big ideas geometry and data analysis · we will be giving you other definitions of geometry as we...

BigIdeasinMathematicsforFutureTeachers

BigIdeasinGeometryandData

JohnBeam,JasonBelnap,EricKuennen,AmyParrott,CarolE.Seaman,andJenniferSzydlik

(UpdatedSummer2019)

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This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.

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Hey!Readthis.Itwillhelpyouunderstandthebook.

Theonlywaytolearnmathematicsistodomathematics. PaulHalmos

Thisbookwaswrittentopreparefutureteachersforthemathematicalworkofteaching.Thefocusofthismoduleisgeometryanddata–andthisdomainencompassesmanydeepandwonderfulmathematicalideas.Thistextisnotintendedtohelpyourelearnyourelementarymathematics;itisaboutteachingyoutothinklikeamathematiciananditisabouthelpingyoutothinklikeamathematicsteacher.TheNationalCouncilofTeachersofMathematics(NCTM,2000)writes:

Teachersneedseveraldifferentkindsofmathematicalknowledge–knowledgeaboutthewholedomain;deep,flexibleknowledgeaboutcurriculumgoalsandabouttheimportantideasthatarecentraltotheirgradelevel;knowledgeaboutthechallengesstudentsarelikelytoencounterinlearningtheseideas;knowledgeabouthowtheideascanberepresentedtoteachthemeffectively;andknowledgeabouthowstudents’understandingcanbeassessed(p.17).

Wearegoingtoworktowardthesegoals.(Readthemagain.Thisisatallorder.Inwhichareasdoyouneedthemostwork?)Throughoutthisbook,wewillaskyoutoconsiderquestionsthatmayariseinyourelementaryclassroom.

Isasquarealwaysarectangle?

Whatdoesthisnumbercalledprepresent?Whatdoesitmeantomeasure“area”?

Cantworectangleswiththesameareahavedifferentperimeters?

Whatissospecialaboutrighttriangles? IfIbuya5-inchpizzaandmyfriendbuysa10-inchpizza,doessheget twiceasmuchtoeat?

HowmanypeopledoyouneedtosampleinordertoreasonablyfindouthowmanyhoursofTVsecondgraderswatchinaweek?

Whatdoesitmeantosaythattheprobabilityofaheadwhenflippingacoinis½?

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Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Weviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjustasimportantasanycontentyouteach.Whenyoustandbeforeyourclass,youarearepresentativeofthemathematicalcommunity;wewillhelpyoutobeagoodone.Noonecandothisthinkingforyou.Mathematicsisn’tasubjectyoucanmemorize;itisaboutwaysofthinkingandknowing.Youneedtodoexamples,gatherdata,lookforpatterns,experiment,drawpictures,think,tryagain,makearguments,andthinksomemore.Thebigideasofgeometryarenotalwayseasy–buttheyarefundamentallyimportantforyourstudentstounderstandandsotheyarefundamentallyimportantforyoutounderstand.EachsectionofthisbookbeginswithaClassActivity.Theactivityisdesignedforsmall-groupworkinclass.Someactivitiesmaytakeyourclassaslittleas20minutestocompleteanddiscuss.Othersmaytakeyoutwoormoreclassperiods.TheReadandStudy,ConnectionstoTeaching,andHomeworksectionsarepresentedwithinthecontextoftheactivityideas.Nosolutionsareprovidedtoactivitiesorhomeworkproblems–youwillhavetosolvethemyourselves.ThemathematicscontentinthisbookpreparesyoutoteachtheCommonCoreStateStandardsforMathematicsforgradesK-8.Thesearethestandardsthatyouwilllikelyfollowwhenyouareanelementaryteacher,sowewillhighlightaspectsofthemthroughoutthetext.Inorderforyoutoseehowthemathematicalworkyouaredoingappearsintheelementarygrades,wehavemadeexplicitconnectionstoBridgesinMathematicsfromTheMathLearningCenter.ThisistheonlineelementarygradesmathematicscurriculumadoptedbytheOshkoshAreaSchoolDistrict.Youwilloftenbeaskedtogotothesitebridges.mathlearningcenter.orgtoreadordoproblems.Yourinstructorwillprovideyouwithacodesothatyoucanaccessthesematerials. Allourbest,

John,Jason,Eric,Amy,Carol&Jen

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TableofContents

Tobeateacherrequiresextensiveandhighlyorganizedbodiesofknowledge.Shulman,1985,p.47

ChapterOne.......................................................................................................9ClassActivity1: TrianglePuzzle.......................................................................................10

WhatisGeometry................................................................................................................11

NatureofMathematicalObjects..........................................................................................11

MathematicalCommunication.............................................................................................13

ClassActivity2: DefiningMoments.................................................................................17

RoleofDefinitions................................................................................................................18

ParallelandPerpendicularLines..........................................................................................19

ClassActivity3: GetitStraight........................................................................................26

LanguageofMathematics....................................................................................................29

DeductiveversusInductiveReasoning.................................................................................29

IdeaofAxioms......................................................................................................................30

MakingConvincingMathematicalArguments.....................................................................32

ClassActivity4: AlltheAngles.........................................................................................39

MeasuringAnglesinDegrees...............................................................................................39

RegularPolygons..................................................................................................................41

ClassActivity5: ALogicalInterlude.................................................................................45

ConverseandContrapositive...............................................................................................46

SummaryofBigIdeas........................................................................................................49

ChapterTwo.....................................................................................................50ClassActivity6: MeasureforMeasure............................................................................51

StandardandNonstandardUnits.........................................................................................53

ApproximationandPrecision...............................................................................................54

ClassActivity7: Part1Triangulating................................................................................57

ClassActivity7: Part2AreaEstimation...........................................................................58

IdeaofArea..........................................................................................................................59

CoveringwithUnitSquares..................................................................................................60

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ClassActivity8: FindingFormulas....................................................................................64

MakingSenseofAreaFormulas...........................................................................................65

ClassActivity9: TheRoundUp........................................................................................70

Whatisπ?............................................................................................................................72

InscribedPolygons................................................................................................................74

ClassActivity10: PlayingPythagoras...............................................................................75

ProofsofthePythagoreanTheorem....................................................................................77

GeometricandAlgebraicRepresentations...........................................................................78

SummaryofBigIdeas........................................................................................................80

ChapterThree..................................................................................................81ClassActivity11: StrictlyPlatonic(Solids)........................................................................82

RegularPolyhedra................................................................................................................83

SpatialReasoning.................................................................................................................84

ClassActivity12: PyramidsandPrisms............................................................................87

CountingVertices,Edges,andFaces....................................................................................87

ClassActivity13: SurfaceArea........................................................................................90

IdeaofSurfaceArea.............................................................................................................91

ClassActivity14: NothingButNet...................................................................................94

NetsforCylindersandPyramids..........................................................................................94

ClassActivity15: BuildingBlocks.....................................................................................95

IdeasofVolume....................................................................................................................97

VolumesofPrismsandCylinders.........................................................................................97

ClassActivity16: VolumeDiscount..................................................................................99

Capacity..............................................................................................................................101

VolumesofPyramids,Cones,andSpheres........................................................................102

ClassActivity17: VolumeChallenge..............................................................................106

BuildingModelstoSpecification........................................................................................106

SummaryofBigIdeas......................................................................................................109

ChapterFour..................................................................................................110ClassActivity18: Slides.................................................................................................111

RigidMotions.....................................................................................................................112

Translations........................................................................................................................112

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ClassActivity19: Turn,Turn,Turn.................................................................................115

Rotations............................................................................................................................117

ClassActivity20: ReflectingonReflecting.....................................................................120

Reflections..........................................................................................................................122

ClassActivity21: Part1ASimilarTask..........................................................................125

ClassActivity21: Part2Zoom.......................................................................................126

SimilarPolygons.................................................................................................................127

ClassActivity22: SearchingforSymmetry.....................................................................130

SymmetriesinthePlane.....................................................................................................131


ChapterFive...................................................................................................135ClassActivity23: DiceSums..........................................................................................136

LanguageofProbability......................................................................................................137

ExperimentalandTheoreticalAnalysis..............................................................................141

ClassActivity24: RatMazes..........................................................................................144

TreeDiagrams....................................................................................................................145

CompoundEvents..............................................................................................................146

ClassActivity25: TheMaternityWard...........................................................................149

LawofLargeNumbers........................................................................................................150

IndependentEvents...........................................................................................................150


ChapterSix.....................................................................................................154ClassActivity26: StudentWeightsandRectangles........................................................155

TheLaguageofSampling....................................................................................................158

RandomSamples................................................................................................................159

ClassActivity27: SnowRemoval...................................................................................165

AnalyzingaSurvey..............................................................................................................166

ClassActivity28: Part1NameGames...........................................................................172

ClassActivity28: Part2IntheBalance..........................................................................174

RedistributionConceptoftheMean..................................................................................176

BalanceConceptoftheMean............................................................................................176

MedianandMode..............................................................................................................176

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FiveNumberSummary.......................................................................................................177

Percentiles..........................................................................................................................178

ClassActivity29: MeasuringtheSpread........................................................................183

Range,IQR,andMeanAbsoluteDeviation........................................................................185

ACriterionforFindingSuspectedOutliers.........................................................................185

Boxplots..............................................................................................................................186

ClassActivity30: TheMatchingGame...........................................................................191

Distributions,Skewness,andSymmetry............................................................................193


References.......................................................................................................................204

APPENDICES...................................................................................................206Euclid’sPostulatesandPropositions................................................................................207

Glossary..........................................................................................................................212

GraphPaper....................................................................................................................227

Tangrams.........................................................................................................................229

PatternBlocks.................................................................................................................230

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ChapterOne

SeeingtheWorldGeometrically

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ClassActivity1: TrianglePuzzle

Geometryisthescienceofcorrectreasoningonincorrectfigures. Originalauthorunknown,butquotedfromG.Polya,HowtoSolveIt.Princeton:

PrincetonUniversityPress.1945.Whathappenedtothemissingsquare?

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ReadandStudy

Geometricfiguresshouldhavethisdisclaimer:

“Noportrayalofthecharacteristicsofgeometricalfiguresorofthespatialpropertiesofrelationshipsofactualbodiesisintended,andanysimilaritiesbetweentheprimitiveconceptsandtheircustomarygeometricalconnotationsarepurely

coincidental.”

"GeometryandEmpiricalScience"inJ.R.Newman(ed.)TheWorldofMathematics,NewYork:SimonandSchuster,1956.

Mathematicalobjects–geometricobjectsincluded–arenotrealobjects.Theyareidealobjects.Thismightseemdisturbingbecausethismeansthatgeometricobjects–liketrianglesandcircles–donotexistinthephysicalworld.Youcandrawsomethingthatlookslikeatriangleoracircle,butitwon’thavethepreciseandperfectpropertiesthattheidealmathematicaltriangleorcirclehas.Thesketchwillsimplycalltomindtheidealobject.Whilewedrawlotsofpicturesingeometry,weneedtokeepinmindthatthepicturescanbemisleading.TaketheTrianglePuzzleasanexample.Thispuzzleiscompellingbecauseitreallylookslikethetopandbottomfiguresarebothtriangles.Theyarenot.Onlybyreasoningabouthowtheidealizedpiecesfittogethercanwediscoverthetruth.Thepuzzleisgovernedbyanunderlyingstructure–thepropertiesoftheshapesinvolveddeterminehowtheywillfittogether.Mathematicianslovetorevealhiddenstructureandtoexplainpatterns.Thisiswhatmathematicsisabout.Andthisiswhatgeometry,inparticular,isabout.Onedefinitionofgeometryisthatitisthestudyofidealshapesandtheirproperties,ofthepatternsthoseshapescanform,andoftheactionsonthoseshapesthatpreservetheirproperties.Wewillbegivingyouotherdefinitionsofgeometryaswegoalong.Watchforthevarietyofwaysofthinkingaboutgeometry.W H A T I S G E O M E T R Y N A T U R E O F M A T H E M A T I C A L O B J E C T S

Thisbookisdesignedsothatyougettodogeometry.Wegenerallydonotteachyoutechniquesandthenhaveyoupractice.Instead,weaskthatyouworkonproblemstohelpyouconstructimportantideas.Theproblemscanbedifficult–whichiswhywehopethatyouwillworkonthemingroupsandthendiscussthemasaclass.Wemadethemthatwayonpurpose.Webelievethataproblemisonlyaproblemifyoudon’tknowhowtosolveit.Ifyoudoknowhowtosolveit,thenitisjustanexercise.Wehopethisbookisfullofproblemsforyouandthatyouwill‘getstuck’alot.Trynottoletbeingstuckdiscourageyou.Itispartofdoingmathematics.Thereisnomagictechniqueforsolvingamathematicsproblem–whichisgood,becauseotherwisemathematicswouldn’tbeanyfun.Basically,youjusthavetowrestlewiththe

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problem.Theprocessmighttakeminutes,orhours,orevenyears.Therearestrategiesthatmayprovehelpful,andapurposeofthisfirstsectionistomakeyouawareofsomeofthethingspeopledotoworkonproblems.Fornow,wewouldlikeyoutoseparateyourthinkingaboutproblemsolvingintofourcategories:

1) Understandingtheproblem.Whatdoesitmeantosolvethisproblem?Doyouunderstandtheconditionsandinformationgiveninthestatementoftheproblem?(FortheTrianglePuzzleabove,thismeansunderstandingthatsinceareamustbepreserved,thepictureshavetobemisleadinginsomeway.Solvingthisproblemmeansshowingexactlyhowthepicturesaremisleading.)

2) Reflectingonyourproblemsolvingstrategies.Whatdidyoudotoworkonthe

problem?(Didyoustudythepiecestoseehowtheyfittogether?Randomlyorinsomesystematicmanner?Didyoukeeptrackofanything?Didyoudrawpictures?Didyoucomputesomething?)

3) Explainingthesolution.Whatistheanswertothequestion?(Exactlywhatiswrong

withthepictures?)

4) Justifyingthatyouarebothdoneandcorrect.Whydoesyoursolutionmakesense?Canyouprovethatyouarecorrectandthattheproblemiscompletelysolved?

Beforewegetintothisbookanyfurther,wemightaswelltellyouthatwe’rebossy.Throughoutthereadingsections(whichyoumustdo–youoweittothechildreninyourfutureclassrooms)wewillaskquestionsandissuecommandsinitalics.Dothethingswesuggestinitalics.Don’tworrythatitslowsthereadingdown.Mathematiciansreadvery,very,agonizinglyslowlyandcarefully,withpencilinhand.Wewriteonourbooks–alloverthem.Weverifyclaims;wedotheproblems;weasknewquestionsandtrytoanswerthem.Sowechallengeyoutodofourthingsthisterm.Firstandsecond,readeverywordofyourtextandworkhardoneachandeveryproblem.Third,makeacontributiontoeachdiscussionofaclassactivity.Andfinally,practicelisteningtoandmakingsenseofotherstudents’mathematicalideas.Asateacher,youwillneedtounderstandthemathematicalthinkingofothers;useyourclasstopracticethatskill.ConnectionstoTeaching

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Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoorganizeandconsolidatetheirmathematicalthinkingthrough

communication;tocommunicatetheirmathematicalthinkingcoherentlyandclearlytopeers,teachers,andothers;toanalyzeandevaluatethemathematicalthinking

andstrategiesofothers;andtousethelanguageofmathematicstoexpressmathematicalideasprecisely. NCTMPrinciplesandStandardsforSchoolMathematics,2000

Learningviaproblemsolvingandcommunicationofideasaretwomajorthreadsinelementarymathematicseducation.TheNationalCouncilofTeachersofMathematics(2000),inadoptingthePrinciplesandStandardsforSchoolMathematics,advocatedthatallstudentsofmathematicsengageinproblemsolvingandcommunication,bothoralandwritten,atallgradelevels.Theywrite:

“Solvingproblemsisnotonlyagoaloflearningmathematicsbutalsoamajormeansofdoingso”(p.52).“Communicationisanessentialpartofmathematicsandmathematicseducation.Itisawayofsharingideasandclarifyingunderstanding….Whenstudentsarechallengedtothinkandreasonaboutmathematicsandtocommunicatetheresultsoftheirthinkingtoothersorallyorinwriting,theylearntobeclearandconvincing”(p.60).

Mathematicseducatorsbelievethatproblemsolvingandwrittencommunicationarealsoessentialcomponentsofyourmathematicalpreparationtobecomeelementaryschoolteachers.Wewillbeaskingyoutowriteaboutmathematicsinthisclass.Wewillaskyoutowriteinterpretationsofproblems,descriptionsofstrategies,explanationsofsolutions,andjustificationsofsolutions.M A T H E M A T I C A L C O M M U N I C A T I O N

Howdowewriteaboutmathematics?Isn’tmathematicsallaboutnumberslike2andp?Andaboutsymbolslike║andÐ?Well,no,it’snot.Mathematiciansusesymbolslikethesetowritestatementsandtosolvesomeproblems,andyoucanusesymbolslikeÐand^and@whenyouwriteaboutgeometryproblems.Butmathematicsismuchmoreaboutexploringpatterns,makingconjectures,explainingresultsandjustifyingsolutions.Theseactivitiesrequireustowritewithwordsincompletesentencesthatusemathematicallanguageandlogicappropriately.Symbolsareatoolwewilluse.Butfornow,youshouldfocusonwritingwithwords–clearly,completely,correctly,andconvincingly.Asfutureteachersyoumustpracticecommunicatinginthelanguageofmathematics.Youwillhaveachancetopracticerightnowinthehomework.

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HomeworkYoualwayspassfailureonthewaytosuccess.

MickeyRooney(MQS)

1) TheTangrampuzzleiscomposedofsevenshapesincludingonesquare,oneparallelogram,twosmallisoscelesrighttriangles,onemedium-sizedisoscelesrighttriangle,andtwolargeisoscelesrighttriangles.Inthediagrambelow,thesevenpiecesarearrangedsothattheyfittogethertoformasquare.a) Tracethepieces,cutthemout,andthenidentifyeachone.Lookuptheterms

isosceles,righttriangleandparallelogramintheglossaryandlearnthosedefinitions.

b) Figureouthowtorearrangeallsevenpiecestoformatrapezoid.Noticethatyoufirstneedtounderstandtheproblem.Lookupthedefinitionofatrapezoidifyouneedtodoso.

c) Reflectonyourproblemsolvingstrategiesandwriteadescriptionofthestrategies

youusedtoworkontheaboveTangrampuzzle.

d) Explainthesolutionbygivingcarefulinstructions,usingwordsonly(nopictures),forarrangingthesevenpiecestoformatrapezoid.

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2) ChildrenintheearlyelementarygradescansolvepuzzlessimilartotheTangrampuzzleusingpatternblocks(asetofflatblocksinsixshapes:regularhexagon,isoscelestrapezoid,tworhombi,square,andequilateraltriangle).Lookupthetermsrhombus(thepluralformofrhombusisrhombi)andhexagonintheglossary,thendotheactivitydescribedbelow.

a) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide(yourprofessorshouldhaveacodeforyoutoviewthis).Spend10-15minuteslookingthroughUnit5Module1.ThenworkthroughPatternBlockPuzzle3.Howmanydifferentwaysarepossibletofillthisshape?Seeifyoucanfindatleast5.

b) Whatmightchildrenlearnabouttherelationshipsamongthepatternblocksbyworkingontheseproblems?

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3) HereisapictureofallsevenTangrampiecesrearrangedtomakeatriangle.

Youaregoingtobeginto“justifythatwearecorrect.”Thisinvolvesarguingthatthepiecesreallydofittogetherasshown.(Rememberthatjust“lookingliketheyfit”isn’tgoodenough.)Yougettoassumethattheoriginalpiecesreallyareallperfectshapesandthattheyoriginallyfitperfectlytoformasquarelikethis:

a) Arguethatthevertexoftheyellowtrianglealongwiththeverticesofthelargeorangeandbluetrianglesreallydomeettomake180degrees(theyformastraightangle)atthebottomedgeofthepuzzle.

b) Arguethattheedgeoftheorangetrianglefitsperfectlywiththeedgesofthesquareandyellowtriangleandthatthebigbluetrianglereallyfitsperfectlyalongtheedgesoftheparallelogramandyellowtriangle.

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ClassActivity2: DefiningMoments

Wherethereismatter,thereisgeometry.JohannesKepler(1571-1630)

Mathematicaldefinitionsareimportanttomathematiciansbecausetheygiveustheexactcriteriaweneedtoclassifyobjects.Usethedefinitionofapolygontodecidewhethereachoftheobjectsthesketchcallstomindarepolygons.Ifanobjectdoesnotmeetthedefinition,explainexactlyhowitfails.Visittheglossaryifyouneedtolookupterms.Apolygonisasimple,closedcurveintheplanecomposedonlyofafinitenumberoflinesegments.

1) 2)3) 4) 5)6) 7)

8) 9)

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ReadandStudy

Everythingyou’velearnedinschoolas“obvious”becomeslessandlessobviousasyoubegintostudytheuniverse.Forexample,therearenosolidsintheuniverse.There’snotevenasuggestionofasolid.Therearenosurfaces.Therearenostraightlines. R.BuckminsterFuller

Mathematicianscaredeeplyaboutthewordsweusetotalkaboutmathematics.Wehavemanyspecialwords,likeisosceles,thatdonotappearineverydaylanguage.Thesewordshaveprecisedefinitionsthatprovidepowerfulknowledgeabouttheobjectstowhichtheyrefer.Evencommonwordslikerighttakeonspecialmeaningwhentheyareusedinmathematicaltalk.R O L E O F D E F I N I T I O N S

Wewillsaylots,andwemeanlots,moreaboutthetermsweuseinmathematicsthroughoutthepagesofthisbook.Everytimeyouencounteratermyoudon’tknow,lookupthedefinitionintheglossaryandmakecertainyouunderstandjusthowthewordisusedinmathematicaltalk.Tohelpyoudothis,wewillcontinuetounderlineandboldfacemathematicaltermsthefirsttimeweusethem.Thiswillletyouknowthatthewordhasaparticularmeaninginmathematicstowhichyouneedtopayattention.Mathematicaldefinitionsaresoimportantbecause:

1) Definitionsprovideprecisecriteriafordescribingandclassifyingtheseidealobjects;

2) Definitionsdescriberelationshipsamongobjects;and

3) Definitionsgiveusthepowertomakemathematicalarguments.

Let’stalkabitmoreabouteachofthese.IntheClassActivityyouhadtheopportunitytomakesenseofadefinitionandtouseittoclassifypolygons.Diditsurpriseyouthatthedefinitionreliedonsomanyotherterms?Afterall,theideaofapolygondoesn’tseemthatcomplicated.Howeveryouwillfindthatyourstudentshavemanydifferentideasinmindaboutpolygons.Somewillthinkthatsolidshapesarepolygons.Somemightclassifyshapeswithcurvededges(likecircles)aspolygons.Inordertobesurethatweareallimaginingthesameidealobjects,wemustallhavethesamedefinition.Thatsaid,wehavetostartsomewherewhenwritingourdefinitions,andthatmeansthatnotalltermscanbepreciselydefined.Inparticular,ingeometry,weacceptthefollowingideasasundefined:point,line,plane,andspace.Thesetermscorrespondtothe0-dimensional,the1-dimensional,the2-dimensional,andthe3-dimensionalobjectstowhichthedefinitionsofgeometryapply.

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Maybeitseemsstrangethatsuchafundamentalobjectasalineisanundefinedterm,buteventhoughwedon’tdefineit,wecanunderstandalinetobeacollectionofpoints(alsoundefined)thatobeysasetofrules.Wehaveintuitiveideasaboutwhatapointorlineis,butwecanbestunderstandortalkaboutpointsorlinesintermsofamodel.Usefulmodelsofalineincludethecreaseinasheetofpaper,thestraightedgewherethewallofaroommeetsthefloor,atautpieceofthinstring,orthepicturebelow.Ofcourseeachofthesemodelsisonlyarepresentationofaline.A“true”linehasnowidthatall–onlylength–anditextendsindefinitelyinbothdirections.Likeallothermathematicalobjects,a“true”lineisanidealobject–itexistsonlyinourminds.Itmayalsoseemstrangethatwecan’tdefinealinebysayingthatitis“straight.”Thepropertyof“straightness”isanotherintuitiveideathatcarrieswithitthenotionof“shortestdistance.”Thatis,wesaythatalineis“straight”ifitismeasuringtheshortestdistancebetweenpoints(the“tautstring”idea).Theseintuitiveideasof“straight”workwellonflatsurfaces(andintheworldofEuclideangeometry),butarenotashelpfuloncurvedsurfacessuchasasphere.Whatistheshortestdistancebetweentwopointsonasphere?Findaball(oranorangeoraglobe)andastringandhavealook.Twocommonmodelsforaplaneareaflatsheetofpaperandthesurfaceofawhiteboard(providedwerememberthateachisonlyaportionoftheplanewhichactuallyextendsinfinitelyinalldirections).Asecondwaythatweusedefinitionsistocreaterelationshipsbetweenobjects.Forexample,wesaythattwolinesareparalleliftheylieinthesameplaneanddonotintersect.Thisdefinitionhelpsusunderstandtherelationshipbetweenlinesthatareparallelprovidedweunderstandwhataplaneisandwhatitmeansforlinestointersect.Alternatively,wecansaythattwolinesareparalleliftheylieinthesameplaneanddonothaveanypointsincommon.Thisseconddefinitionincorporatestheideaofnon-intersectionwithoutusingawordthatmaynotbeknown.P A R A L L E L A N D P E R P E N D I C U L A R L I N E S

Weusedefinitionsinathirdwaywhenwemakearguments.Wewilltalkaboutthisfurtherinthenextsection,butfornow,let’slookatasimpleexample:Supposewewanttoarguethatnotrianglecanalsobeasquare.Sincethedefinitionofatrianglestatesthatitisapolygonwithexactlythreesidesandthedefinitionofasquarestatesthatitisapolygonwithexactlyfourcongruentsidesandfourrightangles,andsincethreedoesnoteverequalfour;wecanconcludethatitisnoteverpossibleforatriangletohavefoursides.Soatrianglecanneveralsobeasquare.Nowthisargumentmayseemtrivial,butthepointhereisthatweusedefinitionstomakearguments.Usethedefinitionsof“parallel”and“perpendicular”toarguethattwolinesthatareparallelcanneveralsobeperpendicular.

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ConnectionstoTeachingInprekindergartenthroughgrade2allstudentsshouldrecognize,name,build,draw,

compare,andsorttwo-andthree-dimensionalshapes. NCTMPrinciplesandStandardsforSchoolMathematics,2000

Inrecentyearsmoststates(includingWisconsin)haveadoptedcommonstandardsforschoolmathematics.Thesestandards,calledtheCommonCoreStateStandards(CCSS),prescribethemathematicalcontentandpracticesthatteachersshouldaddressateachgradelevel.Asafutureteacher,youwillneedtoknowandunderstandthem.Thepracticestandardsdescribeexpectationsforstudentsacrossallgradelevels.Whichofthesestandardshaveyouexperiencedsofarinthisclass?

©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.

Inthisbook,wewillfocusonthecontentstandardsrelatedtogeometryandmeasurement,andwewillbeginnowwithgeometryforchildreninkindergartenandfirstgrade.Takeaminutetoreadthem.

CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

1. Makesenseofproblemsandpersevereinsolvingthem.2. Reasonabstractlyandquantitatively.3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

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Inordertohelpchildrentodistinguishbetweendefiningandnon-definingattributes,youmightaskchildrentosortshapesintocategories.Forexample,youmightaskthattheyidentifyallthetrianglesinthefollowinggroupofshapes:

CCSSKindergarten:GeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres).

1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.

2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.

3. Identifyshapesastwo-dimensional(lyinginaplane,“flat”)orthree-dimensional(“solid”).

Analyze,compare,create,andcomposeshapes.

4. Analyzeandcomparetwo-andthree-dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/“corners”)andotherattributes(e.g.,havingsidesofequallength).

5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.

6. Composesimpleshapestoformlargershapes.Forexample,“Canyoujointhese

twotriangleswithfullsidestouchingtomakearectangle?”

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Whatconversationscouldyouhavewithchildrenregardingthisactivity?InwhatwaysmightyouuseittoaddresstheCCSSforkindergarten?


Anotherideaistoaskchildrentomakeuptherule,sorttheshapes,andthenhaveotherchildrenfigureoutarulethatwillgivethesame“sort.”Childrensometimesattendtoattributesinsolvingsortingproblemsthatwe,asmathematicians,wouldnotpayattentionto.Thusactivitiesliketheseprovideopportunitiestodrawchildren’sattentiontodifferentthings.Forexample,manychildrenwouldsaythatthis

figure▼is“upsidedown”orthattheseare“differentshapes”becausetheyareorienteddifferently.Mathematicianswouldsaythattheabovefiguresarethesameshape.Theydonottakeorientationoftwo-dimensionalshapesintoaccountwhendecidingifthoseshapesare“thesame.”Belowweshowa“studentsort”fromafirstgradeclassroom.CanyoufigureoutDevione’srule?Istheremorethanonerulethatcouldgivethesamesort?

CCSSGrade1:GeometryReasonwithshapesandtheirattributes.

1. Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthree-sided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.

2. Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.

3. Partitioncirclesandrectanglesintotwoandfourequalshares,describetheshares

usingthewordshalves,fourths,andquarters,andusethephraseshalfof,fourthof,andquarterof.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.

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TheseshapesfitMyRule TheseshapesdonotfitMyRule TheCommonCoreStateStandardsaskthatyou,asateacher,alsohelpchildrentotalkaboutthepositionofobjects,toreasonabouthowobjectsarecomposedofotherobjects,andtorecognizewhetheranobjectistwo-dimensional(flat)orthree-dimensional.Whataresomeactivitiesthatyoumightdowithchildrentoaccomplishthesethings?

Homework

Theonlyplacesuccesscomesbeforeworkisinthedictionary. VinceLombardi

1) DoalltheitalicizedthingsintheReadandStudysection.

2) DoalltheitalicizedthingsintheConnectionssection.

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A B

3) Thereareseveraltermsassociatedwithlinesthatyouneedtounderstandandusewithpropernotation.Takeafewminutestostudythese.

Supposewehavethethreepoints,A,B,andC.(Noticethatmathematicianscustomarilyusecapitallettersfromthebeginningofthealphabettodenotepoints.SometimeswearetalkingaboutenoughpointsthatwemakeitallthewaytoZ,butwealmostalwaysstartwithA.)ThelineABistheentiresetofpointsextendingforeverinbothdirections.We

commonlydenotealineas AB andrepresent AB asshownbelow(notethearrowsateachendindicatingthatthelinecontinues):

TherayABisthesetofpointsincludingAandallthepointsonthelineABthatareontheBsideofA.ThepointAiscalledthevertexoftheray.WecommonlydenotearayasAB andrepresent AB asshownbelow:

ThelinesegmentABisthesetofpointsbetweenAandB,includingbothAandB,whicharecalledtheendpointsofthelinesegment.WecommonlydenotealinesegmentasAB andrepresent AB asshownbelow:

4) Visittheglossaryandlearntheprecisedefinitionsforeachofthefollowingterms:square,

parallelogram,rectangle,rhombus,andtrapezoid.Makesureyoucanexplainthedefinitionsusinggoodmathematicallanguage.Sketchanexampleofeach.

5) Whichpropertiesaresufficienttodefinearectangle?Thatis,ifaquadrilateralhasaparticularproperty,doyouknowforcertainthatthequadrilateralmustbearectangle?Explainwhyyouransweris‘yes’or‘no’ineachcase.

a) Ifaquadrilateralhastwosetsofcongruentsides,thenitmustbearectangle.b) Ifaquadrilateralhasoppositeanglescongruent,thenitmustbearectangle.c) Ifaquadrilateralhasdiagonalsthatbisecteachother,thenitmustbearectangle.d) Ifaquadrilateralhastworightangles,thenitmustbearectangle.e) Ifaquadrilateralhascongruentdiagonals,thenitmustbearectangle.f) Ifaquadrilateralhasperpendiculardiagonals,thenitmustbearectangle.g) Ifaquadrilateralhastwosetsofparallelsidesandonerightangle,thenitmustbea

rectangle.h) Ifaquadrilateralhastwosetsofcongruentsidesandonerightangle,thenitmustbe

arectangle.

A B

A B

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Concave PolygonsConvex Polygons

6) Studythefollowingexamplesandformadefinitionofeachoftheseterms:convexandconcave,inyourownwords.Thenlookupthemathematicaldefinitionsintheglossary.Explainthemathematicaldefinitionsinyourownwords.

7) Athirdgradeclassweobservedwaslearningaboutparallellines.Theteacherexplained

thatparallellinesarelinesintheplanethathavenocommonpoints.Thenshedrewthepicturebelowandaskedthechildrenwhetherthelinesshownwereparallelornot.

Severalchildrenarguedthattheywereparallel.Whymighttheyhavesaidthat,andwhatwouldyousaytothemastheirteacher?

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ClassActivity3: GetitStraightGobackalittletoleapfurther. JohnClarke

1) Hereisanactivitytohelpyourupperelementarychildrenmaketheconjecturethattakentogether,theanglesofanytrianglecanformastraightangle.Eachgroupshouldcutoutalargeobtusetriangle,alargeacutetriangleandalargerighttriangle.Foreachtriangle,labelthevertexangles(inanyorder)#1,#2and#3.Then,foreachtriangle,tearoffthethreecornersandputthemtogethersothattheanglesareadjacent.Dothis,anddiscusswhatchildrenmightlearn.Didthisactivityprovethattheanglesofatrianglealwayscanformastraightangle?Whyorwhynot?

2) Again,asinpart1),eachgroupshouldcreatealargeobtusetriangle,alargeacutetriangleandalargerighttriangleand,foreachtriangle,labelthevertexangles#1,#2and#3.Butinsteadoftearingoffthecorners,thistimehaveonepersonuseaprotractortocarefullymeasureeachanglelabeled#1,anotherpersonmeasureeachanglelabeled#2,andanotherpersonmeasureeachanglelabeled#3(eachwithoutlookingattheothers’measurements).Wasthetotalanglemeasureforeachtriangle180degrees?Shouldithavealwaysbeen?Explainanydiscrepancies.

(continuedonthenextpage)

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3) Ourexplorationsinparts1)and2)mighthaveconvincedyouthatthesumofthevertexanglesofanytriangleisthesameasastraightangle,butmathematiciansdonotconsidereitherofthosedemonstrationstobeaproof.Whynot,doyouthink?Nowwearegoingtolearntomathematicallyprovethatsumofthevertexanglesofanytriangleisthesameasastraightangle.

Step1.Startwithanytriangle:Nowwe’llcreatealinethroughonevertexthatisparalleltotheoppositesideofthetriangleandlabelalltheanglessowecantalkaboutthem.Weknowwecanalwaysdothisbecauseitisanaxiomofplanegeometrythatthroughapointnotonalinetherecanbedrawnone(andonlyone)lineparalleltothegivenline.

Bytheway,itshouldn’tbeobvioustoyouwhywe’vedecidedtocreatethisparallelline;itjustturnsouttogiveusagreatwaytobeginourproof.InStep2,wearegoingtoprovethatÐ1iscongruenttoÐ5,andthatÐ2iscongruenttoÐ4.Sofornow,supposingthistobetrue,arguethatangles1,2,and3wouldcombinetoformastraightangle.


m

5

432

1

B

A

C

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Step2.Thisistheonlymissingpieceofourproof.Lookupthedefinitionsofatransversalandofalternateinteriorangles.CanyouseethatÐ1andÐ5arealternateinteriorangles?Whatisthecorrespondingtransversal?

