Big Ideas in Mathematics for Future Elementary Teachers Big Ideas in Geometry and Measurement John Beam, Jason Belnap, Eric Kuennen, Amy Parrott, Carol E. Seaman, and Jennifer Szydlik (Updated Summer 2017)
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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ConnectionstotheElementaryGrades
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.
(continuedonthenextpage)
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
36
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?
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
11) DrawaVenndiagramshowingtherelationshipbetweenallthevariousquadrilateralswehavestudied.HowdoesthisfitwiththeCommonCoreStateStandardsdescribedbelow?
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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:
(continuedonthenextpage)
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3) Findaformulaforthesumofthemeasuresofthevertexanglesofann-gon(apolygonwithnsides).Youmayneedtocollectsomedata.Intheend,makeamathematicalargumentthattheformulayoufindwillworkforaconvexpolygonofanynumberofsides.(Theformulathatyoudevelopedwillworkforconcavepolygonsaswell,buttheargumentistrickier,sowearenotaskingyoutojustifyitatthistime.)
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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.
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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
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Whenyoutalkaboutangleswithchildren,wesuggestthatyoualwaysuseyourhandorarmtoshowthesweepoftheangleinadditiontoshowingthestaticpicture.
Childreningradefourlearntouseaprotractortomeasureanglesindegreesandtoaccuratelyestimatethemeasureofangles.BelowyouwillfindtheCCSSrelatedtoanglemeasureinthisgrade.Readthemcarefully.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
CCSSGrade4:Geometricmeasurement:understandconceptsofangleandmeasureangles.
1. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:
a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommon
endpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
2. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesof
specifiedmeasure.
3. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon-overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.
CCSSGrade4:Drawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.
1. Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo-dimensionalfigures.
2. Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.
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Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module1.PrintoutandthenworkthroughtheMeasuringPatternBlockAnglesactivity.(Notethatinthisactivitystudentsarenotusingprotractorstomeasuretheangles.Youalsowillnotneedtouseaprotractor.).
Homework Energyandpersistenceconquerallthings. BenjaminFranklin
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
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
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ClassActivity5:ALogicalInterlude
Equationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsof geometry.
StephenHawkingInthepicturebelow,eachcardhasacolorononesideandashapeontheotherside.Whichcard(s)wouldyouhavetoturnovertobesurethatthefollowingstatementistrue?
Ifacardisredononeside,thenithasasquareontheotherside.
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ReadandStudy
Whenintroducedatthewrongtimeorplace,goodlogicmaybetheworstenemyofgoodteaching.
GeorgePolya TheAmericanMathematicalMonthly,v.100,no3.HerearetwotheoremsaboutparallellinesthatcanbeprovedfromEuclid’saxioms.
1) Iftwolinesarecutbyatransversalandthealternateinterioranglesarecongruent,thenthelinesareparallel.
2) Iftwoparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.
Drawasketchtomakesurethatyouseewhateachofthesehastosay.
Noticethattheybothhavean‘if-then’statementform.Lotsofmathematicaltheoremsarelikethis.Whenatheoremisstatedin‘if-then’form,whateverfollowsthe‘then’isalwaystruewhenevertheconditionsstatedinthe‘if’partaremet.Youcanthinkofan‘if-then’statementasapromisethatiskeptunlessthe‘if’partistrueandthe‘then’partisnot.Wecallstatement2)theconverseofstatement1).Thesetwotheoremsmaysoundthesametoyou,buttheyarenotsayingthesamething.Tohelpyouseethis,let’schangethecontext.Hereisanotherpairofstatementsinwhichthesecondistheconverseofthefirst(andviceversa):
3) IfIliveinChicago,thenIliveinIllinois.
4) IfIliveinIllinois,thenIliveinChicago.Thinkabouteachofthestatements.Whichoftheseistrue?Sinceoneistrueandtheotherfalse,thesetwostatementscannotbesayingthesamething.Inotherwords,theconverseofastatementisalogicallydifferentstatementfromtheoriginalstatement.Nowhereisastatementthatislogicallyequivalenttotheoriginalstatementmadein3)above.Itiscalledthecontrapositiveofstatement3).
5) IfIdonotliveinIllinois,thenIdonotliveinChicago.
ThisisessentiallywhatyoufoundwhenyouworkedontheClassActivity.Writethecontrapositivetostatement4)above.
