THE HELP, RESOURCES, AND PERSONALSUPPORT YOU AND YOUR STUDENTS NEED!

www.wileyplus.com/resources

Technical Support 24/7FAQs, online chat,and phone support

www.wileyplus.com/support

Student support from an experienced student user

Collaborate with your colleagues,find a mentor, attend virtual and live

events, and view resources

2-Minute Tutorials and allof the resources you and yourstudents need to get started

Your WileyPLUS Account Manager, providing personal training

and support

www.WhereFacultyConnect.com

Pre-loaded, ready-to-use assignments and presentations

created by subject matter experts

StudentPartnerProgram

QuickStart

Courtney Keating/iStockphoto

MathematicsFor Elementary TeachersTENTH EDITION A C O N T E M P O R A R Y A P P R O A C H

Gary L. Musser t Blake E. Peterson t William F. Burger Oregon State University Brigham Young University

FMWileyPlus.indd 3 7/31/2013 2:14:09 PM

To:

Irene, my wonderful wife of 52 years who is the best mother our son could have; Greg, our son, for his inquiring mind; Maranda, our granddaughter, for her willingness to listen; my parents who have passed away, but always with me; and Mary Burger, my initial coauthor's daughter. G.L.M.

Shauna, my beautiful eternal companion and best friend, for her continual support of all my endeavors; my four children: Quinn for his creative enthusiasm for life, Joelle for her quiet yet strong confidence, Taren for her unintimidated ap-proach to life, and Riley for his good choices and his dry wit. B.E.P.

VICE PRESIDENT & EXECUTIVE PUBLISHER Laurie RosatonePROJECT EDITOR Jennifer BradySENIOR CONTENT MANAGER Karoline LucianoSENIOR PRODUCTION EDITOR Kerry WeinsteinMARKETING MANAGER Kimberly KanakesSENIOR PRODUCT DESIGNER Tom KulesaOPERATIONS MANAGER Melissa EdwardsASSISTANT CONTENT EDITOR Jacqueline SinacoriSENIOR PHOTO EDITOR Lisa GeeMEDIA SPECIALIST Laura AbramsCOVER & 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. MathematicsStudy and teaching (Elementary) I. Title. QA39.3.M87 2014510.24372dc23 2013019907

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1


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 PacificJournal of Mathematics, Canadian Journal of Mathematics, The Mathematics Association of America Monthly, the NCTMs The Mathematics Teacher, the NCTMs 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 GeometryA 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 universitys 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 NCTMs 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


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 Szemerdi. 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 Szemerdis significant proofs is found in a paper he wrote in 1975. This paper proved a famous conjecture that had been posed by Paul Erds and Paul Turn in 1936. Szemerdis 1975 paper and the Erds/Turn 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 Szemerdis paper was interesting, his greater contribution was that the tech-nique used in the proof has been subsequently used by many other mathematicians.

Now lets 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


viivii

1 Introduction to Problem Solving 2

2 Sets, Whole Numbers, and Numeration 42

3 Whole Numbers: Operations and Properties 84

4 Whole Number ComputationMental, 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 21.1 The Problem-Solving Process and Strategies 51.2 Three Additional Strategies 21

2 Sets, Whole Numbers, and Numeration 422.1 Sets as a Basis for Whole Numbers 452.2 Whole Numbers and Numeration 572.3 The HinduArabic System 67

3 Whole Numbers: Operations and Properties 843.1 Addition and Subtraction 873.2 Multiplication and Division 1013.3 Ordering and Exponents 116

4 Whole Number ComputationMental, Electronic,and Written 1284.1 Mental Math, Estimation, and Calculators 1314.2 Written Algorithms for Whole-Number Operations 1454.3 Algorithms in Other Bases 162

5 Number Theory 1745.1 Primes, Composites, and Tests for Divisibility 1775.2 Counting Factors, Greatest Common Factor, and Least