CanyouseethatÐ2andÐ4arealternateinteriorangles?Whatisthecorrespondingtransversal?Nowarguethatiftwoparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.Youmayassumethatiftwoparallellinesarecutbyatransversal,thentheinterioranglesonthesamesideofthetransversalformastraightangle.(Note:Thisisabigthingtoassume.ItisoneofEuclid’sfundamentalassumptionsaboutgeometry.)

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ReadandStudy

Iargueverywell.Askanyofmyremainingfriends.Icanwinanargumentonanytopic,againstanyopponent.Peopleknowthisandsteerclearofmeatparties.

Often,asasignoftheirgreatrespect,theydon’teveninviteme. DaveBarry

Mathematicalthinkingalwaysinvolvesreasoningandmakingarguments,andwehaveawholevocabularyfordescribingthatprocess.Inthissection,wehighlightthetermswemathematiciansusetodescribefacetsofdoingmathematics.Theseareimportant.Makesureyouunderstandthem.L A N G U A G E O F M A T H E M A T I C S

1) Anaxiomisastatementthatweagreetoacceptwithoutproof.Itisanassumptionorstartingpoint.(Note:Anotherwordforaxiomispostulate.)

2) Inductivereasoningiscomingtoaconclusionbasedonexamples.Forexample,Iobservethat3,5and7areallprimenumbers.Now,basedontheseexamplesImightreason(incorrectly,bytheway)thatalloddnumbersareprime.OrImightnoticethatthesunrosedaybeforeyesterday,itroseyesterday,itrosetoday.SoImightconcludethatthesunwillrisetomorrow.Thisisinductivereasoning.

3) Deductivereasoningiscomingtoconclusionbasedonlogic.Forexample,Iwillargue

deductivelythatthesunwillcomeuptomorrow:Theearthiscaughtinthesun’sgravitysoitwon’tfloataway,andtheearthisspinning.Wearehereontheearthandsowhenourpartoftheearthturnstowardthesun,wesayit“comesup.”Aslongasnocatastropheoccurstochangethesefacts,thesunwillrisetomorrow.We’llgiveyouanother(moremathematical)exampleinaminute.

D E D U C T I V E V E R S U S I N D U C T I V E R E A S O N I N G

4) Aconjectureisahypothesisoraguessaboutwhatistrue.Forexample,aftersomeexperiencewithcircles,astudentmightconjecturethattwointersectingcirclesalwayssharetwodistinctpointsincommon.Conjecturesareoftenmadebasedoninductivereasoning.

5) Acounterexampleisaspecificexamplethatshowsaconjectureisfalse.Forexample,

thetwocirclesbelowaretangent(theyshareexactlyonepointincommon)andsotheaboveconjectureisshowntobefalsebythecounterexampleshownhere:

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6) proof:amathematicalproofconsistsofadeductiveargumentthatestablishesthetruth

ofaclaim.(Note:Bytruthwemeantruthinthecontextofthemathematicalworldthatiscreatedbytheaxioms.Somethingistrueifitisalogicalimplicationoftheaxioms.)

7) theorem:atheoremisamathematicalstatementthathasbeenprovedtobetrue.For

example,itisatheoremthatverticalanglesarecongruent.YouprovedthisintheClassActivity.(Note:Anotherwordfortheoremisproposition.)

WearegoingtousewhatyoudidintheClassActivitytohighlightthemeaningofsomeoftheseterms.First,youtoreapartavarietyoftrianglesandarrangedthemtoseethateachappearedtohaveastraightangle(180degrees)ofvertexangles.Thiswasinductivereasoning,becauseyouweretestingexamplesoftrianglestoseewhatseemedtrueabouttheirvertexanglemeasure.Atthispointitwouldhavebeenreasonabletoconjecturethatthesumofthevertexanglesofatriangleis180degrees.Thenyouwereaskedtomakeanargumentusingdefinitions,axioms,andlogicthatyouwerecorrect.Inotherwordsyougaveaproofthatthevertexanglesofatrianglesumto180degrees,andnowthatconjectureiscalledatheorem.Oneofthereasonsthatgeometryclass(remembertenthgrade?)hastraditionallyfocusedonproofisthattheaxiomsofgeometryareeasiertostatethantheaxiomsofarithmetic.Butproofispartofallmathematics.Don’tworry,wearenotplanningtofocusontwo-columnproofsoranaxiomaticdevelopmentofgeometry,butwewouldberemissifwedidn’tatleaststatetheoriginalaxioms(assumptions)ofplanegeometry.TheaxiomswerefirstmadeexplicitbyEuclid,aGreekmathematicianwholivedandworkedattheAcademyinAlexandria,Egypt.Heisbestknownforwritinga13-volumebookofmathematicscalledTheElements-thesecondmostpublishedbookintheworld.Ithasbeenusedasamathematicstextbookforover2000years.Inthefirsttwovolumesofthiswork,hedevelopedallofthe(then)knowntheoremsabouttwo-dimensional(plane)geometrystartingwithjustfiveaxioms:I D E A O F A X I O M S

TheAxiomsofEuclideanGeometry:1. Auniquestraightlinesegmentcanbedrawnfromanypointtoanyotherpoint.2. Alinesegmentcanbeextendedtoproduceauniquestraightline.3. Auniquecirclemaybedescribedwithagivencenterandradius.4. Allrightanglesareequaltoeachother.

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5a.Ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.

5b.Throughapointnotonalinetherecanbedrawnexactlyonelineparalleltothegivenline.

Uponassumingaxioms1-4,axioms5aand5bareequivalent.Axiom5aisEuclid’sversion,and5bisamoremoderninterpretationknownasPlayfair’sAxiom.Lookinthisbook’sappendixandyouwillfindtheaxioms(postulates)andtheorems(propositions)fromBook1ofEuclid’sElements,writtenmorethan2,000yearsago.Axiom5aishisfifthpostulate.Youmayfinditinterestingthatuntilthelate1800s,manymathematiciansthoughtEuclid’sfifthaxiomwasredundant–thatitalreadyhadtobetrueifaxioms1-4wereassumedtobetrue–butithassincebeenproventhatthosemathematicianswerewrong.Noticethateachoftheaxiomsdescribessomethingthatcanbeconstructedwiththeexceptionofthefourth.Axiom4saysthattheEuclideanplaneis,insomesense,uniform(nodistortions).Namely,itsaysthatwhereveryouconstructaperpendicularlinesontheplane(andsoformfourangles),thoseangleswillallhavethesamemeasure.ItturnsoutthatEuclidmadesomeimplicitassumptionsthatheshouldhavestatedasadditionalaxiomsinordertodogeometryrigorously–andsomodernmathematicianshaveextendedhislistofaxioms.Butyoudon’tneedtoworryaboutthosetechnicalitiesinthiscourse.Sketchapictureofaxioms1–3and5tohelpyoumakesenseofeach.Then,describehowyoucancreateanequilateraltrianglebyfollowingEuclid’saxioms.WealsowanttonotethatEuclidoftenusedtheword“equal”whenwewouldusetheword“congruent.”Today’smathematiciansuse“equal”whentheywanttocomparetwonumbers.Sowemightsaythat½isequalto0.5.Weusetheword“congruent”whenwewanttosaythattwoobjects(liketwotrianglesortwosegments)arethesamesizeandshape.Thebasicideahereisthattwoobjectsarecongruentinthecasewhereifoneobjectwasmovedtolieontopoftheotherobject,theywouldcorrespondexactly.Wewilldoamorecarefuljobofdefining“congruent”later.

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Whileprovingtheoremsisnotthefocusofthiscourse,wemayoccasionallyaskthatyoutrytoproveaEuclideantheorem.Whenwedo,youshouldturntotheAppendixwhereEuclid’spostulates(axioms)andpropositionsarelistedandfindit.Thenyouarefreetouse(assume)anypostulateandanypropositionlistedbeforetheoneyouaretryingtoprove.Forexample,sayyouwanttoprovethatinanisoscelestriangle,thebaseanglesarecongruent.GototheAppendix(reallydoit)andseeifyoucanfindthattheorem.Thencomerightbackhere.Let’sendthissectionwiththebigidea:Geometryintheplanearisesfromsomeintuitiveidealobjects,theirmathematicaldefinitions,thesefiveaxioms,andalotofdeductivereasoning.Infact,asecondpossibledefinitionofgeometryisthis:anaxiomaticsystemaboutidealobjectscalled“points,”collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.

ConnectionstoTeaching

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstorecognizereasoningandproofasfundamentalaspectsofmathematics;tomakeandinvestigatemathematicalconjectures;todevelopandevaluatemathematicalargumentsandproofs;andtoselectandusevarioustypesofreasoningandmethodsofproof.

ReasoningandProofStandard,NCTMPrinciplesandStandardsforSchoolMathematics,2000

M A K I N G C O N V I N C I N G M A T H E M A T I C A L A R G U M E N T S

Itisimportantthatyouprovidechildreninyourfutureclasseswiththeopportunitiestoreallydomathematics.Theytooneedtouseinductivereasoning,makeconjectures,lookforcounterexamples,andmakedeductivearguments.Inordertounderstandbetterwhatyoushouldexpectfromchildren,readthefollowingfromthediscussionoftheReasoningandProofstandardforthePreK–2gradebandfoundinNCTMPrinciplesandStandardsforSchoolMathematics,2000,pages122–125.Noticetheiruseofmathematicallanguage.

Whatshouldreasoningandprooflooklikeinprekindergartenthroughgrade2?p.122

Theabilitytoreasonsystematicallyandcarefullydevelopswhenstudentsareencouragedtomakeconjectures,aregiventimetosearchforevidencetoproveordisprovethem,andareexpectedtoexplainandjustifytheirideas.Inthebeginning,perceptionmaybethepredominantmethodofdeterminingtruth:ninemarkersspreadfarapartmaybeseenas“more”thanelevenmarkersplacedclosetogether.Later,asstudentsdeveloptheirmathematicaltools,theyshoulduseempiricalapproachessuchasmatchingthecollections,whichleadstotheuseofmore-abstractmethodssuchascountingtocomparethecollections.Maturity,experiences,andincreasedmathematicalknowledgetogetherpromotethedevelopmentofreasoningthroughouttheearlyyears.»

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Creatinganddescribingpatternsofferimportantopportunitiesforstudentstomakeconjecturesandgivereasonsfortheirvalidity,asthefollowingepisodedrawnfromclassroomexperiencedemonstrates.

Thestudentwhocreatedthepatternshowninfigure4.27proudlyannouncedtoherteacherthatshehadmadefourpatternsinone.“Look,”shesaid,“there’striangle,triangle,circle,circle,square,square.That’sonepattern.Thenthere’ssmall,large,small,large,small,large.That’sthesecondpattern.Thenthere’sthin,thick,thin,thick,thin,thick.That’sthethirdpattern.Thefourthpatternisblue,blue,red,red,yellow,yellow.”Herfriendstudiedtherowofblocksandthensaid,“Ithinktherearejusttwopatterns.See,theshapesandcolorsareanAABBCCpattern.ThesizesareanABABABpattern.ThickandthinisanABABABpattern,too.Soyoureallyonlyhavetwodifferentpatterns.”Thefirststudentconsideredherfriend’sargumentandreplied,“Iguessyou’reright—butsoamI!”

Fig.4.27.Fourpatternsinone

Beingabletoexplainone’sthinkingbystatingreasonsisanimportantskillforformalreasoningthatbeginsatthislevel.

Findingpatternsonahundredboardallowsstudentstolinkvisualpatternswithnumberpatternsandtomakeandinvestigateconjectures.Teachersextendstudents’thinkingbyprobingbeyondtheirinitialobservations.Studentsfrequentlydescribethechangesinnumbersorthevisualpatternsastheymovedowncolumnsoracrossrows.Forexample,askedtocoloreverythirdnumberbeginningwith3(seefig.4.28),differentstudentsarelikelytoseedifferentpatterns:“Somerowshavethreeandsomehavefour,”or“Thepatterngoessidewaystotheleft.”Somestudents,seeingthediagonalsinthepattern,willnolongercountbythreesinordertocompletethepattern.Teachersneedtoaskthesestudentstoexplaintotheirclassmateshowtheyknowwhattocolorwithoutcounting.Teachersalsoextendstudents’mathematicalreasoningbyposingnewquestionsandaskingforargumentstosupporttheiranswers.“Youfoundpatternswhencountingbytwos,threes,fours,fives,andtensonthehundredboard.Doyouthinktherewillbepatternsifyoucountbysixes,sevens,eights,ornines?Whataboutcountingbyelevensorfifteensorbyanynumbers?”Withcalculators,studentscouldextendtheirexplorationsoftheseandothernumericalpatternsbeyond100.

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Fig.4.28.Patternsonahundredboard

p.123Students’reasoningaboutclassificationvariesduringtheearlyyears.Forinstance,whenkindergartenstudentssortshapes,onestudentmaypickupabigtriangularshapeandsay,“Thisoneisbig,”andthenputitwithotherlargeshapes.Afriendmaypickupanotherbigtriangularshape,traceitsedges,andsay,“Threesides—atriangle!”andthenput»itwithothertriangles.Bothofthesestudentsarefocusingononlyoneproperty,orattribute.Bysecondgrade,however,studentsareawarethatshapeshavemultiplepropertiesandshouldsuggestwaysofclassifyingthatwillincludemultipleproperties.Bytheendofsecondgrade,studentsalsoshouldusepropertiestoreasonaboutnumbers.Forexample,ateachermightask,“Whichnumberdoesnotbelongandwhy:3,12,16,30?”Confrontedwiththisquestion,astudentmightarguethat3doesnotbelongbecauseitistheonlysingle-digitnumberoristheonlyoddnumber.Anotherstudentmightsaythat16doesnotbelongbecause“youdonotsayitwhencountingbythrees.”Athirdstudentmighthaveyetanotherideaandstatethat30istheonlynumber“yousaywhencountingbytens.”Studentsmustexplaintheirchainsofreasoninginordertoseethemclearlyandusethemmoreeffectively;atthesametime,teachersshouldmodelmathematicallanguagethatthestudentsmaynotyethaveconnectedwiththeirideas.Considerthefollowingepisode,adaptedfromAndrews(1999,pp.322–23):

Onestudentreportedtotheteacherthathehaddiscovered“thatatriangleequalsasquare.”Whentheteacheraskedhimtoexplain,thestudentwenttotheblockcornerandtooktwohalf-unit(square)blocks,twohalf-unittriangular(triangle)blocks,andoneunit(rectangle)block(showninfig.4.29).Hesaid,“Ifthesetwo[squarehalf-units]arethesameasthisoneunitandthesetwo[triangularhalf-units]arethesameasthisoneunit,thenthissquarehastobethesameasthistriangle!”

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Fig.4.29.Astudent’sexplanationoftheequalareasofsquareandtriangularblockfaces

p.124Eventhoughthestudent’swording—thatshapeswere“equal”—wasnotcorrect,hewasdemonstratingpowerfulreasoningasheusedtheblockstojustifyhisidea.Insituationssuchasthis,teacherscouldpointtothefacesofthetwosmallerblocksandrespond,“Youdiscoveredthat»theareaofthissquareequalstheareaofthistrianglebecauseeachofthemishalftheareaofthesamelargerrectangle.”

Whatshouldbetheteacher’sroleindevelopingreasoningandproofinprekindergartenthroughgrade2?Teachersshouldcreatelearningenvironmentsthathelpstudentsrecognizethatallmathematicscanandshouldbeunderstoodandthattheyareexpectedtounderstandit.Classroomsatthislevelshouldbestockedwithphysicalmaterialssothatstudentshavemanyopportunitiestomanipulateobjects,identifyhowtheyarealikeordifferent,andstategeneralizationsaboutthem.Inthisenvironment,studentscandiscoveranddemonstrategeneralmathematicaltruthsusingspecificexamples.Dependingonthecontextinwhicheventssuchastheoneillustratedbyfigure4.29takeplace,teachersmightfocusondifferentaspectsofstudents’reasoningandcontinueconversationswithdifferentstudentsindifferentways.Ratherthanrestatethestudent’sdiscoveryinmore-preciselanguage,ateachermightposeseveralquestionstodeterminewhetherthestudentwasthinkingaboutequalareasofthefacesoftheblocks,oraboutequalvolumes.Oftenstudents’responsestoinquiriesthatfocustheirthinkinghelpthemphraseconclusionsinmore-precisetermsandhelptheteacherdecidewhichlineofmathematicalcontenttopursue.Teachersshouldpromptstudentstomakeandinvestigatemathematicalconjecturesbyaskingquestionsthatencouragethemtobuildonwhattheyalreadyknow.Intheexampleofinvestigatingpatternsonahundredboard,forinstance,teacherscouldchallengestudentstoconsiderotherideasandmakeargumentstosupporttheirstatements:“Ifweextendedthehundredboardbyaddingmorerowsuntilwehadathousandboard,howwouldtheskip-countingpatternslook?”or“Ifwemadechartswithrowsofsixsquaresorrowsoffifteen

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squarestocounttoahundred,wouldtherebepatternsifweskip-countedbytwosorfivesorbyanynumbers?”Throughdiscussion,teachershelpstudentsunderstandtheroleofnonexamplesaswellasexamplesininformalproof,asdemonstratedinastudyofyoungstudents(CarpenterandLevi1999,p.8).Thestudentsseemedtounderstandthatnumbersentenceslike0+5869=5869werealwaystrue.Theteacheraskedthemtostatearule.Annsaid,“Anythingwithazerocanbetherightanswer.”Mikeofferedacounterexample:“No.Becauseifitwas100+100that’s200.”Annunderstoodthatthisinvalidatedherrule,sosherephrasedit,“Isaid,umm,ifyouhaveazeroinit,itcan’tbelike100,becauseyouwantjustplainzerolike0+7=7.”Thestudentsinthestudycouldformrulesonthebasisofexamples.Manyofthemdemonstratedtheunderstandingthatasingleexamplewasnotenoughandthatcounterexamplescouldbeusedtodisproveaconjecture.However,moststudentsexperienceddifficultyingivingjustificationsotherthanexamples.

p.125Fromtheverybeginning,studentsshouldhaveexperiencesthathelpthemdevelopclearandprecisethoughtprocesses.Thisdevelopmentofreasoningiscloselyrelatedtostudents’languagedevelopmentandisdependentontheirabilitiestoexplaintheirreasoningratherthanjust»givetheanswer.Asstudentslearnlanguage,theyacquirebasiclogicwords,includingnot,and,or,all,some,if...then,andbecause.Teachersshouldhelpstudentsgainfamiliaritywiththelanguageoflogicbyusingsuchwordsfrequently.Forexample,ateachercouldsay,“Youmaychooseanappleorabananaforyoursnack”or“Ifyouhurryandputonyourjacket,thenyouwillhavetimetoswing.”Later,studentsshouldusethewordsmodeledforthemtodescribemathematicalsituations:“Ifsixgreenpatternblockscoverayellowhexagon,thenthreebluesalsowillcoverit,becausetwogreenscoveroneblue.”Sometimesstudentsreachconclusionsthatmayseemoddtoadults,notbecausetheirreasoningisfaulty,butbecausetheyhavedifferentunderlyingbeliefs.Teacherscanunderstandstudents’thinkingwhentheylistencarefullytostudents’explanations.Forexample,onhearingthathewouldbe“StaroftheWeek”inhalfaweek,Benprotested,“Youcan’thavehalfaweek.”Whenaskedwhy,Bensaid,“Sevencan’tgointoequalparts.”Benhadtheideathattodivide7by2,therecouldbetwogroupsof3,witharemainderof1,butatthatpointBenbelievedthatthenumber1couldnotbedivided.Teachersshouldencouragestudentstomakeconjecturesandtojustifytheirthinkingempiricallyorwithreasonablearguments.Mostimportant,teachersneedtofosterwaysofjustifyingthatarewithinthereachofstudents,thatdonotrelyonauthority,andthatgraduallyincorporatemathematicalpropertiesandrelationshipsasthebasisfortheargument.Whenstudentsmakeadiscoveryordetermineafact,ratherthantellthemwhetheritholdsforallnumbersorifitiscorrect,theteachershouldhelpthestudentsmakethatdeterminationthemselves.Teachersshouldasksuchquestionsas“Howdoyouknowitistrue?”andshouldalsomodelwaysthatstudentscanverifyordisprovetheirconjectures.Inthisway,studentsgraduallydeveloptheabilitiestodeterminewhetheranassertionistrue,ageneralizationvalid,orananswercorrectandtodoitontheirowninsteadofdependingontheauthorityoftheteacherorthebook.

NCTMPrinciplesandStandardsforSchoolMathematics

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Homework

Euclidtaughtmethatwithoutassumptionsthereisnoproof.Therefore,inanyargument,examinetheassumptions. EricTempleBell

1) DoalltheitalicizedthingsintheReadandStudysection.Writeadescriptionofthe

strategiesyouusedinsolvingtheproblemofcreatinganequilateraltriangleusingonlyEuclid’sfiveaxioms.

2) WhichofthechildrenfromtheConnectionsreadingareusingdeductivereasoningand

whichareusinginductivereasoning?Explain.

3) AccordingtotheNCTM,whatistheteacher’sroleinpromotingreasoningamongchildrenintheearlyelementarygrades?

4) Explainwhyverticalanglesformedbyintersectinglinesarethesame.

5) HavealookatProposition32oftheappendix.Doyouseethatitisthetheoremyou

provedintheClassActivity?Whichpropositionsthatcomebefore#32didyouuseintheproof?

6) DecideifeachofthefollowingstatementsaboutEuclideanlinesandanglesistrueorfalsebyexploringexamplesandlookingupdefinitions.Ifyoudecidethatastatementistrue,writeadeductiveargumentbasedonaxiomsanddefinitions.Ifyoudecidethestatementisfalse,giveacounterexampleoradeductiveargumentthatitisnotpossible.

a) Anytwodistinctlineswilleitherintersectinexactlyonepointortheywillbeparallel.

b) Thereexisttwoacuteangleswhicharesupplementary.c) Everytwolinesthatareeachparalleltoathirdlinemustbeparalleltoeach

other.d) Everytwolinesthatareeachperpendiculartoathirdlinewillbeperpendicular

toeachother.e) Everytwoacuteanglesmustbecomplementary.f) Thereexisttwooppositesidesinanytrapezoidwhichareparallel.g) Ifoneoftwosupplementaryanglesisacute,theotheranglemustbeobtuse.

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7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide(yourprofessorshouldhaveacodeforyoutoviewthis).Spend10-15minuteslookingthroughUnit6Module1.CarefullyreadthroughtheactivityGuessMyShape,thenmakeupyourownriddleforyoursecondgradestudentstosolve.

8) Wehaveaconjecture.Everyrectangleisaparallelogram.Giveaninductiveargumentthatthisconjectureistrue.Now,sincemathematiciansarenotsatisfieduntiltheyhaveadeductiveargument,giveoneofthose.

9) Wehaveanotherconjecture!Everyrectangleisasquare.Isthisconjecturetrueorfalse?

Ifitistrue,giveadeductiveargument.Ifitisfalse,giveacounterexample.

10) HerearesomemoreoftheCommonCoreStateStandardsrelatedtoreasoningaboutshapes.InwhatwaysdoHWproblems4)–7)addressthesestandards?


11) DrawaVenndiagramshowingtherelationshipbetweenallthevariousquadrilateralswehavestudied.HowdoesthisfitwiththeCommonCoreStateStandardsdescribedbelow?


CCSSGrade3:Reasonwithshapesandtheirattributes.

1. Understandthatshapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.

CCSSGrade5:Classifytwo-dimensionalfiguresintocategoriesbasedontheirproperties.

1. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.Forexample,allrectangleshavefourrightanglesandsquaresarerectangles,soallsquareshavefourrightangles.

2. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.

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ClassActivity4: AlltheAngles

DonotworryaboutyourdifficultiesinMathematics.Icanassureyoumineare stillgreater.

AlbertEinstein

1) Hereisthedescription–fromtheCommonCoreStateStandardsforgradefour–ofhowtothinkaboutmeasuringanglesusingdegrees.Readitcarefullyandsketchapicturetohelpyourgroupmakesenseoftheirexplanation.

Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.

Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

M E A S U R I N G A N G L E S I N D E G R E E S

2) Makesureeveryoneinyourgroupcanusetheirprotractortomeasure(indegrees)theangleindicatedbelow:


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3) Findaformulaforthesumofthemeasuresofthevertexanglesofann-gon(apolygonwithnsides).Youmayneedtocollectsomedata.Intheend,makeamathematicalargumentthattheformulayoufindwillworkforaconvexpolygonofanynumberofsides.(Theformulathatyoudevelopedwillworkforconcavepolygonsaswell,buttheargumentistrickier,sowearenotaskingyoutojustifyitatthistime.)

41

ReadandStudy

Theknowledgeofwhichgeometryaimsistheknowledgeoftheeternal. Plato

Aregularpolygonisoneinwhichallofthelinesegmentsarecongruentandallofthevertexanglesarealsocongruent.Sketcharegulartriangleandaregularquadrilateral.Isthebelowrhombusaregularquadrilateral?Explain.

R E G U L A R P O L Y G O N S

Polygonsareoftennamedforthenumberofsidestheycontain.Infact,theprefix“poly”means“many”andtheroot“gon”means“side.”So“polygon”means“many-sided”figure.Inordertonamepolygons,you’llneedtoknowthefollowingprefixes: Five–“penta” Six–“hexa” Seven–“hepta” Eight–“octa” Nine–“nona” Ten–“deca” Twelve–“dodeca”Forexample,five-sidedpolygonsarecalledpentagons.Hereareacoupleexamplesofconvexpentagons.Drawanexampleofaconcavepentagon.Mathematiciansidentifythreetypesofanglesinapolygon:thevertexangles,thecentralangles,andtheexteriorangles.Studythehexagoninthefollowingdiagramandthenexplainthedifferencebetweenthethreetypesofanglesinyourownwords.Theboldarcsmarkthevertexangles;thethinsolidarcsmarkthecentralangles;andthedashedarcsmarktheexteriorangles.PointGisanyinteriorpoint.

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ThinkaboutthesumofthesixcentralanglesformedatpointG.Thissumwillalwaysbe360°regardlessofwhereintheinteriorpointGis,regardlessofhowmanysidesthepolygonhas,andregardlessofwhetherornotthepolygonisregular.Why?Wealsoclaimthatthesumoftheexterioranglesofanypolygonis360°.Wewillaskyoutomakeamathematicalargumenttosupportthisclaiminthehomeworksection.Thecaseofthesumofthevertexanglesofapolygonistheinterestingcase.IntheClassActivityyoufoundthatthissumdoesdependonthenumberofsidesinthepolygon.Doesitdependonwhetherthepolygonisregular?Explain.


Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.

CCSS,p.6

Anglesarenotoriouslydifficultidealobjectsforchildren.Ontheonehand,theyareoftendefinedasafigureformedbytworayswithacommonendpoint(and,infact,thatisexactlyhowwehavedefinedthem).Ontheotherhand,whatisimportantinmeasuringanangleisitsdegreeofturn.

A

B

C

DE

F

G

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Whenyoutalkaboutangleswithchildren,wesuggestthatyoualwaysuseyourhandorarmtoshowthesweepoftheangleinadditiontoshowingthestaticpicture.Childreningradefourlearntouseaprotractortomeasureanglesindegreesandtoaccuratelyestimatethemeasureofangles.BelowyouwillfindtheCCSSrelatedtoanglemeasureinthisgrade.Readthemcarefully.


CCSSGrade4:Geometricmeasurement:understandconceptsofangleandmeasureangles.

1. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:

a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommon

endpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.

b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

2. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesof

specifiedmeasure.

3. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon-overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.

CCSSGrade4:Drawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.

1. Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo-dimensionalfigures.

2. Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.

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Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module1.PrintoutandthenworkthroughtheMeasuringPatternBlockAnglesactivity.(Notethatinthisactivitystudentsarenotusingprotractorstomeasuretheangles.Youalsowillnotneedtouseaprotractor.).

Homework

Energyandpersistenceconquerallthings. BenjaminFranklin



3) ChildreninyourclassmaysaythatÐABCissmallerthanÐDEF.Isthistrue?Astheir

teacher,whatwouldyousaytothesechildren?

4) Usingtheresultsoftheclassactivity,findthemeasureofonevertexangleinan

equilateraltriangle,asquare,aregularpentagon,aregularhexagon,aregularoctagon,aregulardecagon,andaregulardodecagon.

5) Astudentclaimsthatthesumofthevertexanglesofahexagonis6×180becauseeachtrianglehas180degreesofanglesmeasureandshehasshownthat6trianglestomakeupthehexagon.Whatwillyousayasherteacher?

6) IntheReadandStudysectionweclaimthatthesumoftheexterioranglesofanypolygonisalways360°.Makeamathematicalargumenttosupportthisclaim.

B

A

CE

D

F

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ClassActivity5: ALogicalInterludeEquationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsof

geometry. StephenHawking

Inthepicturebelow,eachcardhasacolorononesideandashapeontheotherside.Whichcard(s)wouldyouhavetoturnovertobesurethatthefollowingstatementistrue?

Ifacardisredononeside,thenithasasquareontheotherside.

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ReadandStudy

Whenintroducedatthewrongtimeorplace,goodlogicmaybetheworstenemyofgoodteaching.

GeorgePolya TheAmericanMathematicalMonthly,v.100,no3.

HerearetwotheoremsaboutparallellinesthatcanbeprovedfromEuclid’saxioms.

1) Iftwolinesarecutbyatransversalandthealternateinterioranglesarecongruent,thenthelinesareparallel.

2) Iftwoparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.

Drawasketchtomakesurethatyouseewhateachofthesehastosay.

Noticethattheybothhavean‘if-then’statementform.Lotsofmathematicaltheoremsarelikethis.Whenatheoremisstatedin‘if-then’form,whateverfollowsthe‘then’isalwaystruewhenevertheconditionsstatedinthe‘if’partaremet.Youcanthinkofan‘if-then’statementasapromisethatiskeptunlessthe‘if’partistrueandthe‘then’partisnot.Wecallstatement2)theconverseofstatement1).Thesetwotheoremsmaysoundthesametoyou,buttheyarenotsayingthesamething.Tohelpyouseethis,let’schangethecontext.Hereisanotherpairofstatementsinwhichthesecondistheconverseofthefirst(andviceversa):

3) IfIliveinChicago,thenIliveinIllinois.

4) IfIliveinIllinois,thenIliveinChicago.Thinkabouteachofthestatements.Whichoftheseistrue?Sinceoneistrueandtheotherfalse,thesetwostatementscannotbesayingthesamething.Inotherwords,theconverseofastatementisalogicallydifferentstatementfromtheoriginalstatement.C O N V E R S E A N D C O N T R A P O S I T I V E

Nowhereisastatementthatislogicallyequivalenttotheoriginalstatementmadein3)above.Itiscalledthecontrapositiveofstatement3).

5) IfIdonotliveinIllinois,thenIdonotliveinChicago.

ThisisessentiallywhatyoufoundwhenyouworkedontheClassActivity.Writethecontrapositivetostatement4)above.

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Itishelpfultoknowthatastatementanditscontrapositiveareequivalentbecausethatmeansthatyoucanproveastatementbyprovingitscontrapositive.


Thebeginningofknowledgeisthediscoveryofsomethingwedonotunderstand. FrankHerbert

Nowthatwehavetalkedabitaboutdoingmathematics,wewanttoshowyouthattheCommonCoreStateStandardsrequirethatchildrenalsodomathematics.Giveanexampleofsomethingyouhavedoneinclasssofarthistermthatmeetseachofthesestandards.


CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

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Homework

Amultitudeofwordsisnoproofofaprudentmind. Thales



3) Decideifeachofthefollowingstatementsistrueorfalse.Iftrue,giveamathematical

explanation.Iffalse,giveacounterexample.a) Ifaquadrilateralisasquare,thenitisarectangle.b) Ifaquadrilateralhasapairofparallelsides,thenitmusthaveapairof

oppositesidesthatarecongruent.c) Ifthediagonalsofaquadrilateralareperpendiculartoeachother,thenthe

quadrilateralisarhombus.d) Ifaquadrilateralhasonerightangle,thenallofitsanglesmustberight

angles.e) Writetheconverseofeachofthestatementsina)–d)above.Whichare

true?f) Writethecontrapositiveofeachofthestatementsina)–d)above.Which

aretrue?

4) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module4.ThenworkthroughtheClockAngles&ShapeSketchesfromthehomelinksection.Howisdeductivereasoningusedtosolvetheseproblems?

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SummaryofBigIdeas

Hey!What’sthebigidea? Sylvester

• Geometryisastudyofidealobjects–notrealobjects.

• Definitionsallowustonameandcategorizeidealobjects,tocreaterelationships

betweenobjects,andtomakeargumentsaboutthepropertiesofobjects.• Onedefinitionofgeometryisthatitisthestudyofidealshapesandtheirproperties,of

thepatternsthoseshapescanform,andoftheactionsonthoseshapesthatpreservetheirproperties.

• Aseconddefinitionisthatgeometryisanaxiomaticsystemaboutobjectscalled

“points,”collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.

• Mathematicalthinkingalsoinvolvesdeductivereasoningandmakingarguments–this

meansusinglogicaswellasusingdefinitionsandaxioms.• Congruenceisanimportantrelationshipbetweengeometricobjects.Wesaytwo

objectsarecongruentiftheycoincidewhenplacedontopofeachother.

• InEuclideangeometrythesumoftheanglesinatriangleisalways180degreesandyoucanexplainwhythisisso.

• Youwillrepresentthemathematicalcommunityforyourstudents.Theywilllooktoyoutounderstandwhatwemathematiciansdoandhowwethink.Yourstudentswilltrytodiscernthemeaningsthatyou,yourcurriculummaterials,andotherstudentsgivetoideas,strategies,andsymbolsthroughtheirparticipationindoingmathematicsinyourclassroom.Youmustbeawarethattalkingaboutthemeaningsofwords,ideas,andsymbolsisanimportantpartofyourroleasateacher;andyoumustbetransparentandcarefulinyouruseofwords,symbols,andnotationwhenyouareteachingmathematics.

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ChapterTwo

MeasurementinthePlane

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ClassActivity6: MeasureforMeasureAndtherewentoutachampionoutofthecampofthePhilistines,namedGoliathof

Gath,whoseheightwassixcubitsandaspan. ISamuel17:4

1) Usingyourcubit(lengthfromelbowtofingertips)andhandspan,determinehowtall

GoliathwasbycuttingastringthatisaslongasGoliathwastall.Compareyourstringlengthwiththestringsofothersinyourgroup.Whatarethedifficultiesthatmightarisefromchoosingandusingunitsdeterminedbyeachperson’sownbody?Whydoyouthinkweusethe“foot”asacommonunitoflength?

2) Onthenextpage,arefourcopiesofthesamelinesegment.Usingeachoftherulersprovided,carefullymeasurethelinesegment.

