Big Ideas in Geometry and Measurement · 2018-06-19 · triangle or a circle, but it won’t have...

BigIdeasinMathematics

forFutureElementaryTeachers

BigIdeasinGeometryandMeasurement

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

(UpdatedSummer2017)

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ThisworkislicensedundertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0InternationalLicense.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by-nc-nd/4.0/orsendalettertoCreativeCommons,POBox1866,MountainView,CA94042,USA.

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DearFutureTeacher,Wewrotethisbooktohelpyoutoseethestructurethatunderlieselementarymathematics,togiveyouexperiencesreallydoingmathematics,andtoshowyouhowchildrenthinkandlearn.Wefullyintendthiscoursetotransformyourrelationshipwithmath.Asteachersoffutureelementaryteachers,wecreatedorgatheredtheactivitiesforthistext,andthenwetriedthemoutwithourownstudentsandmodifiedthembasedontheirsuggestionsandinsights.Weknowthatsomeoftheproblemsaretough–youwillgetstucksometimes.Pleasedon’tletthatdiscourageyou.There’smuchvalueinwrestlingwithanidea. Allourbest,

John,Jason,Eric,Amy,Carol&Jen

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Hey!Readthis.Itwillhelpyouunderstandthebook. Theonlywaytolearnmathematicsistodomathematics. PaulHalmosThisbookwaswrittentopreparefutureelementaryteachersforthemathematicalworkofteaching.Thefocusofthismoduleisgeometry–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? Whatisgeometryabout?Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Weviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjustasimportantasanycontentyouteach.Whenyoustandbeforeyourclass,youarearepresentativeofthemathematicalcommunity;wewillhelpyoutobeagoodone.

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Noonecandothisthinkingforyou.Mathematicsisn’tasubjectyoucanmemorize;itisaboutwaysofthinkingandknowing.Youneedtodoexamples,gatherdata,lookforpatterns,experiment,drawpictures,think,tryagain,makearguments,andthinksomemore.Thebigideasofgeometryarenotalwayseasy–buttheyarefundamentallyimportantforyourstudentstounderstandandsotheyarefundamentallyimportantforyoutounderstand.EachsectionofthisbookbeginswithaClassActivity.Theactivityisdesignedforsmall-groupworkinclass.Someactivitiesmaytakeyourclassaslittleas20minutestocompleteanddiscuss.Othersmaytakeyoutwoormoreclassperiods.TheReadandStudy,ConnectionstotheElementaryGrades,andHomeworksectionsarepresentedwithinthecontextoftheactivityideas.Nosolutionsareprovidedtoactivitiesorhomeworkproblems–youwillhavetosolvethemyourselves.ThemathematicscontentinthisbookpreparesyoutoteachtheCommonCoreStateStandardsforMathematicsforgradesK-8.Thesearethestandardsthatyouwilllikelyfollowwhenyouareanelementaryteacher,sowewillhighlightaspectsofthemthroughoutthetext.Inorderforyoutoseehowthemathematicalworkyouaredoingappearsintheelementarygrades,wehavemadeexplicitconnectionstoBridgesinMathematicsfromTheMathLearningCenter.ThisistheonlineelementarygradesmathematicscurriculumadoptedbytheOshkoshAreaSchoolDistrict.Youwilloftenbeaskedtogotothesitebridges.mathlearningcenter.orgtoreadordoproblems.Yourinstructorwillprovideyouwithacodesothatyoucanaccessthesematerials.

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BigIdeasinGeometryandMeasurementTableofContents

Tobeateacherrequiresextensiveandhighlyorganizedbodiesofknowledge.Shulman,1985,p.47

Chapter1:SEEINGTHEWORLDGEOMETRICALLYClassActivity1:TrianglePuzzle………….……………….……………………….……………………………….…p.10 WhatisGeometry?

NatureofMathematicalObjectsMathematicalCommunication

ClassActivity2:DefiningMoments……….…………………………..…………..…………………………......p.17 RoleofDefinitions Lines,Segments,RaysandPolygons ParallelandPerpendicularLinesClassActivity3:GetitStraight…………….…..……………………………….……....…………………………..p.26 LanguageofMathematics

IdeaofAxioms DeductiveversusInductiveReasoning MakingConvincingMathematicalArguments ClassActivity4:AlltheAngles……………………………………………………………………………………..…p.40 MeasuringAnglesinDegrees

VertexAngleSumsRegularPolygons

ClassActivity5:ALogicalInterlude……….………………………………………………………..…….…..……p.46 ConverseandContrapositive ClassifyingQuadrilaterals ClassActivity6:EnoughisEnough...……………………………………….…...…..………………………….…p.50 Congruence TriangleCongruencetheoremsSummaryofBigIdeasfromChapterOne………………...………………………………………………………p.56

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Chapter2:TRANSFORMATIONS,TESSELLATIONS,ANDSYMMETRIESClassActivity7:Slides…………………….……………………………………….…………..………………………….p.60 TranslationsClassActivity8:Turn,Turn,Turn…………………………………………………………..…………………......…p.62 Rotations

ClassActivity9:ReflectingonReflection.………………..……………………………………...……………..p.67 ReflectionsClassActivity10:Zoom…………………………………………………………………………………………………….p.72 Similarity Similarpolygons

ClassActivity11:SearchingforSymmetry……………………………………...……............................p.75 SymmetriesinthePlaneClassActivity12:Tessellations……………………...………….………………………………….………………….p.79 TrianglesandQuadrilaterals

RegularTessellations

SummaryofBigIdeasfromChapterTwo………………………………………………………………………...p.85Chapter3:MEASUREMENTINTHEPLANEClassActivity13:MeasureforMeasure.…………………....……………………………………..….….…….p.87 StandardandNon-StandardUnits ApproximationandPrecisionClassActivity14A:Triangulating……………………………………………………………………….…………….p.93ClassActivity14:AreaEstimation…………………………………………………………………………………...p.94 IdeaofArea CoveringwithUnitSquares ClassActivity15:FindingFormulas………..………………………………………………………………...…..p.100

MakingSenseofAreaFormulasRectangles,Parallelograms,Triangles,andTrapezoids

ClassActivity16:TheRoundUp…………………………………………...…...………...........................p.105 Whatisp? CircleArea InscribedPolygons

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ClassActivity17:PlayingPythagoras……….…………………….……...…………………………….…….…p.110 ProofsofthePythagoreanTheorem GeometricandAlgebraicRepresentationsClassActivity18:Coordination………………………………………………………………………………...……p.115 GeoboardArea CoordinateGeometry UsingthePythogoreanTheoremSummaryofBigIdeasfromChapterThree…………………………….………………………………...…...p.121

Chapter4:THETHIRDDIMENSIONClassActivity19:StrictlyPlatonic(Solids)..………………………………………………………………......p.123 RegularPolyhedra SpatialReasoningClassActivity20:PyramidsandPrisms………………………………………………………………….…..….p.130 CountingVertices,Edges,andFaces ClassActivity21:SurfaceArea…………..…………………………….…………………………………………….p.131 IdeasofSurfaceArea ConstantVolume–ChangingSurfaceArea ClassActivity22:NothingbutNet…………………………………………………………………………….…..p.135 NetsforCylindersandPyramids ClassActivity23:BuildingBlocks……………………………………………………………………………………p.136 IdeasofVolume VolumesofPrismsandCylinders Scalingin3DimensionsClassActivity24:VolumeDiscount…………………..……………..………………………….…………..…….p.140 Capacity

VolumesofPyramids,Cones,&SpheresClassActivity25:VolumeChallenge…………………………………………………………….………….……p.147 BuildingModelstoSpecification VolumeandSurfaceAreaApplications SummaryofBigIdeasfromChapterFour…………………………..………………….……………………...p.150

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APPENDICESEuclid’sElementsExcerpt……………………………………………………………………………………………..p.152Glossary………………………………………………………………………………………………………………….……p.158References……………………………………………………………………………..…………………………………….p.168

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ChapterOne

SeeingtheWorldGeometrically

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ClassActivity1:TrianglePuzzleGeometryisthescienceofcorrectreasoningonincorrectfigures.

Originalauthorunknown,butquotedfromG.Polya,HowtoSolveIt.Princeton:PrincetonUniversityPress.1945.

Whathappenedtothemissingsquare?

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ReadandStudy

Geometricfiguresshouldhavethisdisclaimer:“Noportrayalofthecharacteristicsofgeometricalfiguresorofthespatialpropertiesofrelationshipsofactualbodiesisintended,andanysimilaritiesbetweentheprimitiveconceptsandtheircustomarygeometricalconnotationsarepurelycoincidental.”

"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.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.

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ConnectionstotheElementaryGrades

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoorganizeandconsolidatetheirmathematicalthinkingthroughcommunication;tocommunicatetheirmathematicalthinkingcoherentlyandclearlytopeers,teachers,andothers;toanalyzeandevaluatethemathematicalthinkingandstrategiesofothers;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.Howdowewriteaboutmathematics?Isn’tmathematicsallaboutnumberslike2andp?Andaboutsymbolslike║andÐ?Well,no,it’snot.Mathematiciansusesymbolslikethesetowritestatementsandtosolvesomeproblems,andyoucanusesymbolslikeÐand^and@whenyouwriteaboutgeometryproblems.Butmathematicsismuchmoreaboutexploringpatterns,makingconjectures,explainingresultsandjustifyingsolutions.Theseactivitiesrequireustowritewithwordsincompletesentencesthatusemathematicallanguageandlogicappropriately.

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Symbolsareatoolwewilluse.Butfornow,youshouldfocusonwritingwithwords–clearly,completely,correctly,andconvincingly.Asfutureteachersyoumustpracticecommunicatinginthelanguageofmathematics.Youwillhaveachancetopracticerightnowinthehomework.Homework

Youalwayspassfailureonthewaytosuccess.

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

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Inprekindergartenthroughgrade2allstudentsshouldrecognize,name,build,draw,compare,andsorttwo-andthree-dimensionalshapes.

NCTMPrinciplesandStandardsforSchoolMathematics,2000Inrecentyearsmoststates(includingWisconsin)haveadoptedcommonstandardsforschoolmathematics.Thesestandards,calledtheCommonCoreStateStandards(CCSS),prescribethemathematicalcontentandpracticesthatteachersshouldaddressateachgradelevel.Asafutureteacher,youwillneedtoknowandunderstandthem.Thepracticestandardsdescribeexpectationsforstudentsacrossallgradelevels.Whichofthesestandardshaveyouexperiencedsofarinthisclass?

http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdfInthisbook,wewillfocusonthecontentstandardsrelatedtogeometryandmeasurement,andwewillbeginnowwithgeometryforchildreninkindergartenandfirstgrade.Takeaminutereadthem.

CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

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

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

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http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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?

(http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

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?

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A B

Homework Theonlyplacesuccesscomesbeforeworkisinthedictionary. VinceLombardi

1) DoalltheitalicizedthingsintheReadandStudysection.

2) DoalltheitalicizedthingsintheConnectionssection.

3) Thereareseveraltermsassociatedwithlinesthatyouneedtounderstandandusewithpropernotation.Takeafewminutestostudythese.

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

commonlydenotealineas AB andrepresent AB asshownbelow(notethearrowsateachendindicatingthatthelinecontinues):

TherayABisthesetofpointsincludingAandallthepointsonthelineABthatareontheBsideofA.ThepointAiscalledthevertexoftheray.WecommonlydenotearayasAB andrepresent ABasshownbelow:

ThelinesegmentABisthesetofpointsbetweenAandB,includingbothAandB,whicharecalledtheendpointsofthelinesegment.WecommonlydenotealinesegmentasAB andrepresent AB asshownbelow:

4) Visittheglossaryandlearntheprecisedefinitionsforeachofthefollowingterms:square,

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

A B

A B

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Concave PolygonsConvex Polygons

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.

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

7) Athirdgradeclassweobservedwaslearningaboutparallellines.Theteacherexplained

thatparallellinesarelinesintheplanethathavenocommonpoints.Thenshedrewthepicturebelowandaskedthechildrenwhetherthelinesshownwereparallelornot.

Severalchildrenarguedthattheywereparallel.Whymighttheyhavesaidthat,andwhatwouldyousaytothemastheirteacher?

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ClassActivity3:GetitStraight

Gobackalittletoleapfurther. 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.