Common Multiple 190

6 Fractions 2066.1 The Set of Fractions 2096.2 Fractions: Addition and Subtraction 2236.3 Fractions: Multiplication and Division 233

7 Decimals, Ratio, Proportion, and Percent 2507.1 Decimals 2537.2 Operations with Decimals 2627.3 Ratio and Proportion 2747.4 Percent 283

8 Integers 3028.1 Addition and Subtraction 3058.2 Multiplication, Division, and Order 318

CONTENTS


ix

9 Rational Numbers, Real Numbers, and Algebra 3389.1 The Rational Numbers 3419.2 The Real Numbers 3589.3 Relations and Functions 3759.4 Functions and Their Graphs 391

10 Statistics 41210.1 Statistical Problem Solving 41510.2 Analyze and Interpret Data 44010.3 Misleading Graphs and Statistics 460

11 Probability 48411.1 Probability and Simple Experiments 48711.2 Probability and Complex Experiments 50211.3 Additional Counting Techniques 51811.4 Simulation, Expected Value, Odds, and Conditional

Probability 528

12 Geometric Shapes 54612.1 Recognizing Geometric ShapesLevel 0 54912.2 Analyzing Geometric ShapesLevel 1 56412.3 Relationships Between Geometric ShapesLevel 2 57912.4 An Introduction to a Formal Approach to Geometry 58912.5 Regular Polygons, Tessellations, and Circles 60512.6 Describing Three-Dimensional Shapes 620

13 Measurement 64413.1 Measurement with Nonstandard and Standard Units 64713.2 Length and Area 66513.3 Surface Area 68613.4 Volume 696

14 Geometry Using Triangle Congruence andSimilarity 71614.1 Congruence of Triangles 71914.2 Similarity of Triangles 72914.3 Basic Euclidean Constructions 74214.4 Additional Euclidean Constructions 75514.5 Geometric Problem Solving Using Triangle Congruence

and Similarity 765

15 Geometry Using Coordinates 78015.1 Distance and Slope in the Coordinate Plane 78315.2 Equations and Coordinates 79515.3 Geometric Problem Solving Using Coordinates 807


x16 Geometry Using Transformations 82016.1 Transformations 82316.2 Congruence and Similarity Using Transformations 84616.3 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 SiteResources for Technology Problems

eManipulativesSpreadsheet ActivitiesGeometers Sketchpad Activities

Technology TutorialsSpreadsheetsGeometers SketchpadProgramming in LogoGraphing Calculators

WebmodulesAlgebraic ReasoningChildrens LiteratureIntroduction to Graph Theory

Additional ResourcesGuide to Problem SolvingProblems for Writing/DiscussionResearch ArticlesWeb Links

VideosBook OverviewAuthor Walk-Through VideosChildrens Videos


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) TheMathematical Education of Teachers prepared by the Conference Board of the Mathematical Sciences, (2) the National Council of Teachers of Mathematics StandardsDocuments, 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, [email protected] E. PETERSON, [email protected]

Unique Content FeaturesNumber 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 childs 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 Topicsr Topic 1, Elementary Logic, may be used anywhere

in a course.

r 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 ThemesProblem 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 Geometers

xi

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

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 Geometers Sketchpad are dynamic representations of geomet-ric relationships that allow exploration. Exercises and problems that involve eManipulatives, spreadsheets, and Geometers Sketchpad sketches have been integrated into the problem sets throughout the text.

Course OptionsWe 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 1-7Basic course with logic: Topic 1, Chapters 17Basic course with informal geometry: Chapters 17,

12Basic course with introduction to geometry and mea-

surement: Chapters 17, 12, 13

Summary of Changes to theTenth Edition

r 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.

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

r Chapter 10 has been revised to include a discus-sion of recommendations by the GAISE document and the NCTM Principles and Standards for School Mathematics. These revisions include a discussion of steps to statistical problem solving. Namely, (1) formulate questions, (2) collect data, (3) organize and display data, (4) analyze and interpret data. These steps are then applied in several of the examples through the chapter.

r 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.

r 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.

r Childrens 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.

r 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.

r Childrens Literature and Reflections from Researchmargin notes have been revised/refreshed.

r 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.