53

ReadandStudy

Itisnotenoughtohaveagoodmind.Themainthingistouseitwell. ReneDescartes

Thebigideaofmeasurementisthatofcomparinganattributeofanobjecttoanappropriateunit.Forexample,wemightcomparethelengthofourdesktoafoot-longrulerorwemightcomparetheareaofsheetofpapertotheareaofagreentriangle.Sothekeyquestionregardingmeasurementisthis:Howmanyoftheunitfitintotheobject?Whatmakesaunitappropriate?Well,firstitmusthaveadimensionthatmatchestheattributetobemeasured.Forexample,wemeasurelength(orwidthorheight)usingone-dimensionalunits.Hereisanexampleofaone-dimensionalunit:Wemeasureareausingtwo-dimensionalunitslikethis:Wemeasurevolumeusingthree-dimensionalunitslikethis:Wewilltalkmoreaboutvolumemeasurementinlatersections.Second,ifyouwanttobeabletocommunicatewithothers,ithelpsthattheunitbea‘standard’one.Astandardunitisonethattheculturehasagreedupon.Eachpersonhasamentalmodeloftheunitsoheorshecanpicturehowbigitis.Forexample,inourcultureafootisastandardunitformeasuringlength.InEurope(andmostoftherestoftheworld)ameterisastandardunitformeasuringlength.Ifwewanttotalkamongcultures,weneedtobeabletoconvertfromoneunittoanother.Thereareabout3.28feetinameter.Ifaroomis15feetlong,approximatelyhowmanymetersisthat?S T A N D A R D A N D N O N S T A N D A R D U N I T S

Third,itisusefulthattheunitbeofreasonablesizeinrelationtotheattributetobemeasured.Itwouldbeinconvenienttomeasurethelengthoffootballfieldusingmicrons(areallysmallone-dimensionalunit).Inthemetricsystemsizeisindicatedbytheprefix.Forexample,theprefixkilomeans1000times.Soakilometeris1000meters.Inthecurriculummaterialselementarystudentsareexpectedtousetheprefixesmilli(onethousandth),centi(onehundredth),andkilo.Youshouldmemorizetheseandbeabletomakeconversions.Approximatelyhowmanycentimetersarethereinafoot?Becausemeasurementalwaysinvolvescomparison,itisnecessarilyalwaysanapproximation.Weoftenindicateourdegreeofcertaintyaboutameasurementbasedonthewaywereportit.Forexample,ifIclaimadeskis1.23meterslong,thissuggeststhatIamconfidentintheaccuracytothehundredthofameter(tothecentimeter).Whenworkingwithchildrenyou

j ''

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shouldexplicitlydiscussthesefacetsofmeasurementandyoushouldbesuretoalwaysreporttheunitwithanyactualmeasurement.Sayingthatadeskis1.23longmeansnothing.Sayingthatitis1.23meterslongmakessense.Acommonmeasurementwemakeforaplaneobjectistomeasurethedistancearounditsboundary.Wecallthismeasurementtheperimeteroftheobject.(Whentheobjectisacircle,wecallthislengththecircumference.)Theperimeterofaplaneobjectisaone-dimensionalmeasurement–soweuselinearunitslikeinchesorcentimeters.Wecalculateaperimeterbysimplyaddingupthelengthsofthecurvesthatmakeuptheobject.Wecanmeasurethelengthsoflinesegmentswitharuler.Wewillfindaformulaforthecircumferenceofacircleinafutureactivity.A P P R O X I M A T I O N A N D P R E C I S I O N

ConnectionstoTeachingOnlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.

Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient. EugeneS.Wilson

TheCommonCoreStateStandardsformathematicsrequirethatchildrenbegintostudystandardmeasurementoflengthbeginningingrade2.ReadtheexcerptfromtheCCSSbelow.


NoticethatchildrenaretobelearningbothmetricandEnglishunits.Comeupwithabenchmark(somethingtoimaginethatistherightlength)foreachoftheseunits:inches,feet,centimeters,andmeters.

CCSSGrade2:Measureandestimatelengthsinstandardunits.

1. Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.

2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.

3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.

4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthe

lengthdifferenceintermsofastandardlengthunit.

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TheCommonCoreStateStandardsformeasurementingrade1areshownbelow.Comparethemtothegrade2standards.Howdotheydiffer?


Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide.Spend10-15minuteslookingthroughUnit4Module1Session1TeacherFeet.Inwhatwaysdothechildrenneedtoanalyzetheprocessofmeasurementinthissession?

Homework

Manyoflife'sfailuresarepeoplewhodidnotrealizehowclosetheyweretosuccesswhentheygaveup.

ThomasA.Edison


2) DotheConnectionsproblems.

3) Whataregoodbenchmarksforthemillimeter,centimeter,decimeter,andkilometer?Findsomethingthatisaboutthelengthofeach.

4) UseanappropriateEnglishunittoestimatea)thelengthofyourtable,b)thedistancefromChicagotoDenver,andc)thewidthofapencil.

5) Useanappropriatemetricunittoestimatea)thelengthofyourtable,b)thedistancefromChicagotoDenver,andc)thewidthofapencil.

CCSSGrade1:Measurelengthsindirectlyandbyiteratinglengthunits.

1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.

2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limittocontextswheretheobjectbeingmeasuredisspannedbyawholenumberoflengthunitswithnogapsoroverlaps.

56

6) Findtheperimetersoftheplanefiguresshownbelowbymeasuringthem

a) UsinganappropriateEnglishunit.b) Usinganappropriatemetricunit.c) NowconvertyourmetricunitstoEnglishunitstocheckthatyourmeasurementsin

b)matchyoumeasurementsina).

7) Explainwhyitmakessensethattheperimeterofarectanglecanbefoundbycomputing2l+2wwherelisitslengthandwisitswidth.

8) Iwanttorunaroundthebelowlake.Iplantohugtheshoreline.HowfardoIhavetorun?Thinkaboutvariouswaysyoucoulduseamaptoanswerthisquestion.Howaccurateisyourestimate?Howcanyouimproveitsaccuracy?LakeJen

Scale1cm=3miles

9) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide.Spend10-15minuteslookingthroughUnit4Module1.Then,carefullyexaminetheworksheet4AEstimate&MeasureInchesRecordSheet.Whatconversationscouldyouhavewithyourstudentsaboutthebigideasoflinearmeasurementbasedonthisactivity?

57

ClassActivity7: Part1Triangulating

Godowndeepenoughintoanythingandyouwillfindmathematics.DeanSchlicter

Defineonetriangleunittobetheareaofthegreentrianglepatternblock.Usingthisunit,determinetheareaofthissheetofpaper.Thenusethetriangleunittoestimatetheareaofeachofthepatternblockshapespicturesbelow.

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ClassActivity7: Part2AreaEstimationHereisamapofthegreatstateofWisconsin.Withoutlookinganythinguponline,yourgroupneedstomakeareasonableestimateofitsareainsquaremiles.Firstdiscussacoupleofdifferentwaysofdoingthisusingthemap.Thengoaheadandcarefullycomputeyourestimate.

http://www.nationsonline.org/oneworld/map/USA/wisconsin_map.htm

Wouldyouguaranteeyourestimatetothenearsest10squaremiles?Thenearest100squaremiles?Thenearest1000squaremiles?Somethingelse?Howdidyoudecide?

59

ReadandStudy

Themanignorantofmathematicswillbeincreasinglylimitedinhisgraspofthemainforcesofcivilization.

JohnKemenyMathematiciansdefineareaasthequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigure.Wecommonlymeasurethisareaintermsofsquareunitssuchassquareinchesorsquarecentimeters.Sotofindtheareaofafigureyouneedtofindthenumberofsquareunitsitwouldtaketocoverthefigure.Wewantyoutoliterallydothisnow.Hereisaunitofarea(thesquarecentimeter):Traceitandthenseehowmanyofthoseunits(includingpartsofunits)ittakestocovertheshapebelow: Iftheshapeisarepresentationofalake,andacentimeteroflengthcorrespondsto3miles,thenwhatistheareaoftheactuallake?Explain.I D E A O F A R E A

Weoftenmeasuretheareaofirregularlyshapedobjectsjustbycounting(andestimating)thenumberofsquareunitswithintheboundaryoftheobject.Sometimeswesuperimposeagridtohelpwiththatestimate.Estimatetheareaofthisobjectnowassumingthateachsmallsquareisaunitofarea.Whenwemeasuretheareaofanobjectinsquareunits,noticethatwearetakingadvantageofthefactthatsquarestessellatetheplane–therearenogapsbetweenthesquares.Wehavealreadydiscoveredthatthereareothershapesthattessellatetheplaneaswell;twoofthem

60

are,likethesquare,alsoregularpolygons.Whichones?Whydoyouthinkpeoplechosetousesquareunitsinsteadofsomeothershapethattessellatestheplane?


Teachingcreatesallotherprofessions.AuthorUnknown

ChildrenneedtohaveexperiencesfindingareasbycoveringfigureswithsquareunitslikeyoudidintheReadandStudysection.Ifyoushowthemformulastooearly,theywillsimplytrytorememberthoseformulasandtheymaynotfocusontheideaofarea.Besides,formulasworkonlywithalimitednumberofshapes,andsoestimatingareasusingthedefinitionisausefulskillinitsownright.Wesuggestthatchildreningrades2and3spendmanyweeksestimatingareasusingsquarestickersorsquaregridstocoverfigures.Someofthosefiguresshouldhavecurvedboundariesandsomeshouldhavespecialpolygonshapes.Childrenwillquicklybegintoproposeshortcutsontheirownthatwillleadnaturallytotheareaformulas.Forexample,ifchildrencomputeareasofthebelowrectangleshapesusingstickersoragrid,someofthemwillnoticethatashortcutforfindingareasofrectangleshapesistosimplymultiplythelengthoftherectanglebyitswidth.Thenyoucantalkwiththemaboutwhythisisso.Practicethatnowbydoingthebelowtasks.C O V E R I N G W I T H U N I T S Q U A R E S

Firstuseagridtoestimatehowmanyareaunitsittakestofilleachrectangleshape.Thenexplainwhyitmakessensethattheareaofarectanglecanbefoundbymultiplyingitslengthbyitswidth.Hereisaunitoflength:_____ Hereisaunitofarea:

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TheCommonCoreStateStandardsdescribethefollowingstandardsforchildreningrade3.Readthemtoseethatthethingschildrenshouldlearnaremanyofthethingswehavedescribedinthissection.


Wehavenottalkedexplicitlyabout2.c.and2.d.above.Sketchpicturestohelpyoutomakesenseofwhattheymeanbythose.

CCSSGrade3:Geometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.

1. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“one

squareunit”ofarea,andcanbeusedtomeasurearea.

b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.

c. Measureareasbycountingunitsquares(squarecm,squarem,squarein,square

ft,andimprovisedunits).

2. Relateareatotheoperationsofmultiplicationandaddition.a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,and

showthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.

b. Multiplysidelengthstofindareasofrectangleswithwholenumbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.

c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-

numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.

d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposing

themintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

62

Homework

ThemoreIpractice,theluckierIget.JerryBarber



3) Useagridtoestimatecarefullytheareaofthebelowcircularfigureinsquare

centimeters.

4) Astudentcomestoyouandasks,"Whydoweusesquarecentimeterstomeasuretheareaofthecircle?Acircleisroundandnotsquare."Explaintoherwhywestillusesquarecentimeterstomeasuretheareaofacircle.

5) WewouldliketohavetheabovelakeclassifiedasoneoftheGreatLakes.AspartoftheapplicationtotheDepartmentoftheInterior,wehavetoreportitsarea.Estimatetheareaofthelakeinsquaremiles.

Scale1cm=3miles

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6) Hereisafloorplanforthefirstlevelofahouse.a) Whatisitsareainsquarefeetincludingthegarage?Assumethat1cm

represents6feetoflength.b) Howgoodisyourestimate?Areyouconfidenttothenearestsquarefoot?c) WhatCommonCoreStateStandardismetbythisproblem?Explain.

7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade3TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3.

a. Howareareaandperimeterintroduced?b. PrintoutandthencarefullyworkthroughtheworksheetBayardOwl’sBorrowed

Tables.Whatconversationscouldyouhavewithyourstudentsaboutthebigideasofperimeterandareabasedonthisactivity?

8) AnNBAbasketballcourtmeasures50feetby96feet.Usethisinformationbelowtodeterminehowmanyacresitcovers.

1foot=12inches1yard=3feet1mile=1760yards1acre=4840squareyards1squaremile=640acres

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ClassActivity8: FindingFormulas

Onecannotescapethefeelingthatthesemathematicalformulashaveanindependentexistenceandanintelligenceoftheirown,thattheyarewiserthanwe

are,wisereventhantheirdiscoverers. HeinrichHertz

Wecommonlycomputeareasofsomespecialpolygonshapeswiththeuseofformulas.Theseformulascanbeexplained–theyarebasedongeometricdefinitionsandtheorems–andwewantyoutounderstandwhytheymakesense.Thatisthegoalofthisactivity.

1) Rectangle:Usetheideaofareaasthenumberofsquareunitsittakestocoveranobjecttoexplainwhyitmakessensethattheareaofarectangleissimplytheproductofitslengthandwidth.

2) Parallelogram:Showhowtocutandrearrangeaparallelogramtomakearectanglewith

thesamearea.Arguethattheresultingfigureisinfactarectangle.UsetheformulafortheareaofarectangletofindaformulafortheareaAofaparallelogramusingthebasebandtheheighthoftheparallelogram.

3) Triangle:Showhowtorearrangetwocopiesofthesametriangletomakeaparallelogram.Arguethattheresultingfigureisinfactaparallelogram.UsetheformulafortheareaofaparallelogramtofindaformulafortheareaAofatriangleusingthebasebandtheheighthofthetriangle.

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ReadandStudy

Theessenceofmathematicsisitsfreedom. GeorgCantor

Manyofyourstudents(andmanyoftheirparents)willthinkthatformulastellthewholestoryaboutarea.Infact,somepeoplemistakenlydefineareaas“lengthtimeswidth.”Everyclosedtwo-dimensionalshapehasarea,butaswehaveseeninanearliersection,onlyaveryfewoftheseshapeshaveformulaswecanusetocalculatethearea.Theareaofsomegeometricobjectsismoreeasilydeterminedthroughtheuseofareaformulas.IntheClassActivityyoudevelopedseveralusefulandwell-knownformulasthatarereadilyfoundthroughoutthecurriculum.Notethatalloftheseformulasarebasedonthechoiceofasquareastheunitofarea.Ifwehadusedadifferentshapeastheunit(aswedidintheprevioussection),alloftheformulaswouldhavechanged.Themostfundamentalareaformulaisfortheareaofarectangle:length´width.Alloftheotherformulasforareaarebuiltonthat.Asyouhaveseen,theseformulasarereallytheoremsthathavebeenproventowork.And,thesetheoremsareonlytruewhenweusetheunitsquareasourunit.Explaincarefullywhythatpreviousstatementissoimportant.Whathappensifwedonotusesquaresasourunit?YoushouldhaveyourupperelementarystudentsdoactivitieslikethoseintheClassActivitysothattheycanseethisforthemselves.IntheBridgesinMathematicscurriculumforgrade4,studentsdiscovertheformulasfortheareaandperimeterofrectangles.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module3Session3.Inwhatwaysdoesthisactivityhelpstudentstodevelopformulasfortheperimeterandareaofarectangle. M A K I N G S E N S E O F A R E A F O R M U L A S

ConnectionstoTeaching Knowingisnotenough;wemustapply.Willingisnotenough;wemustdo.

JohannWolfgangvonGoethe

Oncechildrenunderstandtheideasofperimeterandarea,theycansolvesomepracticalproblems.Wewillposesomenowthatspecificallyaddressthestandardsthatfollowthem.Dotheseproblemsandthenreadtoseewhichofthestandardswe’veaddressedwiththem.

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a) Ifarectangularroomhasalengthof5feetandanareaof60squarefeet,whatisitswidth?

b) Ifarectanglehasalengthof10feetandaperimeterof36feet,whatisitswidth?

c) Isittruethatrectangleswithbiggerperimetersalwayshavebiggerareastoo?Explain.

d) Isittruethatrectangleswithbiggerareasalwayshavebiggerperimeterstoo?Explain.

67



Homework

Toliveacreativelife,wemustloseourfearofbeingwrong. JosephChiltonPearce

1) DotheproblemsandthendotheitalicizedstatementintheConnectionssection.

CCSSGrade3:Geometricmeasurement:recognizeperimeterasanattributeofplanefiguresanddistinguishbetweenlinearandareameasures.Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.

CCSSGrade4:Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmallerunit.

1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwocolumntable.Forexample,knowthat1ftis12timesaslongas1in.Expressthelengthofa4ftsnakeas48in.Generateaconversiontableforfeetandincheslistingthenumberpairs(1,12),(2,24),(3,36),...

2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.

3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.Forexample,findthewidthofarectangularroomgiventheareaoftheflooringandthelength,byviewingtheareaformulaasamultiplicationequationwithanunknownfactor.

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2) PracticeexplainingwhyeachoftheformulasfromtheClassActivitymakessense.

3) Oneofyourstudentsisconfusedaboutareacalculations.Adamwonderswhy,ifyoutakearectangleandmultiplythelengthofeachsideby2,theareaofthenewrectangleisn'ttwiceasbigastheareaoftheoldrectangle.Drawsomepicturestohelpyouseewhatisgoingonhere.Whatwillyousaytohim?

4) Explainwhyitmakessensethattheareaofatrapezoidisalways½h(b1+b2)whereb1andb2arethelengthsoftheparallelbasesandhistheheight.Youcandothisbypartitioningthetrapezoidregionintopieces,orbycuttingitapartandrearrangingit,orbyaddingonstructure.Youmayusewhatyouknowaboutfindingareasofrectangles,parallelograms,andtriangles.Seeifyoucanfindmorethanonewaytodothis.

5) Findtheareaofthetrapezoidshownbelowinasmanydifferentwaysasyoucan.

Assumethateachgridsquarerepresentsoneunitofarea.

* * * * * *

* * * * * *

* * * * * *

* * * * * *

6) Supposearectanglehasaperimeterof36units.Whatareallthepossiblewholenumberdimensionsoftherectangle?Makeagraphofwidthvs.area.Whichwidthgivesthegreatestarea?

7) Supposeyouhave100metersofflexiblefencingtomarkapastureoutontheplains.

Howwouldyousetitup(whatshape)toenclosethemostgrazingareaforyourcattle?Whatdimensionswouldyouuseifthepasturehadtobearectangle?

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8) Thetrianglebelowisconstructedona1cmgrid.Findtheareaofthetriangleusingatleastthreedifferentmethods.

9) Assumethatthetriangleareaaboverepresentspartofasignthatneedstobepainted.

Thescaleofthedrawingisthat1cmrepresents10feet.Theinstructionsonthepaintcansaythat1gallonofpaintwillcover100squarefeet.Howmanygallonsofpaintwillyouneedtobuyinordertopaintthetriangle?

1 cm

70

ClassActivity9: TheRoundUp

Donotdisturbmycircles!Archimedes’finalwords

1) Acircleisdefinedasthesetofallpointsintheplanethatareequidistantfromagiven

pointcalledthecenter.Studythisdefinition.Iftheword“all”wasmissing,howwouldthatchangethings?Whatifthewords“intheplane”weremissing?

2) Howmanytimesdoesthediameterofacirclefitintoitscircumference?Gathersomedatatosee.

3) Exploretheideaoftheareaformulaforacircularregionbyrearrangingitintoaparallelogram-shapedfigure.Findtheareaofthe“parallelogram”intermsoftheradiusofthecircle.Whyisthisjusttheideaoftheargument?

71

ReadandStudy

Itisnotoncenortwice,buttimeswithoutnumberthatthesameideasmaketheirappearanceintheworld.

Aristotle CirclesareasfundamentaltoEuclideangeometryasarepointsandlines.RecallthatEuclid’sthirdaxiomassuresusthatwecanalwaysmakeacircleofanysize(radius)wewant.Ofcourse,thecircleswedrawarestillonlyapproximationsofatruemathematicalcircle,justasthelinesegmentswemakearejustapproximationsofatruelinesegment.Acircleisthesetofallpointsintheplanethatareequidistantfromagivenpoint,calledthecenter(Ointhediagrambelow).Thediagramshowssomeoftheotherimportanttermsassociatedwithacircle.Becertainyouunderstandeachtermandcanexplainitsmathematicaldefinition,whichyouwillfindintheglossary.

Central Angle Ð AOC

Tangent

Secant

Chord DE Diameter AB Arc DE

Radius CO

Sector O

C

A

B

D

E

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Itisanamazingfactthat,foranysizecircle,theratioofthecircumferencetothediameteristhesamenumber.Thiswasknowninallearlycivilizations.Wecallthatratio“pi.”Soπ(pi)isthesymbolforthenumberoftimesthediameterofacirclefitsintoitscircumference.Readthatagain;itisimportant.Thisnumberpisanirrationalnumber.Thatmeansithasadecimalrepresentationthatneitherendsnorrepeats.Thiswasprovedin1761byamathematiciannamedJohannHeinrichLambert.Evenwhenweusethepkeyonacalculator,weareusinganapproximatevalue.Elementarystudentscommonlyuseeither3.14or

722 asanapproximatevalueforpwhen

carryingoutcalculationsinvolvingcircles.Alwaysbesuretomakethepointthatthisisjustanapproximation.W H A T I S Π ?


Attheageofeleven,IbeganEuclid,withmybrotherasmytutor.Thiswasoneofthegreateventsofmylife,asdazzlingasfirstlove.

BertrandRussellIntheBridgesinMathematicscurriculumforgrade4,studentsareintroducedtobasiccircleterminology.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module1Session5.Howarethepartsofacircleintroducedtothestudents?Withinthelesson,thestudentsareaskedtowritetheirowndefinitionofacircle.IssaandSuzigavethefollowingdefinitions.Aretheirdefinitionscorrect?Howasateacherdoyourespondtoeachofthesestudents? Issa’sdefinition:Acircleisashapethat’sroundandhas360°. Suzi’sdefinition:Acircleisashapethathasthesamewidthallthewayaround.

73

Homework

Eureka!I’vegotit! Archimedes


2) SolvetheproblemsintheConnectionssection.

3) Studyeachboldandunderlinedtermusedinthissection.Thismeansyoushouldbe

abletoexplainthedefinitionusinggoodmathematicallanguageandthatyoushouldbeablesketchexamplesandnon-examplesofeachterm.

4) HowdoesthedefinitionofπleadtotheformulaC=2πr(whereCisthecircumferenceofacircleandrisitsradius)?

5) Ifacirclehasameasuredradiusof5inches,then,usingtheformulaforfindingtheareaofacircle,wewouldsaythattheareaofthecircleisapproximately78.5squareinches.Giveatleasttwodistinctreasonswhythecalculationisapproximate.

6) Explaintherelationshipbetweenasecantandachordofacircle,betweenaradiusandadiameter,andbetweenasecantandatangentofacircle.

7) Hereisanothersetofpicturesdesignedtogiveanintuitiveargumentthattheareaofacircularregionisπr2.Imaginethatthecircleshownismadeofcircularstringssittingoneinsidethenext.Thenyoutakeascissors,sniparadius,andflattenthestringstomakethetriangularshape.Whatistheareaofthetriangleintermsofr(theradiusofthecircle)?

8) IfIdoublethediameterofacircularregion,whathappenstoitscircumference?

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9) IfIdoublethediameterofacircularregion,whathappenstoitsarea?

10) Ifan8-inch(diameter)pizzacosts$5,howmuchshoulda16-inchpizzacost?Justifyyourresult.

11) Havealookatthecirclesbelow.

a) Carefullymeasuretheperimeterofthesquarethatisinscribedinsidethefirstcircle.

b) Carefullymeasuretheperimeteroftheregularpentagoninscribedinsidethesecondcircle.

c) Carefullymeasuretheperimeteroftheregularhexagoninscribedinsidethethirdcircle.

d) TrueorFalse?Asthenumberofsidesoftheinscribedpolygongrows,sodoestheperimeterofthepolygon.

e) TrueorFalse?Ifweinscribeapolygonwithinfinitymanysides,thentheperimeterofthatpolygonwillbeinfinitelylong.Explainyourthinking.

f) Howcouldyouusethesemeasurementstogetanestimateforπ?Explain.

Morethan2000yearsagoArchimedesfoundapproximateboundsforπusinginscribedandcircumscribed(outside)polygonswith,getthis,120sides.WestilluseArchimedesboundstoday( 7

2271223

<< ).Theaverageofthesetwovaluesis

roughly3.1419whichiscorrecttothreedecimalplaces.Notbadfor350BC.

I N S C R I B E D P O L Y G O N S

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ClassActivity10: PlayingPythagoras

Ihavehadmyresultsforalongtime,butIdonotknowyethowIamtoarriveatthem.

CarlFriedrichGaussThePythagoreanTheoremisthoughttobealmost4000yearsold.TheBabylonians,theEgyptians,andtheChineseallknewit.Thatis,theyknewthatthesumoftheareaofthesquaresonthelegsofarighttriangleequalstheareaofthesquareonthehypotenuse,andtheyusedthisfactnumericallyinconstructionandcommerceandsurveying.

1) Carefullymeasuretheareasofthesquaresinthebelowexampletoseeifthistheoremseemstrue.

2) Whathappensifthetriangleisn’tarighttriangle?Isthesumofthesquaresonthe“legs”(shortersides)ofanobtusetrianglemoreorlessthanthesquareonthelongestside?Whathappensinanacutetriangle?Drawsomediagramstosee.

(Thisactivityiscontinuedonthenextpage.)

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ThefirstproofofthetheoremisattributedtoPythagorasofSamos(it’sinGreece)around600B.C.Sincethen,hundredsofdifferentproofshavebeencreated.Youaregoingtoexploreoneofthem.a) Theproofbeginswithanyrighttriangle.Carefullydrawyourownandlabelthe

lengthofthehypotenusec,thelengthofthelongerlegb,andthelengthoftheshorterlega.

b) Nowmakefourcongruentcopiesofyourtriangle,cutthemout,andarrangethem

intoaquadrilateralasshownbelow.

c) Justifythattheboundaryoftheouterquadrilateralisasquare.

d) Justifythattheinnerquadrilateralisasquare.

e) Nowgiveanalgebraicproofthatc2=a2+b2byusingthefactthatthefivepolygonsformthelargefigure(sotheareaformulasforthefivemustsumtotheareaformulaofthelargesquare).

f) Whatthingsgowrongifthetrianglesarenotrighttriangles?

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ReadandStudy

I’mverywellacquaintedtoowithmattersmathematical,Iunderstandequations,boththesimpleandquadratical,AboutbinomialtheoremI’mteemingwithalotofnews--

Withmanycheerfulfactsaboutthesquareofthehypotenuse.Gilbert&Sullivan,“ThePiratesofPenzance”

ThePythagoreanTheoremisoneofthemostwell-knownandmostimportanttheoremsofallofelementarymathematics.ItisnamedaftertheGreekmathematician,Pythagoras,andEuclidincludeditasthefittingendtovolumeoneofTheElements.InEuclid’swordsthetheoremsaysthis:

Inright-angledtrianglesthesquareonthesideoppositetherightangleequalsthesumofthesquaresonthesidescontainingtherightangle.

Euclidintendedtheword“square”tomeanthephysicalsquaredrawnonthehypotenuseorlegoftherighttriangle.Sohistheoremsaysthattheareaoftheyellowsquare(inthefigureabove)isequaltothesumoftheareasoftheredandbluesquareswhenABCisarighttriangle.Todaywemorecommonlyuseanalgebraicstatement:

Ifarighttrianglehaslegsoflengthsaandbandahypotenuseoflengthc,thenc2=a2+b2.

Noticethathere,a,b,andcarenumbersrepresentingthelengthsofthevarioussidesofthetriangle.Sonowtheword“square”carriesthealgebraicmeaningdenotedbytheexponenttwo.P R O O F S O F T H E P Y T H A G O R E A N T H E O R E M

Therearemany(atonecount,atleast367)proofsofthePythagoreanTheorem.YourecreatedoneoftheproofsintheClassActivity.YouwillbeaskedtoexploretwoothersaspartoftheHomeworkforthissection.

C B

A

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ThePythagoreanTheoremisanexampleofatheoremwhoseconverseisalsoatheorem.StatetheconverseofthePythagoreanTheorem.Ifyouhavetolookup“converse,”visittheglossaryanddoso.TheconverseofthePythagoreanTheoremgivesusawaytodiscoverwhetherornotatriangleisarighttriangleevenwhenweknownothingabouttheanglemeasuresofthetriangle.Forexample,supposeweknowthatthesidesofatriangleareexactly3,4,and5incheslong.Isthisarighttriangle?Let’ssubstitutethevalues3fora,4forb,and5forc(Howdoweknowthatthehypotenusemustbethesideoflength5?)andcheckoutthePythagoreanrelationshipc2=a2+b2.Does52=32+42?Iftheansweris“yes,”thenthetriangleisarighttriangle.


Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstocreateanduserepresentationstoorganize,record,andcommunicate

mathematicalideas;toselect,apply,andtranslateamongmathematicalrepresentationstosolveproblems;andtouserepresentationstomodelandinterpret

physical,social,andmathematicalphenomena.NCTMPrinciplesandStandardsforSchoolMathematics,2000

Havingtwo(ormore)waystointerpretorrepresentamathematicalideaisanimportantcharacteristicofmathematicsforteaching.TheNCTMStandardsrecognizethisandcallforallelementarystudentsto“select,apply,andtranslateamongmathematicalrepresentationstosolveproblems.”Andso,asteachers,itisalsoimportantthatweunderstandamathematicalconceptfrommorethanonepointofviewinordertoassistourstudentstousedifferentrepresentationsofthatconcepttosolveproblems.G E O M E T R I C A N D A L G E B R A I C R E P R E S E N T A T I O N S

ThePythagoreanTheoremisonesuchideathatcanbeunderstoodfrommanyviewpoints:geometric,numeric,andalgebraic.Someofourstudentswillmoreeasilygraspthealgebraicapproachwhileotherswillpreferthemoreconcretegeometricornumericrepresentation.Asteachers,wemustmasterallrepresentationsinordertocarryoutourresponsibilitiestosupporteachstudent’slearning.

79

Homework

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackinwill.

VinceLombardi


2) JamesAbramGarfield(1831-1881),thecountry’stwentiethpresident,createdthisproofofthePythagoreanTheoremin1876,whilehewasamemberoftheHouseofRepresentatives.FindGarfield’sproofbyusingthediagrambelowbyfindingformulasfortheareaofthetrapezoidintwodifferentways.Howdoeshisargumentfailifthetrianglesarenotrighttriangles?

3) Whenthelengthsofthesidesofarighttriangleareallintegersthethreenumbers(a,b,

c)areknownasaPythagoreanTriple.Explainwhy(3,4,5)isaPythagoreanTriple.WhichofthefollowingtriplesofnumbersarePythagoreanTriples?a)(4,5,6) b)(4,6,8) c)(6,8,10)

4) Supposeatrianglehassidesoflength8,15,and17.Isitarighttriangle?

5) Whatistheexactheightofanequilateraltriangleifallsidesareoflength10?Oflength3?Oflength4?Oflengths?

6) Aclosetis3feetdeep4feetwideand12feethigh.Findthedistancefromonecornerat

thefloortothediagonallyoppositecornerattheceiling.Youmightwanttohavealookataboxtohelpyouseewhattodohere.

a

b

c

c b

a

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SummaryofBigIdeas Althoughthismayseemaparadox,allexactscienceisdominatedbytheidea

ofapproximation. BertrandRussell

• Measurementisthecomparisonofanattributeofanobjecttoaunit.Tomeasure

meanstoseehowmanyoftheunitfitintotheobjectyouaremeasuring.

• Lengthisaone-dimensionalmeasurement;areaistwo-dimensional;andvolumeisthree-dimensional.

• Allmeasurementisapproximate.• Thechoiceofaunitisafundamentalpartofthemeasuringprocess.

• Areaisthenumberofsquareunitsittakestocoveranobject.Itisnotdefinedas“length

timeswidth,”and,infact,thatformulaworksonlyinafewlimitedcases.

• Formulasforthecalculationofareacanbeexplainedbythegeometryoftheobjectsbeingmeasured.Asateacher,youwillneedtohelpyourstudentstoseewhereformulascomefromandwhytheymakesense.

• πisdefinedasthenumberoftimesthediameterofacirclefitsintoitscircumference.Itisanirrationalnumber.

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ChapterThree

TheThirdDimension

82

ClassActivity11: StrictlyPlatonic(Solids)

IfIhaveevermadeanyvaluablediscoveries,ithasbeenowingmoretopatient attention,thantoanyothertalent. SirIssacNewton

Hereisadefinitionforyoutostudy:Apolyhedronisafinitesetofpolygon-shapesjoinedpairwisealongtheedgesofthepolygonstoencloseafiniteregionofspacewithinonechamber.Thepolygon-shapedsurfacesarecalledfaces.Itisrequiredthatthesurfacebesimple(notmorethanonechamberisenclosed)andclosed(youcan’tgetinfromtheoutsidewithouttearingit).Thesegmentswherethepolygonsmeetarecallededges.Thepointswhereedgesintersectarecalledvertices.

1) Usethedefinitiontodecidewhichofthebelowimagesof3-dimensionalobjectsrepresentpolyhedra(“polyhedra”isthepluralformofpolyhedron)

Apolyhedronisregularifallofthefacesarethesamecongruent,regularpolygonandalloftheverticeshaveexactlythesamenumberofpolygons.

2) UsethepiecesofaFrameworksÔset(manufacturedbyPolydron)tobuildalloftheregularpolyhedra.Besystematicsoyoucangiveanargumentthatyouhavefoundthemall.(Polyhedraarenamedforthenumberoffacestheycontain;forexample,apolyhedronwithtenfaceswouldbecalledadecahedron.)

3) Canyoubuildapolyhedronthatusesonlyonetypeofcongruentregularpolygonbutisnotregular?Explain.

4) Seeifyoucanfindshortcutwaysofcountingthenumbersofverticesoredgesofregularpolyhedra.

83

ReadandStudy

Everythingshouldbemadeassimpleaspossible,butnotonebitsimpler. AlbertEinstein

IntheClassActivityyouwereaskedtobuildmodelsoftheregularpolyhedra.Thesespecialobjects,alsocalledthePlatonicsolids,havebeenknownsincebeforethedaysoftheGreekmathematics.Approximationsoftheregularpolyhedraevenoccurinnature.Inparticular,thetetrahedron,cube,andoctahedronshapesallappearascrystalstructures.Wealsofindpolyhedralshapesamonglivingthings,suchastheCircogoniaicosahedrashownbelow,aspeciesofRadiolaria,whichisshapedlikearegularicosahedron.

(http://en.wikipedia.org/wiki/Platonic_solids)Manyvirusesalsohavetheshapeofaregularicosahedron.Viralstructuresarebuiltofrepeatedidenticalproteinsubunitsandapparentlytheicosahedronistheeasiestshapetoassembleusingthesesubunits.R E G U L A R P O L Y H E D R A

Netsaretwo-dimensionalfiguresthatcanbefoldedintothree-dimensionalobjects.Belowarenetsfortheregulartetrahedronandforthecube.Imaginehoweachfoldsuptomakethe3-dimensionalobject.

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Beforeyoureadfurther,gotothissiteandbuildyourselfoneofeachoftheregularpolyhedra.Youwillneedthemforthehomework.http://www.mathsisfun.com/platonic_solids.html


Studentsingrades3–5shouldexaminethepropertiesoftwo-andthree-dimensionalshapesandtherelationshipsamongshapes.Theyshouldbeencouragedtoreason

aboutthesepropertiesbyusingspatialrelationships. NCTMPrinciplesandStandards,2000

ThefollowingexcerptfromtheNCTMStandardsforGrades3–5Geometry(p.168)describestheimportanceofvisualizationandspatialreasoningastoolselementarystudentscanusetounderstandthepropertiesofgeometricobjectsandtherelationshipbetweenthesepropertiesandtheshapes.Readtheseparagraphsandstudytheexamples.Thenbuildthebuildingtheydescribe.S P A T I A L R E A S O N I N G

Usevisualization,spatialreasoning,andgeometricmodelingtosolveproblemsStudentsingrades3–5shouldexaminethepropertiesoftwo-andthree-dimensionalshapesandtherelationshipsamongshapes.Theyshouldbeencouragedtoreasonaboutthesepropertiesbyusingspatialrelationships.Forinstance,theymightreasonabouttheareaofatrianglebyvisualizingitsrelationshiptoacorrespondingrectangleorothercorrespondingparallelogram.Inadditiontostudyingphysicalmodelsofthesegeometricshapes,theyshouldalsodevelopandusementalimages.Studentsatthisagearereadytomentallymanipulateshapes,andtheycanbenefitfromexperiencesthatchallengethemandthatcanalsobeverifiedphysically.Forexample,“Drawastarintheupperright-handcornerofapieceofpaper.Ifyouflipthepaperhorizontallyandthenturnit180°,wherewillthestarbe?”Muchoftheworkstudentsdowiththree-dimensionalshapesinvolvesvisualization.Byrepresentingthree-dimensionalshapesintwodimensionsandconstructingthree-dimensionalshapesfromtwo-dimensional»representations,studentslearnaboutthecharacteristicsofshapes.Forexample,inordertodetermineifthetwo-dimensionalshapeinfigure5.15isanetthatcanbefoldedintoacube,studentsneedtopayattentiontothenumber,shape,andrelativepositionsofitsfaces.