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Itishelpfultoknowthatastatementanditscontrapositiveareequivalentbecausethatmeansthatyoucanproveastatementbyprovingitscontrapositive.ConnectionstotheElementaryGrades
Thebeginningofknowledgeisthediscoveryofsomethingwedonotunderstand. FrankHerbert
Nowthatwehavetalkedabitaboutdoingmathematics,wewanttoshowyouthattheCommonCoreStateStandardsrequirethatchildrenalsodomathematics.Giveanexampleofsomethingyouhavedoneinclasssofarthistermthatmeetseachofthesestandards.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
CommonCoreStateStandardsforMathematicalPractice
Childrenshould…
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
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Homework
Amultitudeofwordsisnoproofofaprudentmind. Thales
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
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?
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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°
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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.
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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:
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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.
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Homework
Homecomputersarebeingcalledupontoperformmanynewfunctions,includingthe consumptionofhomeworkformerlyeatenbythedog.
DougLarson
1) DoalltheitalicizedthingsintheReadandStudysection.
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°.
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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.
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ChapterTwo
Transformations,Tessellations,andSymmetries
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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
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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
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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
1) DoalltheitalicizedthingsintheReadandStudysection.
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.
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3) Belowyouwillfindacoordinategrid.Applythefollowingthreetranslationstoatrianglewithverticesinitiallylocatedat(0,0),(-2,-3),and(3,-3).Whatisashortcutwayofdoingpartc)?
a) up5,left3 b)down2,right4 c)up5,left3followedbydown2,right4
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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.
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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
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Homework
Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair. JohnQuincyAdams
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Usethegridtorotatetheplane90degreescounterclockwiseaboutpointP.Showtheimageofthefiguresaftertherotation.
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3) Decideifeachofthestatementsaboutrotationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.
a) Correspondingsidesofanobjectanditsrotatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderarotation,thenthelinesegmentjoining
vertexAtovertex A goesthroughthecenteroftherotation.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe
parallelsidesofitsimagetobeverticalafterarotation.d) IfAand A andBand B arecorrespondingpointsunderarotation,itispossible
forthelines AA and BB tointersect.e) Everypointintheplanemovestoanewpositionunderarotation,i.e.,thereare
nofixedpointsinarotation.f) 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
(Thisactivityiscontinuedonthenextpage.)
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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
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Onthesquaregridbelow,uselinenasthelineofreflectiontoreflectthegivenobjects.Labeleachimageappropriately.
3) Whatrelationshipsdoyouseebetweenoriginalfiguresandtheirreflectedimages?Betweentheobjects,theirimagesandthegivenlineofreflection?Makeasmanyconjecturesasyoucanaboutreflections.
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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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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.
… …
ConnectionstotheElementaryGrades
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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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
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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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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
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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.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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?
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide.Spend10-15minuteslookingthroughUnit4Module1Session1TeacherFeet.Inwhatwaysdothechildrenneedtoanalyzetheprocessofmeasurementinthissession?
Homework
Manyoflife'sfailuresarepeoplewhodidnotrealizehowclosetheyweretosuccesswhentheygaveup.
ThomasA.Edison
1) DoalltheitalicizedthingsintheReadandStudysection.
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.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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.
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Homework
ThemoreIpractice,theluckierIget.JerryBarber
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
3) Useagridtoestimatecarefullytheareaofthebelowcircularfigureinsquare
centimeters.
4) Astudentcomestoyouandasks,"Whydoweusesquarecentimeterstomeasuretheareaofthecircle?Acircleisroundandnotsquare."Explaintoherwhywestillusesquarecentimeterstomeasuretheareaofacircle.
5) WewouldliketohavetheabovelakeclassifiedasoneoftheGreatLakes.AspartoftheapplicationtotheDepartmentoftheInterior,wehavetoreportitsarea.Estimatetheareaofthelakeinsquaremiles.
Scale1cm=3miles
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6) Hereisafloorplanforthefirstlevelofahouse.a) Whatisitsareainsquarefeetincludingthegarage?Assumethat1cm
represents6feetoflength.b) Howgoodisyourestimate?Areyouconfidenttothenearestsquarefoot?c) WhatCommonCoreStateStandardismetbythisproblem?Explain.