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

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

The three dimensions of the cubecognitive levels, representational levels, and mathematical contentare 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).


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 Changesto the Tenth Editionr Student Page Snapshots have been updated.r 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.

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

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

r Mathematical Tasks have been integrated throughout.r Author Walk-Throughs videos have been made avail-

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

r Childrens videos, produced by Blake Peterson and available via QR codes, have been integrated through-out.

Key FeaturesProblem-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.


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, Geometers 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.


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.

Childrens 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.

Bla

ke E

. Pet

erso

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.

Childrens 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.


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 StudentsStudent Activities Manual with Discussion Questions for the Classroom This activity manual is designed to enhance student learning as well as to model effective classroom practices. Since many instructors are working with students to create a personalized journal, this edition of the manual is shrink-wrapped and three-hole punched for easy customization. This supplement is an extensive revi-sion of the Student Resoure Handbook that was authored by Karen Swenson and Marcia Swanson for the first six editions of this book.

ISBN 978-1-118-67904-3

Features Include:

r Hands-On Activities: Activities that help develop initial understandings at the concrete level.r Discussion Questions for the Classroom: Tasks designed to engage students with mathematical

ideas by stimulating communication.r Mental Math: Short activities to help develop mental math skills.r Exercises: Additional practice for building skills in concepts.r Directions in Education: Specially written articles that provide insights into major issues of the day,

including the Standards of the National Council of Teachers of Mathematics.r Solutions: Solutions to all items in the handbook to enhance self-study.r Two-Dimensional Manipulatives: Cutouts are provided on cardstock.

Prepared by Lyn Riverstone of Oregon State University

The ETA Cuisenalre Physical Manipulative Kit A generous assortment of manipulatives (including blocks, tiles, geoboards, and so forth) has been created to accompany the text as well as the Student Activity Manual. lt is available to be packaged with the text. Please contact your local Wiley representative for ordering information.

ISBN 978-1-118-67923-4

Student Hints and Solutions Manual for Part A Problems This manual contains hints and solutions to all of the Part A problems. It can be used to help students develop problem-solving profi-ciency in a self-study mode. The features include:


Preface xvii

r Hints: Give students a start on all Part A problems in the text.r Additional Hints: A second hint is provided for more challenging problems.r Complete Solutions to Part A Problems: Carefully written-out solutions are provided to model one correct solution.

Developed by Lynn Trimpe, Vikki Maurer,and Roger Maurer of Linn-Benton Community College.

ISBN 978-1-118-67925-8

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

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

r eManipulatives mirror physical manipulatives as well as provide dynamic representations of other mathematical situations. The goal of using the eManipulatives is to engage learners in a way that will lead to a more in-depth understanding of the concepts and to give them experience thinking about the mathematics that underlies the manipulatives.

Prepared by Lawrence O. Cannon, E. Robert Heal, and Joel Duffin of Utah State University,Richard Wellman of Westminster College, and Ethalinda K. S. Cannon of A415software.com.

This project is supported by the National Science Foundation.

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

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

Technology Tutorialsr The Geometers Sketchpad tutorial is written for those students who have access to the software and who are

interested in investigating problems of their own choosing. The tutorial gives basic instruction on how to use the software and includes some sample problems that will help the students gain a better understanding of the software and the geometry that could be learned by using it.

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

r The Spreadsheet Tutorial is written for students who are interested in learning how to use spreadsheets to investi-gate mathematical problems. The tutorial describes some of the functions of the software and provides exercises for students to investigate mathematics using the software.

Prepared by Keith Leatham, Brigham Young University.

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

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

Prepared by Keith Leatham, Brigham Young University.

r The Childrens Literature Webmodule provides references to many mathematically related examples of childrens 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 childrens literature in the classroom.

