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Creative Problem Solving in School Mathematics

About the Author

Dr. George Lenchner was formerly the Director of Mathematics for the Valley Stream CentralHigh School District, and Consultant to the three associated Elementary School Districts inLong Island, NY. He was the founder of the Nassau County Interscholastic MathematicsLeague (Mathletes) in 1955. He also organized the Mathematical Olympiads for Elementaryand Middle Schools (MOEMS) in 1979 and served as its Executive Director until hisretirement in 1996. He currentlv continues to serye MOEMS as its Executive DirectorEmeritus.

Dr. Lenchner is the author of many mathematics textbooks and articles appearing in nationalpublications. He brings to this book over 50 years of experience as a mathematics teacher,supervisor, teacher-trainer, and creator of problems. Harvard University has honored himfor Outstanding and Distinguished Secondary School Teaching.

PublisherMathematical Olympiads for Elementary and Middle Schools, Inc., Bellmore, NY Il7l0Web site: www.moems.org Telephone: 866-781-24II

ConsultantRichard S. Kalman

Layout, Graphics, and Cover DesignRichard Kalman, MOEMSCover photograph by James L. Ballard

PrinterTobay Printing Company. Inc., Copiague, NY

Copyright O 2005 by George Lenchner

A11 rights reserved. No part of this work may be reproduced or ffansmitted in any form or by any means, electronic ormechanical, including photocopying and recording, or by any information or retrieval system, except as may be expresslypermitted by the Copyright Act of 1976 or in writing by the Publisher. Requests for permission should be addressed to:George Lenchner, c/o MOEMS, 2154 Bellmore Avenue, Bellmore, NY 11710-5645.

First Edition, printed in U.S.A., published in New York, N.Y. 1983.Revised and expanded edition, printed in U.S.A., published in Bellmore, N.Y. 2005.

ISBN No. l-882144-10-4Library of Congress Control Number: 200 9n296

George Lenchner

Gontents,NTRODUCTTON

About the Author .......... iiPreface

...................... vii

PART A. TEACHING PROBLEM SOLV/,NG.Wuer E Paoateu SowtNc? .......2Uswe e Foua-Srep Meruoo.... ..................... g2,1 Understanding the Problem .................... 32.2 Planning How To Solve The Problem .............. .........42.3 Carrying Out The Plan ..........42.4 Looking Back ....... 5Cuooswa Paoateus ...................7EvntunnNa Pnoateus ...............8PaeseNnNe Paoateus ...............95.1 The Chalkboard ..................... I5.2 The Overhead Projector ........ I5.3 Duplicated Sheets............... .................. 105.4 Oral Presentation ............... ................... 10HetnNe Sruoerurs ..... 11Uswa Cntcutnroas euo CotvtpurERs......... ................. 11

PART B. SOME PROBLEM SOLVING S7RATEG/,E5................_......731. Dnnwwe t Prcruae oa Dtnaanu ................. 142. Menua nN Onaeuzeo bsr.......... ............... lo3. MnrcNa n Tnate

...... 184. SotwNa t Swptra Retnreo PaoateM ........205. FrNowa e Penenn

...226. Guessrrvc nuo Cuecnua .............247. ExprnueNnNc..........

................ 268. Acnxa Our Tue PaoateM .........299. WoaruNa Bncrcwenos ................29

10. WnmNa eu Eaunnott ...............3111. CunNawa Youn Powr or Vnw.. ................ 3312. Mrscrunwen ...........35

1.2.

3.4.5.

6.7.