DaveBarryMathematicalthinkingalwaysinvolvesreasoningandmakingarguments,andwehaveawholevocabularyfordescribingthatprocess.Inthissection,wehighlightthetermswemathematiciansusetodescribefacetsofdoingmathematics.Theseareimportant.Makesureyouunderstandthem.

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.

4) Aconjectureisahypothesisoraguessaboutwhatistrue.Forexample,aftersome

experiencewithcircles,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:

TheAxiomsofEuclideanGeometry:

1. Auniquestraightlinesegmentcanbedrawnfromanypointtoanyotherpoint.

2. Alinesegmentcanbeextendedtoproduceauniquestraightline.

3. Auniquecirclemaybedescribedwithagivencenterandradius.

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4. Allrightanglesareequaltoeachother.

5a.Ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesameside

lessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthat

sideonwhicharetheangleslessthanthetworightangles.

5b.Throughapointnotonalinetherecanbedrawnexactlyonelineparalleltothegiven

line.

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.Thebasicidea

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hereisthattwoobjectsarecongruentinthecasewhereifoneobjectwasmovedtolieontopoftheotherobject,theywouldcorrespondexactly.Wewilldoamorecarefuljobofdefining“congruent”later.Whileprovingtheoremsisnotthefocusofthiscourse,wemayoccasionallyaskthatyoutrytoproveaEuclideantheorem.Whenwedo,youshouldturntotheAppendix(startingonp.168)whereEuclid’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.ConnectionstotheElementaryGrades

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstorecognizereasoningandproofasfundamentalaspectsofmathematics;tomakeandinvestigatemathematicalconjectures;todevelopandevaluatemathematicalargumentsandproofs;andtoselectandusevarioustypesofreasoningandmethodsofproof.

ReasoningandProofStandard,NCTMPrinciplesandStandardsforSchoolMathematics,2000

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

Whatshouldreasoningandprooflooklikeinprekindergartenthroughgrade2?p.122

Theabilitytoreasonsystematicallyandcarefullydevelopswhenstudentsareencouragedtomakeconjectures,aregiventimetosearchforevidencetoproveordisprovethem,andareexpectedtoexplainandjustifytheirideas.Inthebeginning,perceptionmaybethepredominantmethodofdeterminingtruth:ninemarkersspreadfarapartmaybeseenas“more”thanelevenmarkersplacedclosetogether.Later,asstudentsdeveloptheir

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mathematicaltools,theyshoulduseempiricalapproachessuchasmatchingthecollections,whichleadstotheuseofmore-abstractmethodssuchascountingtocomparethecollections.Maturity,experiences,andincreasedmathematicalknowledgetogetherpromotethedevelopmentofreasoningthroughouttheearlyyears.»

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.Teachersneedtoaskthesestudentsto

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explaintotheirclassmateshowtheyknowwhattocolorwithoutcounting.Teachersalsoextendstudents’mathematicalreasoningbyposingnewquestionsandaskingforargumentstosupporttheiranswers.“Youfoundpatternswhencountingbytwos,threes,fours,fives,andtensonthehundredboard.Doyouthinktherewillbepatternsifyoucountbysixes,sevens,eights,ornines?Whataboutcountingbyelevensorfifteensorbyanynumbers?”Withcalculators,studentscouldextendtheirexplorationsoftheseandothernumericalpatternsbeyond100.

Fig.4.28.Patternsonahundredboard

p.123

Students’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,teachersshouldmodelmathematicallanguagethatthe

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studentsmaynotyethaveconnectedwiththeirideas.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!”

Fig.4.29.Astudent’sexplanationoftheequalareasofsquareandtriangularblockfaces

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

Whatshouldbetheteacher’sroleindevelopingreasoningandproofinprekindergarten

throughgrade2?

Teachersshouldcreatelearningenvironmentsthathelpstudentsrecognizethatallmathematicscanandshouldbeunderstoodandthattheyareexpectedtounderstandit.Classroomsatthislevelshouldbestockedwithphysicalmaterialssothatstudentshavemanyopportunitiestomanipulateobjects,identifyhowtheyarealikeordifferent,andstategeneralizationsaboutthem.Inthisenvironment,studentscandiscoveranddemonstrate

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generalmathematicaltruthsusingspecificexamples.Dependingonthecontextinwhicheventssuchastheoneillustratedbyfigure4.29takeplace,teachersmightfocusondifferentaspectsofstudents’reasoningandcontinueconversationswithdifferentstudentsindifferentways.Ratherthanrestatethestudent’sdiscoveryinmore-preciselanguage,ateachermightposeseveralquestionstodeterminewhetherthestudentwasthinkingaboutequalareasofthefacesoftheblocks,oraboutequalvolumes.Oftenstudents’responsestoinquiriesthatfocustheirthinkinghelpthemphraseconclusionsinmore-precisetermsandhelptheteacherdecidewhichlineofmathematicalcontenttopursue.

Teachersshouldpromptstudentstomakeandinvestigatemathematicalconjecturesbyaskingquestionsthatencouragethemtobuildonwhattheyalreadyknow.Intheexampleofinvestigatingpatternsonahundredboard,forinstance,teacherscouldchallengestudentstoconsiderotherideasandmakeargumentstosupporttheirstatements:“Ifweextendedthehundredboardbyaddingmorerowsuntilwehadathousandboard,howwouldtheskip-countingpatternslook?”or“Ifwemadechartswithrowsofsixsquaresorrowsoffifteensquarestocounttoahundred,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.”

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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

Homework

Euclidtaughtmethatwithoutassumptionsthereisnoproof.Therefore,inanyargument,examinetheassumptions. EricTempleBell

1) DoalltheitalicizedthingsintheReadandStudysection.WriteadescriptionofthestrategiesyouusedinsolvingtheproblemofcreatinganequilateraltriangleusingonlyEuclid’sfiveaxioms.

2) WhichofthechildrenfromtheConnectionsreadingareusingdeductivereasoningand

whichareusinginductivereasoning?Explain.

3) AccordingtotheNCTM,whatistheteacher’sroleinpromotingreasoningamongchildrenintheearlyelementarygrades?

4) Explainwhyverticalanglesformedbyintersectinglinesarethesame.

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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.

7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade

2TeachersGuide(yourprofessorshouldhaveacodeforyoutoviewthis).Spend10-15minuteslookingthroughUnit6Module1.CarefullyreadthroughtheactivityGuessMyShape,thenmakeupyourownriddleforyoursecondgradestudentstosolve.

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

9) Wehaveanotherconjecture!Everyrectangleisasquare.Isthisconjecturetrueorfalse?Ifitistrue,giveadeductiveargument.Ifitisfalse,giveacounterexample.

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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.Icanassureyouminearestillgreater.

AlbertEinstein

1) Hereisthedescription–fromtheCommonCoreStateStandardsforgradefour–ofhowtothinkaboutmeasuringanglesusingdegrees.Readitcarefullyandsketchapicturetohelpyourgroupmakesenseoftheirexplanation.

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

Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

2) Makesureeveryoneinyourgroupcanusetheirprotractortomeasure(indegrees)theangleindicatedbelow:


41

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

42

ReadandStudy

Theknowledgeofwhichgeometryaimsistheknowledgeoftheeternal. PlatoAregularpolygonisoneinwhichallofthelinesegmentsarecongruentandallofthevertexanglesarealsocongruent.Sketcharegulartriangleandaregularquadrilateral.Isthebelowrhombusaregularquadrilateral?Explain.

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.

43

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

Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments. CCSS,p.6

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

A

B

C

DE

F

G

44

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.

45

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

Homework Energyandpersistenceconquerallthings. BenjaminFranklin



3) ChildreninyourclassmaysaythatÐABCissmallerthanÐDEF.Isthistrue?Astheirteacher,whatwouldyousaytothesechildren?

4) Usingtheresultsoftheclassactivity,findthemeasureofonevertexangleinanequilateraltriangle,asquare,aregularpentagon,aregularhexagon,aregularoctagon,aregulardecagon,andaregulardodecagon.

5) Astudentsclaimsthatthesumofthevertexanglesofahexagonis6×180becauseeachtrianglehas180degreesofanglesmeasureandshehasshownthat6trianglestomakeupthehexagon.Whatwillyousayasherteacher?

6) IntheReadandStudysectionweclaimthatthesumoftheexterioranglesofanypolygonisalways360°.Makeamathematicalargumenttosupportthisclaim.

B

A

CE

D

F

46

ClassActivity5:ALogicalInterlude

Equationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsof geometry.

StephenHawkingInthepicturebelow,eachcardhasacolorononesideandashapeontheotherside.Whichcard(s)wouldyouhavetoturnovertobesurethatthefollowingstatementistrue?

Ifacardisredononeside,thenithasasquareontheotherside.

47

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.Nowhereisastatementthatislogicallyequivalenttotheoriginalstatementmadein3)above.Itiscalledthecontrapositiveofstatement3).

5) IfIdonotliveinIllinois,thenIdonotliveinChicago.

ThisisessentiallywhatyoufoundwhenyouworkedontheClassActivity.Writethecontrapositivetostatement4)above.

48

Itishelpfultoknowthatastatementanditscontrapositiveareequivalentbecausethatmeansthatyoucanproveastatementbyprovingitscontrapositive.ConnectionstotheElementaryGrades

Thebeginningofknowledgeisthediscoveryofsomethingwedonotunderstand. FrankHerbert

Nowthatwehavetalkedabitaboutdoingmathematics,wewanttoshowyouthattheCommonCoreStateStandardsrequirethatchildrenalsodomathematics.Giveanexampleofsomethingyouhavedoneinclasssofarthistermthatmeetseachofthesestandards.


CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

49

Homework

Amultitudeofwordsisnoproofofaprudentmind. Thales



3) Decideifeachofthefollowingstatementsistrueorfalse.Iftrue,giveamathematicalexplanation.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?

50

ClassActivity6:EnoughisEnough

Ilearnedveryearlythedifferencebetweenknowingthenameofsomethingandknowingsomething.

RichardFeynman

SupposeyouaregivensomeinformationaboutatriangleABC.Inwhichofthefollowingcaseswilltheinformationbeenoughtoallowyoutodeterminetheexactsizeandshapeofthetriangle?Ifyouhaveenoughinformation,drawatriangleguaranteedtobeexactlythesamesizeandshapeasDABC.Ifyoudonothaveenoughinformation,describetheproblemyouencounterinattemptingtodrawDABC.Youwillneedtousearulertomeasurelengthsincentimeters(cm)andaprotractortomeasuretheanglesindegrees.

a) AB =4cmand BC =5cm

b) AB =8cmand AC =6cmandÐBAC=45°

c) AB =8cmand AC =7cmandÐABC=45°

d) ÐABC=75°,ÐBCA=80°,andÐCAB=25°

e) BC =7cm, AC =8cm,and AB =9cm

f) AB =9cm,BC =3cm,and AC =4cm

g) AB =7cm,ÐABC=25°,andÐBAC=105°

h) BC =11cm,ÐABC=75°,andÐBAC=40°

51

ReadandStudy

Puremathematicsis,initsway,thepoetryoflogicalideas. AlbertEinstein

Nowitistimetotalkabouttheideaofcongruence.YouwereworkingwiththisideaintheClassActivity.Fornowwewillusethefollowingdefinition:twogeometricobjectsarecongruentiftheycanbemovedsothattheycoincide(sitontopofoneanotherandfitexactly).Wewillmakethisdefinitionmorepreciselaterinthebook.Theideaofcongruenceisrelatedtotheideaofequality,butitisnotthesamething.Congruenceisarelationshipbetweenobjectswhereasequalityisarelationshipbetweennumbers.Wewouldsaytwolinesegmentsarecongruent(coincide),andwewouldsaythatthemeasuresoftheirlengths(numbers)areequal.Wewouldsaythattwoanglesarecongruent(coincide),andwewouldsaytheirmeasuresindegrees(numbers)areequal.Wedonotusetheequalssign(=)forcongruence.Insteadwehaveaspecialsymbol( )tosaythat AB iscongruenttoCD ,( CDAB ).Besuretouse whenyoumeancongruenceand=whenyoumeanequality.IntheClassActivityyouinvestigatedconditionsthatwillensurethattwotrianglesarecongruent.Youfoundthathavingtwopairsofcongruentsidesisnotsufficient,butthathavingthreepairsofcongruentsidesdoesguaranteethetrianglescoincide.Thecaseofanglesismorecomplex.Havingthreepairsofcongruentanglesisnotsufficientinformation,butifwehavetwopairsofcongruentanglesandonepairofcongruentsides,wedogetcongruenttriangles.Andthenthereisthecasewherewehavetwopairsofcongruentsidesandonepairofcongruentangles–sometimeswehavecongruenttrianglesandsometimesnot–itmakesadifferencewhetherornottheanglesinquestionaretheanglesbetweenthetwopairsofcongruentsides.Euclidcompiledallofthisinformationaboutwhentrianglesarecongruentintothesefourtheorems.Noticethateachofthesetheoremsisinthe“if-then”statementform.Readeachonecarefully–makecertainyouunderstandallthetermsandcanexplaineachoneinyourownwords.Thensketchapicturetoillustratewhateachoneissaying.