Prepared by Joan Cohen Jones, Eastern Michigan University.


xviii Preface

r The Introduction to Graph Theory Webmodule has been moved from the Topics to the companion Web site to save space in the book and yet allow professors the flexibility to download it from the Web if they choose to use it.

The companion Web site also includes:

r Links to NCTM Standardsr Links to Common Core Standardsr A Logo and TI-83 graphing calculator tutorialr Four cumulative tests covering material up to the end of Chapters 4, 9, 12, and 16r Research Article References: A complete list of references for the research articles that are mentioned in the

Reflection from Research margin notes throughout the book

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

r Opening Problem: an introductory problem to motivate the need for a strategy.r Solution/Discussion/Clues: A worked-out solution of the opening problem together with a discussion of the strategy

and some clues on when to select this strategy.

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

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

r Additional Practice Problems and Additional Mixed Strategy Problems: Sections that provide more practice for par-ticular strategies as well as many problems for which students need to identify appropriate strategies.

Prepared by Don Miller, who retired as a professorof 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 Geometers 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 youre doingall at one easy-to-use Web site.

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

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

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

Instructor Resource Manual This manual contains chapter-by-chapter discussions of the text material, student expectations (objectives) for each chapter, answers for all Part B exercises and problems, and answers for all of the even-numbered problems in the Guide to Problem-Solving.

Prepared by Lyn Riverstone, Oregon State UniversityISBN 978-1-118-67924-1


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.


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 ChattanoogaJuli Ratheal, University of Texas Permian BasinMarie Franzosa, Oregon State UniversityMary Beth Rollick, Kent State UniversityLinda Lefevre, SUNY Oswego

Reviewers for the Ninth Edition

Larry Feldman, Indiana University of PennsylvaniaSarah Greenwald, Appalachian State UniversityLeah Gustin, Miami University of Ohio, MiddletonLinda LeFevre, State University of New York, OswegoBethany Noblitt, Northern Kentucky UniversityTodd Cadwallader Olsker, California State University, FullertonCynthia Piez, University of IdahoTammy Powell-Kopilak, Dutchess Community CollegeEdel Reilly, Indiana University of PennsylvaniaSarah Reznikoff, Kansas State UniversityMary Beth Rollick, Kent State University

Ninth Edition Interviewees

John Baker, Indiana University of PennsylvaniaPaulette Ebert, Northern Kentucky UniversityGina Foletta, Northern Kentucky UniversityLeah Griffith, Rio Hondo CollegeJane Gringauz, Minneapolis Community CollegeAlexander Kolesnick, Ventura CollegeGail Laurent, College of DuPageLinda LeFevre, State University of New York, OswegoCarol Lucas, University of Central OklahomaMelanie Parker, Clarion University of PennsylvaniaShelle Patterson, Murray State UniversityCynthia Piez, University of IdahoDenise Reboli, Kings CollegeEdel Reilly, Indiana University of PennsylvaniaSarah Reznikoff, Kansas State UniversityNazanin Tootoonchi, Frostburg State University

Ninth Edition Focus Group Participants

Kaddour Boukkabar, California University of PennsylvaniaMelanie Branca, Southwestern CollegeTommy Bryan, Baylor UniversityJose Cruz, Palo Alto CollegeArlene Dowshen, Widener UniversityRita Eisele, Eastern Washington UniversityMario Flores, University of Texas at San AntonioHeather Foes, Rock Valley College