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Fig.5.15.Ataskrelatingatwo-dimensionalshapetoathree-dimensionalshape

Studentsshouldbecomeexperiencedinusingavarietyofrepresentationsforthree-dimensionalshapes,forexample,makingafreehanddrawingofacylinderorconeorconstructingabuildingoutofcubesfromasetofviews(i.e.,front,top,andside)likethoseshowninfigure5.16.

Fig.5.16.Viewsofathree-dimensionalobject(AdaptedfromBattistaandClements1995,p.61)

86

Homework

Gettingaheadinadifficultprofessionrequiresavidfaithinyourself.Thatiswhysomepeoplewithmediocretalent,butwithgreatinnerdrive,gomuchfurtherthan

peoplewithvastlysuperiortalent. SophiaLoren

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.

2) Writeoutthedetailsofamathematicalargumentthatthereareexactlyfiveregular

polyhedra.

3) Thefollowingpictureisoftengivenasanexampleofaregularicosahedron.Examinethepicturecarefullyanddeterminewhythispictureisclaimingtobesomethingthatitisnot.

4) Isitpossibletobuildapolyhedronwhereallofthefacesarecongruentregularpolygonsbutthepolyhedronisnotregular?Explain.

5) Usethefivemodelsyoubuilttofindaformulathatrelatesnumbersofvertices,edges,andfacesinregularpolyhedra.

6) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade

1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Session2MysteryBagSorting.Makeupyourownmysteryforyourstudentstosolvesimilartothosegiveninthelesson.

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ClassActivity12: PyramidsandPrisms

Itisbettertoknowsomeofthequestionsthanalloftheanswers. JamesThurber

Therearetwospecialcategoriesofpolyhedrawewillexploreinthisactivity:pyramidsandprisms.Apyramidisapolyhedroninwhichonefaceiscalledthebaseandalloftheremainingfacesaretrianglesthatshareacommonvertex(calledtheapex).Apyramidisnamedfortheshapeofitsbase.Forexample,asquarepyramidisonewhosebaseisasquare.(Noticethatthebaseofapyramidneednothavetheshapeofaregularpolygon.Itcouldlooklikethefigurebelow,forexample.)

1) Youhavealreadybuiltapyramidwithanequilateraltrianglebase(thetetrahedron).In

yourgroup,sketchapyramidwithasquarebaseandanotherwithahexagonalbase.

2) Useyourpicturestohelpyoufindformulasforthenumberoffaces,thenumberofvertices,andthenumberofedgesinapyramidwhosebaseisann-gon.Thenprovethatyourformulaswillworkforallpyramids.

Aprismisapolyhedroninwhichtwoofthefaces(calledthebases)arecongruentandlie on parallel (non-intersecting) planes andtheremainingfacesareparallelograms.Theprismisalsonamedafteritsbase.Iftheparallelogramfacesarerectangular,theprismisarightprism.Iftheparallelogramfacesarenon-rectangular,theprismisanobliqueprism.

3) Which(ifany)oftheregularpolyhedraareprisms?Explain.

4) Sketchanobliqueprism.C O U N T I N G V E R T I C E S , E D G E S , A N D F A C E S

5) Sketchaprismwithatriangularbaseandonewithahexagonalbase.Thenuseyour

picturestofindformulasforthenumberoffaces,thenumberofvertices,andthenumberofedgesinaprismwhosebasesarecongruentn-gons.Provethatyourformulaworksinthecaseofallprisms.

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ReadandStudy

Themediocreteachertells.Thegoodteacherexplains.Thesuperiorteacherdemonstrates.Thegreatteacherinspires.

WilliamArthurWard

ThemostfamouspyramidsaretheEgyptianpyramids.Thesehugestonestructuresareamongthelargestman-madeconstructions.InAncientEgypt,apyramidwasreferredtoasthe"placeofascendance."TheGreatPyramidofGizaisthelargestinEgyptandoneofthelargestintheworldwithabasethatisover13acresinarea.ItisoneoftheSevenWondersoftheWorld,andtheonlyoneoftheseventosurviveintomoderntimes.TheMesopotamiansalsobuiltpyramids,calledziggurats.Inancienttimesthesewerebrightlypainted.Sincetheywereconstructedofmud-brick,littleremainsofthem.TheBiblicalTowerofBabelisbelievedtohavebeenaBabylonianziggurat.AnumberofMesoamericanculturesalsobuiltpyramid-shapedstructures.Mesoamericanpyramidswereusuallystepped,withtemplesontop,moresimilartotheMesopotamianzigguratthantheEgyptianpyramid.ThelargestpyramidbyvolumeistheGreatPyramidofCholula,intheMexicanstateofPuebla.Thispyramidisconsideredthelargestmonumenteverconstructedanywhereintheworld,andisstillbeingexcavated.Modernarchitectsalsousethepyramidshapeforbuilding.AnexampleistheLouvrePyramidinParis,France,inthecourtoftheLouvreMuseum.DesignedbytheAmericanarchitectI.M.Peiandcompletedin1989,itisa20.6meter(about70foot)glassstructurewhichactsasanentrancetothemuseum.Themostcommonexampleofaprisminthe“realworld”isitsoccurrenceinoptics,whereaprismisatransparentopticalelementwithflat,polishedsurfacesthatrefractlight.Theexactanglesbetweenthesurfacesdependontheapplication.Thetraditionalgeometricalshapeisthatofatriangularprismwithatriangularbaseandrectangularsides,andincolloquialuse"prism"usuallyreferstothistype.Prismsaretypicallymadeoutofglass,butcanbemadefromanymaterialthatistransparenttothewavelengthsforwhichtheyaredesigned.Aprismcanbeusedtobreaklightupintoitsconstituentspectralcolors(thecolorsoftherainbow).Theycanalsobeusedtoreflectlight,ortosplitlightintocomponentswithdifferentpolarizations.

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Itisthesupremeartoftheteachertoawakenjoyincreativeexpressionandknowledge.

AlbertEinsteinElementarystudentsbegintheirstudyof3-dimensionalshapesinthefirstandsecondgrades.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Sessions4and5.Whatideasaboutprismsandpyramidsareemphasizedinthesesections?NowlookattheHomelinksectionfromSession5.Examinequestions4and5carefully.Determineseveralreasonswhyyourchosenfiguredoesnotbelong.Inquestion5,determineatleasttwodifferentfiguresonwhichtoplacethe“X”.

Homework

Iattributemysuccesstothis:Inevergaveortookanyexcuse. FlorenceNightingale

1) DotheproblemsintheConnectionssection.

2) Makeamathematicalargumentfortheformulasyoufoundforthenumberoffaces,the

numberofvertices,andthenumberofedgesinapyramidwhosebaseisann-gon.

3) Makeamathematicalargumentfortheformulasyoufoundforthenumberoffaces,thenumberofverticesandthenumberofedgesinaprismwhosebasesarecongruentn-gons.

4) Provethattheformulayoufoundrelatingthevertices,edges,andfacesofaregularpolyhedraalsoholdsforthen-gonalpyramidandforthen-gonalprism.

5) Thefollowingaredescriptionsofpyramidsandprisms.Identifytheprismorpyramidandsketchanet.a) Apolyhedronwith5facesand9edges.b) Apolyhedronwithtwohexagonalfacesandtheremainingfacesarerectangles.c) Apolyhedronwith12edgesand8verticies.d) Apolyhedronwith7facesand7verticies.

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ClassActivity13: SurfaceArea

Mathematicsmaybedefinedastheeconomyofcounting.Thereisnoprobleminthewholeofmathematicswhichcannotbesolvedbydirectcounting.

ErnstMach

1) Use14interlockingunitcubestobuildthe3-dimensionalfigurepictured.

a) Whatisthesurfaceareaofthisobject?

b) Whatisitsvolume?

c) Isthisapolyhedron?Explain.

2) Whatareallthepossiblevaluesforthesurfaceareaoffiguresmadewith14interlocked

cubes?Explain.

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ReadandStudy

Numberrulestheuniverse. Pythagoras

Surfaceareaisthemeasureoftheboundaryofathree-dimensionalobjectinthesamewaythatperimeteristhemeasureoftheboundaryofatwo-dimensionalobject.Andjustlikewemeasureperimeterbyaddingupthelengthsofeachsectionoftheboundaryoftheobject,wemeasuresurfaceareabyaddinguptheareasofeachfaceoftheboundaryoftheobject.Forexample,supposewehavearectangularprismthatis3cmlongby5cmwideby8cmtall.Takeaminutetosketchthatfigure.I D E A O F S U R F A C E A R E A

Thismeansthatwehavetwofacesthatarerectanglesthatare3cmby5cm(andsohaveanareaof15squarecmor15cm2),twofacesthatarerectanglesthatare5cmby8cm(andsohaveanareaof40squarecmeach),andtwofacesthatare3cmby8cm(andsohaveanareaof24squarecmeach).Thenthesurfaceareaoftheprismis15+15+40+40+24+24=158squarecm.(Checkourwork.)Noticethatweusesquareunitstomeasuresurfaceareasinceitisameasureoftwo-dimensional(flat)space.Thereisnoneedtodevelopcompletelynewformulasforsurfacearea.Wecanusewhicheverareaformulasareappropriategiventheshapesthatmakeupthefacesoftheobject.

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Theimportantthingisnottostopquestioning.Curiosityhasitsownreasonforexisting.

AlbertEinsteinElementarystudentscanuseinterlockingcubes(suchastheonesweusedintheClassActivity)tocreate“buildings”andthenusethosebuildingstobuildanunderstandingofsurfaceareathroughdrawingthebuildingsfromvariousview-points.Forexample,thebuildingbelowcanbeviewedfromthetop,front,andside.

Buildyourownbuildingoutof5cubes.Thensketchthefigureanditscorrespondingtop,front,andsideviews.Herearethetop,front,andsideviewsofanotherbuilding.Sketchapossiblebuildingwiththeseviews.

Howdotheseactivitiesrelatetothesurfaceareaofthebuilding?Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit1Module2Session4.Theendofthelessonlistsseveralquestionsforyoutoaskyourstudents.Thinkuptwomorequestionsyoucouldaskyourstudentsabouttheactivity.

top front right side

top front right side

93

Homework

Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.

JohnQuincyAdams


2) Supposeyouareaskedtobuildanobjectusinginterlockingunitcubesthathasasurfaceareaof36squareunits.a) Howmanyunitcubesataminimumwouldyouneed?b) Whatisthemaximumnumberofunitcubesyoucoulduse?c) Forwhatthree-dimensionalshapewillthevolumebegreatestforafixedsurface

area?Makeaconjecture.

3) Belowisanetforatetrahedron.Ifeachequilateraltrianglehassidesoflength5cm,whatisthesurfaceareaofthetetrahedron?(Thoselittleextrapartsaretabsthathelpyoutotapeittogether–don’tincludethose.)

4) Whatamountofpaper(area)wouldyouneedtomakeanetforthecubewithanedgelengthof7cm?(Ignoreanypaperneededfortabs.)

5) Sketchanetforarectangularprismthatis4cmby5cmby7cm.Whatisthesurfaceareaofthatprism?

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ClassActivity14: NothingButNet

Puremathematicsis,initsway,thepoetryoflogicalideas. AlbertEinstein

1) Figureouthowtobuildapapermodelofarightcylinderwitharadiusof3cmanda

heightof10cm.Useyourmodeltohelpyoufindaformulaforthesurfaceareaofacylinder.

N E T S F O R C Y L I N D E R S A N D P Y R A M I D S

2) Buildapapermodelofapyramidthathasa6cmby6cmsquarebaseandaheightof4

cm.Whatisitssurfacearea?

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ClassActivity15: BuildingBlocks

Measurewhatismeasurable,andmakemeasurablewhatisnotso. Galileo

Volumeisameasureoftheamountofspaceenclosedbyathree-dimensionalobject.Onewaytodefinevolumeisasthenumberofcubes(cubicunits)thatittakestofilltheenclosedspace.Usingthisdefinition,wecanmeasurevolumebycarefullyestimatingthenumberofcubesthatfitwithintheobject.Someobjectshaveformulasthatwillhelpustocomputevolume.Whatisaformulaforthevolumeofarectangularprismwithlengthl,widthw,andheighth?Whydoesitmakesense?Next,youaregoingtobuildthreeobjectsoutofsmallwoodencubes(withhalf-inchedges)andlargewoodencubes(withone-inchedges).Followthestepsbelow.StepA:Buildsomethingoutof10smallwoodencubes.We’llcallthisfigure,“ObjectA.”StepB:Using10largewoodencubes,buildalargerversionofObjectA.We’llcallthisfigure,“ObjectB.”StepC:Usingasmanysmallcubesasnecessary,reproduceObjectB.(ItshouldbethesamesizeandshapeasObjectB,butbuildoutofsmallcubesinsteadoflargeones.)We’llcallthisfigure,“ObjectC.”

1) HowmuchtallerisObjectBthanObjectA?

2) HowmuchbiggeristhesurfaceareaofObjectBcomparedtoObjectA?

(Thisproblemcontinuesonthenextpage.)

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3) HowmuchbiggeristhevolumeofObjectBcomparedtoObjectA?

4) WhatistherelevanceofObjectCtoansweringthesequestions?

5) WhatifIgaveyoualargerblockthatis3timesthelengthofthesmallcube;howwouldyouranswersto1-3change?Whatabout4timesthelength?

6) WhatifIgaveyouasmallerblockthatis½timesthelengthofthesmallcube;howwouldyouranswersto1-3change?

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ReadandStudy

Mathematicsisnotacarefulmarchdownawell-clearedhighway,butajourneyintoastrangewilderness,wheretheexplorersoftengetlost.

W.S.AnglinIntheClassActivityyoutalkedaboutwhyitmadesensetocalculatethevolumeofarectangularprismbymultiplyingthelengthloftheprismbythewidthwoftheprismbytheheighthoftheprism(V=l´w´h).Theideahereisthatwecanthinkofaprismaslayersofthebasestackedoneuponthenext.Sovolumeistheareaofthebasemultipliedbytheheight(V=AreaofBase×h).Havealookatthepicturebelowtoseewhatwemean.

Willthatsameideaworkforacylinder?Isitsvolumetheareaofitsbasemultipliedbyitsheight?Makeasketchofacylinderanduseittohelpyoutoexplainyourthinking.I D E A S O F V O L U M E

V O L U M E S O F P R I S M S A N D C Y L I N D E R S

ConnectionstoTeaching:

Thecureforboredomiscuriosity.Thereisnocureforcuriosity. DorothyParker

Childrenoftenbegintheirexplorationsofvolumebydeterminingthevolumeofvariousrectangularboxes.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit1Module1Sessions4and5.Howdoesthisactivityhelpstudentstounderstandvolume?

98

Homework

Whentheworldsays,"Giveup,"Hopewhispers,"Tryitonemoretime."

AuthorUnknown


2) DoallthoseproblemsintheConnectionssection.

3) Whenyoudoublethelengthofallthesidesofacube,whathappenstoitsvolume?Whydoesthishappen?Whathappenswhenyoutriplethelength?

4) Belowwehavesketchedanetforacube.a) Buildapapermodelofacubethatistwiceaslongineachlineardimension.b) Buildapapermodelofacubethathastwicethesurfaceareaofthecubesuggested

bythenet.c) Buildapapermodelofacubethathastwicethevolumeofthecubesuggestedby

thenet.

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ClassActivity16: VolumeDiscount

ThelawsofnaturearebutthemathematicalthoughtsofGod.Euclid

1)Reasonwithunitcubestocreatevolumeformulasforthefollowingobjects:

a) arectangularprism

b) atriangularprism

c) acylinder1) Userice--notformulas--toanswerthenextfourquestions:

a) Howdoesthevolumeofthesquarepyramidcomparetothevolumeofthesquareprism?Usethisinformationtocreateavolumeformulaforasquarepyramid.

(Thisactivitycontinuesonthenextpage.)

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b) Howdoesthevolumeofthetriangularpyramidcomparetothevolumeofthetriangularprism?Usethisinformationtocreateavolumeformulaforatriangularpyramid.

c) Howdoesthevolumeoftheconecomparewiththevolumeofthecylinder? Usethisinformationtocreateavolumeformulaforacone.

d) Howdoesthevolumeoftheconecomparewiththevolumeofthesphere?Usethisinformationtocreateavolumeformulaforasphere.

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ReadandStudy

Ifyouwouldbearealseekeraftertruth,itisnecessarythatatleastonceinyourlifeyoudoubt,asfaraspossible,allthings. ReneDescartes

IntheClassActivityyouobservedthatthevolumeofaprismisaboutthreetimesthevolumeofapyramidwiththesameheightandbase,andthatthevolumeofacylinderisthreetimesthevolumeofaconewiththesameheightandradius.Itturnsoutthatfortheidealobjects,thefactorofthreeisexactlycorrect.Thesenicerelationshipsmakeformulasforvolumerelativelystraightforwardifwebuildonformulaswealreadyknow.C A P A C I T Y

InanearlierClassActivity,youexplainedwhyitmakessensethatthevolumeofarectangularprismisV=(l´w´h).Now,sinceyouhaveseenthatthisvolumeisthreetimesthevolumeofthepyramidwiththesameheightandrectangularbase,thevolumeofthepyramidshouldbegivenbytheformulaV= 13 (l´w´h).

Wecangeneralizethesetwoformulassothattheyapplytoallprismsandallpyramids(andtoallcylindersandallcones).Noticeinthevolumeformulafortherectangularprismthatl´wisjusttheareaoftherectangularbase.Ifthebasehasadifferentshape,wejustneedtousetheappropriateareaformulatofindtheareaofthebaseandthenmultiplybytheheighttofindthevolumeoftheprism:V=(AreaofBase)´hforallprismsandcylinders.

FigurefromG.S.Rehill’sInteractiveMathsSeries.

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Likewise,wecanuseV= 13 (Areaofbase´h)forallpyramidsandcones.

Thesphereisanotherthree-dimensionalshapethathasawell-knownvolumeformula, 34

3V r=

,whereristheradiusofthesphere.Thisformulacomesfromcalculus–sowedon’thavethemachinerytoproveitworks–howeveryou(andyourstudents)canobservethattheformulaseemsplausibleusingthewaterexperiment.Here’stheidea.Sinceacylinderhasvolumeπr2×h,andthecylinderyouusedintheClassActivityhasaheightof2r,thismeansthatthecylinderyouusedtopourwaterhasavolumeof2πr3.Takeaminutetocarefullythinkthisthrough.V O L U M E S O F P Y R A M I D S , C O N E S , A N D S P H E R E S

Now,sinceaconehasonethirdthevolumeofacylinderwiththesamebase,theconeyouusedmusthaveavolumeof1/3×(2πr3).Soifittakestwoconestofillasphereitthewaterpouringexperiment,thenitmakessensetoconjecturethatthevolumeofaspheremustbegivenbytheformula, 34

3V r= .Makesurethat

youunderstandwhatwe’resayinghere.Thecorrespondingformulaforthesurfaceareaofasphereis 24SA r= .

Ofcourse,therearemanythree-dimensionalobjectsforwhichwedonothaveformulastocalculatetheirvolume.Forallofthese,wecanusethe“capacity”definitionthatwasdiscussedintheClassActivity.Whenwemeasurevolumeusingcapacitywecommonlyuseunitslikethecup,thequart,thegallon,theliter,etc.Whenwefindvolumeusingaformulawecommonlyuseunitslikecubicinches,cubicyards,andcubiccentimeters.Volumeisathree-dimensionalmeasuresotheunitsusedwillallbecubicunits.

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ConnectionstoTeachingPuremathematicsis,initsway,thepoetryoflogicalideas.

AlbertEinsteinChildreningradethreeshouldhaveexperiencesmeasuringvolumesusingwaterandweighingphysicalobjects.HerearetherelevantCommonCoreStateStandards.Readthem.


Ingrade5,studentsshouldlearntodomanyofthethingswehavetalkedaboutinthelasttwosections.Theyshouldthinkofvolumemeasurementasboththenumberofcubicunitsrequiredtofilla3-dimensionalobject,andasthecapacityoftheobject.Theyshouldunderstandhowtothinkaboutthevolumeofaprismandmakesenseofsomevolumeformulas.YouwillfindtherelevantCommonCoreStateStandardsforgrade5below.Readthem.Whatdotheymeanwhentheysaythatstudentsshouldrecognizevolumeas“additive?”

CCSSGrade3:Solveproblemsinvolvingmeasurementandestimationofintervalsoftime,liquidvolumes,andmassesofobjects.

1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.

2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.

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CCSSGrade5:Geometricmeasurement:understandconceptsofvolumeandrelatevolumetomultiplicationandtoaddition.

1. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.

a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubic

unit”ofvolume,andcanbeusedtomeasurevolume.

b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.

2. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,and

improvisedunits.

3. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.

a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengths

bypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.

b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofind

volumesofrightrectangularprismswithwholenumberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.

c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwo

non-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

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Homework

Toclimbsteephillsrequiresaslowpaceatfirst. WilliamShakespeare


2) DotheitalicizedthingsintheConnectionssection.

3) Supposeyouhavearightcircularcylinderwithheighthandradiusrandanoblique

circularcylinderwithheighthandradiusr.Dothesetwocylindershavethesamevolume?Compareastraightstackofpenniestoa“slanted”stacktosee.(Reallydoit.)Ineachcase,howisheightmeasured?

4) Theclosetinmylivingroomhasanoddshapebecausemyapartmentisonthetopfloorofahousewithaslantedroof.Theclosetis6feettallinfrontbutonly4feettallinback.Itis3feetdeepand12feetwide.

a) Buildascaleddownpapermodelofmycloset.Reallydothis.Itwillhelpyouwiththerestofthisproblem.

b) Howmanycubicfeetofstoragedoesithold?c) Iwanttopainttheinsidewallsandceilingofmycloset.Howmanysquarefeet

willIneedtopaint?

5) Howmanycubicfeetofwaterdoesasemi-cylindrical(halfacylinder)troughholdthatis10feetlongby1footdeep?Howmanycubicinchesisthat?

6) Iused1500cubicinchesofheliumtofillmyballoon.Assumingmyballoonisasphere,tothenearesttenthofaninch,whatisitsdiameter?Whatisitssurfacearea?

7) Amovietheatersellspopcorninaboxfor$2.75andpopcorninaconefor$2.00.Thedimensionsoftheboxandtheconearegiven.Whichisthebetterbuy?Explain.

8) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade

5TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3Session4.Howdoesthisactivityfurtherstudents’understandingofvolume?

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ClassActivity17: VolumeChallengeIfpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonot

realizehowcomplicatedlifeis.JohnLouisvonNeumann

1) Yourgroupshouldworktogethertobuildpapermodelsofeachofthefollowingobjects

insuchawaythatthevolumeofeachis60cm3.(Otherthanthat,youmayuseanydimensionsyoulike.)

a) Cylinderb) Rectangularprismthatisnotacubec) Pyramidwithasquarebased) Prismwithanequilateraltrianglebase

2) Findthesurfaceareaofeachoftheobjectsyoubuiltin1)above.

B U I L D I N G M O D E L S T O S P E C I F I C A T I O N

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ReadandStudyandConnectionstoTeaching

It'snotthatI'msosmart;it'sjustthatIstaywithproblemslonger. AlbertEinstein

Bythetimetheyreachupperelementaryschool,studentscansolvemanypracticalproblemsingeometry.Howeverthesestudentsarenotreadytosimplyapplyformulastosolveproblems;insteadtheyneedtousemodelsinordertomakesenseofproblems.Astheirteacher,yourjobwillbetomakesurethatchildreninyourclassesbuildandhandleappropriatemodels.NoticethattheCommonCoreStateStandardsforgrade6explicitlyrequirethatstudentsusehands-onmodelstoexploreideas.Studentsareaskedtocuttrianglesandothershapesapart,todrawpictures,topackspaceswithunitcubes,tousethecoordinateplane,andtobuildandusenets.Readthesestandards.Havewedoneofthesethingsinthisbooksofarthisterm?


CCSSGrade6:Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.

1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV=lwhandV=bhtofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

108

Homework

Theelevatortosuccessisnotrunning;youmustclimbthestairs.ZigZiglar


2) Atriangleonacoordinateplanehasverticesat(2,0),(7,0)and(7,8).

a) Sketchthepolygononacoordinateplane.b) Whatisthelengthofthehypotenuse?c) Whatistheareaofthepolygon?d) WhatCommonCoreStateStandardisaddressedbythisproblem?

3) AreplicaoftheGreatPyramidstands2feettallandis3.15feetlongonaside(ithasa

squarebase).a) Approximatelyhowmuchvolumedoesthisreplicatakeup?Usethemodelyoubuilt

intheClassActivitytohelpyoutothinkaboutthisproblem.b) Whatisthesurfaceareaofthereplica?c) Supposethescaleofthereplicatotherealthingisinis1footto240feet.Whatis

thevolumeofGreatPyramid?Whatisitssurfacearea?d) WhichoftheCommonCoreStateStandardsaremetbythisseriesofproblems?

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SummaryofBigIdeas

Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes

• Thereareexactlyfiveregularpolyhedra–andchildrencanunderstandwhythisisso.

• Surfaceareaisameasureofthesumofareasofthefacesofathree-dimensionobject.

Itisthenumberofsquareunitsittakestocoverthefacesoftheobject.Aunitofsurfaceareaisflatlikethis:

• Volumeisameasureofthenumberofcubicunitsittakestofillathreedimensionalobject.Aunitofvolumeisthree-dimensionalandlookslikethis:

• Volumecanalsobemeasuredbytheamountofliquid(orsand)ittakestofillathreedimensionalobject.

• Thevolumeofaprismorcylindercanbefoundbycomputingtheareaofthebaseandmultiplyingthatbyitsheight(thenumberoflayersofthebase).Thisideaworksforbothrightandobliqueobjects.

• Ittakesthevolumeofthreepyramidstofillaprismwiththesamebaseandheight,andittakesthevolumeofthreeconestofillacylinderwiththesamebaseandheight.

• Childrenneedtohavemanyyearsofexperiencesbuildingandusingavarietyofmodels.

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ChapterFour

Transformations,Tessellations,andSymmetries

111

ClassActivity18: Slides

Forthethingsofthisworldcannotbemadeknownwithoutaknowledgeofmathematics.

RogerBacon

Informally,thinkofatranslationasamotionwhich“slides”theentireplaneinonedirectionaparticulardistance.Inordertotranslateanobjectwemustknowhowfartoslideitandwemustknowthedirectiontouse.Thesetwopiecesofinformationareusuallygiventousintheformofatranslationvector(alsocalledatranslationarrow).

1) OnthesquaregridbelowyouaregiventranslationvectorRSandseveralgeometricobjectsontheplane.ShowwhereeachobjectendsupaftertheplaneistranslatedbyvectorRS.

2) Hereisadefinitiontostudy:AtranslationAA’isarigidmotionoftheplanethattakesAtoA’,andforallotherpointsPontheplane,PgoestoP’wherevectorPP’andvectorAA’havethesamelengthanddirection.Discuss,inyourgroups,howthisdefinitionfitswiththeaboveidea.

3) Whatrelationshipsdoyouseebetweentheoriginalfiguresandtheirtranslatedimages?

Betweentheobjects,theirimages,andthetranslationvector?Makeasmanyconjecturesasyoucanabouttranslations.

4) IfwetranslatetheplaneusingRSandthenperformasecondtranslation,say,ST,whatistheresultingrigidmotion?Explain.

H

K

JC

IF G

S

D E

A RB

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ReadandStudy

Onlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient.

EugeneS.WilsonTransformationsareabigcategoryofmotionswecanapplytothepointsofaplanewhichcauseobjectsintheplanetochangetheirposition,ortheirsize,oreventheirshape.Sometransformations,likescaling,onlychangethesizeofanobject.Othertransformations,likeashearingcanchangeboththesizeandshapeofanobject.Theobjectthatistheresultofatransformationappliedtoanobjectiscalledtheimageoftheobjectunderthattransformation.Forexample,therectanglebelowisenlarged1½timestoproducethescaledimageandisshearedhorizontallytoproducetheshearedimage.

Inthisandthenexttwosections,wewillstudythreetransformationsthatchangethepositionofanobjectbutdonotchangeitssizeoritsshape.Thesetypesoftransformationsarecalledrigidmotions.R I G I D M O T I O N S

T R A N S L A T I O N S

Rigidmotionsarethetransformationsoftheplaneforwhichthedistancebetweenpointsispreserved.Inotherwords,iftwopointswereacertaindistanceapartbeforethemotion,thentheyarestillthatsamedistanceapartafterthemotion.(Whydoesthename“rigidmotion”makesense?)Usingtheideaofrigidmotions,wecanmorepreciselydefinecongruence:twoobjectsarecongruentifthereexistsaseriesofrigidmotionswhereoneobjectistheimageoftheother.Rigidmotionsincludetranslations,rotations,andreflections.Eachofthesetransformationswillmovetheplaneinauniqueway.Translationsslidetheplaneaparticulardistanceinaparticulardirection.Rotationsturntheplaneeitherclockwiseorcounterclockwisearoundafixedcenter.Reflectionsmovetheplanebyflippingitacrossaline.Itturnsoutthatallrigidmotionsoftheplanearecombinationsofjustthesemoves.Asyoudiscoveredintheclassactivity,atranslationvectorisusedtodescribethedistanceanddirectioneachpointismovedinatranslation.Iftheendpointsofthevectoraregivenascoordinatesonasquaregrid,wecandescribethedistanceanddirectioneachpointismovedassomanyunitsupordownandsomanyunitsrightorleft.UsethislanguagetodescribethetranslationvectorRSonthepreviouspage.

sheared image

scaled imageoriginalrectangle

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Weuseastandardnotationtolabeltheverticesoftheimageofanobjectunderatranslation.Forexample,iftheoriginalobjectistherectangleABCD,thenitsimageislabeled DCBA .Herevertex A oftheimagecorrespondingtovertexAoftheoriginalrectangle,vertex B tovertexB,etc.ThepointsAand A arecalledcorrespondingpoints.ThesidesABand BA arecalledcorrespondingsides.


Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoapplytransformationsandusesymmetrytoanalyzemathematical

situations. NCTMPrinciplesandStandardsforSchoolMathematics,2000

Studentsintheelementarygradescanpredictanddescribetheresultsofsliding,flipping,andturningtwo-dimensionalshapes.Variouselementarycurriculausedifferentapproaches.Somemakeuseofmanipulativesandothershavestudentscutoutshapesinordertophysicallyperformtheslide,flip,orturntheshapes.

Homework

Youmaybedisappointedifyoufail,butyouaredoomedifyoudon’ttry.BeverlySills


2) Decideifeachofthestatementsabouttranslationsistrueorfalse.Iftrue,givea

mathematicalexplanation;iffalse,explainwhyorgiveacounterexample.a) Correspondingsidesofanobjectanditstranslatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderatranslation,thenthelinesegment

joiningvertexAtovertex A isthetranslationvector.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe

parallelsidesofitsimagetobeverticalafteratranslation.d) IfAand A andBand B arecorrespondingpointsunderatranslation,itis

possibleforthelines AA and BB tointersect.e) Everypointintheplanemovestoanewpositionunderatranslation,i.e.,there

arenofixedpointsinatranslation.f) Underarotation,linesegmentsthatjoinpointstotheirimagesareallcongruent.

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3) Belowyouwillfindacoordinategrid.Applythefollowingthreetranslationstoatrianglewithverticesinitiallylocatedat(0,0),(-2,-3),and(3,-3).Whatisashortcutwayofdoingpartc)?

a) up5,left3 b)down2,right4 c)up5,left3followedbydown2,right4

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ClassActivity19: Turn,Turn,Turn

Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple.

S.GudderInformally,arotationinvolves“turning”theplaneaboutafixedpoint.Inordertospecifyarotation,weneedanangle(withdirection,clockwiseorcounterclockwise)andafixedpoint(calledthecenteroftherotation).Ineachcaseyourgroupshouldusetracingpaperandacompassandprotractortofigureoutwhereeachshapeendsupafterthegivenrotation.(Therearequestionsonthenextpagetoo.)

1) Rotatetheplane60degreescounterclockwiseaboutpointA. A

2) Rotatetheplane140degreesclockwiseaboutpointP.

P.


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3) Rotatetheplane90degreesclockwiseaboutapointQintheexactcenterofthesquare.

4) Studythethreerotationsinthisactivity.Whatrelationshipsdoyouseebetweenthe

originalfiguresandtheirrotatedimages?BetweencorrespondingpointsandthepointP?Makeasmanyconjecturesasyoucanaboutrotations.

5) Hereisthedefinition.Studyittoseehowitfitswiththeideaofrotation.ArotationaboutapointPthroughanangleqisatransformationoftheplaneinwhichtheimageofPisPand,iftheimageofAis 'A ,then PA @ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation.

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ReadandStudy

Wedon’tseethingsastheyare.Weseethingsasweare. AnaisNin

Arotationisa“turn”aboutagivenpointcalledthecenterthroughagivenangleofturn.(Theturncanbemadeclockwiseorcounterclockwise.Thisistypicallyindicatedintheproblem.)R O T A T I O N S

Formally,arotation(aboutapointPthroughanangleq)isatransformationoftheplaneinwhichtheimageofPisPand,iftheimageofAis 'A ,thenPA @ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation.HavealookatthefollowingillustrationofthemotionofturningtriangleABCclockwise240°aroundpointP.

NoticehoweachvertexofthetrianglemovesalongacirclewhosecenterisP.Whatpartofthedefinitionsaysthatthismusthappen?Howcanweseethe240°angleinthepictureabove?HowarethesegmentsAPand PA related?ThesegmentsBPand PB ?ThesegmentsCPandPC ?

C'

A'B'

B

A

C

P

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Homework

Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.

JohnQuincyAdams


2) Usethegridtorotatetheplane90degreescounterclockwiseaboutpointP.Showtheimageofthefiguresaftertherotation.

K

PJ

H

IG F

AD E

C

B

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3) Decideifeachofthestatementsaboutrotationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.

a) Correspondingsidesofanobjectanditsrotatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderarotation,thenthelinesegmentjoining

vertexAtovertex A goesthroughthecenteroftherotation.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe

parallelsidesofitsimagetobeverticalafterarotation.d) IfAand A andBand B arecorrespondingpointsunderarotation,itispossible

forthelines AA and BB tointersect.e) Everypointintheplanemovestoanewpositionunderarotation,i.e.,thereare

nofixedpointsinarotation.f) Underarotation,linesegmentsthatjoinpointstotheirimagesareallcongruent.