F$Itt

Creative Problem Solving in School Mathematics

PART C. SOME TOPICS IN PROBLEM SOLyTTVG ,.a.......,...............'371. Nunraea Pn::eaNs .'.38

'1.1 Addition Patterns .........""....381.2 Multiplication Patterns ".."'-' 421.3 More Addition Patterns ."-'--' 441.4 Unusual Patterns .'.'.....".'.." 461.5 Patterns And Sums........ """ 49

2. Fncroas Auo MustpLEs......... ..'. 522.1 Factors '.-'-'..".'." 522.2 Factors and Primes '."...'.....542.3 Greatest Common Factor (GCF) -'.'."'-' 572.4 Least Common Multiple (LCM) ".'.'..'....60

3. Drvtstsurv .........'.'...653.1 Divisibility By 2, 5,10, and 100 ........... .................... 653.2 Divisibility Principle For Sums And Differences .'."."' .'.""....".' 673.3 Divisibility by Powers ot 2........... '......'... 683.4 Divisibility by 3 and 9............... ""."'.""-703.5 Divisibility by 11 ............. .....:'......'.". --.."713.6 Combined Divisibility............ -""."".."".743.7 Divisibility by 7,11, and 13 '-""..'.".-"".'75

4. FnecrtoNs '....'..'...'.774.1 Unit Fractions ............. "".'..'774.2 Complex Fractions ............... "'-"""'.'.'.-794.3 Extended Finite Fractions ..."'.'."".'...... 814.4 Fractional Parts .'.'...............82

5. Geonterny nno MensuaEMENT ..'.' 845.1 Squares and Rectangles........... ..'.".".'. 845.2 Triangles............. .................875.3 Circles ............"" 895.4 Perimeter............. ................ 925.5 Circumference .....................945.6 Area of Rectangles and Squares .......... 965] Area of Circles ..................... 985.8 Geometric Patterns............... .'..'.."""' 101

6. TnntNs, Booxs, Ctocxs, evo Tnuas ......... 1036.1 Motion Problems ........'....'. 1036.2 Book problems .................. 1056'3 work Problems """"""""" 1076.4 Clock Problems .'..'..""'..... 1096.5 Related Problems .'.'...'......'111

George Lenchner

7. Loarc ..... lls

7.1 Cryptarithms............... ....... 115

7.2 Certainty Prob1ems............... ....."......... 118

7.3 Venn Diagram Problems ...1207.4 Whodunits............

..............124

SOLUTIOA'SPART D' SOLUTIO,wS 70 PART A PROBLEMS

'..'.'.'..'1.....'...'."...732SOLUTIO,'\'S 7(, PART B PROBLEMS ,...r... | 34

1 . DanwNa e Prcrune oR A DTAGRAM .............. .............. l g42. MnrcNa nN OacnNzeo Lrcr.......... ............ lgs3. MenNa e Teare

... l 974. SotwNa e Swprcn Rrtnreo Paoateu ...... l7g5. Fttt;'wa t Pntenn .l7g6. GuEsslvc nNo CuecnNG.............. ............ l4l7. ExprnueNrNc.......... .............. 1428. AcnNa Our rue Paoarcu... ... 1429. WoanNe Bncxwnnos ............. 143

10. WnmNc eN EauenoN ............. 14411. CunNawa Youa Portr or Vpw.. ............. 14512. MrscetteNte

.......... 140

PART E. SOLUTTOTVS TO PART C PROBLEMS...... ,....,,.,.1501. Nulara Paoarcus

. lS01.1 Addition Patterns

............... 1501.2 Multiplication Patterns ....... 1511.3 More Addition Patterns

......1Sz1.4 Unusual Patterns

............... 1591.5 Patterns and Sums ............ 156

2. Fncrons nuo MurnptEs............ ............... 160

2.1 Factors ............. 160

2.2 Factors and Primes ........... 161

2.3 Greatest Common Factor ................... 1622.4 Least Common Multiple .... 164

3. DtwsBturY ............. 1663.1 Divisibility by 2,5,10, 100 ................... 1663.2 Divisibility Principle for Sums and Differences ......... ..............1673.3 Divisibility by Powers ot 2........... ......... 1693.4 Divisibility by 3 and L.............. ............ 1703.5 Divisibility by 11 ............. ....1723.6 Combined Divisibility............ ............... 1733.7 Divisibility by 7, 11, and 13 ............. ..... 179

Creative Problem Solving in School Mathematics

4. Fnncnous ............... 1764.1 Unit Fractions ............. .......1764.2 Complex Fractions ............... ............... 1794.3 Extended Finite Fractions ................... 1804.4 Fractional Parts ................. 181