Mary Forintos, Ferris State UniversityMarie Franzosa, Oregon State UniversitySonia Goerdt, St. Cloud State UniversityRalph Harris, Fresno Pacific UniversityGeorge Jennings, California State University, Dominguez HillsAndy Jones, Prince Georges Community CollegeKarla Karstens, University of VermontMargaret Kidd, California State University, FullertonRebecca Metcalf, Bridgewater State CollegePamela Miller, Arizona State University, WestJessica Parsell, Delaware Technical Community CollegeTuyet Pham, Kent State UniversityMary Beth Rollick, Kent State UniversityKeith Salyer, Central Washington UniversitySherry Schulz, College of the CanyonsCarol Steiner, Kent State UniversityAbolhassan Tagavy, City College of ChicagoRick Vaughan, Paradise Valley Community CollegeDemetria White, Tougaloo CollegeJohn 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 UniversityElayne Bowman, University of OklahomaAnne Brown, Indiana University, South BendDavid C. Buck, ElizabethtownAlison Carter, Montgomery CollegeJanet Cater, California State University, BakersfieldDarwyn Cook, Alfred UniversityChristopher Danielson, Minnesota State University, MankatoLinda DeGuire, California State University, Long BeachCristina Domokos, California State University, SacramentoScott Fallstrom, University of OregonTeresa Floyd, Mississippi CollegeRohitha Goonatilake, Texas A&M International UniversityMargaret Gruenwald, University of Southern IndianaJoan Cohen Jones, Eastern Michigan UniversityJoe Kemble, Lamar UniversityMargaret Kinzel, Boise State UniversityJ. Lyn Miller, Slippery Rock UniversityGirija Nair-Hart, Ohio State University, NewarkSandra Nite, Texas A&M UniversitySally Robinson, University of Arkansas, Little RockNancy Schoolcraft, Indiana University, BloomingtonKaren E. Spike, University of North Carolina, WilmingtonBrian Travers, Salem StateMary Wiest, Minnesota State University, MankatoMark A. Zuiker, Minnesota State University, Mankato

Student Activity Manual Reviewers

Kathleen Almy, Rock Valley CollegeMargaret Gruenwald, University of Southern Indiana

xx

FMAcknowledgments.indd 20 7/31/2013 12:26:16 PM

Acknowledgments xxi

Kate Riley, California Polytechnic State UniversityRobyn Sibley, Montgomery County Public Schools

State Standards Reviewers

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

Reviewers

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

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

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

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

Questionnaire Respondents

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

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


xxii Acknowledgments

Gregory Davis, University of Wisconsin, Green BayJennifer Davis, Ulster County Community CollegeDennis De Jong, Dordt CollegeMary De Young, Hop CollegeLouise Deaton, Johnson Community CollegeShobha Deshmukh, College of Saint Benedict/St.

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

Jack Lombard, Harold Washington CollegeBetty Long, Appalachian State UniversityAnn Louis, College of the CanyonsC. A. Lubinski, Illinois State UniversityPamela Lundin, Lakeland CollegeCharles R. Luttrell, Frederick Community CollegeCarl Maneri, Wright State UniversityNancy Maushak, William Penn CollegeEdith Maxwell, West Georgia CollegeJeffery T. McLean, University of St. ThomasGeorge F. Mead, McNeese State UniversityWilbur Mellema, San Jose City CollegeClarence E. Miller, Jr. Johns Hopkins UniversityDiane Miller, Middle Tennessee State UniversityKen Monks, University of ScrantonBill Moody, University of DelawareKent Morris, Cameron UniversityLisa Morrison, Western Michigan UniversityBarbara Moses, Bowling Green State UniversityFran Moss, Nicholls State UniversityMike Mourer, Johnston Community CollegeKatherine Muhs, St. Norbert CollegeGale Nash, Western State College of ColoradoT. Neelor, California State UniversityJerry Neft, University of DaytonGary Nelson, Central Community College, Columbus CampusJames A. Nickel, University of Texas, Permian BasinKathy Nickell, College of DuPageSusan Novelli, Kellogg Community CollegeJon ODell, Richland Community CollegeJane Odell, Richland CollegeBill W. Oldham, Harding UniversityJim Paige, Wayne State CollegeWing Park, College of Lake CountySusan Patterson, Erskine College (retired)Shahla Peterman, University of MissouriGary D. Peterson, Pacific Lutheran UniversityDebra Pharo, Northwestern Michigan CollegeTammy Powell-Kopilak, Dutchess Community CollegeChristy Preis, Arkansas State University, Mountain HomeRobert Preller, Illinois Central CollegeDr. William Price, Niagara UniversityKim Prichard, University of North CarolinaStephen Prothero, Williamette UniversityJanice Rech, University of NebraskaTom Richard, Bemidji State UniversityJan Rizzuti, Central Washington UniversityAnne D. Roberts, University of UtahDavid Roland, University of Mary HardinBaylorFrances Rosamond, National UniversityRichard Ross, Southeast Community CollegeAlbert Roy, Bristol Community CollegeBill Rudolph, Iowa State UniversityBernadette Russell, Plymouth State CollegeLee K. Sanders, Miami University, HamiltonAnn Savonen, Monroe County Community CollegeRebecca Seaberg, Bethel CollegeKaren Sharp, Mott Community CollegeMarie Sheckels, Mary Washington CollegeMelissa Shepard Loe, University of St. ThomasJoseph Shields, St. Marys College, MN