52

1) Angle-Side-AngleTriangleCongruence(ASA):Iftwoanglesandtheincludedsideofonetrianglearecongruenttotwoanglesandtheincludedsideofanothertriangle,thenthetrianglesarecongruent.

2) Side-Angle-SideTriangleCongruence(SAS):Iftwosidesandtheincludedangleofone

trianglearecongruenttotwosidesandtheincludedangleofanothertriangle,thenthetrianglesarecongruent.

3) Side-Side-SideTriangleCongruence(SSS):Ifthreesidesofonetrianglearecongruent

tothreesidesofanothertriangle,thenthetrianglesarecongruent.

4) Angle-Angle-SideTriangleCongruence(AAS):Iftwoanglesandthesideoppositeoneoftheminonetrianglearecongruenttothecorrespondingpartsofanothertriangle,thenthetrianglesarecongruent.

YoumighthavenoticedthatTheorem4)isredundanttoTheorem1),inlightofafactwe’vealreadyestablishedabouttrianglesinaprevioussection.Towhatfactarewereferring?WewantyoutotaketimetorelatethesetheoremstothecasesyouinvestigatedintheTriangleExplorationActivity.Infact,wearegoingtogiveyouspaceheretorevisiteachsetofconditionsanddecidewhich,ifany,oftheabovefourtheoremsapplytothegivencases.Reallydothis.

a) AB =4cmand BC =5cm

b) AB =8cmand AC =6cmandÐBAC=45°

c) AB =8cmand AC =7cmandÐABC=45°

d) ÐABC=75°,ÐBCA=80°,andÐCAB=25°

e) BC =7cm, AC =8cm,and AB =9cm

f) AB =9cm,BC =3cm,and AC =4cm

g) AB =7cm,ÐABC=25°,andÐBAC=105°

h) BC =11cm,ÐABC=75°,andÐBAC=40°Okay,nowthatyouknowthetrianglecongruencetheorems,let’stakealookatsomeoftheothernicetheoremsabouttriangles:

53

5) Thesumoftheanglemeasuresinanytriangleis180degrees.(Youalreadyproved

this.)

6) Inatriangle,anglesoppositecongruentsidesarecongruent.

7) Inatriangle,sidesoppositecongruentanglesarecongruent.

Theorems6)and7)areoftencalledtheIsoscelesTriangleTheorems–andtheyarequiteuseful.Drawasketchofeachtobecertainyouunderstandwhattheysay.Noticethat6)and7)arenottheoremsabouttwodifferenttrianglesbeingcongruent.Boththeoremstalkaboutasingletriangleinwhicheithertwosidesofthattrianglearecongruentortwoanglesofthattrianglearecongruent.Hereisonemoreideathatiscommonlyusedaspartoftrianglecongruenceproofs:

8) Correspondingpartsofcongruenttrianglesarecongruent.Wewillusetheorem8)almosteverytimewemakeanargumentinvolvingtrianglesfromnowon.Since,byourdefinitionofcongruentobjects,twocongruenttrianglescoincide,everypairofcorrespondingpartsormeasurementsofanattributemustbeidentical.So,intwocongruenttriangles,thesmallestangles(iftherearesmallestangles)arecongruent,andthelongestsides(iftherearelongestsides)arecongruent,andsoon.Youmay(fondly)recalltheanagramforthistheorem,CPCTC,fromahighschoolgeometrycourse.

54

Homework

Homecomputersarebeingcalledupontoperformmanynewfunctions,includingthe consumptionofhomeworkformerlyeatenbythedog.

DougLarson


2) Studyeachboldandunderlinedtermusedinthissection.Thismeansyoushouldbeabletoexplainthedefinitionusinggoodmathematicallanguageandthatyoushouldbeablemakeexamplesandnon-examplesofeachterm.

3) ForeachofTheorems1),2),3),5),6),and7)ofReadandStudy,findEuclid’scorrespondingpropositionintheAppendix.(EucliddidnotstateTheorems4)or8).)

4) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewearedescribing?Explainyouranswer.

I. Oneofthesidesis8cmlong.II. Oneofthesidesis4cmlong.III. Theanglebetweenthesidesmentionedaboveis60degrees.IV. Thetrianglehasa90degreeangle.

5) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewearedescribing?Explainyouranswer.

I. Oneofthesidesis3cmlongandanotheris7cmlong.II. Theanglebetweenthe7cmsideandtheunknownsideis20degrees.III. Theunknownsideisthelongestside.IV. Thetrianglehasanobtuseangle.

6) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewe

aredescribing?Explainyouranswer.

I. Oneoftheanglesmeasures140degrees.II. Anotherofmyanglesmeasures25degrees.III. Oneofmysidesmeasures7cm.IV. Mylongestsidemeasures7cm.

7) Makeamathematicalargumentthatatrianglecanonlyhaveoneobtuseangle.

55

8) Makeamathematicalargumentthatthetwoacuteanglesofarighttrianglearecomplementary

9) MakeamathematicalargumentforTheorem7),thatinasingletriangle,theanglesthatareoppositethecongruentsidesmustbecongruent.

10) Makeamathematicalargumentthattwoacuteanglesofanisoscelesrighttriangleareeach45°.

11) Makeamathematicalargumentthateachangleinanequilateraltriangleis60°.

56

SummaryofBigIdeasfromChapterOne Hey!What’sthebigidea? Sylvester

• Geometryisastudyofidealobjects–notrealobjects.

• Definitionsallowustonameandcategorizeidealobjects,tocreaterelationshipsbetweenobjects,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.

57

ChapterTwo

Transformations,Tessellations,andSymmetries

58

ClassActivity7:Slides

Forthethingsofthisworldcannotbemadeknownwithoutaknowledgeofmathematics. RogerBacon

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

59

ReadandStudy

Onlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient.

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

Inthisandthenexttwosections,wewillstudythreetransformationsthatchangethepositionofanobjectbutdonotchangeitssizeoritsshape.Thesetypesoftransformationsarecalledrigidmotions.Rigidmotionsarethetransformationsoftheplaneforwhichthedistancebetweenpointsispreserved.Inotherwords,iftwopointswereacertaindistanceapartbeforethemotion,thentheyarestillthatsamedistanceapartafterthemotion.(Whydoesthename“rigidmotion”makesense?)Usingtheideaofrigidmotions,wecanmorepreciselydefinecongruence:twoobjectsarecongruentifthereexistsaseriesofrigidmotionswhereoneobjectistheimageoftheother.Therigidmotionsarethetranslation,therotation,andthereflection.Eachofthesetransformationswillmovetheplaneinauniqueway.Thetranslationwillslidetheplaneaparticulardistanceinaparticulardirection.Therotationwillturntheplaneeitherclockwiseorcounterclockwisearoundafixedcenter.Thereflectionwillmovetheplanebyflippingitacrossaline.Itturnsoutthatallrigidmotionsoftheplanearecombinationsofjustthesemoves.Asyoudiscoveredintheclassactivity,atranslationvectorisusedtodescribethedistanceanddirectioneachpointismovedinatranslation.Iftheendpointsofthevectoraregivenascoordinatesonasquaregrid,wecandescribethedistanceanddirectioneachpointismovedassomanyunitsupordownandsomanyunitsrightorleft.UsethislanguagetodescribethetranslationvectorRSonthepreviouspage.

sheared image

scaled imageoriginalrectangle

60

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

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoapplytransformationsandusesymmetrytoanalyzemathematicalsituations.

NCTMPrinciplesandStandardsforSchoolMathematics,2000Studentsintheelementarygradescanpredictanddescribetheresultsofsliding,flipping,andturningtwo-dimensionalshapes.Variouselementarycurriculausedifferentapproaches.Somemakeuseofmanipulativesandothershavestudentscutoutshapesinordertophysicallyperformtheslide,flip,orturntheshapes.

Homework

Youmaybedisappointedifyoufail,butyouaredoomedifyoudon’ttry.

BeverlySills


2) Decideifeachofthestatementsabouttranslationsistrueorfalse.Iftrue,giveamathematicalexplanation;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.

61

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

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

62

ClassActivity8:Turn,Turn,Turn

Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple.

S.GudderArotationinvolves“turning”theplaneaboutafixedpoint.Inordertospecifyarotation,weneedanangle(withdirection,clockwiseorcounterclockwise)andthefixedpoint(calledthecenteroftherotation).Ineachcaseyourgroupshouldusetracingpaperandacompassandprotractortofigureoutwhereeachshapeendsupafterthegivenrotation.(Therearequestionsonthenextpagetoo.)

1) Rotatetheplane60degreescounterclockwiseaboutpointA. A

2) Rotatetheplane140degreesclockwiseaboutpointP.

P.

(Thisactivityiscontinuedonthenextpage.)

63

3) Rotatetheplane90degreesclockwiseaboutapointQintheexactcenterofthesquare.

4) Studythethreerotationsinthisactivity.Whatrelationshipsdoyouseebetweenthe

originalfiguresandtheirrotatedimages?BetweencorrespondingpointsandthepointP?Makeasmanyconjecturesasyoucanaboutrotations.

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

64

ReadandStudy

Wedon’tseethingsastheyare.Weseethingsasweare. AnaisNinArotationisa“turn”aboutagivenpointcalledthecenterthroughagivenangleofturn.(Theturncanbemadeclockwiseorcounterclockwise.Thisistypicallyindicatedintheproblem.)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

65

Homework

Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair. JohnQuincyAdams


2) Usethegridtorotatetheplane90degreescounterclockwiseaboutpointP.Showtheimageofthefiguresaftertherotation.

K

PJ

H

IG F

AD E

C

B

66

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) Thelinesegmentsjoiningcorrespondingverticesarecongruent.

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ClassActivity9:ReflectingonReflection Mathematics,rightlyviewed,possessesnotonlytruth,butsupremebeauty–abeautycoldandaustere,likethatofsculpture. BertrandRussell

Areflectionflipstheentireplaneaboutagivenlineresultinginitsmirrorimage.OfficiallyareflectioninalinelisarigidmotionoftheplaneinwhichtheimageofapointPonlisP,andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA .

1) Studythatdefinition.Makesureeveryoneinyourgroupunderstandshowitfitswiththeideaofareflection.SeeifyoucanusethedefinitiontohelpyoutosketchthereflectionoftriangleABCinlinel.

l


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2) Carefullyusethedefinitionofareflectiontosketchthereflectionofthetrapezoidshown.Firstyouwillreflectitinlinemandthenyouwillreflectwhatyougetinlinen(thatisparalleltolinem).

m n

Whatisthesinglerigidmotionthatwouldtaketheinitialfiguredirectlytothefinalfigure?Explain.

3) Whatwouldhappenifthelinesintersected?Tryitandthenuseyourobservationstomakeaconjecture.

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ReadandStudy

WecoulduseuptwoEternitiesinlearningallthatistobelearnedaboutourownworldandthethousandsofnationsthathavearisenandflourishedandvanishedfromit.Mathematicsalonewouldoccupymeeightmillionyears.

MarkTwainThereareseveralwaystohelpchildrenpicturetheresultsofareflection.Thinkaboutthereflectionoftheparallelograminlineshownbelow.

Onewaytoseewheretheimageshouldbelocatedistotracetheparallelogramandthelineofreflectiononasheetofthinpaperandthenphysicallyflipthepaperoverandplaceitbackontopoftheoriginalpapersothatthetwolinescoincide.Thecopyoftheparallelogramonthetracingpaperisnowpositionedastheimageofthereflection.Useasheetofpaperandtrythismethod.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) Thelinesegmentsjoiningcorrespondingverticesarecongruenttoeachother.

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ClassActivity10:Zoom

Onecanstate,withoutexaggeration,thattheobservationofandthesearchforsimilaritiesanddifferencesarethebasisofallhumanknowledge.

