Rediscovering Mathematics: You Do the Math

240 Pages·2011·2.165 MB·English

by  Shai Simonson

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Rediscovering Mathematics
You Do the Math
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c 2011bytheMathematicalAssociationofAmerica,Inc.
(cid:13)
LibraryofCongressCatalogCardNumber2010939888
PrinteditionISBN978-0-88385-770-0
ElectroniceditionISBN978-0-88385-912-4
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
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Rediscovering Mathematics
You Do the Math
Shai Simonson
StonehillCollege
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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CommitteeonBooks
FrankFarris,Chair
ClassroomResource MaterialsEditorialBoard
GeraldM.Bryce,Editor
MichaelBardzell
WilliamC.Bauldry
DianeL.Herrmann
BarbaraEReynolds
WayneRoberts
SusanG.Staples
PhilipD.Straffin
HollyS.Zullo
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CLASSROOMRESOURCEMATERIALS
Classroom Resource Materials is intended to provide supplementary classroom material
forstudents—laboratoryexercises,projects,historicalinformation,textbookswithunusual
approachesforpresentingmathematicalideas,careerinformation,etc.
101CareersinMathematics,2ndeditioneditedbyAndrewSterrett
Archimedes:WhatDidHeDoBesidesCryEureka?,ShermanStein
Calculus: An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz,
StanSeltzer,JohnMaceli,andEricRobinson
TheCalculusCollection:AResource forAPandBeyond,editedbyCarenL.Diefenderfer
andRogerB.Nelsen
CalculusMysteriesandThrillers,R.GrantWoods
ConjectureandProof,Miklo´sLaczkovich
CounterexamplesinCalculus,SergiyKlymchuk
CreativeMathematics,H.S.Wall
EnvironmentalMathematicsintheClassroom,editedbyB.A.FusaroandP.C.Kenschaft
ExcursionsinClassicalAnalysis:PathwaystoAdvancedProblemSolvingandUndergrad-
uateResearch, byHongweiChen
ExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.Weller
GeometryFromAfrica:MathematicalandEducationalExplorations,PaulusGerdes
HistoricalModulesfortheTeachingandLearningofMathematics(CD),editedbyVictor
KatzandKarenDeeMichalowicz
IdentificationNumbersandCheckDigitSchemes,JosephKirtland
InterdisciplinaryLivelyApplicationProjects,editedbyChrisArney
InverseProblems:ActivitiesforUndergraduates,CharlesW.Groetsch
LaboratoryExperiencesinGroupTheory,EllenMaycockParker
Learn from the Masters, Frank Swetz, John Fauvel, OttoBekken, Bengt Johansson, and
VictorKatz
MathMadeVisual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaand
RogerB.Nelsen
OrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.Sa´nchez
OvalTrackandOtherPermutationPuzzles,JohnO.Kiltinen
APrimerofAbstractMathematics,RobertB.Ash
ProofsWithoutWords,RogerB.Nelsen
ProofsWithoutWordsII,RogerB.Nelsen
RediscoveringMathematics:YouDotheMath,ShaiSimonson
SheDoesMath!,editedbyMarlaParker
SolveThis:MathActivitiesforStudentsandClubs,JamesS.Tanton
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StudentManualforMathematicsforBusiness DecisionsPart1:ProbabilityandSimula-
tion, DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic
StudentManualforMathematicsforBusiness Decisions Part2:CalculusandOptimiza-
tion,DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic
TeachingStatisticsUsingBaseball,JimAlbert
VisualGroupTheory,NathanC.Carter
Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go,
AnnalisaCrannell,GavinLaRose,ThomasRatliff,ElynRykken
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1MAA FAX:[phone]
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Acknowledgments
Thankstothedozensofteachersandhundredsofstudentswhoinspiremypassionfordo-
ingandteachingmathematics.Thankstoallmycolleagueswhocontributedtothiseffort.
FernandoGouveacontributedtothechapteronHowtoReadMathematics.TimWoodcock
andRalphBravacoreadthroughmanypartsofthemanuscript,offeringexcellentsugges-
tions, and correcting many of my errors. Bruce Bukiet, Tara Holm, Chris Page, Klaus
Peters,andAndreaSimonsonreviewedsectionsofearlydrafts.NeilComins,ChetRaymo
and Elizabeth Loew shared their experience and wisdom about writing and publishing.
HendrikLenstra,ArtBenjamin,andRonRivestprovidedspecificmathematicalexpertise,
andVictorKatzintroducedmetothemathematicsofLevibenGershon.JerryBryceand
theanonymousCRMreviewboardattheMathematicalAssociationofAmericaprovided
many good suggestionsand wise guidance. Thanks also to the expert MAA editingand
productionteamofDonAlbers,ElainePedreira,CarolBaxter,andespeciallyBevRuedi.
Forsevenyears,thestaffattheSouthAreaSolomonSchechterDaySchool(SASSDS)
courageously gave me carte blanche to design and test a curriculum that diverges from
commonpractice.Muchofthisbookgrewoutofthatexperience.Mystudentsthereblessed
mewitheffort,dedication,respect,andcontinuingrelationships.
TheadministrationatStonehillCollegegrantedmeayear-longsabbaticaltowritethis
book,andtheEducationdepartmentencouragedtheuseofthebookinthedesignofanew
mathcourseforpre-serviceK–12teachers,calledMathematicalReasoningforEducation
(MA143).ThecoursefollowstheMassachusettsSTEM(science,technology,engineering,
andmathematics)initiative,whichemphasizestheNationalCouncilofTeachers ofMath-
ematics(NCTM)recommended“student-centered”and“active-learning”approaches.My
students in this course create an atmosphere of challenge and excellence, providing me
withvaluablefeedback.
Finally,thankstowifeAndrea,andmychildrenZosh,Yair,andYonawhohumormy
passionsandfeedmewithlove.
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TomywifeandcompanionAndrea,alwayswillingtoshare
my joy in solving puzzles large and small, who taught me
that love is a puzzle best enjoyed unsolved.
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Contents
Acknowledgements vii
AGuidefortheReader xiii
Introduction:HowtoReadMathematics xvii
1 AnExampleofMathematicalWriting—TheBirthdayParadox . . . . . . . . xxii
2 OurReaderTacklestheBirthdayParadox . . . . . . . . . . . . . . . . . . . xxiii
3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
1 MathematicalDiscoveryintheClassroom 1
1.1 ASimpleLesson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 TheMathTeacher asConductor. . . . . . . . . . . . . . . . . . . . . . . 2
1.3 TheEffectiveTeacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Don’tReachforYourCalculator(Yet) 13
2.1 AMagicTrick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 CleverCalculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 PythagoreanTriples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 HaveAnotherPieceofPie,Zeno? 31
3.1 ASimplerWay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Euclid,Proofs,andWritingMathematics . . . . . . . . . . . . . . . . . . 35
3.3 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 ThinkingLikeaMathematician—LessonsfromaMedievalRabbi 41
4.1 RabbiLevibenGershonandhisSums . . . . . . . . . . . . . . . . . . . 44
4.2 TheNumberofSquaresandRectanglesinaGrid . . . . . . . . . . . . . 52
4.3 TheTrianglePuzzle—AModelMathematicalProblem . . . . . . . . . . 55
4.4 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5 WhatisMathematicsGoodFor? 63
5.1 KnockHockey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2 Basketball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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