Acknowledgments xxiii

Lawrence Shirley, Towson State UniversityKeith Shuert, Oakland Community CollegeB. Signer, St. Johns UniversityRick Simon, Idaho State UniversityJames Smart, San Jose State UniversityRon Smit, University of PortlandGayle Smith, Lane Community CollegeLarry Sowder, San Diego State UniversityRaymond E. Spaulding, Radford UniversityWilliam Speer, University of Nevada, Las VegasSister Carol Speigel, BVM, Clarke CollegeKaren E. Spike, University of North Carolina, WilmingtonRuth Ann Stefanussen, University of UtahCarol Steiner, Kent State UniversityDebbie Stokes, East Carolina UniversityRuthi Sturdevant, Lincoln University, MOViji Sundar, California State University, StanislausAnn Sweeney, College of St. Catherine, MNKaren Swenson, George Fox CollegeCarla Tayeh, Eastern Michigan UniversityJanet Thomas, Garrett Community CollegeS. Thomas, University of OregonMary Beth Ulrich, Pikeville CollegeMartha Van Cleave, Linfield CollegeDr. Howard Wachtel, Bowie State UniversityDr. Mary Wagner-Krankel, St. Marys UniversityBarbara Walters, Ashland Community CollegeBill Weber, Eastern Arizona CollegeJoyce Wellington, Southeastern Community CollegePaula White, Marshall UniversityHeide G. Wiegel, University of GeorgiaJane Wilburne, West Chester UniversityJerry Wilkerson, Missouri Western State CollegeJack D. Wilkinson, University of Northern IowaCarole Williams, Seminole Community CollegeDelbert Williams, University of Mary HardinBaylorChris Wise, University of Southwestern LouisianaJohn L. Wisthoff, Anne Arundel Community College (retired)Lohra Wolden, Southern Utah UniversityMary Wolfe, University of Rio GrandeVernon E. Wolff, Moorhead State University

Maria Zack, Point Loma Nazarene CollegeStanley L. Zehm, Heritage CollegeMakia Zimmer, Bethany College

Focus Group Participants

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

Content Connections Survey Respondents

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

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 Mussers 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 Geometers Sketchpad Tutorial, to Joan Cohen Jones for the Childrens 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.


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 withJohn Wiley would have been proud of them.

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

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


1There 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 ofRecommendations 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 Childrens 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 Childrens 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

FMANotetoOurStudents.indd 1 8/1/2013 8:39:16 PM

2G eorge Plya 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 in1942.

A

P/W

ide

Wor

ld P

hoto

s

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.

Plya 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 Plya 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 cometry to disclose the general pattern that lies behind the present concrete situation.