AlfredNobel

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

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ReadandStudy

Wehavedifferentnotionsof“sameness”ingeometry.Inthestrongestsense,ifwesaytwoobjectsarethesame,wemeantheyarecongruent.Butwemightalsorefertotwoobjectshavingthesameshapeeveniftheyaren’tcongruent.Forinstance,thetwotrianglesinourClassActivityhavethesameshape,buttheyarenotthesamesize.Observethattheircorrespondinganglesarecongruent,andthattheircorrespondingsidesareproportional.Makesurethatyouunderstandwhatthismeans.Thesetwotrianglesaren’tcongruent,buttheyarewhatwecall“similar”.Formally,wesaythattwoobjectsintheplanearesimilarifonecanbeobtainedfromtheotherbycomposingarigidmotion(tochangetheobject’spositionifnecessary)witha“dilation”.Adilationisamotionoftheplaneinwhichonepoint,P,remainsfixed,andallotherpointsarepushedradiallyoutwardfromPorpulledradiallyinwardtowardPsothatalldistanceshavebeenmultipliedbysomescalefactor.Itcanbeshownthattwopolygonsaresimilarifandonlyiftheircorrespondingvertexanglesarecongruent,andtheircorrespondingsidesareproportional.Euclidprovedatheoremaboutsimilartriangles:

1) Angle-AngleTriangleSimilarityTheorem(AA):Iftwoanglesofonetriangleare

congruenttotwoanglesofanothertriangle,thenthetrianglesaresimilar.ThistheoremappearedinhisBookVI(aboutsimilargeometricfiguresandproportionalreasoning)ratherthanhisBookIthatwehavestudiedpreviously.Homework

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

a) drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointCand

scalefactor2. b) instead,drawthesimilarshapethatistheresultofadilationoftheplanewithfixed

pointPandscalefactor2. c) Whatcommonalitiesanddifferencesdoyouobserveaboutthethreeshapesyouhave

drawn?

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2) Beginningwiththetrianglepicturedbelow,drawthesimilartrianglethatistheresultofadilationoftheplanewithfixedpointPandscalefactor1/2.

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

4) Supposetwoobjectsaresimilarandthescalefactorofthedilationis1.Whatelsecanyousayabouttherelationshipbetweenthosetwoobjects?

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

P

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ClassActivity11: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

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ReadandStudy

Mathematicsisthesciencewhichuseseasywordsforhardideas. E.KasnerandJ.Newman

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

Manyobjectsinnaturedisplaythiskindofbilateralsymmetry.ThelettersinATOYOTAalsoformasymmetricpattern:ifyoudrawaverticallinethroughthecenterofthe“Y”andthenreflecttheentirephraseacrosstheline,theleftsidebecomestherightsideandviceversa.Thepicturedoesn’tchange.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. ChristaMcAuliffeTheCommonCoreStateStandardsformathematicsmakeinformalideasofsymmetryatopicforgrade4.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

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.

2) Classifythesymmetriesforthefollowingtrafficsigns–(considertheentiresign–notjusttheinteriordesign).

3) Isitpossibleforanobjecttohaverotationsymmetrieswithouthavingreflectionsymmetries?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.

4) Isitpossibleforanobjecttohavetworeflectionsymmetrieswithouthaving180degreerotationsymmetry?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.

5) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module3Sessions1and2.Whatideasaboutrotationalsymmetryareemphasizedinthesesections?

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ClassActivity12:Tessellations Themathematician’spatterns,likethepainter’sorthepoet’smustbebeautiful;theideas,likethecolorsorthewordsmustfittogetherinaharmoniousway.Beautyisthefirsttest:thereisnopermanentplaceinthisworldforuglymathematics.

GodfreyH.HardyTherearemanynewdefinitionsinvolvedinthisactivity.Takealittletimetostudyeachofthem.Atilingisanarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplanewithnogapsandnooverlaps.Atessellationisatilinginwhichallverticesmeetonlyothervertices.Aregulartessellationisatessellationthatusesonlyoneregularpolygon.Findallthepossibleregulartessellations,makeasketchofeachone,andthenmakeanargumentthatyouhavefoundthemall.

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Traceablecopiesofmanyregularpolygons.Allofthemhavebeenscaledsothattheyhavethesamesidelength.

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ReadandStudy

Everythinghasbeauty,butnoteveryoneseesit.Confucius

Atilingisanarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplanewithnogapsandnooverlaps.Anexampleusinga“T”shapepolygonisshownbelow.

Atessellationisatilinginwhichallverticesonlymeetothervertices.Istheabovetilingatessellation?Explain.Somecurriculummaterialsusethewordstilingandtessellationinterchangeably.Wewillnotdoso,butwewantyoutobeawareofthatfact.Mathematically,weareinterestedininterpretingthedefinitionofatessellation.Howcanweknowitwillhavenogapsandnooverlaps?Howcanwedeterminethatagivenarrangementwillextendindefinitelyinalldirections?Firstwewillexaminewhathappensatasinglevertexpointwithinatessellation.TakeacloselookatverticesAandBinthefollowingarrangementcomposedofregularhexagonsandequilateraltriangles.

TherearetwohexagonsandtwotrianglesmeetingatvertexAandsixtrianglesmeetingatvertexB.Sinceweknowthatthehexagonsandthetrianglesareregular,weknowthatthevertexanglesofthehexagonare120°each.(HavealookbackatClassActivity4,ifyouhave

D

B C

E

A

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forgottenhowtothinkaboutthis.)Likewise,weknowthatthevertexanglesoftheequilateraltriangleare60°each.Thismeansthatthereare2*120°+2*60°=360°atvertexAandthereare6*60°=360°atvertexB.(Checkthis.)Now,sincethesumoftheanglesateachvertexisexactly360°,wehaveprovedthattherearenogapsornooverlapsinthisarrangement.Nowwewillnotethatifwemadeinfinitelymanycopiesofthisarrangement,wecouldslide(translate)themaroundontheplane(e.g.,slideonesothatAgoestoC)tocovertheentireplane.Wewilltellyouthattherearemanytessellationsofregularpolygonsthatarenotregular.Forexample,havealookbackatthetessellationmadeofhexagonsandtrianglesthatwewerejustdiscussingabove.Whyisthistessellationnotregular?Therearemanywebsitesthatpresenttessellations-ortiling-typeactivities.IntheHomeworksectionyouwillbeaskedtoexploreseveralthatblurthelinebetweengeometryandart.ConnectionstotheElementaryGrades Learningisnotcompulsory...neitherissurvival. W.EdwardsDemingCreatingtessellationsisanactivitythatcanbeadaptedtoeverygradelevel–veryyoungchildrencancreatepatchworkquiltsfromconstructionpaperusingonlyrectanglesorsquaresorisoscelesrighttriangles.Olderstudentscanmakemorecomplexartworkusingthemathematicalconceptsofcongruencyandtransformations.Onlineresourcescanallowstudentstocreateavarietyofmorecomplicateddesignsquicklyandaccurately.Spend15minutesatthewebsiteathttp://www.mathcats.com/explore/tessellations/tesspeople.htmltoseeanexampleofaninteractiveweb-basedexplorationoftessellationsusedinthefifthgradeTrailblazerscurriculuminNorthCarolina.

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Homework

Creativityisallowingyourselftomakemistakes.Artisknowingwhichonestokeep. ScottAdams,‘TheDilbertPrinciple’1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.Makesureto

spendsometimeatthewebsite.

2) Anytriangle(ifyouhaveenoughcopiesofit)canbeusedtotessellatetheplane.Toexplorethis,foldapaperupsoyoucancutout8congruentscalenetrianglesallatonce.Thenhavealook.Payparticularattentiontothetransformationsyouareusingtomovethetriangleintonewpositions.

a. Makeamathematicalargumentthatyourtrianglewould,infact,tessellatetheplane.

b. Whereinyourproofdidyouusetransformationsoftheplane?

3) Trueorfalse?Anyquadrilateral(ifyouhaveenoughcopiesofit)canbeusedtotessellatetheplane.Toexplorethis,foldapaperupsoyoucancutout8congruentquadrilateralsallatonce.Thenhavealook.Whataboutconcavequadrilaterals?Makeanargumenttosupportyourchoice.

4) Trueorfalse?It’spossibletofindapentagonthatcanbeusedtotessellatetheplane(ifyouhaveenoughcopiesofit).Makeanargumenttosupportyourchoice.

5) Makeamathematicalargumentthatthenumberofregulartessellationsyoufoundin

theclassactivityistheexactnumberpossible.

6) Apolygonwithmorethansixsideswillnottileifitisconvex.Explainwhynot.Thefollowingpolygonshavemorethansixsides,buttheyareconcave.Sketchaportionofatilingforeachpolygon.

a) b)

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7) Describesomereflectional,rotationalandtranslationalsymmetriesforthetessellationbelow.

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SummaryofBigIdeasfromChapterTwo

Ifanidea’sworthhavingonce,it’sworthhavingtwice. TomStoppard

• Therearethreedistinctwaysto“move”theplanewithoutchangingtheshapeorsizeofobjects:thetranslation,therotation,andthereflection.

• Atranslationslidestheplaneagivendistanceinagivendirection.

• Arotationturnstheplanearoundagivenpointthroughagivenamountofrotation

(usuallygivenindegrees).• Areflectionflipsanobjecttoitsmirrorimageacrossalineofreflection.

• Oneofthegoalsofelementaryschoolgeometryinstructionisthatstudentslearnto

visualizeandapplytransformations.

• Twoobjectsaresimilarifonecanbeobtainedfromtheotherbyarigidmotionandadilation.

• Symmetryisaphenomenonofthenaturalandartisticworldsthatcanbeexplained

withthelanguageofrigidmotions.

• Mathematiciansmostoftentalkabouttwotypesofsymmetry:reflectionsymmetry,inwhichanobjectisdividedbyalineofreflectionintotwopartsthataremirrorimagesofeachother,androtationsymmetry,whereanobjectisrotatedaroundacenterpointthroughacertainangleandendsupoccupyingthesamepositionintheplane.

• Creatingtilingsandtessellationsisanactivitythatcanbeadaptedtoeverygradelevel–

veryyoungchildrencancreatepatchworkquiltsfromconstructionpaperusingonlyrectanglesorsquaresorisoscelesrighttriangles.Olderstudentscanmakemorecomplexartworkusingthemathematicalconceptsofcongruencyandtransformations.

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ChapterThreeMeasurementinthePlane

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ClassActivity13:MeasureforMeasure

AndtherewentoutachampionoutofthecampofthePhilistines,namedGoliathofGath,whoseheightwassixcubitsandaspan.

ISamuel17:4

1) Usingyourcubit(lengthfromelbowtofingertips)andhandspan,determinehowtallGoliathwasbycuttingastringthatisaslongasGoliathwastall.Compareyourstringlengthwiththestringsofothersinyourgroup.Whatarethedifficultiesthatmightarisefromchoosingandusingunitsdeterminedbyeachperson’sownbody?Whydoyouthinkweusethe“foot”asacommonunitoflength?

2) Onthenextpage,arefourcopiesofthesamelinesegment.Usingeachoftherulersprovided,carefullymeasurethelinesegment.

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

j ''

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accuracytothehundredthofameter(tothecentimeter).Whenworkingwithchildrenyoushouldexplicitlydiscussthesefacetsofmeasurementandyoushouldbesuretoalwaysreporttheunitwithanyactualmeasurement.Sayingthatadeskis1.23longmeansnothing.Sayingthatitis1.23meterslongmakessense.Acommonmeasurementwemakeforaplaneobjectistomeasurethedistancearounditsboundary.Wecallthismeasurementtheperimeteroftheobject.(Whentheobjectisacircle,wecallthislengththecircumference.)Theperimeterofaplaneobjectisaone-dimensionalmeasurement–soweuselinearunitslikeinchesorcentimeters.Wecalculateaperimeterbysimplyaddingupthelengthsofthecurvesthatmakeuptheobject.Wecanmeasurethelengthsoflinesegmentswitharuler.Wewillfindaformulaforthecircumferenceofacircleinafutureactivity.ConnectionstotheElementaryGrades

Onlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient.

EugeneS.WilsonTheCommonCoreStateStandardsformathematicsrequirethatchildrenbegintostudystandardmeasurementoflengthbeginningingrade2.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.