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ClassActivity20: ReflectingonReflecting

Mathematics,rightlyviewed,possessesnotonlytruth,butsupremebeauty–abeautycoldandaustere,likethatofsculpture.

BertrandRussellInformally,areflectionflipstheentireplaneaboutagivenlineresultinginitsmirrorimage.OfficiallyareflectioninalinelisarigidmotionoftheplaneinwhichtheimageofapointPonlisP,andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof

'AA .

1) Studythatdefinition.Makesureeveryoneinyourgroupunderstandshowitfitswiththeideaofareflection.SeeifyoucanusethedefinitiontohelpyoutosketchthereflectionoftriangleABCinlinel.

l


121

2) Carefullyusethedefinitionofareflectiontosketchthereflectionofthetrapezoidshown.Firstyouwillreflectitinlinemandthenyouwillreflectwhatyougetinlinen(thatisparalleltolinem).

m n

Whatisthesinglerigidmotionthatwouldtaketheinitialfiguredirectlytothefinalfigure?Explain.

3) Whatwouldhappenifthelinesintersected?Tryitandthenuseyourobservationstomakeaconjecture.

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ReadandStudy

WecoulduseuptwoEternitiesinlearningallthatistobelearnedaboutourownworldandthethousandsofnationsthathavearisenandflourishedandvanished

fromit.Mathematicsalonewouldoccupymeeightmillionyears. MarkTwain

Thereareseveralwaystohelpchildrenpicturetheresultsofareflection.Thinkaboutthereflectionoftheparallelograminlineshownbelow.

Onewaytoseewheretheimageshouldbelocatedistotracetheparallelogramandthelineofreflectiononasheetofthinpaperandthenphysicallyflipthepaperoverandplaceitbackontopoftheoriginalpapersothatthetwolinescoincide.Thecopyoftheparallelogramonthetracingpaperisnowpositionedastheimageofthereflection.Useasheetofpaperandtrythismethod.R E F L E C T I O N S

Anotherwaytovisualizeareflectionimageistophysicallyfoldtheoriginalsheetofpaperalongthelineofreflection.Theoriginalobjectanditsimageunderreflectionshouldnowcoincide,asinan“ink-blot”drawing.Trythis.Reallydoit.RecallthatareflectioninalinelisatransformationoftheplaneinwhichtheimageofapointPonlisP,andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA .Thisdefinitionofareflectionprovidesinsightintohowwecansketchtheimageofareflection.Eachpointmusttravelalongalineperpendiculartothelineofreflectionsothatthatlineisthemidpointbetweenofthelinesegmentconnectingcorrespondingpoints.Usethatmethodtosketchtheimageoftheparallelogramabove.

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Therearemanyinstancesofreflectioninphysicalphenomena.Commonexamplesincludethereflectionoflight,sound,andwaterwaves.Weareallfamiliarwiththephenomenaoflightreflection–takealookinamirror.

Homework

Weallhaveafewfailuresunderourbelt.It’swhatmakesusreadyforthesuccesses. RandyK.Milholland,Webcomicpioneer


2) Onthesquaregridbelow,uselinenasthelineofreflectiontoreflectthegivenobjects.

Labeleachimageappropriately.

3) Whatrelationshipsdoyouseebetweenoriginalfiguresandtheirreflectedimages?Betweentheobjects,theirimagesandthegivenlineofreflection?Makeasmanyconjecturesasyoucanaboutreflections.

nH

K

J

IF G

AC

D E

B

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4) Usethedefinitionofareflectiontoreflectthebelowobjectinline.

5) Decideifeachofthestatementsaboutreflectionsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.

a) Correspondingsidesofanobjectanditsreflectedimagearealwaysparallel.b) If DCBA istheimageofABCDunderareflection,thenthelinesegment

joiningvertexAtovertex A isperpendiculartothelineofreflection.c) If DCBA istheimageofABCDunderareflection,thenthelinesegment

joiningvertexAtovertex A isbisectedbythelineofreflection.d) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe

parallelsidesofitsimagetobeverticalafterareflection.e) IfAand A andBand B arecorrespondingpointsunderareflection,itispossible

forthelines AA and BB tointersect.f) Everypointintheplanemovestoanewpositionunderareflection,i.e.,there

arenofixedpointsinareflection.g) Underareflection,linesegmentsthatjoinpointstotheirimagesareall

congruent.

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ClassActivity21: Part1ASimilarTask

Onecanstate,withoutexaggeration,thattheobservationofandthesearchforsimilaritiesanddifferencesarethebasisofallhumanknowledge.

AlfredNobel

Hereisthedefinitionofanewtypeoftransformation:

Adilation(withcenterPandscalefactork>0)isamotionoftheplaneinwhichtheimageofPisPandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.

Studythisdefinitionanduseittodothefollowing:

1. DrawtheimageoftheseobjectsunderadilationwithcenterPandscalefactor2

2. DrawtheimageoftheseobjectsunderadilationwithcenterCandscalefactor½

3. Writedownsomeconjecturesthatyouhaveaboutdilationsandtheeffecttheyhaveonobjects.

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ClassActivity21: Part2Zoom

Inthepictureabove,westartedbydrawingthesmallertriangleonacomputerscreen,andthenwezoomedin.Thetrianglegotbiggerandmovedtothenewposition.Onepointonourscreenremainedfixedwhenwedidthiszoom.Findthatpoint.Whatwasthescalefactorofthezoom?Inasmanywaysasyoucan,findevidenceforyouranswer.

127

ReadandStudy

Oursimilaritiesbringustoacommonground;ourdifferencesallowustobefascinatedbyeachother.

TomRobbinsWehavedifferentnotionsof“sameness”ingeometry.Inthestrongestsense,ifwesaytwoobjectsarethesame,wemeantheyarecongruent.Butwemightalsorefertotwoobjectshavingthesameshapeeveniftheyaren’tcongruent.Forinstance,thetwotrianglesinourClassActivityhavethesameshape,buttheyarenotthesamesize.Observethattheircorrespondinganglesarecongruent,andthattheircorrespondingsidesareproportional.Makesurethatyouunderstandwhatthismeans.Thesetwotrianglesaren’tcongruent,buttheyarewhatwecall“similar”.S I M I L A R P O L Y G O N S

Formally,wesaythattwoobjectsintheplanearesimilarifonecanbeobtainedfromtheotherbycomposingarigidmotion(tochangetheobject’spositionifnecessary)witha“dilation”.Adilation(withcenterPandscalefactork>0)isamotionoftheplaneinwhichtheimageofPisPandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.Inotherwords,PisafixedpointandallotherpointsarepushedradiallyoutwardfromPorpulledradiallyinwardtowardP,sothateachpoint’sdistancefromPhasbeenmultipliedbythescalefactor.Itcanbeshownthattwopolygonsaresimilarifandonlyiftheircorrespondingvertexanglesarecongruent,andtheircorrespondingsidesareproportional.Euclidprovedatheoremaboutsimilartriangles:

2) Angle-AngleTriangleSimilarityTheorem(AA):Iftwoanglesofonetriangleare

congruenttotwoanglesofanothertriangle,thenthetrianglesaresimilar.ThistheoremappearedinhisBookVI(aboutsimilargeometricfiguresandproportionalreasoning)ratherthanhisBookIthatwehavestudiedpreviously.

Homework

Therearenosecretstosuccess.Itistheresultofpreparation,hardwork,andlearningfromfailure.

ColinPowel

1) Usingyourcompass,drawacircle.PlaceapointCattheexactcenterofyourcircle,andapointPsomewhereoutsideofthecircle.Thenbeginningwiththatcircle,

a) drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointCandscalefactor2.

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b) Instead,drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointPandscalefactor2.

c) Whatcommonalitiesanddifferencesdoyouobserveaboutthethreeshapesyouhavedrawn?

2) Beginningwiththetrianglepicturedbelow,drawthesimilartrianglethatistheresultofadilationoftheplanewithfixedpointPandscalefactor1/2.

3) Decideifeachofthesestatementsaboutdilationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.

a) Correspondingsidesofanobjectanditsdilatedimagearealwaysparallel.b) Dilationspreserveangles(i.e.anyangleanditsdilatedimagearealways

congruent).c) IfA’andB’aretheimagesoftwopointsAandBunderadilationwithscale

factork,thenthedistanceA’B’isktimesthedistanceAB.d) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe

parallelsidesofitsimagetobeverticalafteradilation.e) IfAand A andBand B arecorrespondingpointsunderadilation,itispossible

forthelines AA and BB tointersect.f) Everypointintheplanemovestoanewpositionunderadilation,i.e.,thereare

nofixedpointsinadilation.g) IfA’istheimageofAunderadilationwithfixedpointPandscalefactork,then

thedistanceAA’isktimesthedistancePA.

4) Onenight,a6-foottallmanstood10feetfromalamppost.Thelightfromthelamppostcasta12footshadowoftheman.Howtallwasthelamppost?

P

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5) Supposetwoobjectsaresimilarandthescalefactorofthedilationis1.Whatelsecanyousayabouttherelationshipbetweenthosetwoobjects?

6) LookagainatEuclid’strianglesimilaritytheorem(AA).Giventheotherthingsyouhavealreadylearnedabouttriangles,itisequivalenttosaying:ifallthreeanglesofonetrianglearecongruenttothecorrespondinganglesofanothertriangle,thenthetrianglesaresimilar.Considerthefollowingconjectureaboutquadrilaterals:ifallfouranglesofonequadrilateralarecongruenttothecorrespondinganglesofanotherquadrilateral,thenthequadrilateralsaresimilar.Isthisconjecturetrueorfalse?Giveanargumenttosupportyouranswer.

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ClassActivity22: SearchingforSymmetry

Themathematicalsciencesparticularlyexhibitorder,symmetry,andlimitation;andthesearethegreatestformsofthebeautiful.

AristotleAsymmetryofanobjectisarigidmotionoftheplaneinwhichtheimagecoincideswiththeoriginalobject.Therearetwoprimarytypesofsymmetry.Intuitively,anobjecthasreflectionsymmetryifitcanbecutbyalineofreflectionintotwopartsthataremirrorimagesofeachother.Thisbutterflyhasreflectionsymmetryandsodoesthisarrow.Sketchthelineofreflectionineachcase.

Anobjecthasrotationsymmetryifitcanberotatedaroundacenterpointthroughacertainangleandendupwiththeimagecoincidingwiththeoriginal.Wewouldsaythattherecyclingsignbelowhas120,240and360degreerotationalsymmetry.

1) Findallthesymmetriesofthecapitallettersinthefollowingtypeface:

ABCDEFGHIJKLMNOPQRSTUVWXYZ

2) Ineachcase,sketchapolygonwiththegivensymmetries,orexplainwhysuchapolygoncannotexist.

a) nolinesofreflectionsymmetry,but180°(and360°)rotationsymmetriesb) 90°(and360°)rotationsymmetriesandnoothersymmetries.c) 2linesofreflectionsymmetry,360°rotationsymmetry,andnoother

symmetriesd) 6linesofreflectionsymmetrye) nolinesofreflectionsymmetry,but90°,180°,270°(and360°)rotational

symmetryf) anytranslationsymmetry

131

ReadandStudy

Mathematicsisthesciencewhichuseseasywordsforhardideas. E.KasnerandJ.Newman

Afigure,picture,orpatternissaidtobesymmetricifthereisatleastonerigidmotionoftheplanethatleavesthefigureunchanged.Forexample,thisleafis(prettymuch)symmetricbecausethereisalineofreflectionsymmetry.

Manyobjectsinnaturedisplaythiskindofbilateralsymmetry.ThelettersinATOYOTAalsoformasymmetricpattern:ifyoudrawaverticallinethroughthecenterofthe“Y”andthenreflecttheentirephraseacrosstheline,theleftsidebecomestherightsideandviceversa.Thepicturedoesn’tchange.S Y M M E T R I E S I N T H E P L A N E

Theorderofarotationsymmetryisdeterminedbycountingthenumberofturnstheobjectcanmakeandcoincidewithitselfbeforereturningtoitsoriginalposition.Theanglemeasureofthesmallestturnisdeterminedbydividing360°bythatnumberofturns.Whydoesthismakesense?Forexample,anequilateraltrianglehas“order3rotationsymmetry.”B A CTheturnof120°takesvertexAtovertexB;theturnof240°takesvertexAtovertexC;andtheturnof360°takesvertexAbacktovertexA.Whataretherotationsymmetriesofthesquare?Oftheregularhexagon?Oftheregularoctagon?Oftheregularn-gon?

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Onlyrepeatinginfinitepatternshavetranslationsymmetry.Theyaretheonlytypeofobjectthatcanbeslidandstillfallbackonthemselves.Imaginethatthispatternstripcontinuesforeverinbothdirectionssoifyouslideitonepatterntotheright(ortwoorthree…)itlooksjustthesame.

… …


Itouchthefuture.Iteach. ChristaMcAuliffe

TheCommonCoreStateStandardsformathematicsmakeinformalideasofsymmetryatopicforgrade4.Whiletheymentiononlyreflectionsymmetry,childrenatthisage(andevenyounger)arecapableofexploringandrecognizing“turn”symmetryaswell.Readthisstandard.


Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2.CarefullyexaminetheMosaicGame.Playthegameonetimeandrecordyourresult.Explainhowyouknowthatyourarrangementproducesthemostlinesofsymmetry.Whatmightstudentslearnaboutsymmetryfromcompletingthisactivity?Playthegameagain,butthistime,trytoproduceafigurewiththelargestorderofrotationalsymmetry.Again,explainyourreasoning.

CCSSGrade4:

1. Recognizealineofsymmetryforatwo-dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline-symmetricfiguresanddrawlinesofsymmetry.

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Homework

Byperseverancethesnailreachedtheark. CharlesHaddonSpurgeon


2) Classifythesymmetriesforthefollowingtrafficsigns–(considertheentiresign–not

justtheinteriordesign).

3) Isitpossibleforanobjecttohaverotationsymmetrieswithouthavingreflection

symmetries?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.

4) Isitpossibleforanobjecttohavetworeflectionsymmetrieswithouthaving180°rotationsymmetry?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.

5) Onlineatbridges.mathlearningcenter.org,findtheBridgesinMathematicsGrade1

TeachersGuide.Unit5Module3Session1introducesNinePatchInventions,whichSession2usestocreateNinePatchMini-quilts.Spend10-15minutesstudyingthesesessions.Whatideasaboutrotationalsymmetryareemphasizedinthesesections?

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SummaryofBigIdeas

Ifanidea’sworthhavingonce,it’sworthhavingtwice. TomStoppard

• Therearethreedistinctwaysto“move”theplanewithoutchangingtheshapeorsizeof

objects:thetranslation,therotation,andthereflection.• Atranslationslidestheplaneagivendistanceinagivendirection.

• Arotationturnstheplanearoundagivenpointthroughagivenamountofrotation

(usuallygivenindegrees).• Areflectionflipsanobjecttoitsmirrorimageacrossalineofreflection.

• Oneofthegoalsofelementaryschoolgeometryinstructionisthatstudentslearnto

visualizeandapplytransformations.

• Twoobjectsaresimilarifonecanbeobtainedfromtheotherbyarigidmotionandadilation.

• Symmetryisaphenomenonofthenaturalandartisticworldsthatcanbeexplained

withthelanguageofrigidmotions.

• Mathematiciansmostoftentalkabouttwotypesofsymmetry:reflectionsymmetry,inwhichanobjectisdividedbyalineofreflectionintotwopartsthataremirrorimagesofeachother,androtationsymmetry,whereanobjectisrotatedaroundacenterpointthroughacertainangleandendsupoccupyingthesamepositionintheplane.

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ChapterFive

Probability

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ClassActivity23: DiceSums

Theexcitementthatagamblerfeelswhenmakingabetisequaltotheamounthemightwintimestheprobabilityofwinningit.

BlaisePascalRolltwodiceatonceandifthesumis2,3,4,5,10,11or12,yourgroupwins.Ifthesumis6,7,8,or9,theinstructorwins.Isthisafairgame?Dothisproblemtwoways.First,actuallyplaythegamelotsoftimesandcollectdatatodecide.Then,doatheoreticalanalysisoftheproblemtodecide.(Youwillneedtobeginbydecidingwhatitmeansthatagameofchanceis“fair.”)

137

ReadandStudy

Areasonableprobabilityistheonlycertainty.E.W.Howe

Gamesofchancehavebeenaroundforatleastaslongaspeoplehaverecordedhistory,butitwasn’tuntilthe16thcenturythatmathematiciansfirstbegantostudytherulesthatmighthelpanswerquestionssuchas“Isthisgamefair?”or“HowlikelyamItodrawaroyalflushinagameofpoker?”Wecalltheareaofmathematicsthatcontainstheserulesprobabilitytheory.Simplyput,probabilityisthestudyofchanceorrandomevents.Supposewerollastandardsix-sideddieonce.Wedon’tknowtheresultoftherollinadvance,sowesaythattheoutcomeisrandom.However,theoutcomeisnotcompletelyunpredictable.Ourdiehassixsidesandsoweknowtheoutcomewillbeoneofsixnumbers:1,2,3,4,5,or6.Wealsoknow(assumingwehaveafairdie)thatanyoneoftheseoutcomesisequallylikely;thatis,ifwerollthediemany,many,manytimesandrecordalltheoutcomes,wewillgetapproximatelythesamepercentageof‘1’saswedo‘2’saswedo‘3’sandsoforth.Supposewewouldliketobeabletoassignanumbertoexpresshowlikelyitisthatwegeta5ononeroll.Inotherwords,wewanttoknowtheprobabilityofgettinga5ononerollofastandard6-sideddie.Bytheway,theuseofgoodmathematicallanguageisahabit.Ifyouwanttousemathematicallanguagecorrectlywithyourelementarystudents,youmustpracticedoingsonow.Youwillfindunderlinedvocabularyandideasdefinedforyouintheglossary.Gothereandhavealookatthedefinitionof“probability”now.L A N G U A G E O F P R O B A B I L I T Y

Inordertotalkabouttheanswertothisquestion,wemustagreeonawaytomeasureprobability.Fourmainruleshavebeenadopted:

1) Theprobabilityofanoutcomeisanumberfrom0to1.Ifanoutcomeiscertaintooccur(suchas,attheequatorthesunwillrisetomorrow),thenitsprobabilityis1.Ifanoutcomecannothappen(suchas,themoonwillfallintothePacificOceantonight),thenitsprobabilityis0.Ifanoutcomeisneithercertainnorimpossible,thentheprobabilitywillbesomenumberbetween0and1.

2) Thesumoftheprobabilitiesofallpossibleoutcomesofarandomexperimentis1.

3) Theprobabilitythataneventwilloccuris1minustheprobabilitythatitwillnotoccur.Thatis,aneventwilleitheroccuroritwillnotoccur,andthesumofthosetwoprobabilitiesmustbe1.

4) Iftwoeventsaredisjoint(theyhavenooutcomesincommon),theprobabilitythatoneortheotherwilloccuristhesumoftheprobabilitiesthateachwilloccurbyitself.

Sowemustthinkabouthowwecanassignanumberfrom0to1totheoutcome“obtaininga5whenrollingonedie”thatwewillcallitsprobability.Todosomathematiciansusethelanguageofsets.Herecometheofficialdefinitionsofrandomexperiment,outcome,event,andsample

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space.Arandomexperimentisanactivity(suchasrollingour6-sideddie)wheretheoutcome(1,2,3,4,5,or6)cannotbeknowninadvance.Asetofallpossibleoutcomes,{1,2,3,4,5,6},isasamplespaceforthatexperimentandaneventisanysubsetofthesamplespace(suchas“rollinga5”or“rollinganevennumber”).Noteasubtledistinctionherebetweenanoutcomeandanevent:anoutcomeisanelementofthesamplespace,whereasaneventisasubsetofthesamplespace.Soifthesamplespaceis{1,2,3,4,5,6}then‘3’iscalledanoutcome,whereas{3}iscalledanevent.{1,3,5}isanotherevent.Listafewmoreeventsforthissamplespaceusinggoodcurly-bracketsetnotation.Doestheemptysetmeetthedefinitionofan“event?”Whyorwhynot?Okay,nowbyrulenumber1,theprobabilityofeachoutcomeinthesamplespacewillbeanumberbetweenzeroandone,andbyrulenumber2,thesumofalltheprobabilitiesoftheoutcomesinthesamplespacemustequalone.Sohowdowedeterminetheprobabilityof“rollinga5?”Wellthatturnsouttobeprettyeasyinthecasewherealltheoutcomesareequallylikelytooccur.(Wewillassumethatourdieisevenlybalancedandaperfectcube-wecallsuchadiefair–andsowearejustaslikelytoobtainanyoneofthenumbersfrom1to6.)Whenoutcomesareequallylikely,theprobabilitythatanyoneoccursisjusttheratioof1tothetotalnumberofoutcomes.Inthecaseofrollingonefairdie,theprobabilityofrollinga‘5’is .Whenaneventismorecomplicated(suchasobtainingasumof3whenrollingtwodice),wecanusethesameapproachbutweneedtochooseoursamplespacewisely.Supposeweareinterestedinthesumobtainedwhenrollingtwofairdice.Ifwelistthepossibleoutcomesforthesum,wegetthesamplespaceof{2,3,4,5,6,7,8,9,10,11,12},whichcontainselevenoutcomes.Unliketheearlierexampleofrollingonediehowever,theseoutcomesarenotequallylikely.Forexample,thereareseveralwayswecouldobtainasumof7(1+6,2+5,etc.),butonlyonewaywecouldobtainasumof12(6+6).Whentheoutcomesarenotequallylikely,weneedadifferentwaytodeterminetheprobabilityofeachoutcome.Inthisexample,theprobabilityofeachsumisnot .Onewaytodeterminetheprobabilitiesofeachsumistoconsiderasamplespaceoftherandomexperimentofrollingtwodicewheretheoutcomesareequallylikely.Forexample,wecanthinkoftheoutcomesinthissamplespaceasorderedpairs(x,y),wherexisthenumberon

61

111

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thefirstdieandyisthenumberontheseconddielikeyoumayhavedonewhenyouworkedontheclassactivity.Thissamplespaceisshowninthetablebelow.Theadvantageofusingthissamplespaceisthatalloutcomeshereareequallylikely.

1 2 3 4 5 6

1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)

4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)

5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)

6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

Apossiblesamplespaceforarolloftwodice

Noticethatthereare36equallylikelyoutcomesinthissamplespaceandsotheprobabilityof

eachoutcomeis .Wecanusethissamplespacetocomputetheprobabilityoftheevent“obtainingasumof3”(that’stheevent{(1,2),(2,1)})bycountingthenumberofpossibleoutcomesthathaveasumof3.Wemustbecarefultomakecertainthatweactuallydocountallpossibleoutcomesthathaveasumof3.Forexample,inthesamplespace,theoutcome(1,2)whichgives1+2=3andtheoutcome(2,1)whichgives2+1=3aretwoseparateelements.Thatis,therearetwowaystogeneratethesumof3:1)wecanrolla1onthefirstdieandthena2ontheseconddieor2)wecanrolla2onthefirstdieandthena1ontheseconddie.Tohelpyourstudentsthinkaboutthiswesuggesttwothings.First,havethemdoanexperimentwhereeveryonerollsapairofdiceforfiveminutesandtalliesthenumberoftimestheyrollasumof2andseparatelythenumberoftimestheyrollasumof3.Inthisway,theywillseethatthesumof3istwiceaslikelytooccur.Second,havethemeachholdonereddieandonegreendie.Askthattheymakeasumof3.Thenaskthattheymakeasumofthreeadifferentwaysotheycanseethattheoutcome(1,2)giving1+2=3andtheoutcome(2,1)giving2+1=3aretwoseparateelementsinthesamplespace.Trythisyourself.Nowmakeasumof2andnoticethatthereisnootherwaytodothis.

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Rollinga1onthereddieanda2onthegreendieisadifferentoutcomefromrollinga2onthereddieanda1onthegreendie,butthereisonlyonewaytohavea1oneachdie.Okay,backtooursamplespacewith36outcomes.Usingprobabilityrule4,theprobabilityofrollingtwodicewithasumof3,writtenP(Sum=3),isthesumoftheprobabilitiesofeach

outcomewhichequals + = = .WecanalsocomputeP(Sum=3)astheratioofthenumberofoutcomesthathaveasumof3tothenumberoftotaloutcomessinceallthe

outcomesareequallylikely.SoP(Sum=3)= .

Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.056),asapercent(5.6%)orasodds(2:34,thechancesofhavingasumofthreetothechancesofhavingasumotherthanthree).Checkallthistobesureitmakessensetoyou.Let’sconsideranotherexample:Whataremychancesofdrawingafacecardfromastandarddeckof52playingcards?Beforeyoureadfurther,takeaminutetoseeifyoucanfigureitout.

Hereisourthinking.Payattentiontothelanguageweuseaswellastothewaywesolvetheproblem.Astandarddeckofcardscontains52cards,13ofeachoffoursuits.Eachsuitcontains3facecards,thejack,thequeen,andtheking,soIhave4×3=12waysofdrawingafacecardand52waysofdrawinganycard.Sotheprobabilityofdrawingafacecardfromadeckofcards

is .Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.231),asapercent(23.1%)orasodds(12:40,thechancesofdrawingafacecardtothechancesofnotdrawingafacecard).Inthisexample,theexperimentis“drawingacardfromadeckofcards,”theeventis“drawingafacecard,”andthesamplespacecontains52equallylikelyoutcomes(the52cardsinthedeck).Let’slookatonemoreexample:Whatistheprobabilityofspinningredoneachofthespinnersbelow?Takeamomenttothinkthisthrough.

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5212

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Whatisthesamplespaceforeachspinner?Arealltheeventsinthesamplespaceequallylikely?Inbothofthesepictures,wehaveasamplespaceof{red,blue,green,yellow}.However,inthespinnerontheright,theeventsarenotequallylikely.Tocalculatetheprobabilityofspinningredoneitherspinner,wearereallylookingattheareaofthecirclethatisreddividedbytheentireareaofthecircle.Inotherwords,thefractionofthecirclethatisshadedredortheratiooftheareaofredtotheareaoftheentirecircle.Areaisonewaywecanthinkaboutprobabilitiesgeometrically.Dependinguponthesituation,wemightwanttothinkaboutafractionallength,area,orvolume.


Ingrades3–5,allstudentsshouldunderstandthatthemeasureofthelikelihoodofaneventcanberepresentedbyanumberfrom0to1.

NCTMPrinciplesandStandards,p.176InbothoftheReadandStudyscenarioswehavebeendiscussingwhatwecalltheoreticalprobabilities,thatisprobabilitiesthatareassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieoradeckofcards).Thismayhaveleftyouwiththeimpressionthatallrandomexperimentscanbeanalyzedinatheoreticalmanner.Notso.Considertheexperimentof“flipping”alargemarshmallow.Nowitcouldlandonaflatcircularendoronitsside(thecurvedpartofthecylinder),sowe’llidentifythesamplespaceastheset:{end,side}.Childrenmaythinkthatthesetwooutcomeseachhaveaprobabilityof½sincetherearetwochoices,butsomeexperimentingwithactualmarshmallowswillshowthatthesearenotequallylikelyoutcomes.Furthermore,atheoreticalanalysisoftheprobabilitiesofeachoutcomewouldrequireknowledgeofphysicsandgeometry,andmightdependonthesurface,thetemperature,orotherfactors.Incaseslikethat,itisbettertoperformanexperimenttodeterminetheprobability.Inotherwords,itisbesttohavethechildrenflipthemarshmallowalargenumberoftimesandtousethedatatoestimatetheprobabilityofeachoutcome.Probabilitiesthatarebasedondatacollectedbyconductingexperiments,playinggames,orresearchingstatisticsarecalledexperimentalprobabilities.Asateacher,itisimportantthatyougiveyourstudentsplentyofexperiencecalculatingnotjusttheoreticalprobabilities,butalsoexperimentalprobabilities.E X P E R I M E N T A L A N D T H E O R E T I C A L A N A L Y S I S

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Homework

You'llalwaysmiss100%oftheshotsyoudon'ttake.WayneGretzky


2) Ifyoudecidedthatthegame“DiceSums”wasunfair,changetherulessothatthegame

isfair.Istheremorethanonesetofrulesthatwouldgiveafairgame?Ifso,howmanydifferentsetsofrulescanyoumake?

3) Consideranewgamecalled“DiceDifferences.”Twodicearetossedandthesmallernumberissubtractedfromthelargernumber.PlayerIscoresonepointifthedifferenceisodd.PlayerIIscoresonepointifthedifferenceiseven.(Note:Zeroisanevennumber.)Isthisafairgame?Ifso,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.

4) Readthe“TheseBeansHaveGottoGo!”activityfromUnit2Module1Session1oftheBridgesinMathematicsGrade2HomeConnections,onpage29-34.

a. Howwouldyouplaceyourbeansifyouwereplayingthisgame(andwantedtowin).

b. Howwouldyouexpectatypical2ndgradertoplacetheirbeans?c. Whatanswerswouldyouexpectthatyourfuture2ndgradestudentsmightwrite

toquestions3,4and5onpage34.

5) Abagcontainstwowhitemarbles,fourgreenmarbles,andsixredmarbles.Therandomexperimentistodrawamarblefromthebag.

a) Writeareasonablesamplespaceforthisexperiment.Usesetnotation.b) Whatistheprobabilityofdrawingawhitemarble?Aredmarble?Agreen

marble?Explainyouranswers.c) Whatistheprobabilityofdrawingablackmarble?Explain.d) Whatistheprobabilityofnotdrawingagreenmarble?Explain.e) Howmanymarblesofwhatcolorsmustbeaddedtothebagtomakethe

probabilityofdrawingagreenmarbleequalto0.5?

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6) Abagcontainsseveralmarbles.Somearered,somearewhite,andtherestaregreen.

Theprobabilityofdrawingaredmarbleis andtheprobabilityofdrawingawhite

marbleis .

a) Whatistheprobabilityofdrawingagreenmarble?Explain.b) Whatisthesmallestnumberofmarblesthatcouldbeinthebag?Explain.c) Couldthebagcontain48marbles?Ifso,howmanyofeachcolor?Explain.d) Ifthebagcontainsfourredmarblesandeightwhitemarbles,howmany

greenmarblesdoesitcontain?Explain.

7) Decidewhetherthefollowinggameisfairorunfair:Playershavethreeyellowchips,eachwithanAsideandaBside,andonegreenchipwithanAsideandaBside.Flipallfourchips.PlayerIwinsifallthreeyellowchipsshowA,ifthegreenchipshowsA,orifallchipsshowA.Otherwise,PlayerIIwins.Ifthegameisfair,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.

8) The6outcomesofrollingafairdieareequallylikely;soarethe52outcomesofa

randomdrawofonecardfromadeckofcards.Nameatleastthreemorerandomexperimentsthatgenerateequallylikelyoutcomes.Nameatleastthreethatproduceoutcomesthatarenotequallylikely.

9) Supposeyoubreaka25cmlongspaghettinoodlerandomly.Whatistheprobabilitythatoneofthetwopiecesislessthan5cmlong?

10) Supposeapointwasrandomlychosenwithintheinteriorofthelargestsquarebelow,whatistheprobabilitythatthepointisonwhite?Now,supposeapointwasrandomlychosenwithintheinteriorofthelargestcircle.Whatistheprobabilitythatthepointisonblack?

11) Considertherandomexperimentoftossingthreecoinsatonce.Writeasamplespaceforthisexperimentthathasequallylikelyoutcomes.Nowwriteanothersamplespaceforthissameexperimentthatdoesnothaveequallylikelyoutcomes.

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ClassActivity24: RatMazesThetroublewiththeratraceisthatevenifyouwin,you'restillarat.

LilyTomlin

1) Supposethataratissentintoeachofthebelowmazesatthe‘start.’Iftheratcannotgobackwards,butotherwisemakesallthedecisionsatrandom,whatistheprobabilitythatshefindsacheeseineachcase?Explainyouranswersasyouwouldtoanupperelementaryschoolstudent.

2) Makearatmazetoillustrateeachoftheseproblems:a) Youflipacoinandthenrolladie.WhatistheprobabilitythatyougetaHead

andthenamultipleofthree?b) Youflipacointhreetimes,whatistheprobabilitythatyougetexactlytwo

Heads?

Start

Start

Start

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ReadandStudy

Chancefavorsonlythepreparedmind. LouisPasteur

Manysituationsrequiresuccessivechoices–likeintheratmazeproblemswheretherathadtomakeatleasttwodecisionsaboutwhichpathtotaketogettothecheese.Hereisanotherexample:gettingdressedforschoolinthemorningrequiresseveraldecisions.Imustchooseoneoftwopairsofjeans(onlytwoareclean,abluepairandablackone),oneofthreesweaters(red,purple,blue),andoneoffourpairsofshoes(let’sjustcallthemA,B,C,&D)Let’ssaythatshoepairsBandDdon’tcoordinatewiththeblackjeans,butshoepairsA&Ccanbewornwitheitherpairofjeans.1)WhatistheprobabilitythatIwearbluejeans,aredsweater,andCshoes?2)Whatistheprobabilitythatmyoutfitcontainsthecolorblue?3)WhatistheprobabilitythatIweartheBpairsofshoes?T R E E D I A G R A M S

Treediagrams(orratmazes)areusefulinfindinganswerstoquestionslikethese.Belowwehavemadeatreethatcountsallthepossibleoutfits(orpathsinthemaze).Inotherwords,thechoicesforgettingdressedlooklikethis.

Nowassumingthatallchoicesaremadeatrandom,thereareacouplewaystocomputetheprobabilitiesinquestions1-3above.Takeaminutetodosobeforereadingfurther.Mypersonalfavoriteisthesending-rats-into-the-mazetechnique.I’llsend,say,24ratsintothetopofthemaze.ThenIexpect12togoleftdownthebluejeanchoice,4ofthosetogodowneachsweatercolor,and1tocomeoutofeachoftheshoechoicesA,B,CandD.Ofthe12thatwillgorightdowntheblackjeanchoice,4willgodowneachsweaterrouteand2willemergefromeachshoetypeAandC.Nowwecanusethisinformationtoanswertheabovequestions.1)Since1outof24comeoutofthebluejeans,redsweater,Cshoespath,theprobabilityof

C

C

A

A

A

D

C B

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C B

C

B A

A

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A

blue sweater

blue sweater

red sweater red sweater

black jeans blue jeans

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thatis .2)Sincealltheblue-jeansratsandalsotheblue-sweaterratscontainblue(that’s12+

4), or istheprobabilitythatmyoutfitcontainsblue.3)Since3ratsendupwithBshoes,

theprobabilitythatIwearthoseshoesis or .C O M P O U N D E V E N T S

Okay,nowyoumightbethinking,buthowdidsheknowtosend24ratsintothemaze?Doesthatnumbermatter?Tosee,youtryit.Firstuse48rats.Thentry100toseewhathappens.Reallydoit.Whatdidyoulearn?Anothersolutionistousetheprobabilityequivalentofthemultiplicationcountingprincipletofindtheprobabilitythatwereachtheendofeachbranch.Forexample,theprobabilitythatIwearbluejeans,aredsweater,andshoepairAis!

"×!

%× !&= !

"&.Whydowesayweareusing

themultiplicationcountingprinciplehere?FindtheprobabilitiesofeachoftheoutfitsIcouldchoose.Doyouneedtocalculateeachprobabilityindividually?Orcanyoumakeanargumentthatalloftheoutfitswithbluejeansareequallylikely?Whatabouttheoutfitswithblackjeans?Finally,let’sthinkaboutthelasttwoquestionsweposedinthefirstparagraphusingthissecondapproach:Whatistheprobabilitythatmyoutfitcontainsthecolorblue?MyoutfitwillcontainblueifIwearbluejeansorifIwearthebluesweater.Halfofthepossibleoutfitsincludethebluejeansandonethirdoftheremaininghalfoftheoutfitsusethebluesweater.Soatotalof!