5. Geouerav nNo MensuaEMENT .. 1835.1 Squares and Rectangles........... .......... 1835.2 Triangles..,.......... ............... 1845.3 Circles .............. 1855.4 Perimeter............. .............. 1875.5 Circumference ................... 1905.6 Area of Rectangles and Squares ........ 1925.7 Area of Circles ................... 1955.8 Geometric Patterns............... .............. 198

6. TnnNs, Booxs, Ctocxs, nNa Tnxas .........2006.1 Motion Problems ...............2006.2 Book Problems ..................2016.3 Work Problems ..................2036.4 Clock Problems .................2046.5 Related Problems ..............206

7. Loarc .....2097.1 Cryptarithms............... .......2097.2 Certainty Prob1ems............... ...............2117.3 Venn Diagram Problems ...2127.4 Whodunits............ ..............216

PART F. APPEND'CES 221AppeNorx 1: Besrc lxroaunnox .... 222AppeNorx 2: Auete-Meesunes w PotyaoNs .....227Solutions

.................294Notes .....297

AppeNotx 3: PwunaoREAN Tueonem ..............298Solutions .................246

AppeNorx 4: Wonrcxa Wrn ExponevTs ............ ...............249Solutions .................254AppeNorx 5: Jusnrwtua Sor',te DMsBttw Rurcs ............257Solutions .................262AppeNotx 6: Seouexces e'vo Semes ...............264Solutions .................273

PART G, INDEX

George Lenchner

Preface

Creative Problem Solving in School Mathematics is a problem solving handbook forteachers of mathematics, parents, students, and other interested people. Although it waswritten primarily for elementary and middle schools, part of the material in the book is alsoappropriate at the secondary level.

The writing of this book was inspired by an in-service course designed for elementaryand middle school teachers. The purpose of the course was to acquaint the participatingteachers with rich and exciting problem solving experiences related to the core mathematicsof the school curriculum. The teachers found that solving interesting and significant math-ematical problems enhanced their interest and curiosity in mathematics, and they learnedthat by approaching problem solving creatively in the mathematics classroom they couldsimilarly arouse the interest and curiosity of their students.

The basic text of Creative Problem Solving in School Mathematics consists of threeparts. Part A is a brief discussion of teaching techniques that have been found to be espe-cially effective in the introduction of problem soiving strategies. Part B highlights somestrategies that are commonly used in school mathematics and provides some practice prob-lems for each strategy. In Part C, problern solving is examined in relation to many standardtopics of the school mathematics curriculum, and the discussion of each topic is followedby a comprehensive set of reiated practice problems. At the end of the book, six appendicesprovide additional material that supplement the topics in Part C. An index is also included.

The purpose of Creative Problem Solving in School Mathematics is to help teachersand parents improve each student's ability to solve problems. However, it is a well-knownsaying that we learn by doing. Therefore, this book affords teachers, parents, and studentsan opportunity to learn more about problem solving strategies by trying out some new ap-proaches and new techniques for themselves. Through these experiences it is hoped thatthey will be able to bring to their classrooms and homes some fresh insights and ideas.

Richard Kalman, Executive Director of MOEMS, voluntarily created and supervisedevery element of the production of the book. In addition, he supplied many suggestionsrelative to content and problems. The result is an unusually attractive appearance and aprofessional book of the highest order.

I wish to recognize my good friend and distinguished colleague, Lawrence J.Zimmerman for his significant contributions to this book. His excellent advice and recom-mendations enriched the content of this book beyond expectations.

Finally, I gratefully acknowiedge the thoughtful contributions of Gilbert W. Kessler,Wendy Hersh, and Elliott Bird, whose input enhanced several sections of the book, and ofCurt Boddie, Michael Carlson, Sandy Cohen, Grant Duffrin, John Lufrano, Betty Minson,Lori Nimmo, Cheryl Novick, Eric O'Brien, and Dot Steinert for their diligent and carefulreviewing of the manuscript.

George Lenchner

January,2005