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6) Findtheperimetersoftheplanefiguresshownbelowbymeasuringthema) 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?

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ClassActivity14A:Triangulating

Godowndeepenoughintoanythingandyouwillfindmathematics.DeanSchlicter

Defineonetriangleunittobetheareaofthegreentrianglepatternblock.Usingthisunit,determinetheareaofthissheetofpaper.Thenusethetriangleunittoestimatetheareaofeachofthepatternblockshapespicturesbelow.

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ClassActivity14B:AreaEstimation HereisamapofthegreatstateofWisconsin.Withoutlookinganythinguponline,yourgroupneedstomakeareasonableestimateofitsareainsquaremiles.Firstdiscussacoupleofdifferentwaysofdoingthisusingthemap.Thengoaheadandcarefullycomputeyourestimate.

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

Wouldyouguaranteeyourestimatetothenearsest10squaremiles?Thenearest100squaremiles?Thenearest1000squaremiles?Somethingelse?Howdidyoudecide?

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ReadandStudy

Themanignorantofmathematicswillbeincreasinglylimitedinhisgraspofthemainforcesofcivilization.

JohnKemenyMathematiciansdefineareaasthequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigure.Wecommonlymeasurethisareaintermsofsquareunitssuchassquareinchesorsquarecentimeters.Sotofindtheareaofafigureyouneedtofindthenumberofsquareunitsitwouldtaketocoverthefigure.Wewantyoutoliterallydothisnow.Hereisaunitofarea(thesquarecentimeter):Traceitandthenseehowmanyofthoseunits(includingpartsofunits)ittakestocovertheshapebelow: Iftheshapeisarepresentationofalake,andacentimeteroflengthcorrespondsto3miles,thenwhatistheareaoftheactuallake?Explain.Weoftenmeasuretheareaofirregularlyshapedobjectsjustbycounting(andestimating)thenumberofsquareunitswithintheboundaryoftheobject.Sometimeswesuperimposeagridtohelpwiththatestimate.Estimatetheareaofthisobjectnowassumingthateachsmallsquareisaunitofarea.Whenwemeasuretheareaofanobjectinsquareunits,noticethatwearetakingadvantageofthefactthatsquarestessellatetheplane–therearenogapsbetweenthesquares.Wehavealreadydiscoveredthatthereareothershapesthattessellatetheplaneaswell;twoofthem

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are,likethesquare,alsoregularpolygons.Whichones?Whydoyouthinkpeoplechosetousesquareunitsinsteadofsomeothershapethattessellatestheplane?ConnectionstotheElementaryGrades

Teachingcreatesallotherprofessions.AuthorUnknown

ChildrenneedtohaveexperiencesfindingareasbycoveringfigureswithsquareunitslikeyoudidintheReadandStudysection.Ifyoushowthemformulastooearly,theywillsimplytrytorememberthoseformulasandtheymaynotfocusontheideaofarea.Besides,formulasworkonlywithalimitednumberofshapes,andsoestimatingareasusingthedefinitionisausefulskillinitsownright.Wesuggestthatchildreningrades2and3spendmanyweeksestimatingareasusingsquarestickersorsquaregridstocoverfigures.Someofthosefiguresshouldhavecurvedboundariesandsomeshouldhavespecialpolygonshapes.Childrenwillquicklybegintoproposeshortcutsontheirownthatwillleadnaturallytotheareaformulas.Forexample,ifchildrencomputeareasofthebelowrectangleshapesusingstickersoragrid,someofthemwillnoticethatashortcutforfindingareasofrectangleshapesistosimplymultiplythelengthoftherectanglebyitswidth.Thenyoucantalkwiththemaboutwhythisisso.Practicethatnowbydoingthebelowtasks.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.

98

Homework

ThemoreIpractice,theluckierIget.JerryBarber



3) Useagridtoestimatecarefullytheareaofthebelowcircularfigureinsquare

centimeters.

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

5) WewouldliketohavetheabovelakeclassifiedasoneoftheGreatLakes.AspartoftheapplicationtotheDepartmentoftheInterior,wehavetoreportitsarea.Estimatetheareaofthelakeinsquaremiles.

Scale1cm=3miles

99

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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ClassActivity15:FindingFormulas

Onecannotescapethefeelingthatthesemathematicalformulashaveanindependentexistenceandanintelligenceoftheirown,thattheyarewiserthanweare,wisereventhantheirdiscoverers.

HeinrichHertzWecommonlycomputeareasofsomespecialpolygonshapeswiththeuseofformulas.Theseformulascanbeexplained–theyarebasedongeometricdefinitionsandtheorems–andwewantyoutounderstandwhytheymakesense.Thatisthegoalofthisactivity.

1) Rectangle:Usetheideaofareaasthenumberofsquareunitsittakestocoveranobjecttoexplainwhyitmakessensethattheareaofarectangleissimplytheproductofitslengthandwidth.

2) Parallelogram:Showhowtocutandrearrangeaparallelogramtomakearectanglewiththesamearea.Arguethattheresultingfigureisinfactarectangle.UsetheformulafortheareaofarectangletofindaformulafortheareaAofaparallelogramusingthebasebandtheheighthoftheparallelogram.

3) Triangle:Showhowtorearrangetwocopiesofthesametriangletomakeaparallelogram.Arguethattheresultingfigureisinfactaparallelogram.UsetheformulafortheareaofaparallelogramtofindaformulafortheareaAofatriangleusingthebasebandtheheighthofthetriangle.

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ReadandStudy

Theessenceofmathematicsisitsfreedom. GeorgCantorManyofyourstudents(andmanyoftheirparents)willthinkthatformulastellthewholestoryaboutarea.Infact,somepeoplemistakenlydefineareaas“lengthtimeswidth.”Everyclosedtwo-dimensionalshapehasarea,butaswehaveseeninanearliersection,onlyaveryfewoftheseshapeshaveformulaswecanusetocalculatethearea.Theareaofsomegeometricobjectsismoreeasilydeterminedthroughtheuseofareaformulas.IntheClassActivityyoudevelopedseveralusefulandwell-knownformulasthatarereadilyfoundintheelementaryschoolcurriculum.Themostfundamentalareaformulaisfortheareaofarectangle:length´width.Alloftheotherformulasforareaarebuiltonthat.Asyouhaveseen,theseformulasarereallytheoremsthathavebeenproventowork.And,thesetheoremsareonlytruewhenweusetheunitsquareasourunit.Explaincarefullywhythatpreviousstatementissoimportant.Whathappensifwedonotusesquaresasourunit?YoushouldhaveyourupperelementarystudentsdoactivitieslikethoseintheClassActivitysothattheycanseethisforthemselves.IntheBridgesinMathematicscurriculumforgrade4,studentsdiscovertheformulasfortheareaandperimeterofrectangles.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module3Session3.Inwhatwaysdoesthisactivityhelpstudentstodevelopformulasfortheperimeterandareaofarectangle?ConnectionstotheElementaryGrades

Knowingisnotenough;wemustapply.Willingisnotenough;wemustdo.

JohannWolfgangvonGoethe

Oncechildrenunderstandtheideasofperimeterandarea,theycansolvesomepracticalproblems.Wewillposesomenowthatspecificallyaddressthestandardsdiscussedbelow.

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Dotheseproblemsandthenreadtoseewhichofthestandardswe’veaddressedwiththem.

a) Ifarectangularroomhasalengthof5feetandanareaof60squarefeet,whatisitswidth?

b) Ifarectanglehasalengthof10feetandaperimeterof36feet,whatisitswidth?c) Isittruethatrectangleswithbiggerperimetersalwayshavebiggerareastoo?Explain.d) Isittruethatrectangleswithbiggerareasalwayshavebiggerperimeterstoo?Explain.

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.

103

Homework Toliveacreativelife,wemustloseourfearofbeingwrong. JosephChiltonPearce

1) DotheproblemsandthendotheitalicizedstatementintheConnectionssection.

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.

* * * * * *

* * * * * *

* * * * * *

* * * * * *

104

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

7) Supposeyouhave100metersofflexiblefencingtomarkapastureoutontheplains.

Howwouldyousetitup(whatshape)toenclosethemostgrazingareaforyourcattle?Whatdimensionswouldyouuseifthepasturehadtobearectangle?

8) Thetrianglebelowisconstructedona1cmgrid.Findtheareaofthetriangleusingatleastthreedifferentmethods.

9) Assumethatthetriangleareaaboverepresentspartofasignthatneedstobepainted.

Thescaleofthedrawingisthat1cmrepresents10feet.Theinstructionsonthepaintcansaythat1gallonofpaintwillcover100squarefeet.Howmanygallonsofpaintwillyouneedtobuyinordertopaintthetriangle?

1 cm

105

ClassActivity16: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?

106

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

107

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

722 asanapproximatevalueforpwhen

carryingoutcalculationsinvolvingcircles.Alwaysbesuretomakethepointthatthisisjustanapproximation.ConnectionstotheElementaryGrades

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.

108

Homework

Eureka!I’vegotit! Archimedes1) DoalltheitalicizedthingsintheReadandStudysection.

2) SolvetheproblemsintheConnectionssection.

3) Studyeachboldandunderlinedtermusedinthissection.Thismeansyoushouldbeabletoexplainthedefinitionusinggoodmathematicallanguageandthatyoushouldbeablesketchexamplesandnon-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?

9) IfIdoublethediameterofacircularregion,whathappenstoitsarea?

109

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

11) Havealookatthecirclesbelow.

a) Carefullymeasuretheperimeterofthesquarethatisinscribedinsidethefirst

circle.b) Carefullymeasuretheperimeteroftheregularpentagoninscribedinsidethe

secondcircle.c) Carefullymeasuretheperimeteroftheregularhexagoninscribedinsidethe

thirdcircle.d) TrueorFalse?Asthenumberofsidesoftheinscribedpolygongrows,sodoes

theperimeterofthepolygon.e) TrueorFalse?Ifweinscribeapolygonwithinfinitymanysides,thenthe

perimeterofthatpolygonwillbeinfinitelylong.Explainyourthinking.f) Howcouldyouusethesemeasurementstogetanestimateforπ?Explain.

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

2271223

<< ).Theaverageofthesetwovaluesis

roughly3.1419whichiscorrecttothreedecimalplaces.Notbadfor350BC.

110

ClassActivity17:PlayingPythagoras

Ihavehadmyresultsforalongtime,butIdonotknowyethowIamtoarriveat

them. CarlFriedrichGauss

ThePythagoreanTheoremisthoughttobealmost4000yearsold.TheBabylonians,theEgyptians,andtheChineseallknewit.Thatis,theyknewthatthesumoftheareaofthesquaresonthelegsofarighttriangleequalstheareaofthesquareonthehypotenuse,andtheyusedthisfactnumericallyinconstructionandcommerceandsurveying.

1) Carefullymeasuretheareasofthesquaresinthebelowexampletoseeifthistheoremseemstrue.

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


111

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?

112

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.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.ConnectionstotheElementaryGrades

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstocreateanduserepresentationstoorganize,record,andcommunicatemathematicalideas;toselect,apply,andtranslateamongmathematicalrepresentationstosolveproblems;andtouserepresentationstomodelandinterpretphysical,social,andmathematicalphenomena.

NCTMPrinciplesandStandardsforSchoolMathematics,2000Havingtwo(ormore)waystointerpretorrepresentamathematicalideaisanimportantcharacteristicofmathematicsforteaching.TheNCTMStandardsrecognizethisandcallforallelementarystudentsto“select,apply,andtranslateamongmathematicalrepresentationstosolveproblems.”Andso,asteachers,itisalsoimportantthatweunderstandamathematicalconceptfrommorethanonepointofviewinordertoassistourstudentstousedifferentrepresentationsofthatconcepttosolveproblems.ThePythagoreanTheoremisonesuchideathatcanbeunderstoodfrommanyviewpoints:geometric,numeric,andalgebraic.Someofourstudentswillmoreeasilygraspthealgebraicapproachwhileotherswillpreferthemoreconcretegeometricornumericrepresentation.Asteachers,wemustmasterallrepresentationsinordertocarryoutourresponsibilitiestosupporteachstudent’slearning.