"+ !

%× !"= !

"+ !

*= "

%oftheoutfitscontainblue;theprobabilitythatIwearblueis66 "

%%.

UsethisideatofindtheprobabilitythatIweartheBpairofshoes.

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Ingrades6–8allstudentsshouldcomputeprobabilitiesforsimplecompoundevents,usingsuchmethodsasorganizedlists,treediagrams,andareamodels.

NCTMPrinciplesandStandards,p.248Treediagramsarepartoftheelementarygradescurriculumbecausetheygivechildrenawayto“see”allofthepossibleoutcomesandtokeeptrackoftheprobabilitiesofeachchoicethatmightbemadeinacompoundevent.Bytheway,itisnotenoughtoknowjustonewaytodothingsanymore.Ifyouaregoingtobeateacher,youwillneedtomakesenseofthethinkingofyourstudentseveniftheirmethodsaredifferentfromyours.Practicingdifferentwaysofthinkingaboutandexplainingproblemswillmakeyouabetterandmoreflexibleteacher.Homework

Onceyoulearntoquit,itbecomesahabit.VinceLombardi


2) Decideifeachofthefollowingstatementsistrueorfalse.Ifitistrue,explainwhy.Ifitis

false,rewritethestatementsothatitistrue.

a) Iftheprobabilityofaneventhappeningis ,thentheprobabilitythatitdoesnothappenis .

b) Theprobabilitythatanoutcomeinthesamplespaceoccursis1.c) Ifaneventcontainsmorethanoneoutcome,thentheprobabilityofthe

eventisthesumoftheprobabilitiesofeachoutcome.d) IfIhavea60%chanceofmakingafirstfreethrowanda75%chanceof

makingasecond,thentheprobabilitythatImissbothshotsis10%.

3) Makearatmazetomodeleachoftherandomexperiments.a) Ihaveabagcontainingthreechips.TheredchiphasasideAandasideB.

TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.Idrawonechipatrandomfromthebagandtossit.

b) Threechipsaretossed.TheredchiphasasideAandasideB.TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.

4) Supposethechipsin#3abovearealltossed.Useyoursecondratmazetodetermine

theprobabilityofgettingexactlya)oneC b)oneB c)oneAd)twoCs e)twoBs f)twoAs

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5) Supposethatthreechipsaretossedasin#3.PlayerIscoresapointifapairofAsturnup,andplayerIIscoresapointifapairofBsoraCturnsup.Isthisafairgame?Ifso,explainwhy.Ifnot,changetherulestomakeitfair.

6) Abasketballplayerhasa70%freethrowshootingaverage.Theplayergoesupforaone-and-onefreethrowsituation(thismeansthattheplayershootsonefreethrow,andonlyifshemakesit,shegetstoattemptasecondshot).Whatistheprobabilitytheplayerwillmake0shots?1shot?2shots?

7) SupposethatItossafaircoinuptofourtimesoruntilIgetaHead(whichevercomesfirst).

a) Makeamazetomodelthisrandomexperiment.b) Writeoutthesamplespaceusinggoodnotation.c) WhatistheprobabilitythatyougetTTH?

8) Place2blackcountersand1whitecounterintoapaperbagandshakethebag.Without

looking,draw1counter.Thendrawasecondcounterwithoutputtingthefirstcounterbackintothebag.Ifthe2countersdrawnarethesamecolor,youwin.Otherwise,youlose.Whatistheprobabilityofwinning?Doestheprobabilityofwinningchangeifyoureplacethefirstcounterbeforedrawingthesecondcounter?Whyorwhynot?Whatistheprobabilityofwinningifyoureplace?

9) Supposeyouspinbothspinnersbelow.Whatistheprobabilityyouspinred-red?Whataboutayellowononeandgreenontheother?

10) Here’showyouplayourlottery:youwritedownanysixnumbersfromtheset{1,2,3,…35,36}.Thensixwinningnumbersaredrawnfromthatsetatrandom(withoutreplacement–soallofthemwillbedifferentnumbers).

a) Ifyoumatchallthewinnersinorder,youwin$5,000,000.00.Whatistheprobabilityofthat?

b) Ifyoumatchallthewinnersbutnotnecessarilyinorder,youwin$1,000,000.00.Whatistheprobabilityofthat?

c) HowaretheseproblemsrelatedtoourPhotographandCommitteeproblems?Explain.Bespecific.Howwouldyouexplainthistosomeone?

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ClassActivity25: TheMaternityWard

WhenIwasakid,Iwassurroundedbygirls:oldersisters,oldergirlcousinsjustdownthestreet….EachyearIwouldwishforababybrother.Itneverhappened.

WallyLamb

Isitmorelikelythat70%ormoreofthebabiesbornonagivendayareboysinasmallhospitalorinalargehospital(ordoesn’titmatter)?Yourclassshoulddecideonasimulationtodoinordertohelpyoutoanswerthisquestion.

150

ReadandStudy

“Ithinkyou’rebeggingthequestion,”saidHaydock,“andIcanseeloomingaheadoneofthoseterribleexercisesinprobabilitywheresixmenhavewhitehatsandsixmenhaveblackhatsandyouhavetoworkitoutbymathematicshowlikelyitisthatthehatswillgetmixedupandinwhatproportion.Ifyoustartthinkingaboutthings

likethat,youwouldgoroundthebend.Letmeassureyouofthat!”AgathaChristieinTheMirrorCrack’d

Ourintuitionoftenmisleadsuswhenwethinkaboutprobabilities.Initially,youmayhavethoughtthatthechancesofhaving70%ormoreofthebabiesbornbeboyswouldbebetterinalargehospitalbecauseitismorelikelythatmorebabiesareborninalargehospitalononedayandthereforemorelikelytohavemoreboys.Itismorelikelythatthenumberofbabiesbornonagivendayislargerinalargehospitalthanitisinasmallone,butthisdoesnotmeanthatitismorelikelythatthepercentageofboybabieswillbe70%ormoreinthelargehospital.L A W O F L A R G E N U M B E R S

Infact,thelargerthehospital(andthereforethelargerthenumberofbabiesbornonagivenday),themorelikelyitisthatthepercentageofboysbornwillbe50%(theapproximateprobabilityofhavingababyboyinasinglebirth).Mathematically,wecallthisresultthelawoflargenumbers,whichstatesthatinrepeated,independenttrialsofarandomexperiment,(suchasbabiesbeingborninahospital),asnumberoftrials(births)increases,theexperimentalprobabilitiesobserved(theprobabilitythatababybornisaboy)willconvergetothetheoreticalprobability(inthiscase,50%).Inotherwords,thelargerthesample,themorelikelyyouaretogetclosetothetheoreticalresult.Sothemorebirthsthereareonagivenday(thelargerthehospital),themorelikelyitisthatthepercentageofboysbornisnear50%.Thefactthatthetrialsshouldbeindependentisimportant.(Besuretousetheglossaryfornewmathematicalterms;don’tassumethatthemathematicalmeaningofawordisthesameasthemeaningthewordmighthaveincommonusage.)Forourpurposesconsidertwoeventstobeindependentiftheoccurrenceornon-occurrenceofoneeventhasnoeffectontheprobabilityoftheothereventhappening.Inthehospitalproblem,thebirthofababyboytoonemotherhasnoeffectontheboy/girloutcomeofthenextbirthinthehospital.Theoutcomesofthetwobirthsareindependentofoneanother.Whatabouttherolloftwodice?Aretheoutcomesoneachdieindependent?Whataboutthreeflipsofacoin?Isthehead/tailoutcomeoneachflipindependent?I N D E P E N D E N T E V E N T S

Notalleventsareindependent.Considertheevent“itwillraintomorrow”andtheevent“theweddingwillbeheldoutsidetomorrow.”Theprobabilitythattheweddingwillbeheldoutsideisaffectedbywhetheritrains.Writedownanotherexampleoftwoeventsthatarenotindependent.Onewaytoteachchildrenaboutthelawoflargenumbersistorunlotsofsimulationslikeyoudidinclasswiththehospitalproblem.Thatis,wemodeledthenumberofboyandgirlbirthsonagivendayinvarioussizehospitals,calculatedtheaveragepercentageofboybirthsforeach

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sizehospitalandthencomparedtheresults.Inordertohavethissimulationworkweneededamodel(acoinperhaps)thathastheapproximatelythesameprobabilityofsuccessasdoestheeventweareinvestigating.Andweneedtobecertainthatthenumberof‘births’weconsiderlargeislargeenoughforthelawoflargenumberstoapply.Apracticalwaytodesigncollectingdataonalargenumberoftrialsinyourclassroomistopooleveryone’sdata.Youwereaskedtorunasimulationintheclassactivity.Howdidyoudesignyoursimulation?Howmightyouhaveusedtherollofonefairdie?Howdidyoudecidethenumberofbirthsthatwouldrepresentasmallhospitalandthenumberofbirthsthatwouldrepresentalargehospital?Whatwastheeffectofpoolingalltheclasses’data?

Homework

IamalwaysdoingthatwhichIcannotdo,inorderthatImaylearnhowtodoit.PabloPicasso


2) Acoinhasbeentossedfivetimesandhascomeupheadseachtime.Whichofthe

followingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.

3) Acoinhasbeentossedfivehundredtimesandhascomeupheadseachtime.Whichof

thefollowingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.

4) Explainhowthelawoflargenumbersisrelatedtoyouranswerstothequestioninproblems2)and3)above.

5) Explainhowthelawoflargenumbersisrelatedtothedeterminationofexperimentalprobabilities.Whatdoesthislawtellyouaboutsimulationsyouwilldoinyourelementaryclassrooms?

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6) Afaircoinhasbeentossedathousandtimesandthenumberofheadsandtailsrecorded.Whichofthefollowingstatementsismostlikelytrue?

a) Thenumberoftailsis601andthenumberofheadsis399.b) 47%ofthetosseswereheads.c) Therewere20moreheadstossedthantails.

7) Supposeweflipacoin10,000,000,000times(justsuppose)andnoticethat

5,801,595,601areheads.Whatcanyousayaboutthecoin?Explain.

8) Designasimulationtodeterminetheexperimentalprobabilityofdrawingaspadefromastandarddeckofcards.Howcanyoubereasonablycertainyourresultisclosetothetheoreticalprobability?Carryoutyoursimulationandcompareyourresulttothetheoreticalprobabilityofdrawingaspadefromadeckofcards.

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SummaryofBigIdeas

Anideaisalwaysageneralization,andgeneralizationisapropertyofthinking.Togeneralizemeanstothink.

GeorgHegel

• Arandomexperimentisanactivitywheretheoutcomecannotbeknownin

advance.

• Theprobabilityofaneventhappeningistheproportionoftimesthateventwouldoccuriftherandomexperimentwasperformedaverylargenumberoftimes.

• Itisimportantforyoutogiveyourstudentsexperiencescalculatingprobabilities

experimentallyaswellastheoretically.• Treediagramshelpstudentsthinkthroughprobabilityideas.• Thelawoflargenumberstellsusthatthelargerthesample,themorelikelyweare

togetclosetothetheoreticalresult.

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ChapterSix StatisticsandDealingwithData

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ClassActivity26: StudentWeightsandRectangles

Themathematicsisnotthere‘tilweputitthere. SirArthurEddington(MQS)

Whatistheaverageweightofallthestudentsregisteredforclassesatyourschoolthissemester?Inyourgroups,makeaplanforsomewaytogatherinformationtoanswerthisquestion.Bespecificaboutwhatyouwilldoandwhenyouwilldoit.Atthispoint,timeandcostisnoobject.(Waittodiscussthisquestionasaclassbeforeyouturnthepage.)

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Now,havealookatthepopulationof100rectanglesonthenextpage.Noticethateachonehasanarea(asize).Forexample,rectanglenumber74hassize6squareunitsbecauseitiscomposedof6smallsquares.Carefullyselectasampleoftenrectanglesthatyouthinkwillhaveanaveragesizethatisclosetoaveragesizeofalltherectanglesonthepage.Makesuretopersonallychooseeachoftheten.Thenfindtheaveragesizeofyourself-selectedten.Yourclasswillthenmakeafrequencygraphshowingeveryone’saverages.Next,eachpersonshouldselect10rectanglesatrandom.(Yourinstructorwillhaveaplanfordoingthis.)Thenfindtheaveragesizeofyourrandomly-selectedten.Yourclassshouldmakeanotherfrequencygraphshowingeveryone’saverages.Finally,estimate,basedonthefrequencygraphs,whatistheaveragesizeofall100rectanglesonthepage?Explainyouranswer.Whatdoesallthishavetodowiththestudent-weightproblem?

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A Population of Rectangles

158

ReadandStudy

Heusesstatisticsasadrunkenmanuseslampposts–forsupportratherthanillumination.

AndrewLang Thisrectanglesizeproblemcontainsahugelessoninstatistics.Huge.Itisthis:humansarenogoodatselectingsamples.Ifyouwantunbiasedinformation,randomsamplesarethewaytogo.Inordertotalkmoreaboutthis,wearegoingtointroducethelanguageofsampling.Rememberthatwewantyoutolearnthesedefinitionsandtousethem.First,asamplereferstoasubsetofapopulation.Inourstudentweightproblem,thepopulationofinterestwasallstudentsregisteredforclassesatyourschoolthissemester.Inanidealworld,wejustwouldhaveeverysinglememberofthepopulationcometogetweighedandwewouldcalculatetheaverageweight.Unfortunately,thiswouldlikelyproveimpossible.Lots(most?)ofthestudentswouldnotshowuptobeweighed,andforcingthemtodosowouldbeunethical.Worse,thosewhowouldbewillingtoshowupmightbedifferentthanthosewhowouldrefuse.Peoplewhowereweight-consciousoroverweightmightbemorelikelytoavoidsteppingonyourscale.Sointheend,youwouldendupwithabiasedsample.Anothersolutionmightbetosendoutanonymoussurveystotheentirepopulationaskingeachpersontorecordhisorherweight.Butwouldyoureturnthatsurvey?Mostpeoplewouldnot(masssurveysmailedoutlikethattendtohaveverylowresponserates)andthosepeoplewhoreturnthemareagaindifferentfromthepopulation.Additionally,whenasurveyreliesonparticipantstoself-reporttheirowndata(likeweight),sometimesthatdataisn’taccurate.Okay,wellwemighttrytocalleveryoneorgodoor-to-doortocollectinformationonstudentweights.Thattypeofsamplingyieldsmuchhigherresponseratesbutcouldproveveryexpensiveandtakealotoftime(particularlyifyouhappentohavethousandsofstudentsinthepopulation). T H E L A G U A G E O F S A M P L I N G

So,whatarewetodo?Hereisthestatistician’ssolution.Forgetaboutgatheringdata(information)fromeachandeverymemberofthepopulationandfocusyourtime,energyandresourcesinsteadongettinginformationfromjustarandomsampleofthepopulation.Thatwayyoucanaffordtogodoor-to-doorandgetdatafrommostmembersofthesample.Itstillwon’tbeperfect.Butitturnsoutthatthisistheverybestyoucandoinafreesociety.Whyshouldthesampleberandom?Whynotsimplycollectdataontheweightsofyourfriends,ormaybeeveryoneinyourdorm,orwhynotstandinfrontofthecampuslibraryandaskeveryonewhowalksbyfortheirweights?Becauseofthethreatof…samplingbias.Asamplingprocedureisbiasedifittendstoover-sample(orunder-sample)populationmemberswithcertaincharacteristics.Thiswillmostsurelyhappenifyouself-selectyour

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sample–afterall,thestudentsyouknowhavedifferentcharacteristicsthanthepopulationofyourcampus.Whataresomeofthosedifferences?Stopandwritedownatleastthree.Butyoursamplewillalsobebiased,inamoresubtleway,ifyoustandinfrontofthelibraryandaskpeople“atrandom.”Thisisbecauseyouarenotreallyaskingrandompeople.Youareaskingthepeoplewhowalkbythelibraryatthattimeofday(maybeoversamplingpeoplewhostudyoftenorunder-samplingthosewhohaveaclassatthattime).Youareaskingpeoplewhohavealookatyouanddecidetheywanttoansweryourquestion(unwittinglyoversamplingpeoplelikeyourself).Giveanotherreasonwhythelibrarysamplemightbebiased.Samplingbiasissneaky.Itcreepsinbasedonwhenyousampleandwhereyousampleandhowyousample(aswellaswhoyousample).Watchforitasyouanalyzestudies.Sothesolutionistogeneratearandomsampleandthentouseyourresourcestogetthebestpossibleresponseratefromthem.Herearetwotypesofrandomsamplingthatareconsideredacceptable.Thefirstisthebest.Itiscalledsimplerandomsamplinganditismathematicallyequivalenttopullingnamesoutofahat.Everymemberofthepopulationhasthesamechanceofbeinginthesampleasanyothermemberofthepopulationandanysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesize.Stop.Readthatlastsentenceagainandthinkaboutit.Ifyouhadyourregistrargeneratealistofallstudentsenrolledinclassesthissemesterandyouassignedeachstudentanumberandthenhadyourcalculatorgenerate100randomnumbersandusedthosenumberstogetnames,thenthatwouldbesimplerandomsampling.IfIlinedupyourclassinarowandthentookeverythirdpersontomakemysample,wouldthatbesimplerandomsampling?Explain.R A N D O M S A M P L E S

Thereareotherformsofacceptablesampling.Forexample,clustersamplingisrandomsamplinginstages.Forexample,ifyourandomlychosetenpagenumbersfromtheregistrar’slistofstudentsandthenrandomlychose10peoplefromeachofthosepages,thenthatwouldbeclustersampling.TheU.S.Censushasmadeuseofclustersamplingwhenithasattemptedtofindthosepeoplewhodidnotreturntheircensusforms.We’lltellthatstoryinalatersection,fornowletussaythatacensusmeansthatyougatherinformationfromeachandeverymemberofapopulation.TheU.S.Censusisbutoneexampleofanattemptatacensus.Ifyoutriedtogetweightinformationfromeverystudentregisteredatyourcampus,thenyoutoowouldbeattemptingacensus.Itisprettymuchimpossibletoconductacensuswithalargepopulationbecauseyounevergetdatafromeveryone.Ifyoucollectdatafromasubsetofthepopulation,youarereallyconductingasurvey.Okay,herearetwomoretermstoknow:thenumericinformationthatyouwanttoknowaboutapopulation(ifyoucouldgetit)iscalledaparameter;thenumericinformationyougetfromasampleiscalledastatistic.Inourexample,theparameterwewereafterwastheaverage

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weightofallthestudentsregisteredforclassesatyourschool.Ifwecoulddoacensusofthatpopulation,wecouldfindthatparameter.However,wehavediscussedsomeofreasonswhywe’renotlikelytogetthatparameter.Instead,wewillsettleforperformingasurveyofasampleofthepopulation,andwhatwewillgetisastatistic.Hereisourilluminatingpicture:

Population Sample(fromthepopulation)

Askeveryone,it’scalledacensus. Askasubset,it’scalledasurvey.#computedfromthepopulation↔parameter. #computedfromthesample↔statistic.Thedifferencebetweentheparameterandthestatisticiscalledsamplingerror.Samplingerrorisanidea.Inpractice,whenconductingasurvey,youaren’tgoingtoknowwhatitisbecauseallyouwillhaveisastatistic.You’llprobablyneverknowtheexactparameter.Butwestillusethesewordstotalkabouttheideas.Onesourceofsamplingerrorwehavealreadydiscussed:samplebias.Anothersourceofsamplingerrorischanceerror.Chanceerroriserrorduetothediversityofthemembersofthepopulationandthefactthatyouarejustcollectingdatafromasubsetofthem.Ifthepopulationisdiverse(different)intheirweights,forexample,thenifyoupickarandomsampleofthem,youmayendupwithanaverageweightthatisabitdifferentfromthepopulationweight.Youmayeven,duetochancealone,getsomepeoplewhoareverythinorveryheavyinthesample.ThinkbacktoyourrandomsampleoftenrectanglesfromtheClassActivity.Theaveragesizeofyourrandomrectangleswaslikelycloseto,butnotexactly,theaveragesizeofall100rectanglesonthesheet.Thedifferencewasduetochanceerror.Nowthinkbacktotheaveragesizeofyourtenself-selectedrectangles.Thedifferencebetweenthatnumberandtheaveragesizeofall100rectanglesonthesheetisduetobothchanceerrorandsamplebias.Howdoyoureducesamplebias?Makeyoursamplerandom.

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Howdoyoureducechanceerror?Makeyoursamplebigger.Sojusthowbigdoesasamplehavetobe?Itseemslikeyouwouldneedsomesignificantsamplingrate(likemaybe25%ofthepopulation),butitturnsoutthatgoodsurveyscanbedonewithquitesmallsamples.Forexample,nationalpollingtopredicttheoutcomeofthepresidentialelectionisoftendonewithonlyafewthousandpeople.That’safewthousandoutofmorethan100millionlikelyvoters.Samplingrate[(#inthesample)÷(#inthepopulation)]canbesosmallbecausepeoplearenotallthatdiversefromasamplingperspective.Thinkofitthisway:supposeyouaregoingtosampleabigpotofsouptoseehowittastes.Aslongasthatsoupisstirredupreallywell,andthechunksofmeatsandvegetablesintherearen’ttoobigortoodifferent,allyouneedisasmallbitetoknowtheflavor.Bytheway,whatwasthesamplingrateofyourrandomsamplefromtherectanglesheet?Figureitout.

ConnectionstoTeachingIngrades3-5allstudentsshould–proposeandjustifyconclusionsandpredictionsthatarebasedondataanddesignstudiestofurtherinvestigatetheconclusionsand

predictions. NationalCouncilofTeachersofMathematics

PrinciplesandStandardsforSchoolMathematic,p.176Ideasofsamplingaresurprisinglycomplex–butgroundworkforthemcanbelaidintheelementaryschool.TheNCTMStandardsadvocatesthatstudentsingrades3-5should

“…begintounderstandthatmanydatasetsaresamplesoflargerpopulations.Theycanlookatseveralsamplesdrawnfromthesamepopulation,suchasdifferentclassroomsintheirschool,orcomparestatisticsabouttheirownsampletoknownparametersforalargerpopulation…theycanthinkabouttheissuesthataffecttherepresentativenessofasample–howwellitrepresentsthepopulationfromwhichitisdrawn–andbegintonoticehowsamplesfromthepopulationcanvary(p.181).”

Noticetheuseofthetechnicallanguage.Whatdotheymeanby“knownparametersforalargerpopulation?”Givesomeexamples.Supposethatyourthirdgradershavecollecteddatafromtheirclassonthenumberoftimeslastweekthattheyeathotlunchatschoolandfoundthefollowingdatadisplayedbelowasafrequencyplot(eachXrepresents1childintheclass):

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X X X X X X X X X X X X X X X X X X X X 0 1 2 3 4 5 #ofHotLunchDayslastweekOfcourse,therearemanyquestionsyoumightaskyourstudentsaboutthesedata–buttherearesomequestionsparticularlyrelatedtosamplingthatshouldbeaskedanddiscussed.Questions1:Wasthereanythingspecialaboutlastweekthatmighthavemadethesedatadifferentthanifwe’daskedthesamequestionnextweekordoyouthinklastweekwasatypicalweek?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Question2:Howdoyouthinkthedatamighthavebeendifferentifwe’daskedthefifthgradersthesamequestion?

Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Questions3:Woulditbeokaytolabelthisgraph“NumberofDaysEachWeekthatChildrenEatHotLunchatOurSchool?”Whyorwhynot?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Noticethatthesequestionshelpchildrenthinkaboutwhetherthedataisrepresentativeofapopulationlargerthanthesampleof20third-gradersshownonthegraph,andithelpsthemto

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begintothinkcriticallyaboutwhatthedatasaysandwhatitdoesnotsay.Samplingbecomesanexplicittopicinthemiddlegrades.

Homework

Weknowthatpollsarejustacollectionofstatisticsthatreflectwhatpeoplearethinkingin‘reality.’Andrealityhasawellknownliberalbias.

StephenColbert

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.2) Explainthedifferencebetweenacensusandasurvey.

3) Explainthedifferencebetweenaparameterandastatistic.

4) TrueorFalse?Thebiggerthesamplethesmallerthesamplebias.Explainyourthinking.

5) Sometimesawebsiteorpublicationwilladvertiseapollaskingitsviewersto‘callin’or

‘clicktorespond’toquestionsonatopic.Inthiscasewesaythatthesampleisself-selected.Whattypesofbiasesarelikelyinthistypeofsurvey?Explain.

6) Itispossibletobiasresultsofasurveybyaskingquestionsusing“charged”language.ConsiderthesequestionsfromtheRepublicanNationalCommittee’s2010ObamaAgendaSurvey,andfromaDemocraticStateSenator’sdistrictsurvey.(Canyoutellwhichiswhich?)Explainhoweachquestionmightbeaskedinalessleadingmanner.

a) “Doyoubelievethatthebestwaytoincreasethequalityandeffectivenessof

publiceducationintheU.S.istorapidlyexpandfederalfundingwhileeliminatingperformancestandardsandaccountability?”

b) “DoyousupportthecreationofanationalhealthinsuranceplanthatwouldbeadministeredbybureaucratsinWashington,D.C.?”

c) “DoyoubelievethatBarackObama’snomineesforfederalcourtsshouldbeimmediatelyandunquestionablyapprovedfortheirlifetimeappointmentsbytheU.S.Senate?”

d) “Doyousupportoropposelegislationtorestoreindexingofthehomesteadtaxcredit,whichwouldboostoureconomyandhelppeopletostayintheirhomes?”

e) “DoyousupportoropposelimitinglocalcontrolandweakeningenvironmentalstandardstoencourageironoremininginWisconsin?”

7) TheColumbusCityCouncilwantedtoknowwhetherthe65,000citizensofthecityapprovedofaproposaltospend$1,000,000torevitalizethedowntown.Thefollowingstudywasconductedforthispurpose.Asurveywasmailedtoevery10thpersononthe

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listofregisteredvotersinthecity(itwasmailedto3500people)askingabouttheproposal.CitizenswereaskedtocompletethesurveyandreturnittotheCouncilbymailinaself-addressed,stampedenvelope.3100peopleresponded;ofthese,75%supportedtheexpenditureand25%didnot.

a) Thepopulationforthisstudyis

A)allregisteredvotersinthecity B)allcitizensofthecityofColumbus C)the3500peoplewhoweresentasurvey D)the3100peoplewhoreturnedasurvey E)noneoftheabove

b) Whatwasthesamplingrateforthissurvey?

c) The"75%"reportedaboveisa A)parameter B)population C)sample D)statistic E)noneoftheabove

d) Thissurveysuffersprimarilyfrom A)samplingbias B)chanceerror C)havingasamplesizethatistoosmall D)non-responsebias

e) TrueorFalse?Theresultsofthissurveymaybeunreliablebecauseregisteredvotersmaynotberepresentativeofallthecitizens.Explainyourthinking.

f) TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribedas

simplerandomsampling.Explainyourthinking.

g) TrueorFalse?Samplebiascouldbereducedbytakingalargersampleofregisteredvoters.Explainyourthinking.

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ClassActivity27: SnowRemoval

Errorsusinginadequatedataaremuchlessthanthoseusingnodataatall.CharlesBabbage

Inordertostudyhowsatisfiedthe120,000citizensofGreenBay,WIarewithsnowremovalfromcitystreets,Iconductedthefollowingstudy.IstoodinfrontoftheGreenBayWalmartonaMondaymorninginFebruaryfrom9:00untilnoonandIaskedeverythirdpersonwhowalkedinwhethertheyweresatisfiedwithsnowremovalfromstreetsinGreenBay.Onehundredandeightpeoplesaidtheyweresatisfied.Twenty-fivepeoplesaidtheywerenot,and18refusedtoanswermyquestion.Answerthebelowquestioninyourgroups.

1) Describethepopulationforthestudy.

2) DidIconductacensusorasurvey?Explain.

3) Whatwasthesamplingrate?(Writethisinseveraldifferentways.)

4) Whatwastheresponserate?(Writethisinseveraldifferentways.)

5) TrueorFalse?Thisstudysuffersprimarilyfromnon-responsebias.Explain.

6) TrueorFalse?Thisstudysuffersprimarilyfromsamplingbias.Explain.

7) TrueorFalse?Theresultsofmystudymaynotbevalidbecausecityemployeesmayhavebeenpartofthesample.Explain.

8) TrueorFalse?Amainproblemwithmystudyisthatmysamplesizeistoosmall.Explain.

9) TrueorFalse?PeopleshouldconcludefrommystudythatGreenBaycitizensare

generallyhappywithsnowremovalfromcitystreets.

10) Describeatleastthreewaystoimprovemystudy.

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ReadandStudy

Youcanuseallthequantitativedatayoucanget,butyoustillhavetodistrustitanduseyourownintelligenceandjudgment.

AlvinToffler

“Fouroutoffivedentistssurveyedrecommendsugarlessgumfortheirpatientswhochewgum.”You’veheardthatclaim.Itisperhapsthemostwellrememberedstatistical‘fact’inthehistoryofmarketingresearch.Butwhatdoesittellyou?Themakersofsugarlessgumhopethisstatisticconvincesyoutopurchasetheirproduct.Butbeforeyourunoutforapackofsugarlessgum,youshouldaskafewquestionsaboutthis‘fact.’Howwerethedentistsinthesamplechosen?Whatwastheresponserate?Whatwasthesamplesize?Whatwastheexactquestiontowhichthedentistswereaskedtorespond?Andevenifitturnsoutthatthesurveyusedexemplarymethodology,notethattheresultdoesnotsuggestyouthatyoushouldtakeupchewingsugarlessgum.Itonlysuggeststhatifyoualreadychewgum,thesedentistssuggestyouchewsugarlessgum.Allsurveysarenotcreatedequal.Infact,somearedownrightmisleading.Itisimportantforyoutothinkcriticallyaboutinformationbasedonsurveydata.A N A L Y Z I N G A S U R V E Y

Surveysaremeanttoprovideinformationaboutapopulation.Theycanhelpcompaniesunderstandhowtomarketproducts;theycanhelpgovernmentalagenciesunderstandconstituents;theycanhelpcampaignspredictpublicreactionstoideasorevents;theycanhelpsecondgradeteachersunderstandwhichmoviestheirstudentsprefer.Hereisanexampleofonerealsurvey.TheUnitedStateCensusBureauusessurveystounderstandthepopulationofourcountry.(Bytheway,asteachersyoucangetgooddataforyourstudentstoanalyzefromtheU.S.CensusBureauwebsite.Keepthatinmind.)Forexample,theyconductasurveyofU.S.housingunitseveryotheryearinordertomakedecisionsaboutfederalhousingprojects.Intheirwords,theiraimisto“[p]rovideacurrentandongoingseriesofdataonthesize,composition,andstateofhousingintheUnitedStatesandchangesinthehousingstockovertime.”Hereisafigureshowingsomeoftheresultsofthe2007AmericanHousingSurvey(AHS).Takeafewminutestostudyit.

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U.S.CensusBureau,CurrentHousingReports,SeriesH150/07AmericanHousingSurveyfortheUnitedStates:2007.U.S.GovernmentPrintingOffice,Washington,DC.

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Theseresultsdrivenationalpolicy,sowehopethattheAHSiswelldesigned.Let’shavealookatthemethodology.ReadthefollowingparagraphbasedontheinformationfromtheUSCensusBureauwebsite:TheAHShassampledthesame53,000addressesineachodd-numberedyearsince1985.Thoseaddresseswerechoseninthefollowingmanner.

“FirsttheUnitedStateswasdividedintoareasmadeupofcountiesorgroupsofcountiesandindependentcitiesknownasprimarysamplingunits(PSUs).AsampleofthesePSUswasselected.ThenasampleofhousingunitswasselectedwithinthesePSUs”(U.S.CensusBureau,2007,p.B-1).

TheAHSincludesbothoccupiedandvacanthousingunitsandallpeopleinthosehousingunits.Dataiscollectedusingcomputer-assistedtelephoneandpersonal-visitinterviews.Whenahousingunitwasvacant,informationwasgatheredfromneighbors,rentalagents,orlandlords.TheAHSisavoluntarysurveymeaningthathouseholdscandeclinetoanswerquestions.Interviewsforthedatapresentedabovewereconductedfrommid-ApriltoSeptember2007.

Okay,shouldwebelievetheAHSresults?Let’stalkthroughananalysisofthesurveydesign.First,whatisthepopulationforthissurveyandwhatisthesample?Accordingtothe2001census,thereareabout200,000,000housingunitstotalintheUnitedStates–thatistheapproximatepopulation.Thesamplecontainedabout53,000ofthesehomes.Thatgivesasamplingrateofabout0.0027%.Thismightseemsmall,butthesamplewascarefullyselectedusingclustersamplingandsothesamplingerrorwillbefairlysmalltoo.(Whywasthisclustersampling?Explainwhybasedonthedefinition.)Inotherwords,thereisreasontobelievethatthehouseholdsinthesamplearerepresentativeofhouseholdsintheUnitedStates.Let’sgoonestepfurtherinthinkingaboutthequalityofthedata.Itcouldbethatthesamplewaswell-chosen,butthatmanyhouseholdsgavefaultyinformation(orsimplyrefusedtogiveinformationatall).LuckilytheCensusBureauusedthebestpossiblemethodforcollectingdata–theycalledfirst,andiftheycouldnotreachahouseholdbyphone,theysentsomeonetotheaddresstoconductaninterviewwitheithertheoccupants,orinthecaseofanemptyunit,aneighbororlandlordorrealestateagent.Thislikelyproducedafairlyhighresponserate.(Infact,theBureaureportsan89%responserate.Intheothercaseseithernoonewasathomeevenafterrepeatedvisits,theoccupantsrefusedtobeinterviewed,ortheaddresscouldnotbelocated.)So,ourtakeisthatthemethodologyfortheAHSissoundandtheresultsarebelievable.Wewantyoutolearntothinklikethiswheneveryouneedtojudgethebelievabilityofaclaimbasedondata.

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Mathematicsisasmuchasaspectofcultureasitisacollectionofalgorithms.CarlBoyer

The Common Core standards in grades 6 and 7 for surveys are listed below. Read them carefully and then determine which standards we have addressed so far.


CCSSGrade6:Developunderstandingofstatisticalvariability.1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedata

relatedtothequestionandaccountsforitintheanswers.Forexample,"HowoldamI?"isnotastatisticalquestion,but"Howoldarethestudentsinmyschool?"isastatisticalquestionbecauseoneanticipatesvariabilityinstudents'ages.

CCSSGrade6:Summarizeanddescribedistributions.

5.Summarizenumericaldatasetsinrelationtotheircontext,suchasby:A.Reportingthenumberofobservations.B.Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.

CCSSGrade7:Userandomsamplingtodrawinferencesaboutapopulation.

1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.

2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.Forexample,estimatethemeanwordlengthinabookbyrandomlysamplingwordsfromthebook;predictthewinnerofaschoolelectionbasedonrandomlysampledsurveydata.Gaugehowfarofftheestimateorpredictionmightbe.

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Homework

USATodayhascomeoutwithanewsurvey--apparently3outofevery4peoplemakeup75%ofthepopulation.

DavidLetterman


2) DoalltheitalicizedthingsintheConnectionssection.3) Spendafewminuteslookingatthefigureonoccupiedhomesfromthe2007AHS.Write

aparagraphhighlightingthethingsthatyoufindmostinterestingornotableabouttheseresults.