9. Do not give away your whole secret at oncelet the students guess before you tell itlet 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 SOLVINGGeorge PlyaThe Father of Modern Problem Solving

c01.indd 2 7/30/2013 2:36:04 PM

3Problem-SolvingStrategies 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 ProblemPlace 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.

c01.indd 3 7/30/2013 2:36:05 PM

AUTHOR

WALK-THROUGH

4

I N T R O D U C T I O NOnce, at an informal meeting, a social scientist asked a mathematics professor, Whats 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 NCTMs 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 NCTMs 2000 Principles and Standards for School Mathematics identified problem solving as one of the processes by which all mathematics should be taught.

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

Key Concepts from the NCTM Principles and Standards for School Mathematics

r PRE-K-12PROBLEM SOLVINGBuild new mathematical knowledge through problem solving.Solve problems that arise in mathematics and in other contexts.Apply and adapt a variety of appropriate strategies to solve problems.Monitor and reflect on the process of mathematical problem solving.

Key Concepts from the NCTM Curriculum Focal Points

r KINDERGARTEN: Choose, combine, and apply effective strategies for answering quantitative questions.r GRADE 1: Develop an understanding of the meanings of addition and subtraction and strategies to solve such

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

r GRADE 3: Apply increasingly sophisticated strategies to solve multiplication and division problems.r GRADE 4 AND 5: Select appropriate units, strategies, and tools for solving problems.r GRADE 6: Solve a wide variety of problems involving ratios and rates.r GRADE 7: Use ratio and proportionality to solve a wide variety of percent problems.

Key Concepts from the Common Core State Standards for Mathematics

r ALL GRADESMathematical 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.

c01.indd 4 7/30/2013 2:36:05 PM

Section 1.1 The Problem-Solving Process and Strategies 5

Plyas Four StepsIn 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 solvers 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 Plya, devoted much of his teaching to helping students become better problem solvers. His major contribution is what has become known as Plyas four-step process for solving problems.

Step 1 Understand the Problem

r Do you understand all the words?r Can you restate the problem in your own words?r Do you know what is given?r Do you know what the goal is?r Is there enough information?r Is there extraneous information?r 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 ResearchMany 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 Practice1)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 Practice1)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 classmates. Are there any similarities in your approaches to solving this problem?

Lins 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 Lins 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.

c01.indd 5 7/30/2013 2:36:05 PM

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

r Implement the strategy or strategies that you have chosen until the problem is solved or until a new course of action is suggested.

r 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!)

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

Step 4 Look Back

r Is your solution correct? Does your answer satisfy the statement of the problem?r Can you see an easier solution?r 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 Plyas 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 Plyas four steps as well as the strategies and clues as you continue to develop your expertise in solving problems.

Common Core Grades K-12 (MathematicalPractice1)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 (MathematicalPractice1)Mathematically proficient stu-dents monitor and evaluate their progress and change course if necessary.

Reflection from ResearchResearchers 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 StandardInstructional 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.

c01.indd 6 7/30/2013 2:36:06 PM

7From Chapter 6, Lesson Problem Solving from My Math, Volume 1 Common Core State Standards, Grade 2, copyright 2013 by McGraw-Hill Education.

c01.indd 7 7/30/2013 2:36:09 PM

8 Chapter 1 Introduction to Problem Solving

Problem-Solving StrategiesThe remainder of this chapter is devoted to introducing several problem-solving strategies.

Guess and Test

ProblemPlace 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


Tear off six pieces of paper and mark the numbers 1 through 6 on them and then try combinations until one works.


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


Rather than randomly moving the numbers around, begin by placing the smallest numbersnamely, 1, 2, 3in the corners. If that does not work, try increasing the numbers to 1, 2, 4, and so on.


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


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


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

c01.indd 8 7/30/2013 2:36:10 PM

Section 1.1 The Problem-Solving Process and Strategies 9

be in a corner. Finally, suppose that 3 is in a corner. Then we must satisfy Figure 1.5. However, only 5 4 9+ = of the remaining numbers. Thus, if there is a solu-tion, 4, 5, and 6 will have to be in the corners (Figure 1.6). By placing 1 between 5 and 6, 2 between 4 and 6, and 3 between 4 and 5, we have a solution.

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