114

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.Findthedistancefromonecorneratthefloortothediagonallyoppositecornerattheceiling.Youmightwanttohavealookataboxtohelpyouseewhattodohere.

a

b

c

c b

a

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ClassActivity18:Coordination

Thoughtisonlyaflashbetweentwolongnights,butthisflashiseverything. HenriPoincare

Whichofthetrianglespicturedabovearesimilartoeachother?Justifyyouranswerinasmanywaysasyoucan.(Youmayassumethatthedotsareequallyspacedonthegridinboththehorizontalandverticaldirections,andthattheverticesofthetriangleslieexactlyonthedotsastheyappearto.)

116

ReadandStudy

Teachersopenthedoor.Youenterbyyourself.ChineseProverb

Wearegoingtousethissectiontodiscusstheideaof“coordinategeometry.”Inthe1700’sRenéDescartes(pronouncedDay-cart)hadtheideathathecouldsolvesomegeometricproblemsmoreeasilybytranslatingthemintoalgebraicproblems.Hisideawastoplaceastructure(agrid)ontopoftheplaneandtogivenames(like(-3,-1))tothepoints.Thecoordinateplanefeaturestwoperpendicularaxes,thehorizontalx-axisandtheverticaly-axis,thatintersectatapointcalledtheorigin.Welabeleachpointontheplanewithanorderedpairofcoordinates(x,y).Thex-coordinatetellsushowfarthepointisfromtheorigin(0,0)inthehorizontaldirectionandthey-coordinategivesthedistancefromtheoriginintheverticaldirection.Forexample,thepoint(-3,-1)islocated3unitsleftand1unitdownfromtheorigin.

UsingthePythagoreanTheoremwecanfindthedistancebetweenanytwopointsonthecoordinateplane.Forexample,let’sfindthedistancebetweenpointsAandDinthepictureabove.ThelinesegmentADisthehypotenuseofarighttriangle(sketchitonthepictureabove)withahorizontallegoflength5=(2–(-3))andaverticallegoflength2=((-1)–(-3)).Sothe

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

D: (2, - 3)

C: (4, 0)

B (-2, 5)

A (- 3, -1)

origin

y-axis

x-axis

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squareofthedistancebetweenAandDis52+22=25+4=29andthedistancebetweenAandDis 29 .Findthedistancebetween(2,-3)and(4,0).

Therearetwofactsaboutlinesonthecoordinateplanethatareusefultorecall.Oneisthateverylinehasaslope,whichisameasureofitsinclinationwiththex-axis.Slopeistheamountyouneedtomoveinthey-directiontostayonthelineforaoneunitchangeinthex-direction.Sothinkaboutthis.Whatdoesaslopeof3mean?Sketchalinewiththatslope.Whatdoesaslopeof-¼mean?Sketchalinewiththatslopeontheaxisabove.Wecancalculatetheslope(m)ofalinebyusingthecoordinatesoftwopointsthatlieonthelinewiththeformula:

12

12

xxyym = where ),( 11 yx and ),( 22 yx arethecoordinatesofthetwopoints.

Howdoestheformularelatetothedefinitionofslopeasstatedabove?Explain.Computetheslopeofthelinecontainingthepoints(4,0)and(-2,5).Iftwolinesareparallel,thentheywillmakethesameanglewiththex-axisandsowillhavethesameslope–andviceversa,iftwolineshavethesameslope,thentheyareparallel.Thinkabouthowyoucouldmakeanargumentforthisfact.Thisturnsouttobeaveryusefulobservation.Ifweneedtoshowthattwolinesareparallel,wecansimplycalculatetheirslopesandshowthattheyareequal.Whatiftwolinesareperpendicular?Howaretheirslopesrelated?Itturnsoutthattheslopesofperpendicularlinesalsohaveanumericalrelationship.Theproductoftheslopesofperpendicularlinesisalways-1.Makeanargumentforthisfact.Whatistheslopeofthelinethatisperpendiculartothelinecontainingthepoints(4,0)and(-2,5)?

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IntheClassActivity,didyoutryusingthePythagoreanTheoremtogetherwithatrianglecongruencetheorem?DidyoutryusingslopesoflinestocompareanglesandapplyEuclid’strianglesimilaritytheoremfromoursectiononsimilarity?Ifnot,youshouldgobackanddothosethingsnow!ConnectionstotheElementaryGrades

Often,whenIamreadingagoodbook,Istopandthankmyteacher.Thatis,Iusedto,untilshegotanunlistednumber. AuthorUnknown

Coordinategeometryisatopictobeintroducedingrade5accordingtotheCommonCoreStateStandardsformathematics.Readthedescriptionbelowtobesurethatitfitswithwhatwehavediscussedinthissection.


Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit6Module1.Payparticularattentiontohowthecoordinateplaneisutilizedinthesessions.PrintoutRita’sRobotanddotheactivity.

CCSSGrade5:Graphpointsonthecoordinateplanetosolvereal-worldandmathematicalproblems.

1. Useapairofperpendicularnumberlines,calledaxes,todefineacoordinatesystem,withtheintersectionofthelines(theorigin)arrangedtocoincidewiththe0oneachlineandagivenpointintheplanelocatedbyusinganorderedpairofnumbers,calleditscoordinates.Understandthatthefirstnumberindicateshowfartotravelfromtheorigininthedirectionofoneaxis,andthesecondnumberindicateshowfartotravelinthedirectionofthesecondaxis,withtheconventionthatthenamesofthetwoaxesandthecoordinatescorrespond(e.g.,x-axisandx-coordinate,y-axisandy-coordinate).

2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.

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Homework

Energyandpersistenceconquerallthings.

BenjaminFranklin


2) Ageoboardisaboardcontainingalatticeofpoints,thatis,thepointsarearrangedinageometricpattern.Themostcommonarrangementistohavepointsevenlyspacedinhorizontalandverticalcolumns,formingasquaregriddesignlikethedotpaperyouusedintheClassActivityorthecoordinategriddiscussedintheReadandStudysection.Childrencanformpolygonsonageoboardbyplacingrubberbandsaroundthepegs.Ageoboardpolygon(oradot-paperpolygon)islikeanyotherpolygoninthatitisasimpleclosedcurvemadeupoflinesegments.However,werequirethattheverticesofageoboardpolygoncoincidewithpointsonthegeoboard.

http://www.artfulparent.com/2012/07/kids-art-activities-with-geoboards.html

Whichregularpolygonscanbemadeonthegeoboard?Ifitisnotpossibletomakeaparticularregularpolygon,explainwhyitisnotpossible.

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3) Findtheperimetersandareasofthefiguresonthegeoboardsquaregridsbelow.YoumightfindthePythagoreanTheoremhelpful.

4) Thereare14differentsegmentlengthsthatarepossibletomakeona5by5geoboard.Findthemall.

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SummaryofBigIdeasfromChapterThree

Althoughthismayseemaparadox,allexactscienceisdominatedbytheideaof approximation. BertrandRussell

• Measurementisthecomparisonofanattributeofanobjecttoaunit.Tomeasuremeanstoseehowmanyoftheunitfitintotheobjectyouaremeasuring.

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

• Allmeasurementisapproximate.• Thechoiceofaunitisafundamentalpartofthemeasuringprocess.

• Areaisthenumberofsquareunitsittakestocoveranobject.Itisnotdefinedas“length

timeswidth,”and,infact,thatformulaworksonlyinafewlimitedcases.

• Formulasforthecalculationofareacanbeexplainedbythegeometryoftheobjectsbeingmeasured.Asateacher,youwillneedtohelpyourstudentstoseewhereformulascomefromandwhytheymakesense.

• πisdefinedasthenumberoftimesthediameterofacirclefitsintoitscircumference.Itisanirrationalnumber.

• Sometimesitisusefultoplaceastructure(agrid)ontopoftheplaneandtogivenamestothepoints.Wecallthiscoordinategeometry.

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ChapterFour

TheThirdDimension

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ClassActivity19:StrictlyPlatonic(Solids) IfIhaveevermadeanyvaluablediscoveries,ithasbeenowingmoretopatient attention,thantoanyothertalent. SirIssacNewtonHereisadefinitionforyoutostudy: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.

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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.

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

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

Studentsingrades3–5shouldexaminethepropertiesoftwo-andthree-dimensionalshapesandtherelationshipsamongshapes.Theyshouldbeencouragedtoreasonaboutthesepropertiesbyusingspatialrelationships. NCTMPrinciplesandStandards,2000

ThefollowingexcerptfromtheNCTMStandardsforGrades3–5Geometry(p.168)describestheimportanceofvisualizationandspatialreasoningastoolselementarystudentscanusetounderstandthepropertiesofgeometricobjectsandtherelationshipbetweenthesepropertiesandtheshapes.Readtheseparagraphsandstudytheexamples.Thenbuildthebuildingtheydescribe.

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)

127

Homework

Gettingaheadinadifficultprofessionrequiresavidfaithinyourself.Thatiswhysomepeoplewithmediocretalent,butwithgreatinnerdrive,gomuchfurtherthanpeoplewithvastlysuperiortalent.

SophiaLoren


2) Writeoutthedetailsofamathematicalargumentthatthereareexactlyfiveregularpolyhedra.

3) Thefollowingpictureisoftengivenasanexampleofaregularicosahedron.Examinethepicturecarefullyanddeterminewhythispictureisclaimingtobesomethingthatitisnot.

4) Isitpossibletobuildapolyhedronwhereallofthefacesarecongruentregularpolygonsbutthepolyhedronisnotregular?Explain.

5) Usethefivemodelsyoubuilttofindaformulathatrelatesnumbersofvertices,edges,andfacesinregularpolyhedra.

6) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Session2MysteryBagSorting.Makeupyourownmysteryforyourstudentstosolvesimilartothosegiveninthelesson.

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ClassActivity20:PyramidsandPrisms Itisbettertoknowsomeofthequestionsthanalloftheanswers. JamesThurberTherearetwospecialcategoriesofpolyhedrawewillexploreinthisactivity:pyramidsandprisms.Apyramidisapolyhedroninwhichallbutoneofthefacesaretrianglesthatshareacommonvertex(calledtheapex).Theremainingfacemaybeanypolygonandiscalledthebase.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.

5) Sketchaprismwithatriangularbaseandonewithahexagonalbase.Thenuseyourpicturestofindformulasforthenumberoffaces,thenumberofvertices,andthenumberofedgesinaprismwhosebasesarecongruentn-gons.Provethatyourformulaworksinthecaseofallprisms.

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ReadandStudy

Themediocreteachertells.Thegoodteacherexplains.Thesuperiorteacherdemonstrates.Thegreatteacherinspires.

WilliamArthurWardThemostfamouspyramidsaretheEgyptianpyramids.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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ConnectionstotheElementaryGrades Itisthesupremeartoftheteachertoawakenjoyincreativeexpressionandknowledge. AlbertEinsteinElementarystudentsbegintheirstudyof3-dimensionalshapesinthefirstandsecondgrades.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Sessions4and5.Whatideasaboutprismsandpyramidsareemphasizedinthesesections?

NowlookattheHomelinksectionfromSession5.Examinequestions4and5carefully.Determineseveralreasonswhyyourchosenfiguredoesnotbelong.Inquestion5,determineatleasttwodifferentfiguresonwhichtoplacethe“X”.

Homework

Iattributemysuccesstothis:Inevergaveortookanyexcuse. FlorenceNightingale

1) DotheproblemintheConnectionssection.

2) Makeamathematicalargumentfortheformulasyoufoundforthenumberoffaces,thenumberofvertices,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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ClassActivity21: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.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.Forexamplethebuildingbelowcanbeviewedfromthetop,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

134

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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ClassActivity22:NothingButNet Puremathematicsis,initsway,thepoetryoflogicalideas. AlbertEinstein

1) Figureouthowtobuildapapermodelofarightcylinderwitharadiusof3cmandaheightof10cm.Useyourmodeltohelpyoufindaformulaforthesurfaceareaofacylinder.

2) Buildapapermodelofapyramidthathasa6cmby6cmsquarebaseandaheightof4cm.Whatisitssurfacearea?

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ClassActivity23:BuildingBlocks

Measurewhatismeasurable,andmakemeasurablewhatisnotso. GalileoVolumeisameasureoftheamountofspaceenclosedbyathree-dimensionalobject.Onewaytodefinevolumeisasthenumberofcubes(cubicunits)thatittakestofilltheenclosedspace.Usingthisdefinition,wecanmeasurevolumebycarefullyestimatingthenumberofcubesthatfitwithintheobject.Someobjectshaveformulasthatwillhelpustocomputevolume.Whatisaformulaforthevolumeofarectangularprismwithlengthl,widthw,andheighth?Whydoesitmakesense?Next, you are going to build three objects out of small wooden cubes (with half-inch edges) and large wooden cubes (with one-inch edges). Follow the steps below. 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?