4) Inordertofindouthowitsreadersfeltaboutitscoverageofthe2008Presidential

Election,alocalnewspaper(withacirculationofabout10,000)ranafront-pagerequestintheSundaypaperthatreadersgototheirwebsiteandcompleteanonlineanonymoussurveythatcontainedthefollowingtwoquestions:

I. Whatisyourpoliticalaffiliation?____Democrat____Republican____Independent____Other

II. Howwouldyourateourcoverageofthe2008presidentialelection?____slantedleft____slantedright____unbiased____noopinion

Sixhundredreadersresponded.Ofthose,70%ofDemocrats,40%ofRepublicans,66%ofIndependents,and20%of“Other”saidthatthecoveragewasunbiased.

a) Analyzethesamplingmethod.Isthesampleofrespondentslikelytoberepresentativeofthepopulationthepaperwantstorepresent?Givespecificreasonswhyorwhynot.

b) Whatdoesthedatatellusaboutthepaper’scoverageofthe2008PresidentialElection?Explain.

c) Describesomewaystoimprovethissurvey.

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5) GototheUnitedStatesCensusBureauwebsite(http://www.census.gov/schools/)andfindtheTeachersandStudentslink.Spend15minutesbrowsingtheactivitiesthere.ThendesignamathematicslessonforGrade3usingtheStateFactsforStudentslink.

6) AresearcherwantstopredicttheoutcomeoftheFrostbiteFallsmayoralelection.In

ordertodothis,shemailedasurveytoeverytenthpersononalistofall40,980registeredvotersinthecity.Onethousandfifty-sixpeoplereturnedthesurvey.34%ofrespondentssaidtheysupportedSnidelyWhiplash;48%saidtheysupportedDudleyDoRight;theremaining18%wereundecided.

I. Thepopulationforthisstudyis

A)allmayoral-racevotersinthecityofFrostbiteFalls B)allcitizensofthecityofFrostbiteFalls C)thepeoplewhoweresentasurvey D)thepeoplewhoreturnedasurvey E)noneoftheabove

II. Theresponseratewas___________.III. Thesamplingratewas___________.

IV. The"48%"reportedaboveisa

A)parameter B)population C)sample D)statistic E)noneoftheabove

V. Thissurveysuffersprimarilyfrom

A)selectionbias B)nonresponsebias C)chanceerror D)havingasamplesizethatistoosmall

VI. TrueorFalse?Theresultsofthissurveymaybeunreliablebecausemembersofthecitygovernmentmayhavebeenincludedinthesample.Explainyourthinking.

VII. TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribed

assimplerandomsampling.Explainyourthinking.

VIII. TrueorFalse?Biascouldbereducedbytakingevery5thpersononthelistofregisteredvotersratherthanevery10th.Explainyourthinking.

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ClassActivity28: Part1NameGames

Ifeellikeafugitivefromthelawofaverages. WilliamH.Mauldin

1) Eachpersonintheclassshouldwritehisorherfirstnameonstickynotessothatthere

isoneletteroneachoftheperson’snotes.(Soforexample,IwouldwriteJononenote,Eonanother,andNonthethird.)Putyournamesinstacksontheboardlikewedidintheexamplebelow.Nowyourclassneedstorearrangeyourclasses’stickynotes(orevenpartsofstickynotes)sothateverystackendsupwiththesamenumberofletters.Dothatnow.

AmiJen Mo Omar…

Themean(oraverage)ofasetofnumericalobservationsisthesumofalltheirvaluesdividedbythenumberofobservations.Explainwhyitmakessensethatyourmethodjustcomputedthemeannamelengthforyourclass.

2) Nowwewillmakeadifferenttypeofgraph,abargraph.Eachpersonshouldgetonestickynotetoplaceonthegraphbasedonthelengthofhisorherfirstname.ForexampleiftherewereeightpeopleintheclasswithnamesMo,Ami,Jen,Omar,Karen,Steve,Sienna,andJuanitathebargraphwouldlooksomethinglikethis:

frequencies

____________________________________ 234567 NameLengths

Figureoutawaytorearrangethenotestohelpyoutofindthemeannamelengthonagraphlikethis.Whydoesitmakesensethatyourmethodjustcomputedthemeannamelengthforyourclass?Howisthismethoddifferentfromthemethodyouusedinquestion1?Discussthis.


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3) Themedianofasetofnumericalobservationsisthemiddleobservationwhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”observations,themedianistheaverageofthem.Decidehowyoucoulduseeachtypeofgraphabovetohelpyoufindthemediannamelength.

4) Whichislikelytobelargerforyourclass?Themeannumberofsiblingsorthemediannumber?Explain.Then,collectthedataandseeifyouarecorrect.

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ClassActivity28: Part2IntheBalance

Sodivinelyistheworldorganizedthateveryoneofus,inourplaceandtime,isinbalancewitheverythingelse.

JohannWolfgangvonGoethe

Yourtaskistoputweightsonthenumberedpegsofabalancescaletokeepthesystembalanced.

1) Findseveraldifferentarrangementsof3weightssothatthesystemwillbalance.Makeageneralconjectureaboutallthearrangementsthatwillbalancethesystem.

2) Ifyouhadoneweight1totherightandtwoweights2totheleft,predictwhereyouwouldneedtoplaceafourthweighttobalancethesystem.

3) Findseveralwaystobalancefourweightswhereonlyoneweightisontheleftside.Makeageneralconjectureaboutallthearrangementsofthistypethatwillbalancethesystem.

4) Predictsomearrangementsthatwillbalancefor5weights.Makeageneralconjecture

aboutallthearrangementsof5weightsthatwillbalancethesystem.

5) Supposethereare4catswithmeanweightof10pounds.Youknowtheweightsof3ofthecats.Thoseweightsare:4pounds,7pounds,and15pounds.Howcanyouusethebalancetofigureouttheweightofthe4thcat?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)

6) Usethebalancetofigureoutthisproblem:Youhavetakenfourmathquizzes.Your

scoreswere82,67,85,and72.Whatdoyouneedtoscoreonthe5thquizforyouraveragequizscoretobe75?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)

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ReadandStudy

Theaverageadultlaughs15timesaday;theaveragechild,morethan400times. MarthaBeck

Therearethreestatisticsthatarecommonlyusedtodescribethecenterofadistributionofnumericaldata:mean,medianandmode.Themeaniscomputedbysummingthevaluesanddividingbythenumberofthem.Butabetterwayforchildrentothinkaboutmeanisusingtheideaof“eveningoff.”Themeanistheaveragevalue,thevalueeachpersonwouldgetifallthedatawassharedevenly.Forexample,supposethepicturebelowshowsthecookiesthatbelongtoeachchild.

Keesha’scookies:

Andi’scookies:

Tim’scookies:

Myosia’scookie:

Carlos’cookie: Asyousawintheclassactivity,inordertofindthemeannumberofcookies,thechildrencouldimaginesharingcookies(orevenpartsofcookies)sothattheyeachchildhasexactlythesamenumber.Bytheway,isthisdatadisplayliketheoneinproblem1)ortheoneinproblem2)fromtheClassActivity?Explain.

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Keesha’scookies:

Andi’scookies:

Tim’scookies:

Myosia’scookies:

Carlos’cookies: Sothechildrenseethatthemeannumberofcookiesis3.Inthiscasewedidn’tneedtobreakupcookiesinordertosharethem,butyoucertainlyshoulddoexampleslikethatwithyourstudents.R E D I S T R I B U T I O N C O N C E P T O F T H E M E A N

Whatwehaveillustratedhereistheredistributionconceptofthemean:thatthemeanvalueofasetofnumbersiswhateachobservationwouldgetifthedatawasredistributedtothateachobservationhadthesameamount.Thisconceptisessentiallythesameasthepartitiveconceptofdivision:takingatotalamountandsharingitequallyintoanumberofgroups,andseeinghowmucheachgroupgets.B A L A N C E C O N C E P T O F T H E M E A N

Inclassactivity12b,wealsodiscoveredthebalancepointconceptofthemean,namely,thatthemeanisthepointinadistributionwherethesumofthedistancesthatthedataisabovethemeanbalanceswiththesumofthedistancesthatthedataisbelowthemean.Themodeofasetofdataisthevaluethatoccursmostfrequently.Whatisthemodeforthecookiedata?Sometimesadatasetwillhavemanymodesbecauselotsofvalueswilloccurwiththesamemaximumfrequency.M E D I A N A N D M O D E

Theideaofmedianistheideaofmiddleorcentervalue.Sowesawthatthenumberofcookiesheldbythefivepeopleis4,3,7,1,and1.Ifweputthesevaluesinnumericalorder,weget: 1 1 3 4 8

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Whydoweneedtoputtheminnumericalorderfirst?Howwouldyouexplainthistoachild?Weseethatthemiddlevalueis3(we’llwriteM=3.)Inthecaseofanevennumberofvalues,wehavenoonemiddlevalue,andsothemedianistheaverageofthetwomiddlevalues.Findthemean,mode,andmedianofmybowlingscores:

112 114 120 123 127 127 130 167 220Whydoesitmakesensethatthemeanislargerthanthemedianforthesedata?Explain.F I V E N U M B E R S U M M A R Y

WewilldefineQ1(thefirstquartile)asthemedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorder.Forexampleusingmybowlingscores,127isthemedian.

112 114 120 123|127| 127 130 167 220Sothescores112,114,120and123arepositionedbeforethemedian.Themedianofthosescoresis(114+120)/2=117.SoQ1=117.Q3(thethirdquartile)isthemedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder:127,130,167and220.Q3=148.5.Nowwecanreportsomethingwecallafive-numbersummaryforthesescores:Minimum=112, Q1=117, M=127, Q3=148.5, andMaximum=220Bythewaychildrenmaywanttoknowifduplicatesinthedatacanbethrownout.Whatwouldyousaytothem?Why?Whataboutvaluesofzero–couldtheybediscarded?Whatwouldyousaytothechildrenaboutthat?

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Weusedasmalldatasetforthepurposeofillustratingtheseideas.Wenotethatforlargedatasets,thesestatistics(mean,median,mode,quartiles,andpercentiles)aretypicallycomputedusingacalculatororcomputer.We’llgiveyoutheopportunitytousetechnology-ifyouhaveit-andworkwithalargerdatasetinalaterclassactivity.Asateacher,youwillneedtodiscusswithparentsthemeaningoftheirchild’sscoreonastandardizedexam,solet’sspendafewminutesontheideasthatyoumightfindusefulinthatcontext.Typicallythesescoresarereportedaspercentiles.Ifachildinyourclassscoresinthenthpercentilethismeansthat“npercent”ofthechildrenwhotookthetestscoredatorbelowthatscore.Forexample,ifachildscoresinthe80thpercentileonatest,thatmeansthat80%ofthechildrenwhotookthetestscoredatorbelowherscore.Anotherwaytothinkaboutthisistoimagineliningup100childrenaccordingtotheirtestscores,thischildwouldbenumber80fromthebottomintheline(ornumber20fromthetop).Inthiscontextthemedianisthe50thpercentile.WhatpercentileisQ3?P E R C E N T I L E S


Ingrades3-5allstudentsshouldusemeasuresofcenter,focusingonthemedian,andunderstandwhateachdoesanddoesnotindicateaboutthedataset.

NationalCouncilofTeachersofMathematics

PrinciplesandStandardsforSchoolMathematics,p.176Childreningrades3-5caneasilyunderstandandcomputemedianandmode,buttypicallytheideaofmeanisamoredifficulttopic.Thisisnotbecauseitishardtocalculateamean;ratheritisbecauseitismoredifficulttounderstandtheideaofamean.Belowisanexampleofanactivityappropriateforchildreningrade4thatrepresentsanopportunityfortheteachertodiscussmeanasaredistributionprocessratherthanasacomputation.Explainhowyoucouldusestickynotestohavechildrenuseevening-offtofindthemeanofthebelowdatawithouteverdoingthestandardcomputation.Reallydoit.

Thefamilysizesforchildreninourclassareshownbelow.Makeabargraphforthesedataandthenfindthemeannumberoffamilymembers.

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Family sizeJenson 8Lewis 3McDuff 7EarlesSeaman

45

Ortiz 4PetersToddMiene

653

Weknowthatthestandardcomputation(addthevalues,thendividebythenumberofvalues)isquickerand‘easier’todo.Butasanelementarymathematicsteacher,yourjobistohelpyourchildrenunderstandbigideas–andnottogivethemshortcutsthatallowthemtogetrightanswerswithoutunderstandingtheideas.Childrenmustlearnthatmathematicsmakessense,andthatthinking(insteadofmemorizing)isthewaywedomathematics. Infifthgrade,studentsarefirstexposedtotheideaofthemean.Whatconceptofthemeanisusedhere?


Homework

Wecannotteachpeopleanything;wecanonlyhelpthemdiscoveritwithinthemselves.

GalileoGalilei

1) DoalltheitalicizedthingsintheReadandStudysection.2) DoalltheitalicizedthingsintheConnectionssection.

CCSSGrade5:Representandinterpretdata.1. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,

1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.Forexample,givendifferentmeasurementsofliquidinidenticalbeakers,findtheamountofliquideachbeakerwouldcontainifthetotalamountinallthebeakerswereredistributedequally.

180

3) Lookatthe“FulcrumInvestigation”inUnit8,Module1,Session3ofthe4thgradeBridgesinMathematicscurriculum.Inthisactivity,studentsaregivenapencil,a12-inchrulerandseveral“tiles”(weightsorpennies)allofthesameweight.Byplacing6tileson“0”ontheruler,theycansimulatea60-poundfourthgradestudent(Sarah)sittingontheendofaseesaw.Anotherweightisplacedontheotherendoftheseesaw(at12inches).Ifthefulcrumisinthemiddleofthepencil(at6”),thenitwouldtake6tiles(representing60pounds)attheotherendto“lift”orbalanceSara.Usinganactualpencil,ruler,andweights(suchaspennies),predictthenexperimenthowmuchweightittakestobalanceSarahifthefulcrumisplacedatthe4-inchmark,the5-inchmark,the7-inchmark,andthe8-inchmark.

a. Answerquestions1-6inthe“FulcrumInvestigation”.b. HowcanyouusetheconceptoftheMeantoanswerquestions5and6inthis

investigation?c. A“ChallengeQuestion”inthenextsessionasks“IfSarahcan’tmovethefulcrum

fromthemiddleoftheseesaw,whatcanshedotobalancewiththe40-poundfirstgrader?Whydoesthismethodwork?”Answerthisquestion.Howdoyouthinkfourthgraderswouldbeabletothisout?Howcanyouusetheconceptofthemeantofigurethisout?

d. Repeatpartc,butnowsupposeshewastobalancewiththe100-pound7thgraderwithoutmovingthefulcrum.

4) Hereisadatasetofthenumberofyearsmembersofthemathdepartmenthavebeen

employedatmyschool:7,25,7,15,8,23,27,6,1,3,28,22,24,9,15,14,18,8.Computethemean,medianandmodeforthesedata.Whatdothesedatatellyouaboutthegroupofmathematicsfacultyatthisschool?

5) DecidewhethereachofthefollowingisTrueorFalse.Ineachcase,arguethatyouare

right.

a) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemean.

b) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemedian.c) T F Iftenpeopletooka100-pointexam,itispossiblethatthe meanandmedianare90pointsapart.d) T F Iftenpeopletookathousand-pointexam,itispossiblethe themeanandthemedianare90pointsapart.

e) T F Themeanincomeofyourtownislargerthanthemedian income.

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6) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyourschool

andnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.

7) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyourschool

andnowtheirmedianannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Isitmoreconvincingorlessconvincingthanthescenariointhepreviousproblem?Explain.

8) Supposethatonehundredpeoplegraduatedwithadegreeincomputersciencefrom

yourschoolandnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.Isthismoreorlessconvincingthanthescenariosintheprevioustwoproblems?Explain.

9) Annneedsameanof80onherfiveexamsinordertoearnaBinherclass.Herexam

scoressofarare78,90,64and83.WhatdoessheneedtogetonherfifthexamtoearntheB?Dothisprobleminatleasttwodifferentways.Onewayshouldusetheredistributionconceptofthemean.Anothershouldusethebalancepointconceptofthemean.

10) Awaytothinkaboutthemedianofasetofnumbersisthatitisthevaluethatsplitsthe

distributionintotwoequal‘counts.’Considerthisfrequencygraphof18scoresonaquiz:

X X

X X X X X X X

X X X X X X X X X 1 2 3 4 5 6 7 8 9 10

ScoreonaQuiz(outoftenpossiblepoints)

a) Findthemedianscoreandexplain,asyouwouldtoastudent,whyitmakessense.b) Wesaythatthisdistributionisskewedleftbecauseithasanasymmetrictail(or

evenoutliers)totheleft.Inskewedleftdata,doyouexpectthemeantobehigherorlowerthanthemedian?Explainyourthinking.

c) Nowimaginethatthedistributionaboveshows18equalweightsarrangedonaseesaw.Ifyouwantedtheseesawtobalance,youwouldneedtoplacethefulcrumatthemean.Inotherwords,youcanthinkaboutmeanasthebalancingpointofthedistribution.Findthemeanandseeifitlookslikethebalancingpoint.Whydoesitmakesensethatthemeanissmallerthanthemedianforthesedata?

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11) Achildinyourfourth-gradeclassscoreda200outof500onastandardizedmathematicsassessmentinwhichtheaveragescorewas240.Thisstudentisreportedasscoringinthe35thpercentile.Youneedtoexplaintotheparentsexactlywhatthismeans.Whatdoyoutellthem?Areyouconcernedaboutthischild’sperformance?Explainyourthinking.

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ClassActivity29: MeasuringtheSpreadIt’snotthatGooddoesn’ttriumphoverEvil,it’sthatthepointspreadistoosmall.

BobThavesHerearetheexamscoresfortwostudents,AndreasandBelle:Andreas:45,57,69,69,80,94 Belle: 54,70,71,72,72,77Whichofthesestudentshadbetterscores?Explainhowyoudecidedbeforeyoureadfurther.Noticethatthesestudents’scoresareverysimilarbasedonthemeasuresofcenter(meanandmedian)–buttheyarenotsosimilarbasedonthespreadoftheirscores.Spreadisanideathatiscapturedbydifferentstatistics.Herearethreecommonmeasuresofspread:

1) Range=maximum-minimum.Computetherangeforeachstudent’sscores.Whatisitthatrangemeasures?Beasspecificasyoucan.

2) Inter-QuartileRange(IQR)=Q3–Q1.ComputetheIQRofeachstudent’sscores.What

isitthattheIQRmeasures?Beasspecificasyoucan.

3) MeanAbsoluteDeviation(MAD)=!-

𝑥! − 𝑥 + 𝑥" − 𝑥 + 𝑥% − 𝑥 +⋯+ 𝑥- − 𝑥 Inthisformulanisthenumberofvalues,𝑥isthemean,andeach“x”isaspecificvalue.Recallthat,forexample|-3|means“theabsolutevalueof-3.”ComputethemeanabsolutedeviationofAndreas’scoresandthenBelle’sasagroup.Whatisitthatthemeanabsolutedeviationmeasures?Beasspecificasyoucan.


184

Nowwehavethemachineryweneedtoidentifyanoutlierinthedata.First,adefinition:Anoutlierisaspecificobservationthatlieswelloutsidetheoverallpatternofthedata.Youmightask,whatdoesitmeantobe“welloutside”theoverallpatternofthedata?Gladyouasked.Wehaveacriterionforthat.Itiscalledthe1.5×IQRcriterion:avalueisconsideredanoutlierifitfallsmorethan1.5×IQRaboveQ3orifitfallsmorethan1.5×IQRbelowQ1.Let’susethiscriteriontotestAndreas’scoresforoutliers:45,57,69|69,80,94M=(69+69)/2=69andsitsinthepositionheldbythebarintheabovedataset,sothescores45,57and69arepositionedbelowthemedian(M).Q1=57andlikewise,69,80and94arepositionedabovethemediansoQ3=80.

IQR=Q3-Q1=80–57=23 1.5×IQR=34.5 Soavalueisanoutlierifitismorethan34.5pointsaboveQ3ormorethan34.5pointsbelowQ1.Inotherwords,outliersinAndreas’dataarebiggerthan80+34.5=114.5orsmallerthan57–34.5=22.5.Sincetherearenoscoresthatbigorthatsmallinhisdata,hehasnooutliersbasedonthecriterion.CheckBelle’sscoresforoutliersusingourcriterion.HowcanitbethatBellehasanoutlieramongherscoreswhenAndreasdoesnot?Explain.

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ReadandStudy

Americaisanoutlierintheworldofdemocracieswhenitcomestothestructureandconductofelections.

ThomasMann

IntheClassActivityyouworkedwiththreedifferentstatisticsthatdescribethespreadofthedata.Therangeshowsyoutheentirespreadofthedataset.TheIQRisameasureofthespreadofthemiddlehalfofthedata.Themeanabsolutedeviationmeasureshowspreadoutthedatais(onaverage)arounditsmean.Wewillusetwoprimarystatisticsfordeterminingthecenterofadistributionofdata:meanandmedian,andtwoprimarymeasuresofspreadofthedata:meanabsolutedeviationandinterquartilerange(IQR).So,howdotheyfittogetherandwhywouldwechoosetoreportoneoveranother?Ourfirstconsiderationisthis:doesthedatacontainoutliers?See,weknowthatanoutliercanhaveabigeffectonthemean(butnotonthemedian).Forexample,considerthesetestscoreswithmean75.44andmedian78:

67,68,68,71,78,79,80,83,85Beforeyougoanyfarther,checkthesescoresforoutliersusingthe1.5×IQRcriterion.R A N G E , I Q R , A N D M E A N A B S O L U T E D E V I A T I O N

A C R I T E R I O N F O R F I N D I N G S U S P E C T E D O U T L I E R S

Now,supposethatthescoreof67wasreplacedbyascoreof12.(Noticethat’sanoutlier.)Thischangehasnoeffectonthemedian(whichisstill78)butitdoesaffectthevalueofthemean.Computethenewmeantocheck.Wesaythatthemedianis“resistanttooutliers”andthatthemeanis“affectedbyoutliers.”Ifyourdatacontainsanoutlier(orseveraloutliers)thataffectthemean,thenthemedianisoftenthebetter(morehonest?)measureofcenterforyoutoreportforthosedata.Whichofthemeasuresofspreadisresistanttooutliersandwhichisaffectedbythem?Takeafewminutestofigureitout.

186

Nowwearegoingtointroduceadisplayofthefive-number-summaryofadataset:theboxplot.Recallthatthefive-number-summaryconsistsoftheminimum,Q1,themedian,Q3,andthemaximumvaluewhenthedataisarrangedinnumericorder.Hereissomefictitiousdataonthenumberofyearsmathematicsfacultymembershavebeeninourdepartment.Takeaminutetofindthemedianandthequartilesforthesedata. 1,3,6,7,7,8,8,9,14,15,15,18,22,23,24,25,27,28B O X P L O T S

Hereisanexampleofaboxplotdisplayofthedataonnumberofyearsmathematicsfacultymembershavebeeninourdepartment.

NumberofYearsofService

Noticethataboxplotisnothingmorethanapictureofthefive-numbersummary.The“box”partshowsQ1,M,andQ3,andthe“arms”stretchoutthereachtheminandmaxoneitherside.Whenmakingaboxplot,besurethatyournumberlinehasaconsistentscale,thatyoulabelyourdisplay,andthatyouprovideinformationaboutthenumberofvalues(inthiscaseN=18)shownontheplot.

187


Realistsdonotfeartheresultsoftheirstudy.FyodorDostoevsky

The Common Core expects students to analyze and interpret data. Read the standards below. Which questions or activities have you done so far address each of these standards?


Homework

Ohwhocantelltherangeofjoyorsettheboundsofbeauty? SaraTeasdale


2) Answerallofthequestionsinthe“Mean,Mode&Range”probleminUnit4,Module4,

Session2intheBridgesinMathematicsGrade4StudentBook.Then,asafutureteacher,considerthefollowingquestions:Whymightachildanswer17inchesonquestion6?Whyachildanswer36inches?Whymightachildanswer42inches?Whymightachildanswer52inches?

CCSSGrade6:Developunderstandingofstatisticalvariability.2.Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.3.Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.

CCSSGrade6:Summarizeanddescribedistributions.

4.Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.5.Summarizenumericaldatasetsinrelationtotheircontext,suchasby:

C.Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.D.Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.

188

3) Consideragainthislinegraphof18scoresonaquiz:

X X

X X X X X X X

X X X X X X X X X 1 2 3 4 5 6 7 8 9 10

ScoreonaQuiz(outoften)

a) Whatistherangeofthesedata?b) WhatistheIQR?c) TrueorFalse?(Dothiswithoutperformingacalculation.)Themeanabsolute

deviationofthesedataisabout5.Explainyourthinking.d) Areeitherofthescores1or2anoutlier?Useourcriteriontocheck.e) Whatmeasureofcenterandwhatmeasureofspreadwouldyoureportforthese

dataandwhy?f) Makeaboxplotofthesedata.g) Howcouldyouusetheboxplottoquicklyspotoutliersinthedata?Explain.

4) Herearethosebowlingscoresonceagain:

112 114 120 123 127 127 130 167 220

a) Which(ifany)ofthesescoresareoutliersbasedonourcriterion?b) NowgototheintroductiontothistextandreadtheCommonCoreState

StandardsforGrade6.Noticethattheyaskthatchildren“summarizenumericaldatasetsinrelationtotheircontext.”Dothethingstheysuggestforthesetofbowlingscores.

i. Reportingthenumberofobservations.ii. Describingthenatureoftheattributeunderinvestigation,includinghow

itwasmeasuredanditsunitsofmeasurement.iii. Givingquantitativemeasuresofcenter(medianand/ormean)and

variability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.

iv. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.”

189

5) EditorsofEntertainmentWeeklyrankedeverysingleepisodeevermadeofStarTrek:TheNextGenerationfrombest(ranked#1)toworst(ranked#178).Thentheycompiledtherankingsbasedontheseasoninwhichtheepisodeaired.Belowareboxplotsshowingtherankingsofepisodesineachofthesevenseasons(fromWorkshopStatistics,p.89):

StarTrekRankingsplottedbySeason

a) Whichwasthebestoverallseasonandwhichwastheworstbasedontheserankings?Makeanargumentineachcase.

b) Doanyoftheseseasonshaverankingsthatareoutliers(forthatseason)?Explain.

c) Whichseason(s)hasadistributionofrankingsthatisskewedleft?Explain.d) Createaseriesofgoodquestionsyoucouldaskupperelementarystudents

abouttheseboxplots;thenanswerthemyourself.

6) Thisisameanabsolutedeviationcontest.Youmayuseonlynumbersintheset{1,2,3,4,5}(Youcanusethesamenumberasmanytimesasyoulike).

a) Selectfournumberswiththelowestmeanabsolutedeviation.(Actuallycompute

it).Arethosetheonlyfournumbersthatgivethelowestmeanabsolutedeviation?

b) Selectfournumberswiththehighestmeanabsolutedeviation.(Actuallycomputeit).Arethosetheonlyfournumbersthatgivethehighestmeanabsolutedeviation?

7) Whichislikelybigger?

190

a) Themeannew-housecostinSeattle,WAorthemediannew-housecost?Explain.

b) Themeanorthemedianinaskewed-leftdistribution?Explain.c) Therangeofasetofnumbersorthemeanabsolutedeviationoftheset?

Explain.d) Thesecondquartileorthemedian?Explain.e) Now,giveanexampleofasetof8numberswhosemeanis1000biggerthanits

medianorexplainwhyitisimpossibletodoso.

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ClassActivity30: TheMatchingGame

Letthepunishmentmatchtheoffense. Cicero

Onthefollowingpageyouwillfindbargraphsdisplayingdistributionsforeightdifferentsetsofexamscores.Thenumericalsummariesforthosedatasetsarelistedbelow.

DataSet Mean Median Mean

AbsoluteDeviation

A

44.5 44.0 7.7

B

37.0 36.0 4.0

C

32.5 31.5 12.0

D

40.0 41.5 25.5

E

11.5 3.0 20.0

F

40.0 38.5 11.0

G

33.5 34.5 10.0

H

20.5 20.5 10.0

1) Thesamenumberofstudentstookeachexam.Howmanywasthat?Explainhowyouknow.

2) Matcheachsummarywithadistribution.Makesuretowriteexplanationsforyourchoices.

3) Oneofthebargraphsismissing.Figureoutwhichdatasethasthemissingbargraph,thencreateapossiblebargraphforthatdataset.

4) Whatarethingsthatanupperelementarystudentcouldlearnbyworkingonthisactivity?Bespecific.

192

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ReadandStudy

Anapprenticecarpentermaywantonlyahammerandsaw,butamastercraftsmanemploysmanyprecisiontools.

RobertL.KruseInthissectionwearegoingtotalkspecificallyaboutsomewaystodisplaydataandaboutfeaturesofdatadistributions.First,adistributionofdataissimplyadisplaythatshowsthevaluesofthevaluesandtheirfrequencies(orrelativefrequencies).Oftenwhenwerefertoadistribution,wemeanthepatternofthedatawhenitismadeintoabargraphoralinegraphshowingvaluesofthedatasetonthehorizontalaxis,andeitherfrequencyofvaluesorrelativefrequency(percentsofvalues)ontheverticalaxis.D I S T R I B U T I O N S , S K E W N E S S , A N D S Y M M E T R Y

AllthreeofthedistributionsyouareabouttoseewerefoundattheQuantitativeEnvironmentalLearningProjectwebsite(http://www.seattlecentral.edu/qelp/Data_MathTopics.html).ForexamplethebargraphbelowshowsthedistributionofozoneconcentrationinSanDiegointhesummerof1998asmeasuredbytheCaliforniaAirResourcesBoard.TheEPAhasestablishedthestandardthatozonelevelsover0.12partspermillimeter(ppm)areunsafe.

Approximatelyhowmanydaysaredisplayedalltogether?InhowmanydayswastheozonelevelinSanDiegoclassifiedasunsafebytheEPA?Istheozonedistributionskewedrightorskewedleftorneither?Explain.

0

5

10

15

20

25

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 >0.13

Freq

uenceyinDays

Ozone(ppm)

Summer1998SanDiegoOzone

194

Nowtakeaminutetomakesenseofthereservoirdatabelow.

Abouthowmanyreservoirsareshownonthisbargraph?Whatpercentofreservoirsarefilledtoatleast90%capacity?Doyouexpectthemeanofthereservoirdatatobehigherthanthemedianortheotherwayaround?Explain.ThegraphsincludesreservoirsinbothOregonandArizona.WheredoyouthinkArizona’sreservoirsarelikelyappearingonthebargraph?Why?

0510152025303540

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

>100

>105

Freq

uency

PercentofCapacity

ReservoirLevels12/31/98

195

Finally,belowwehaveonemoregraphshowingafairlysymmetricdistributionofthenumberofhurricanesoccurringineachtwo-weekperiodbeginningJune1.

Onwhichdatesarehurricanesmostlikelybasedonthesedata?Inabargraphtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxiscouldrepresentvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights).Ifyouhavecollecteddataonthenumberofseatswonbyvariouspartiesinparliamentaryelections,youcouldmakeabargraphofthatdatabylistingpartiesalongthehorizontalaxisandmakingthebar-heightsrepresenteitherfrequencyorrelativefrequencyofyoursamples’responses.Thisbargraph(datafromwikipedia.org)showsboththeresultsofa2004election,andthoseof1999election.Takeamomenttomakesenseofthesedata.

020406080100120140160

2 4 6 8 10 12 14 16 18 20 22 24 26

Num

berofH

urric

ane

WeeksBeginningJune1

USAtlanticHurricanes:1851-2000

0

50

100

150

200

250

300

EUL PES EFA EDD ELDR EPP UEN Other

#ofSeats

Group

EuropeanParliamentElections

1999

2004

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Instructionalprogramsfromprekindergartenthroughgrade12shouldenablestudentstoformulatequestionsthatcanbeaddressedwithdataandcollect,organize,anddisplayrelevantdatatoanswerthem.

NationalCouncilofTeachersofMathematicsPrinciplesandStandardsforSchoolMathematics,p.176

Childreninelementaryschoollearntomakeandinterpretpictograms,frequencyplots(sometimescalledlinegraphs),bargraphs,andpiecharts.Onordertohelpyoudistinguishamongthesetypesofdisplays,let’sbeginwithanexampleofeach.ThedatabelowshowhowchildreninMr.Jensen’sclasstraveltoschool.

Student Transportation Student Transportation Student TransportationAbby Bus Sasha Foot Xia FootWinton Bicycle Lucy Bus Kim FootDexter Car Joe Car Ben CarTrevor Car Aiden Bus Leah CarAudrey Car Justin Foot Jeffery FootMavis Foot Campbell Car Queztal CarKate Foot Sonja Car Cheyenne BicycleLim Bicycle Jen Bus Steve Bicycle

First,hereisapictogramofthedata. TransportationPictogramforMr.Jensen’sClass

Bus

Bike

Car

Foot

=1childonfoot =1childdriventoschool =1biker =1busrider

197

Apictogramrepresentseachvalueasameaningfulicon.Thedisplaycanbeorientedeitherverticallyorhorizontally(ascanlineplotsandbargraphs).Noticehowthegraphiscarefullylabeledandthatakeyisprovidedfortheicons.Makesuretohelpchildrengetinthehabitoflabelinganddefiningtheirsymbols.Alsostressthatitisimportantthatalltheiconsbethesamesizeorthegraphcouldbemisleading.Apictogramcanbeunderstoodbychildrenasyoungasthoseinkindergarten.Afrequencyplot(sometimescalledalineplotifthehorizontalaxisisanumberline)ofthesamedataisshownbelow.Notethatthisplotisaprecursortothebargraphinthesensethatitdisplaysfrequencyofresponseinamoreabstractmannerthandoesthepictogram. TransportationFrequencyPlotforMr.Jensen’sClass X X X X X X X X X X X X X X X X X X X X X X X X Bus Bike Car FootCreateabargraphforthedatafromMr.Jensen’sclass.CreateapiegraphforthedatafromMr.Jensen’sclass.Explainhowyoudecidedhowbigtomakeeachsector.Itisimportantnotonlythatchildrenlearntodisplaydatabutalsothattheylearntomakesenseofchartsandgraphstheyfindinbooksandothermedia,andtobecriticalofmisleadingdisplays.Considertheexamplesbelow:

198

Amyclaimsthatthechildreninherkindergartenclasslikecatsmorethandogs.Shehasmadethefollowinggraphofherclassmates’petpreferencestosupportherclaim.Whyisthispictogrammisleading?Explain.Whatwouldyousaytothischild?

Childrenwholikecats

Childrenwholikedogs TuffTruckCompanyclaimsthattheirtrucksarethemostreliableandshowsthebelowgraphtomaketheircase.Whyisthisbargraphmisleading?Explain.(Bytheway,thisisareal-lifeexample.Onlythenameshavebeenchanged.)

Infact,criticalanalysisofdatadisplaysispartofthecurriculumforchildreninupperelementaryschool.

Perc entoftruc ks s tillontheroad,fiveyears afterpurc has e

93%

94%

95%

96%

97%

98%

99%

TuffTruck B randB B randC

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Homework

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butalackofwill. VinceLombardi

1) DoalloftheitalicizedthingsintheReadandStudysection.2) DoalltheproblemsintheConnectionssection.Whataretheexactpercentagesforthe

piechart?Howcanyoucomputetheanglestothenearestdegree?Doitandthenexplainyourwork.

3) HereisatableofdatashowingthepetsownedbychildreninMs.Green’sthirdgrade

class.