138

ReadandStudy

Mathematicsisnotacarefulmarchdownawell-clearedhighway,butajourneyintoastrangewilderness,wheretheexplorersoftengetlost.

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

Willthatsameideaworkforacylinder?Isitsvolumetheareaofitsbasemultipliedbyitsheight?Makeasketchofacylinderanduseittohelpyoutoexplainyourthinking.ConnectionstotheElementaryGrades:

Thecureforboredomiscuriosity.Thereisnocureforcuriosity. DorothyParker

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

139

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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ClassActivity24:VolumeDiscount ThelawsofnaturearebutthemathematicalthoughtsofGod.

Euclid1)Reasonwithunitcubestocreatevolumeformulasforthefollowingobjects:

a) arectangularprism

b) atriangularprism

c) acylinder2) 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.

ReneDescartesIntheClassActivityyouobservedthatthevolumeofaprismisaboutthreetimesthevolumeofapyramidwiththesameheightandbase,andthatthevolumeofacylinderisthreetimesthevolumeofaconewiththesameheightandradius.Itturnsoutthatfortheidealobjects,thefactorofthreeisexactlycorrect.Thesenicerelationshipsmakeformulasforvolumerelativelystraightforwardifwebuildonformulaswealreadyknow.InanearlierClassActivityyouexplainedwhyitmakessensethatthevolumeofarectangularprismisV=(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.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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Puremathematicsis,initsway,thepoetryoflogicalideas.AlbertEinstein

Childreningradethreeshouldhaveexperiencesmeasuringvolumesusingwaterandweighingphysicalobjects.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.

146

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.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3Session4.Howdoesthisactivityfurtherstudents’understandingofvolume?

147

ClassActivity25:VolumeChallenge

Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeis.

JohnLouisvonNeumann

1) Yourgroupshouldworktogethertobuildpapermodelsofeachofthefollowingobjectsinsuchawaythatthevolumeofeachis60cm3.(Otherthanthat,youmayuseanydimensionsyoulike.)

a) Cylinderb) Rectangularprismthatisnotacubec) Pyramidwithasquarebased) Prismwithanequilateraltrianglebase

2) Findthesurfaceareaofeachoftheobjectsyoubuiltin1)above.

148

ReadandStudyandConnections It'snotthatI'msosmart;it'sjustthatIstaywithproblemslonger. AlbertEinsteinBythetimetheyreachupperelementaryschool,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.

149

Homework

Theelevatortosuccessisnotrunning;youmustclimbthestairs.ZigZiglar


2) Atriangleonacoordinateplanehasverticesat(2,0),(7,0)and(7,8).a) Sketchthepolygononacoordinateplane.b) Whatisthelengthofthehypotenuse?c) Whatistheareaofthepolygon?d) WhatCommonCoreStateStandardisaddressedbythisproblem?

3) AreplicaoftheGreatPyramidstands2feettallandis3.15feetlongonaside(ithasasquarebase).

a) Approximatelyhowmuchvolumedoesthisreplicatakeup?UsethemodelyoubuiltintheClassActivitytohelpyoutothinkaboutthisproblem.

b) Whatisthesurfaceareaofthereplica?c) Supposethescaleofthereplicatotherealthingisinis1footto240feet.Whatis

thevolumeofGreatPyramid?Whatisitssurfacearea?d) WhichoftheCommonCoreStateStandardsaremetbythisseriesofproblems?

150

SummaryofBigIdeasfromChapterFour

Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes

• Thereareexactlyfiveregularpolyhedra–andchildrencanunderstandwhythisisso.

• Surfaceareaisameasureofthesumofareasofthefacesofathree-dimensionobject.Itisthenumberofsquareunitsittakestocoverthefacesoftheobject.Aunitofsurfaceareaisflatlikethis:

• Volumeisameasureofthenumberofcubicunitsittakestofillathreedimensionalobject.Aunitofvolumeisthree-dimensionalandlookslikethis:

• Volumecanalsobemeasuredbytheamountofliquid(orsand)ittakestofillathreedimensionalobject.

• Thevolumeofaprismorcylindercanbefoundbycomputingtheareaofthebaseandmultiplyingthatbyitsheight(thenumberoflayersofthebase).Thisideaworksforbothrightandobliqueobjects.

• Ittakesthevolumeofthreepyramidstofillaprismwiththesamebaseandheight,andittakesthevolumeofthreeconestofillacylinderwiththesamebaseandheight.

• Childrenneedtohavemanyyearsofexperiencesbuildingandusingavarietyofmodels.

151

APPENDICES

152

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.

153

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.

154

Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.

Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.

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.

155

Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.

Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.

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

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.

156

Proposition34.Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.

Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.

Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.

Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.

Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.

Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

157

Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.

Proposition46.Onagivenstraightlinetodescribeasquare.

Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.

Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.

158

Glossary

"WhenIuseaword,"HumptyDumptysaid,inaratherscornfultone,"itmeansjustwhatIchooseittomean-neithermorenorless.""Thequestionis,"saidAlice,"whetheryoucanmakewordsmeansomanydifferentthings.""Thequestionis,"saidHumptyDumpty,"whichistobemaster-that'sall." LewisCarroll,ThroughtheLookingGlass

Acuteangle–ananglethatmeasureslessthan90degrees

Acutetriangle–atrianglewiththreeacuteangles

Adjacentangles–twonon-overlappinganglesthatshareavertexandacommonray

Alternateexteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines

thatlieoutsidethelinesandonoppositesidesofthetransversal

Alternateinteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines

thatliebetweenthelinesandonoppositesidesofthetransversal

Angle–thefigureformedbytworayswithacommonendpoint

Anglebisector–thelinethroughthevertexofananglethatdividestheangleintotwo

congruentangles

Apex(ofapyramid)–thecommonpointofthenon-basefacesofapyramid

Apex(ofacone)–thecommonpointofthelinesegmentsthatcreateacone

Arc–thesetofpointsonacirclebetweentwogivenpointsofthecircle(Thereareactuallytwo

arcsbetweenanytwogivenpoints;theshorteroneiscalledtheminorarcandthe

longeroneiscalledthemajorarc.)

Area–thequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigure

Attribute–apropertyofageometricobjectthatcanbemeasured(suchaslength)or

categorized(suchascolor)

Axiom–astatementthatweagreetoacceptastruewithoutproof

Axiomaticsystem–asetofundefinedterms,definitions,axioms,andtheoremsthatcreatea

mathematicalstructure

Axis(ofacone)–thelinejoiningtheapextothecenterofthe(circle)base

Axisofsymmetry–alineinspacearoundwhichathree-dimensionalobjectisrotated

159

Baseangles(ofanisoscelestriangle)–theanglesthatareoppositethecongruentsidesofan

isoscelestriangle

Bilateralsymmetry–anobjecthasbilateralsymmetrywhenithasexactlyonelineof

reflectionalsymmetry

Bisect–todivideageometricobjectsuchasalinesegmentoranangleintotwocongruent

pieces

Boundary–thesetofpointsthatseparatetheinsideofaclosedplanarobjectfromtheoutside

Center(ofacircle)–thepointthatisequidistantfromallpointsonthecircle

Centralangle–ananglewhosevertexisacenterofageometricobject

Chord–alinesegmentwhoseendpointsaredistinctpointsonagivencircle

Circle–thesetofallpointsintheplanethatarethesamedistancefromagivenpoint,called

thecenter

Circumcenter–thepointofintersectionofthethreeperpendicularbisectorsofatriangle

Circumference–thecircumferenceofacircleisitperimeter

Circumscribedcircle–thecirclethatcontainsalltheverticesofapolygon

Closedcurve–acurvethatstartsandstopsatthesamepoint

Coincide–twoobjectsaresaidtocoincideiftheycorrespondexactly(areidentical)

Collinearpoints–pointsthatlieonthesameline

Compass–aninstrumentusedtoconstructacircle

Complementaryangles–twoangleswhosemeasuressumto90degrees

Concavepolygon–apolygonforwhichatleastonediagonalliesoutsidethepolygon

Concurrentlines–threeormorelinesthatintersectinthesamepoint

Cone(circular)-athree-dimensionalgeometricobjectconsistingofalllinesegmentsjoininga

singlepoint(calledtheapex)toeverypointofacircle(calledthebase)

Congruentobjects–twogeometricobjectsthatcoincidewhensuperimposed

Conjecture–aguessorahypothesis

Contrapositive(of“IfA,thenB.”)–“IfnotB,thennotA,”whereAandBarestatements

Converse(of“IfA,thenB.”)–“IfB,thenA,”whereAandBarestatements

Convexpolygon–apolygonallofwhosediagonalslieinsidethepolygon

160

Coordinateplane–aplaneonwhichpointsaredescribedbasedontheirhorizontalandvertical

distancesfromapointcalledtheorigin

Coplanarlines–linesthatlieinthesameplane

Correspondingangles-twoanglesformedbyatransversalofapairoflinesthatlieonthe

samesideofthetransversalandalsolieonthesamesideofthepairoflines

Correspondingpoints–apairofpoints,oneofwhichistheoriginalpointandtheotherof

whichistheimageofthatpointunderatransformation

Counterexample–anexamplethatdemonstratesthatastatement(conjecture)isfalse

Curve–asetofpointsdrawnwithasinglecontinuousmotion

Cylinder(circular)–athree-dimensionalgeometricobjectconsistingoftwoparalleland

congruentcircles(andtheirinteriors)andtheparallellinesegmentsthatjoin

correspondingpointsonthecircles

Decagon–apolygonwithexactlytensides

Deductivereasoning–theprocessofcomingtoaconclusionbasedonlogic

Degree–aunitofanglemeasureforwhichafullturnaboutapointequals360degrees

Diagonal–thelinesegmentjoiningtwonon-adjacentverticesofapolygon

Diameter–alinesegmentthroughthecenterofacirclewhoseendpointslieonthecircle

Dilation–amotionoftheplaneinwhichonepointPremainsfixedandallotherpointsare

pushedradiallyoutwardfromPorpulledradiallyinwardtowardPsothatalldistances

havebeenmultipliedbysomescalefactor

Dimension(ofarealspace)–thenumberofmutuallyperpendiculardirectionsneededto

describethelocationofthesetofpointsinthatspace

Distance(onacoordinateplane)–thesizeoftheportionofastraightlinethatliesbetweenthe

twopointsonthecoordinateplaneasmeasuredbythedistanceformula: 22 bad +=

,whereaisthehorizontaldistancebetweenthepoints(asmeasuredonthex-axis)and

bistheverticaldistance(asmeasuredonthey-axis)

Distinct(geometricobjects)-twoobjectsthatdonotsharealltheirpointsincommon

Dodecagon–apolygonwithexactlytwelvesides

161

Dual(ofaregularpolyhedron)–thepolyhedronwhoseverticesareexactlythemidpointsofthe

facesoftheregularpolyhedron

Edge–thelinesegment(side)thatissharedbytwofacesofapolyhedron

Endpoint(ofalinesegment)–oneoftwopointsthatdeterminesalinesegment

Equiangularpolygon–apolygonallofwhosevertexanglesarecongruent

Equilateralpolygon–apolygonallofwhosesidesarecongruent

Example(ofadefinition)–ageometricobjectthatsatisfiestheconditionsofthedefinition

Exteriorangle–theangleformedbyasideofapolygonandtheextensionofanadjacentside

Face–apolygon(withinterior)thatformsaportionofthetwo-dimensionalsurfaceofa

polyhedron

Fixedpoint–apointwhoselocationremainsthesameunderatransformation

Generalization–theextensionofastatement(aboutapattern)thatistrueforspecificvalues

ofn(anaturalnumber)toastatement(aboutthatpattern)thatistrueforallvaluesofn

Geoboard–amanipulativetypicallycomposedofaboardwith25pegsarrangedina5x5

squarearray

Geoboardpolygon–apolygonwhoseverticesareallpoints(pegs)onageoboard

Height(ofatriangle)–lengthofthelinesegmentfromavertexperpendiculartotheopposite

side.Thislinesegmentisoftencalledthealtitudeofthetriangle

Heptagon–apolygonwithexactlysevensides

Hexagon–apolygonwithexactlysixsides

Homogeneous(verticesinatessellation)–verticesthathaveexactlythesamepolygons

meetinginexactlythesamearrangement

Homogeneous(verticesinapolyhedron)–verticesthathaveexactlythesamepolygonfaces

meetinginexactlythesamearrangement

Hypotenuse–thesideofarighttriangleoppositetherightangle

Image(ofatransformation)–thesetofpointsthatresultfromthemotionofanobjectbya

translation,arotation,orareflection

Inductivereasoning–theinformalprocessofcomingtoaconclusionbasedonexamples

Inscribedpolygon–thepolygoninsideacirclewhoseverticesalllieonthecircle

162

Interiorangle–anyoneofthealternateinterioranglesformedbyatransversaltotwolines

Intersection(oftwolines)–thepointthelineshaveincommon

Intersection(oftwosets)–thesetofelementsthatarecommontobothsets

Isosceles–havingatleastonepairofcongruentsides

Justification–anargumentbasedonaxioms,definitions,andpreviouslyprovenresultstoshow

thataconjectureistrue

Leg–asideofarighttriangleoppositeanacuteangle

Length–thedistancebetweentwopointsonaone-dimensionalcurve

Line–anundefinedone-dimensionalsetofpointsunderstoodcovertheshortestdistanceand

toextendinoppositedirectionsindefinitely

Lineofreflection–thelineaboutwhichanobjectisreflectedtoformitsmirrorimage

Linesegment–thesetofallpointsonalinebetweentwogivenpointscalledtheendpoints

Mass–aconceptofphysicsthatcorrespondstotheintuitiveideaof“howmuchmatterthereis

inanobject;”unlikeweight,themassofanobjectdoesnotdependupontheobject’s

locationintheuniverse

Measure–todeterminethequantityofanattribute(orofafundamentalconceptsuchastime)

usingagivenunit

Metricsystemofmeasurement–thesystemofmeasurementunitsinwhichthereisone

fundamentalunitdefinedforeachquantity(attribute)withmultiplesandfractionsof

theseunitsestablishedbyprefixesbasedonpowersoften

Midpoint–thepointonalinesegmentthatdividesitintotwocongruentlinesegments

Model–arepresentationofanaxiomsysteminwhicheachundefinedtermisgivenaconcrete

interpretationinsuchawaythattheaxiomsallhold

Net–atwo-dimensionalfigurethatcanbefoldedintoathree-dimensionalobject

Nonagon–apolygonwithexactlyninesides

Noncollinear(points)–asetofpointsnotallofwhichlieonthesameline

Non-example(ofadefinition)–anexamplethatdemonstratemostoftheconditionsofa

definitionbutthatfailstosatisfyatleastonecondition

163

Non-standardunitofmeasure–aunitofmeasurewhosevalueisnotestablishedbyreference

toanacceptedstandard;forexample,ablock

Obliqueprism(pyramid,cylinder)–aprism(pyramid,cylinder)thatisnotright

Obtuseangle–ananglewithmeasuregreaterthan90degrees

Obtusetriangle–atrianglewithoneobtuseangle

Octagon–apolygonwithexactlyeightsides

Order(ofarotationalsymmetry)–thenumberofdifferentrotationsthatareasymmetryofan

object

Orientation–thedirection,clockwiseorcounterclockwise,ofthereadingoftheverticesofa

polygoninalphabeticalorder

Parallellines–coplanarlineswithnopointsincommon

Parallelogram–aquadrilateralinwhichbothpairsofoppositesidesareparallel

Partition–adivisionofageometricobjectintoasetofnon-overlappingobjectswhoseunionis

theoriginalobject

Pentagon–apolygonwithexactlyfivesides

Perimeter(ofaplaneobject)–thelengthoftheboundaryoftheobject

Perpendicularbisector–thelinethroughthemidpointofalinesegmentthatisalso

perpendiculartothelinesegment

Perpendicularlines–twolinesthatintersecttoformfourrightangles

Pi(p)–theexactnumberoftimesthediameterofacirclefitsintoitscircumference(orthe

ratioofthecircumferenceofacircletoitsdiameter);thisratioisanirrationalnumber

thatisconstantforallsizecirclesandisapproximatelyequalto3.14159

Planarcurve–acurvethatliesentirelywithinaplane

Plane–anundefinedtwo-dimensionalsetofpointsunderstoodtobe“flat”andtoextendinall

directionsindefinitely

Planeofsymmetry–aplaneinspaceaboutwhichathree-dimensionalobjectisreflected

Platonicsolid–aregularpolyhedronplusitsinterior

Point–anundefinedzero-dimensionalobject;alocationwithnosize

Polygon–afinitesetoflinesegmentsthatformasimpleclosedplanarcurve

164

Polyhedron(plural:polyhedra)–afinitesetofpolygon-shapedfacesjoinedpairwisealongthe

edgesofthepolygonstoencloseafiniteregionofspacewithinonechamber

Prism–apolyhedroninwhichtwoofthefaces(calledthebases)arecongruentandlie on parallel

(non-intersecting) planes andtheremainingfacesareparallelograms.

Proof–adeductiveargumentthatestablishesthetruthofaclaim

Protractor–aninstrumentusedtomeasureangles

Pyramid–apolyhedroninwhichallbutoneofthefacesaretrianglesthatshareacommon

vertex(calledtheapex);theremainingfacemaybeanypolygonandiscalledthebase

Pythagoreantriple–threepositiveintegersthatsatisfythePythagoreantheorem

Quadrilateral–apolygonwithexactlyfoursides

Quantifier(inlogic)–awordorphrase(suchas“all”or“atleastone”)thatindicatesthesizeof

thesettowhichthestatementapplies

Radius(plural:radii)–thelinesegmentjoiningapointonacircletothecenterofthecircle

Ray–thesetofpointsonalinebeginningatagivenpoint(calledtheendpoint)andextending

inonedirectiononthelinefromthatpoint

Rectangle–aquadrilateralwithfourrightangles

Reflection–(inalinel)isatransformationoftheplaneinwhichtheimageofapointPonlisP,

andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisector

of 'AA

Reflectionsymmetry(2-dimensional)–areflectioninwhichanobjectisdividedbythelineof

reflectionintotwopartsthataremirrorimagesofeachother

Reflectionsymmetry(3-dimensional)–areflectioninwhichanobjectisdividedbytheplaneof

reflectionintotwopartsthataremirrorimagesofeachother

Regularpolygon–apolygonwithallsidescongruentandallvertexanglescongruent

Regularpolyhedron–apolyhedronwhosefacesareeachthesameregularpolygonwiththe

samenumberoffacesmeetingateachvertex

Regulartessellation–atessellationthatcontainsonlyoneregularpolygon

Rhombus(plural:rhombi)–aquadrilateralwithfourcongruentsides

Rightangle–ananglethatisexactlyonefourthofacompleteturnaboutapoint

165

Rightprism(pyramid,cylinder)–aprism(pyramid,cylinder)inwhichthelinejoiningthe

centersofthebases(theapexofthepyramidtothecenterofitsbase)isperpendicular

tothebase

Righttriangle–atrianglewithonerightangle

Rigidmotions-transformationsoftheplanethatpreservedistancesbetweenpoints(theydo

notdistorttheshapeorsizeofobjects)

Rotation(aboutapointPthroughanangleq)–atransformationoftheplaneinwhichthe

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

calledthecenteroftherotation

Rotationsymmetry(2-dimensional)–arotationaboutapointinwhichtheimagecoincides

withtheoriginalobject

Rotationsymmetry(3-dimensional)–arotationaboutanaxisofsymmetryinwhichtheimage

coincideswiththeoriginalobject

Scalenetriangle–atrianglenoneofwhosesidesarecongruent

Scaling–atransformationoftheplanethatcauseseitheramagnificationorashrinkingofan

objectinwhichtheimageremainssimilartotheoriginalobject

Secant–alinethatintersectsacircleintwodistinctpoints

Sector–theportionofacircleanditsinteriorbetweentworadii

Semiregularpolyhedron–apolyhedronthatcontainstwoormoreregularpolygonsasfaces

whicharearrangedsothatallverticesarehomogeneous

Semiregulartessellation–atessellationthatcontainstwoormoreregularpolygonsarranged

sothattheverticesarehomogeneous

Shearing–atransformationoftheplanethatchangestheshapeofanobject

Side–oneofthelinesegmentsthatmakeupapolygon

Similarobjects–objectswhereonecanbeobtainedfromtheotherbycomposingarigid

motionwithadilation

Simplecurve–acurvethatdoesnotintersectitself

Slope(ofaline)–theverticaldistancerequiredtostayonalineforaoneunitchangein

horizontaldistance

166

Space–anundefinedtermthatdenotesthesetofpointsthatextendsindefinitelyinthree

dimensions

Sphere–thesetofpointsin(three-dimensional)spacethatareequidistantfromagivenpoint,

calledthecenter

Square–aquadrilateralwithfourrightanglesandfourcongruentsides

Standardunitofmeasure–aunitofmeasurewhosevalueisestablishedbyreferencetoan

acceptedstandard;forexample,themeterisdefinedtobeoneten-millionthofthe

distancefromtheequatortothenorthpole

Straightangle–ananglethatmeasureshalfaturn(ortworightangles)

Straightedge–aninstrumentusedtoconstructlinesegments

Supplementaryangles–twoangleswhosemeasuressumto180degrees

Surface–thesetofpointsthatformtheboundaryofasolidthree-dimensionalobject

Surfacearea–thesumoftheareasofthefacesofaclosed3-dimensionalobject

Symmetry(ofanobject)–atransformationoftheobjectinwhichtheimagecoincideswiththe

original

Tangent(toacircle)–alinethatintersectsacircleinexactlyonepoint

Tessellation–anarrangementofpolygonsthatcanbeextendedinalldirectionstocoverthe

planewithnogapsandnooverlapsinsuchawaythatverticesonlymeetothervertices

Theorem–astatementthathasbeenproventrue

Tiling–anarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplane

withnogapsandnooverlaps

Time–ameasurablepartofthefundamentalstructureoftheuniverse,adimensioninwhich

eventsoccurinsequence

Transformation–amovementofthepointsofaplanethatmaychangethepositionorthesize

andshapeofobjects

Translation(byavectorAA’)–arigidmotionoftheplanethattakesAtoA’,andforallother

pointsPontheplane,PgoestoP’wherevectorPP’andvectorAA’havethesame

lengthanddirection

167

Translationsymmetry–atranslationoftheplanesuchthattheimagecorrespondstothe

originalobject

Translationvector–anarrowthatgivesthedirectionanddistance(itslength)thatapointis

movedduringatranslation

Transversal–alinewhichintersectstwoormorelines

Trapezoid–aquadrilateralwithexactlyonepairofparallelsides

Triangle–apolygonwiththreesides

Trivialrotation–therotationof360°;itisarotationalsymmetryofeveryobject

Undefinedterm–atermwhichhasanintuitivemeaning,butnoformaldefinition

Union(ofsets)–thesetcontainingeveryelementofeachset

Unitofmeasure–anobjectisusedforcomparisonwithanattribute

Unitsquare–asquarethatisoneunitbyoneunitandthushasanareaofonesquareunit

Unitcube–acubethatisoneunitbyoneunitbyoneunitandhasavolumeofonecubicunit

Venndiagram–apictureinwhichtheobjectsbeingstudiedarerepresentedaspointsona

planeandsimpleclosedcurvesaredrawntogroupthepointsintodifferent

classifications.Venndiagramsareusedtovisualizerelationshipsamongsetsofobjects.

Vertex(plural:vertices)–thecommonendpointoftwoadjacentsidesofapolygon

Vertexangle–theangleformedbyadjacentsidesofapolygon

Vertex(ofapolyhedron)–theintersectionoftwoormoreedgesofapolyhedron

Verticalangles–anonadjacentpairofanglesformedbytwointersectinglines

Volume–ameasureofthecapacityofa3-dimensionalobjector,alternatively,thequantityof

spaceenclosedbya3-dimensionalobject

Weight–ameasureoftheforceofgravityonanobject;oftenusedinterchangeablywithmass;

differencesinthemeasuresofweightandmassarenegligibleatsealevelonearth

168

GridPaper

170

References

Givecreditwherecreditisdue. Authorunknown

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

• CommonCoreStateStandardsformathematicsasfoundin2012athttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

• 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.

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SchoolMathematics.Reston,VA:NCTM.

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

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Big Ideas in Geometry and Measurement · 2018-06-19 · triangle or a circle, but it won’t have...

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