Student PetsintheHouseAnna NoneJames 1catand1dogCleo 3dogsViolet 1catAndre 6fishJose 7fish,1dog,2catsAidos NoneYvonne NoneTyrise 1hamsterEllen 2dogsTomas 1dogYani 3fishand2catsOwen NoneAudrey 1cat

a) Makeafrequencyplotofthesedatawithchildren’snamesonthehorizontalaxis.b) Makealineplotwith‘numberofpets’onthehorizontalaxis.c) Makeabargraphofthesedatawith‘numberofpets’onthehorizontalaxis.d) Makeapiechartofthesedata.e) Whatproblemsmightchildrenencounterintryingtorepresentthesedatainthe

aboveways?4) Answerthequestionsinthe“FavoriteBooks”probleminUnit2Module4Session1in

theGrade3BridgesinMathematicscurriculum.Howcanyouusethisproblemtodiscussthesimilaritiesanddifferencesbetweenpicturegraphsandbargraphs?

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5) Answerthequestionsinthe“GiftWrapFundraiser”probleminUnit2Module4Session2intheGrade3BridgesinMathematicscurriculum.Here’ssomeadditionalquestionstothinkaboutasafutureteacher:

a) Explainwhyachildmightgivetheanswerof“5”toquestion2abouthowmanystudentssold7rollsofgiftwrap.

b) Explainwhyachildmightgivetheanswerof“7”toquestion4abouthowmanyrollsofgiftwrapSarahsold.

c) Whyistheanswertoquestion5NOTthetotalnumberofXsonthelineplot?d) MakeafrequencyplotforthesamedatawhereeachXrepresentsarollofgift

wrap.Thenexplaintousethegraphtoanswerquestions1-5.

6) ThisgraphshowslengthsofthewingsofhousefliesfromtheQuantitativeEnvironmentalLearningProject.(OriginaldatafromSokal,R.R.andP.E.Hunter.1955.AmorphometricanalysisofDDT-resistantandnon-resistanthouseflystrainsAnn.Entomol.Soc.Amer.48:499-507)

a) Howwouldyoudescribetheshapeofthisdistributionintermsofskewnessand

symmetry?b) Approximatelyhowmanyfliesweremeasuredforthisstudy?Explain.c) TrueorFalse?Themeanofthesedataisequaltothemedian.Explain.d) TrueorFalse?About90%offlieshavewingssmallerthan51.5(×0.1mm).

Explain.e) Makeupagoodquestiontoaskanelementaryschoolchildaboutthisdataset.

7) Hereisagraphshowing100yearsofdataonWorldwideEarthquakesbiggerthan7.0on

theRichterScalefromtheQuantitativeEnvironmentalLearningProject(datafromtheUSGeologicalSurvey)Soforexample,therewere4yearswhenthenumberofglobalquakes(above7.0)wasabout35.

02468101214161820

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HouseflyWingLengths

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a) Whatdoesthisgraphtellyou?Bespecific.b) Estimatethemedianannualnumberofquakesabove7.0.c) Howwouldyoudescribethisshapeofthedistributionintermsofskewnessor

symmetry?Whatdoesthatshapetellyouaboutearthquakes?

8) Considerthisgraphmakingthecasethatwearehavinganational‘crimewave!’Whatdoyouthink?Doesthegraphprovidecompellingevidence?Explain.

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20

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WorldwideEarthquakes1900-1999

Reportedburg laries intheUnitedS tates ,2001-2006

S ource: S tatis ticalAbs tractoftheUnitedS tates ,2008

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2.13

2.14

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9) Astemandleafplot(orsimplystemplot)isalistingofallthedatatypicallyarrangedsothetensplacemakesthestemandtheonesarethe‘leaves.’Forexample,herearescoresonaGeometryFinalExamarrangedasastemplot:

ExamsScoresinGeometry

2|03|2,74|5|6,8,9,96|2,4,5,5,7,87|0,0,2,4,5,6,78|0,1,1,2,4,49|2,4,510|0

Wewouldreadthescoresas20,32,37,56,58,59,59,62etc.

a) Describetheshapeofthedistributionofscoresintermsofskewnessandsymmetry.b) Whatisthemedianscore?c) Makeaboxplotofthesedata.d) Givesomeexamplesofdatasetsforwhichastemplotwouldbeimpractical.

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SummaryofBigIdeas

Hey!What’sthebigidea?BugsBunny

• Random sampling helps reduce sample bias.

• There are two ways we can conceptualize the mean – as a balancing point and as a

redistribution.

• The mean, median, and mode are statistics that describe the center of data.

• A five-number summary, the IQR, and the mean absolute deviation all describe the spread of a data set.

• There are many different representations for data displays such as pictograms, line plots, and pie charts. You should choose the chart that best displays the data without misrepresenting the data.

204

References

Givecreditwherecreditisdue. Authorunknown

• Bauersfeld,H.(1995).Thestructuringofstructures:Developmentandfunctionof

mathematizingasasocialpractice.InL.Steffe&J.Gale(Eds.),ConstructivisminEducation(pp.137-158).Hillsdale,NJ:Erlbaum.

• BridgesinMathematicsusedthroughoutwithpermissionfromtheMathLearningCenter,http://www.mathlearningcenter.org/

• CommonCoreStateStandardsformathematicsasfoundin2012at©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.

• Dawson,L.(2004).TheSalkpoliovaccinetrialof1954:risks,randomizationandpublicinvolvement,ClinicalTrials,1,pp.122-130.

• EducationalDevelopmentCenter.(2007)ThinkMath!Orlando,FL.HarcourtSchool

Publishers.

• Halmos,P.(1985).IWanttobeaMathematician,Washington:MAASpectrum,1985.

• Gottfried,W.L.,NewEssaysConcerningHumanUnderstanding,IV,XII.

• Ma,L.(1999).KnowingandTeachingElementaryMathematics:Teachers’UnderstandingofFundamentalMathematicsinChinaandtheUnitedStates.Mahwah,N.J.:LawrenceErlbaum.

• MathematicalQuotationsServer(MQS)atmath.furman.edu.

• NationalCouncilofTeachersofMathematics.(2000).PrinciplesandStandardsfor

SchoolMathematics.Reston,VA:NCTM.

• Shulman,L.S.(1985).Onteachingproblemsolvingandsolvingtheproblemsofteaching.InE.A.Silver(Ed.),TeachingandLearningMathematicalProblemSolving:multipleresearchperspectives(pp.439-450).Hillsdale,NJ:Erlbaum.

• Steen,L.A.andD.J.Albers(eds.).(1981).TeachingTeachers,TeachingStudents,Boston:

Birkhïser.

205

• Theearlydevelopmentofmathematicalprobability.p.1293-1302ofCompanionEncyclopediaoftheHistoryandPhilosophyoftheMathematicalSciences,editedbyI.Grattan-Guinness.Routledge,London,1993.

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APPENDICES

207

Euclid’sPostulatesandPropositions

Euclid'sElementsThispresentationofElementsistheworkofJ.T.Poole,

DepartmentofMathematics,FurmanUniversity,Greenville,SC.©2002J.T.Poole.Allrightsreserved.

BookI

POSTULATES

Letthefollowingbepostulated:1.Todrawastraightlinefromanypointtoanypoint.2.Toproduceafinitestraightlinecontinuouslyinastraightline.3.Todescribeacirclewithanycenteranddistance.4.Thatallrightanglesareequaltooneanother.5.That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.

COMMONNOTIONS1.Thingswhichareequaltothesamethingarealsoequaltooneanother.2.Ifequalsbeaddedtoequals,thewholesareequal.3.Ifequalsbesubtractedfromequals,theremaindersareequal.4.Thingswhichcoincidewithoneanotherareequaltooneanother.5.Thewholeisgreaterthanthepart.

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BOOKIPROPOSITIONSProposition1.

Onagivenfinitestraightlinetoconstructanequilateraltriangle.Proposition2.

Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.Proposition3.

Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.

Proposition4.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhaveanglescontainedbytheequalstraightlinesequal,theywillalsohavethebaseequaltothebase,thetrianglewillbeequaltothetriangle,andtheremainingangleswillbeequaltotheremaininganglesrespectively,namelythosewhichtheequalsidessubtend.

Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.

Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.

Proposition7.Giventwostraightlinesconstructedonastraightline(fromitsextremities)andmeetinginapoint,therecannotbeconstructedonthesamestraightline(fromitsextremities),andonthesamesideofit,twootherstraightlinesmeetinginanotherpointandequaltotheformertworespectively,namelyeachtothatwhichhasthesameextremitywithit.

Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.

Proposition9.Tobisectagivenrectilinealangle.

Proposition10.Tobisectagivenfinitestraightline.

Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.

Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.

Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.

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Proposition14.Ifwithanystraightline,andatapointonit,twostraightlinesnotlyingonthesamesidemaketheadjacentanglesequaltotworightangles,thetwostraightlineswillbeinastraightlinewithoneanother.

Proposition15.Iftwostraightlinescutoneanother,theymaketheverticalanglesequaltooneanother.

Proposition16.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisgreaterthaneitheroftheinteriorandoppositeangles.

Proposition17.Inatriangletwoanglestakentogetherinanymannerarelessthantworightangles.

Proposition18.Inanytrianglethegreatersidesubtendsthegreaterangle.

Proposition19.Inanytrianglethegreaterangleissubtendedbythegreaterside.

Proposition20.Inanytriangletwosidestakentogetherinanymanneraregreaterthantheremainingone.

Proposition21.Ifononeofthesidesofatriangle,fromitsextremities,therebeconstructedtwostraightlinesmeetingwithinthetriangle,thestraightlinessoconstructedwillbelessthantheremainingtwosidesofthetriangle,butwillcontainagreaterangle.

Proposition22.Outofthreestraightlines,whichareequaltothreegivenstraightlines,toconstructatriangle:thusitisnecessarythattwoofthestraightlinestakentogetherinanymannershouldbegreaterthantheremainingone.[I.20]

Proposition23.Onagivenstraightlineandatapointonittoconstructarectilinealangleequaltoagivenrectilinealangle.

Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.

Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.

Proposition26.Iftwotriangleshavethetwoanglesequaltotwoanglesrespectively,andonesideequaltooneside,namely,eitherthesideadjoiningtheequalangles,ofthatsubtendingoneoftheequalangles,theywillalsohavetheremainingsidesequaltotheremainingsidesandtheremainingangletotheremainingangle.

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Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.

Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.

Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.

Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.

Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.

Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.

Proposition33.Thestraightlinesjoiningequalandparallelstraightlines(attheextremitieswhichare)inthesamedirections(respectively)arethemselvesalsoequalandparallel.

Proposition34.

Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.

Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.

Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.

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Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.

Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.

Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.

Proposition46.Onagivenstraightlinetodescribeasquare.

Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.

Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.

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Glossary"WhenIuseaword,"HumptyDumptysaid,inaratherscornfultone,"itmeansjust

whatIchooseittomean-neithermorenorless.""Thequestionis,"saidAlice,"whetheryoucanmakewordsmeansomanydifferent

things.""Thequestionis,"saidHumptyDumpty,"whichistobemaster-that'sall."

LewisCarroll,ThroughtheLookingGlass Acuteangle–ananglethatmeasureslessthan90degreesAcutetriangle–atrianglewiththreeacuteanglesAdjacentangles–twonon-overlappinganglesthatshareavertexandacommonrayAlternateexteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines

thatlieoutsidethelinesandonoppositesidesofthetransversalAlternateinteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines

thatliebetweenthelinesandonoppositesidesofthetransversalAngle–thefigureformedbytworayswithacommonendpointAnglebisector–thelinethroughthevertexofananglethatdividestheangleintotwo

congruentanglesApex(ofapyramid)–thecommonpointofthenon-basefacesofapyramidApex(ofacone)–thecommonpointofthelinesegmentsthatcreateaconeArc–thesetofpointsonacirclebetweentwogivenpointsofthecircle(Thereareactuallytwo

arcsbetweenanytwogivenpoints;theshorteroneiscalledtheminorarcandthelongeroneiscalledthemajorarc.)

Area–thequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigureAttribute–apropertyofageometricobjectthatcanbemeasured(suchaslength)or

categorized(suchascolor)Axiom–astatementthatweagreetoacceptastruewithoutproofAxiomaticsystem–asetofundefinedterms,definitions,axioms,andtheoremsthatcreatea

mathematicalstructure

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Axis(ofacone)–thelinejoiningtheapextothecenterofthe(circle)baseAxisofsymmetry–alineinspacearoundwhichathree-dimensionalobjectisrotatedBargraph:adatadisplayinwhichtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxisrepresentsvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights),groupedintointervals.Baseangles(ofanisoscelestriangle)–theanglesthatareoppositethecongruentsidesofan

isoscelestriangleBilateralsymmetry–anobjecthasbilateralsymmetrywhenithasexactlyonelineof

reflectionalsymmetryBisect–todivideageometricobjectsuchasalinesegmentoranangleintotwocongruent

piecesBoundary–thesetofpointsthatseparatetheinsideofaclosedplanarobjectfromtheoutsideCategoricalData:Dataforwhichitmakessensetoplaceanindividualobservationintooneofseveralgroups(orcategories).Census:Anydatacollectionmethodinwhichdataiscollectedfromeachandeverymemberofthepopulation.Center(ofacircle)–thepointthatisequidistantfromallpointsonthecircleCentralangle–ananglewhosevertexisacenterofageometricobjectChord–alinesegmentwhoseendpointsaredistinctpointsonagivencircleChanceError:Errorinsamplingduesimplytothefactthatasampleisnotexactlythesameasapopulationduetochancealone.Circle–thesetofallpointsintheplanethatarethesamedistancefromagivenpoint,called

thecenterCircumference–thecircumferenceofacircleisitperimeterCircumscribedcircle–thecirclethatcontainsalltheverticesofapolygonClosedcurve–acurvethatstartsandstopsatthesamepoint

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Clustersampling:Randomsamplinginstages.Coincide–twoobjectsaresaidtocoincideiftheycorrespondexactly(areidentical)Collinearpoints–pointsthatlieonthesamelineCompass–aninstrumentusedtoconstructacircleComplementaryangles–twoangleswhosemeasuressumto90degreesConcavepolygon–apolygonforwhichatleastonediagonalliesoutsidethepolygonConcurrentlines–threeormorelinesthatintersectinthesamepointCone(circular)-athree-dimensionalgeometricobjectconsistingofalllinesegmentsjoininga

singlepoint(calledtheapex)toeverypointofacircle(calledthebase)Congruentobjects–twogeometricobjectsthatcoincidewhensuperimposedConjecture–aguessorahypothesisContrapositive(of“IfA,thenB.”)–“IfnotB,thennotA,”whereAandBarestatementsConverse(of“IfA,thenB.”)–“IfB,thenA,”whereAandBarestatementsConvexpolygon–apolygonallofwhosediagonalslieinsidethepolygonCoordinateplane–aplaneonwhichpointsaredescribedbasedontheirhorizontalandvertical

distancesfromapointcalledtheoriginCoplanarlines–linesthatlieinthesameplaneCorrespondingangles-twoanglesformedbyatransversalofapairoflinesthatlieonthe

samesideofthetransversalandalsolieonthesamesideofthepairoflinesCorrespondingpoints–apairofpoints,oneofwhichistheoriginalpointandtheotherof

whichistheimageofthatpointunderatransformationCounterexample–anexamplethatdemonstratesthatastatement(conjecture)isfalseCurve–asetofpointsdrawnwithasinglecontinuousmotion

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Cylinder(circular)–athree-dimensionalgeometricobjectconsistingoftwoparallelandcongruentcircles(andtheirinteriors)andtheparallellinesegmentsthatjoincorrespondingpointsonthecircles

Data:AnyinformationcollectedfromasampleDecagon–apolygonwithexactlytensidesDeductivereasoning–theprocessofcomingtoaconclusionbasedonlogicDegree–aunitofanglemeasureforwhichafullturnaboutapointequals360degreesDiagonal–thelinesegmentjoiningtwonon-adjacentverticesofapolygonDiameter–alinesegmentthroughthecenterofacirclewhoseendpointslieonthecircleDilation(withcenterPandscalefactork>0)–amotionoftheplaneinwhichtheimageofPis

PandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.

Dimension(ofarealspace)–thenumberofmutuallyperpendiculardirectionsneededto

describethelocationofthesetofpointsinthatspaceDistance(onacoordinateplane)–thesizeoftheportionofastraightlinethatliesbetweenthe

twopointsonthecoordinateplaneasmeasuredbythedistanceformula:22 bad += ,whereaisthehorizontaldistancebetweenthepoints(asmeasuredon

thex-axis)andbistheverticaldistance(asmeasuredonthey-axis)Distinct(geometricobjects)-twoobjectsthatdonotsharealltheirpointsincommonDistributionofdata:adisplaythatshowsthevaluesthedatatakesalongwiththefrequency(orrelativefrequency)ofthosevaluesDisjoint(events):TwoeventsaredisjointiftheyhavenooutcomesincommonDodecagon–apolygonwithexactlytwelvesidesEdge–thelinesegment(side)thatissharedbytwofacesofapolyhedronEndpoint(ofalinesegment)–oneoftwopointsthatdeterminesalinesegmentEquiangularpolygon–apolygonallofwhosevertexanglesarecongruent

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Equallylikelyoutcomes:Alltheoutcomesofarandomexperimentaresaidtobeequallylikelyif,inthelongrun,theyalloccurwiththesamefrequency(theyallhavethesameprobabilityofoccurring)

Equilateralpolygon–apolygonallofwhosesidesarecongruentEvent:anysubsetofthesamplespaceofarandomexperiment Example(ofadefinition)–ageometricobjectthatsatisfiestheconditionsofthedefinitionExperimentalprobabilitiesareprobabilitiesthatarebasedondatacollectedbyconducting

experiments,playinggames,orresearchingstatisticsExteriorangle–theangleformedbyasideofapolygonandtheextensionofanadjacentsideFace–apolygon(withinterior)thatformsaportionofthetwo-dimensionalsurfaceofa

polyhedronFirstquartile(Q1):themedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorderFiveNumberSummary:ofnumericaldataconsistsofthefollowingfivestatistics:Minimumvalue,Q1,Median,Q3,MaximumvalueFixedpoint–apointwhoselocationremainsthesameunderatransformationGeneralization–theextensionofastatement(aboutapattern)thatistrueforspecificvalues

ofn(anaturalnumber)toastatement(aboutthatpattern)thatistrueforallvaluesofnGeoboard–amanipulativetypicallycomposedofaboardwith25pegsarrangedina5x5

squarearrayGeoboardpolygon–apolygonwhoseverticesareallpoints(pegs)onageoboardHeight(ofatriangle)–lengthofthelinesegmentfromavertexperpendiculartotheopposite

side.ThislinesegmentisoftencalledthealtitudeofthetriangleHeptagon–apolygonwithexactlysevensidesHexagon–apolygonwithexactlysixsidesHistogram:Adatadisplayinwhichtheverticalaxisrepresentsarateandthehorizontalaxisisactuallythex-axis-inotherwords,itisacontinuouspieceoftherealnumberlineandassuchitcontainsallrealnumbersinaninterval.Valuesofthevariablearebrokenintointerval

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‘classes.’(Note:manytextsusetermshistogramandbargraphinterchangeably.Wewillnotdosointhistext.)Homogeneous(verticesinatessellation)–verticesthathaveexactlythesamepolygons

meetinginexactlythesamearrangementHomogeneous(verticesinapolyhedron)–verticesthathaveexactlythesamepolygonfaces

meetinginexactlythesamearrangementHypotenuse–thesideofarighttriangleoppositetherightangleImage(ofatransformation)–thesetofpointsthatresultfromthemotionofanobjectbya

translation,arotation,orareflectionIndependent(events):Twoeventsaresaidtobeindependentifknowingthatoneoccurs(orknowingitdoesnotoccur)doesnotchangetheprobabilityoftheotheroccurringInductivereasoning–theinformalprocessofcomingtoaconclusionbasedonexamplesInscribedpolygon–thepolygoninsideacirclewhoseverticesalllieonthecircleInteriorangle–anyoneofthealternateinterioranglesformedbyatransversaltotwolinesInter-QuartileRange(IQR)=Q3–Q1.ItisthespreadofthemiddlehalfofthedataIntersection(oftwolines)–thepointthelineshaveincommonIntersection(oftwosets)–thesetofelementsthatarecommontobothsetsIsosceles–havingatleastonepairofcongruentsidesJustification–anargumentbasedonaxioms,definitions,andpreviouslyprovenresultstoshow

thataconjectureistrueLawoflargenumbers:Inrepeated,independenttrialsofarandomexperiment,asnumberoftrialsincreases,theexperimentalprobabilitiesobservedwillconvergetothetheoreticalprobability. Inotherwords,theaverageoftheresultsobtainedfromalargenumberoftrialsshouldbeclosetotheexpectedvalue,andwilltendtobecomecloserasmoretrialsareperformed.Leg–asideofarighttriangleoppositeanacuteangleLength–thedistancebetweentwopointsonaone-dimensionalcurve

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Line–anundefinedone-dimensionalsetofpointsunderstoodcovertheshortestdistanceandtoextendinoppositedirectionsindefinitely

Lineofreflection–thelineaboutwhichanobjectisreflectedtoformitsmirrorimageLinesegment–thesetofallpointsonalinebetweentwogivenpointscalledtheendpointsMass–aconceptofphysicsthatcorrespondstotheintuitiveideaof“howmuchmatterthereis

inanobject;”unlikeweight,themassofanobjectdoesnotdependupontheobject’slocationintheuniverse

Mean:Themean(oraverage)valueofasetofnumericdataisthesumofallthevaluesdividedbythenumberofvaluesMeanAbsoluteDeviation(MAD):MAD=!

-𝑥! − 𝑥 + 𝑥" − 𝑥 + 𝑥% − 𝑥 +⋯+ 𝑥- − 𝑥 ,

whichgivestheaveragedistancebetweenthemeanandeachofthendatapoints.Measure–todeterminethequantityofanattribute(orofafundamentalconceptsuchastime)

usingagivenunitMedian:Themedianofasetofnumericvaluesisthemiddlevaluewhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”values,themedianistheaverageofthetwoMetricsystemofmeasurement–thesystemofmeasurementunitsinwhichthereisone

fundamentalunitdefinedforeachquantity(attribute)withmultiplesandfractionsoftheseunitsestablishedbyprefixesbasedonpowersoften

Midpoint–thepointonalinesegmentthatdividesitintotwocongruentlinesegmentsMode:thevaluethatoccursmostfrequentlyModel–arepresentationofanaxiomsysteminwhicheachundefinedtermisgivenaconcrete

interpretationinsuchawaythattheaxiomsallholdMultiplicationprinciple:IfyouhaveAdifferentwaysofdoingonethingandBdifferentwaysofdoinganother(andonechoicedoesnotaffecttheother)thenthetotalnumberofwaystodobothAandBisA×BNet–atwo-dimensionalfigurethatcanbefoldedintoathree-dimensionalobjectNonagon–apolygonwithexactlyninesides

219

Noncollinear(points)–asetofpointsnotallofwhichlieonthesamelineNon-example(ofadefinition)–anexamplethatdemonstratemostoftheconditionsofa

definitionbutthatfailstosatisfyatleastoneconditionNonresponsebias:BiasedinformationthatresultsfromthefactthatsomegroupsinthesampleweremorelikelytorespondthanothersNon-standardunitofmeasure–aunitofmeasurewhosevalueisnotestablishedbyreference

toanacceptedstandard;forexample,ablockNumericalData:numberdataforwhichitmakessensetoperformarithmeticoperations(suchasaveraging).Obliqueprism(pyramid,cylinder)–aprism(pyramid,cylinder)thatisnotrightObtuseangle–ananglewithmeasuregreaterthan90degreesObtusetriangle–atrianglewithoneobtuseangleOctagon–apolygonwithexactlyeightsidesOrder(ofarotationalsymmetry)–thenumberofdifferentrotationsthatareasymmetryofan

objectOrientation–thedirection,clockwiseorcounterclockwise,ofthereadingoftheverticesofa

polygoninalphabeticalorderOutcome:Anyonethingthatcouldhappeninarandomexperiment.Outlier:aspecificvaluethatlieswelloutsidetheoverallpatternofthedata.Parallellines–coplanarlineswithnopointsincommonParallelogram–aquadrilateralinwhichbothpairsofoppositesidesareparallelParameter:Somenumericalinformationthatisdesiredfromanentirepopulation.Usuallyaparameterisanunknownvaluethatisestimatedbyastatistic.Partition–adivisionofageometricobjectintoasetofnon-overlappingobjectswhoseunionis

theoriginalobjectPentagon–apolygonwithexactlyfivesides

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Percentile:Ascoreinthenthpercentilemeansthat“npercent”ofthepeoplewhotookthetestscoredatorbelowthatscorePerimeter(ofaplaneobject)–thelengthoftheboundaryoftheobjectPerpendicularbisector–thelinethroughthemidpointofalinesegmentthatisalso

perpendiculartothelinesegmentPerpendicularlines–twolinesthatintersecttoformfourrightanglesPi(p)–theexactnumberoftimesthediameterofacirclefitsintoitscircumference(orthe

ratioofthecircumferenceofacircletoitsdiameter);thisratioisanirrationalnumberthatisconstantforallsizecirclesandisapproximatelyequalto3.14159

Planarcurve–acurvethatliesentirelywithinaplanePlane–anundefinedtwo-dimensionalsetofpointsunderstoodtobe“flat”andtoextendinall

directionsindefinitelyPlaneofsymmetry–aplaneinspaceaboutwhichathree-dimensionalobjectisreflectedPlatonicsolid–aregularpolyhedronplusitsinteriorPoint–anundefinedzero-dimensionalobject;alocationwithnosizePolygon–afinitesetoflinesegmentsthatformasimpleclosedplanarcurvePolyhedron(plural:polyhedra)–afinitesetofpolygon-shapedfacesjoinedpairwisealongthe

edgesofthepolygonstoencloseafiniteregionofspacewithinonechamberPopulation:thelargestgroupaboutwhichtheresearcherorsurveyorwouldlikeinformationPrism–apolyhedroninwhichtwoofthefaces(calledthebases)arecongruentandlie on parallel

(non-intersecting) planes andtheremainingfacesareparallelogramsProbability:referstotheproportionoftimesaneventwouldoccuriftherandomexperimentwasperformedaverylargenumberoftimesProof–adeductiveargumentthatestablishesthetruthofaclaimProtractor–aninstrumentusedtomeasureanglesPyramid–apolyhedroninwhichonefaceiscalledthebaseandalloftheremainingfacesare

trianglesthatshareacommonvertex(calledtheapex)

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Pythagoreantriple–threepositiveintegersthatsatisfythePythagoreantheoremQuadrilateral–apolygonwithexactlyfoursidesQuantifier(inlogic)–awordorphrase(suchas“all”or“atleastone”)thatindicatesthesizeof

thesettowhichthestatementappliesRadius(plural:radii)–thelinesegmentjoiningapointonacircletothecenterofthecircleRandomExperiment:anyexperimentwheretheoutcomedependsonchanceandcannotbeknownbeforehand.Whileinarandomexperiment,thespecificoutcomecannotbeknown,thereisnonethelessaregulardistributionofoutcomesafteraverylargenumberoftrials.Range=maximumvalue–minimumvalueRay–thesetofpointsonalinebeginningatagivenpoint(calledtheendpoint)andextending

inonedirectiononthelinefromthatpointRectangle–aquadrilateralwithfourrightanglesReflection–(inalinel)isatransformationoftheplaneinwhichtheimageofapointPonlisP,

andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA

Reflectionsymmetry(2-dimensional)–areflectioninalineinwhichtheimageoftheobject

coincideswiththeoriginalobjectReflectionsymmetry(3-dimensional)–areflectioninaplaneinwhichtheimageoftheobject

coincideswiththeoriginalobjectRegularpolygon–apolygonwithallsidescongruentandallvertexanglescongruentRegularpolyhedron–apolyhedronwhosefacesareeachthesameregularpolygonwiththe

samenumberoffacesmeetingateachvertexRegulartessellation–atessellationthatcontainsonlyoneregularpolygonResponseRate:Theratioofthenumberofrespondents(peoplewhoactuallytookpartinthestudy)tothenumberwhowereinvitedtoparticipate.Responsevariable:avariablewhosevariationisexplainedbyanothervariable(calledanexplanatoryvariable).

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Rhombus(plural:rhombi)–aquadrilateralwithfourcongruentsidesRightangle–ananglethatisexactlyonefourthofacompleteturnaboutapointRightprism(pyramid,cylinder)–aprism(pyramid,cylinder)inwhichthelinejoiningthe

centersofthebases(theapexofthepyramidtothecenterofitsbase)isperpendiculartothebase

Righttriangle–atrianglewithonerightangleRigidmotions-transformationsoftheplanethatpreservedistancesbetweenpoints(theydo

notdistorttheshapeorsizeofobjects)Rotation(aboutapointPthroughanangleq)–atransformationoftheplaneinwhichthe

imageofPisPand,iftheimageofAis 'A ,then PA@ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation

Rotationsymmetry(2-dimensional)–arotationaboutapointinwhichtheimagecoincides

withtheoriginalobjectRotationsymmetry(3-dimensional)–arotationaboutanaxisofsymmetryinwhichtheimage

coincideswiththeoriginalobjectSample:asubsetofapopulationfromwhichdataiscollectedSamplebias:Biased(inaccurate)informationthatresultsfromapoorlychosensample.Samplingerror:thedifferencebetweenthevalueofaparameterandthevalueofthestatisticthatrepresentsit.Samplingerrorcanincludechanceerror,samplingbias,andnonresponsebias.Samplingrate:[(#inthesample)÷(#inthepopulation)]Samplespace:ThesamplespaceofarandomexperimentisthesetofallpossibleoutcomesScalefactor,avaluebywhichwemultiplyeachoriginaldimensiontofindthenewlengthsScalenetriangle–atrianglenoneofwhosesidesarecongruentScaling–atransformationoftheplanethatcauseseitheramagnificationorashrinkingofan

objectinwhichtheimageremainssimilartotheoriginalobjectScatterplot:Adisplaythatshowstherelationshipbetweentwonumericalvariablesmeasured

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onthesamesampleofindividuals.Thevaluesofonevariableareshownonthehorizontalaxis,andvaluesoftheothervariableareshownontheverticalaxis.Eachindividualisplottedasapointrepresentinganorderedpair(variable1,variable2).

Secant–alinethatintersectsacircleintwodistinctpointsSector–theportionofacircleanditsinteriorbetweentworadiiSelf-selectedsample:asampleinwhichrespondentsvolunteeredtoparticipate.Semiregularpolyhedron–apolyhedronthatcontainstwoormoreregularpolygonsasfaces

whicharearrangedsothatallverticesarehomogeneousSemiregulartessellation–atessellationthatcontainstwoormoreregularpolygonsarranged

sothattheverticesarehomogeneousShearing–atransformationoftheplanethatchangestheshapeofanobjectSide–oneofthelinesegmentsthatmakeupapolygonSimilarobjects–objectswhereonecanbeobtainedfromtheotherbycomposingarigid

motionwithadilationSimplecurve–acurvethatdoesnotintersectitselfSimpleRandomSample(SRS):AsampleinwhicheverysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesizeSkewedleft:AdistributionofdataisskewedleftifithasanasymmetrictailtotheleftSkewedright:AdistributionofdataisskewedrightifithasanasymmetrictailtotherightSlope(ofaline)–theverticaldistancerequiredtostayonalineforaoneunitchangein

horizontaldistanceSpace–anundefinedtermthatdenotesthesetofpointsthatextendsindefinitelyinthree

dimensionsSphere–thesetofpointsin(three-dimensional)spacethatareequidistantfromagivenpoint,

calledthecenterSquare–aquadrilateralwithfourrightanglesandfourcongruentsides

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Standardunitofmeasure–aunitofmeasurewhosevalueisestablishedbyreferencetoanacceptedstandard;forexample,themeterisdefinedtobeoneten-millionthofthedistancefromtheequatortothenorthpole

Straightangle–ananglethatmeasureshalfaturn(ortworightangles)Straightedge–aninstrumentusedtoconstructlinesegmentsStatistic:Anynumericalinformationcomputedfromasample.(Forexample,ameancalculatedfromasampleisastatistic,whichcanbeusedasanestimateforthepopulationmean,whichwouldbeaparameterStemandleafplot(orsimplystemplot):alistingofallthedatatypicallyarrangedsothetensplacemakesthestemandtheonesarethe‘leaves.’Supplementaryangles–twoangleswhosemeasuressumto180degreesSurface–thesetofpointsthatformtheboundaryofasolidthree-dimensionalobjectSurfacearea–thesumoftheareasofthefacesofaclosed3-dimensionalobjectSurvey:Anydatacollectionmethodinwhichdataiscollectedfromjustasubsetofthe

populationSymmetricDistribution:Adistributionthatissymmetricaboutthemedian.Symmetry(ofanobject)–atransformationoftheobjectinwhichtheimagecoincideswiththe

originalTangent(toacircle)–alinethatintersectsacircleinexactlyonepoint Tessellation–anarrangementofpolygonsthatcanbeextendedinalldirectionstocoverthe

planewithnogapsandnooverlapsinsuchawaythatverticesonlymeetotherverticesTheorem–astatementthathasbeenproventrueTheoreticalProbabilities:probabilitiesassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieorcoinorbagofcandies).Thirdquartile(Q3):themedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder.Tiling–anarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplane

withnogapsandnooverlaps

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Time–ameasurablepartofthefundamentalstructureoftheuniverse,adimensioninwhicheventsoccurinsequence

Transformation–amovementofthepointsofaplanethatmaychangethepositionorthesize

andshapeofobjectsTranslation(byavectorAA’)–arigidmotionoftheplanethattakesAtoA’,andforallother

pointsPontheplane,PgoestoP’wherevectorPP’andvectorAA’havethesamelengthanddirection

Translationsymmetry–atranslationoftheplanesuchthattheimagecorrespondstothe

originalobjectTranslationvector–anarrowthatgivesthedirectionanddistance(itslength)thatapointis

movedduringatranslationTransversal–alinewhichintersectstwoormorelinesTrapezoid–aquadrilateralwithexactlyonepairofparallelsidesTriangle–apolygonwiththreesidesTrivialrotation–therotationof360°;itisarotationalsymmetryofeveryobjectUndefinedterm–atermwhichhasanintuitivemeaning,butnoformaldefinitionUnion(ofsets)–thesetcontainingeveryelementofeachsetUnitofmeasure–anobjectisusedforcomparisonwithanattributeUnitsquare–asquarethatisoneunitbyoneunitandthushasanareaofonesquareunitUnitcube–acubethatisoneunitbyoneunitbyoneunitandhasavolumeofonecubicunitVenndiagram–apictureinwhichtheobjectsbeingstudiedarerepresentedaspointsona

planeandsimpleclosedcurvesaredrawntogroupthepointsintodifferentclassifications.Venndiagramsareusedtovisualizerelationshipsamongsetsofobjects.

Vertex(plural:vertices)–thecommonendpointoftwoadjacentsidesofapolygonVertexangle–theangleformedbyadjacentsidesofapolygonVertex(ofapolyhedron)–theintersectionoftwoormoreedgesofapolyhedron

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Verticalangles–anonadjacentpairofanglesformedbytwointersectinglinesVolume–ameasureofthecapacityofa3-dimensionalobjector,alternatively,thequantityof

spaceenclosedbya3-dimensionalobjectWeight–ameasureoftheforceofgravityonanobject;oftenusedinterchangeablywithmass;

differencesinthemeasuresofweightandmassarenegligibleatsealevelonearth

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GraphPaper

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Tangrams

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PatternBlocks

big ideas geometry and data analysis · we will be giving you other definitions of geometry as we...

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