7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade3TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3.
a. Howareareaandperimeterintroduced?b. PrintoutandthencarefullyworkthroughtheworksheetBayardOwl’sBorrowed
Tables.Whatconversationscouldyouhavewithyourstudentsaboutthebigideasofperimeterandareabasedonthisactivity?
8) AnNBAbasketballcourtmeasures50feetby96feet.Usethisinformationbelowtodeterminehowmanyacresitcovers.
1foot=12inches1yard=3feet1mile=1760yards1acre=4840squareyards1squaremile=640acres
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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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http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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.
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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.
* * * * * *
* * * * * *
* * * * * *
* * * * * *
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6) Supposearectanglehasaperimeterof36units.Whatareallthepossiblewholenumberdimensionsoftherectangle?Makeagraphofwidthvs.area.Whichwidthgivesthegreatestarea?
7) Supposeyouhave100metersofflexiblefencingtomarkapastureoutontheplains.
Howwouldyousetitup(whatshape)toenclosethemostgrazingareaforyourcattle?Whatdimensionswouldyouuseifthepasturehadtobearectangle?
8) Thetrianglebelowisconstructedona1cmgrid.Findtheareaofthetriangleusingatleastthreedifferentmethods.
9) Assumethatthetriangleareaaboverepresentspartofasignthatneedstobepainted.
Thescaleofthedrawingisthat1cmrepresents10feet.Theinstructionsonthepaintcansaythat1gallonofpaintwillcover100squarefeet.Howmanygallonsofpaintwillyouneedtobuyinordertopaintthetriangle?
1 cm
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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?
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ReadandStudy
Itisnotoncenortwice,buttimeswithoutnumberthatthesameideasmaketheirappearanceintheworld. Aristotle
CirclesareasfundamentaltoEuclideangeometryasarepointsandlines.RecallthatEuclid’sthirdaxiomassuresusthatwecanalwaysmakeacircleofanysize(radius)wewant.Ofcourse,thecircleswedrawarestillonlyapproximationsofatruemathematicalcircle,justasthelinesegmentswemakearejustapproximationsofatruelinesegment.
Acircleisthesetofallpointsintheplanethatareequidistantfromagivenpoint,calledthecenter(Ointhediagrambelow).Thediagramshowssomeoftheotherimportanttermsassociatedwithacircle.Becertainyouunderstandeachtermandcanexplainitsmathematicaldefinition,whichyouwillfindintheglossary.
Central Angle Ð AOC
Tangent
Secant
Chord DE Diameter AB
Arc DE
Radius CO
Sector O
C
A
B
D
E
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Itisanamazingfactthat,foranysizecircle,theratioofthecircumferencetothediameteristhesamenumber.Thiswasknowninallearlycivilizations.Wecallthatratio“pi.”Soπ(pi)isthesymbolforthenumberoftimesthediameterofacirclefitsintoitscircumference.Readthatagain;itisimportant.Thisnumberpisanirrationalnumber.Thatmeansithasadecimalrepresentationthatneitherendsnorrepeats.Thiswasprovedin1761byamathematiciannamedJohannHeinrichLambert.Evenwhenweusethepkeyonacalculator,weareusinganapproximatevalue.Elementarystudentscommonlyuseeither3.14or
722 asanapproximatevalueforpwhen
carryingoutcalculationsinvolvingcircles.Alwaysbesuretomakethepointthatthisisjustanapproximation.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.
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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?
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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.
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ClassActivity17:PlayingPythagoras
Ihavehadmyresultsforalongtime,butIdonotknowyethowIamtoarriveat
them. CarlFriedrichGauss
ThePythagoreanTheoremisthoughttobealmost4000yearsold.TheBabylonians,theEgyptians,andtheChineseallknewit.Thatis,theyknewthatthesumoftheareaofthesquaresonthelegsofarighttriangleequalstheareaofthesquareonthehypotenuse,andtheyusedthisfactnumericallyinconstructionandcommerceandsurveying.
1) Carefullymeasuretheareasofthesquaresinthebelowexampletoseeifthistheoremseemstrue.
2) Whathappensifthetriangleisn’tarighttriangle?Isthesumofthesquaresonthe“legs”(shortersides)ofanobtusetrianglemoreorlessthanthesquareonthelongestside?Whathappensinanacutetriangle?Drawsomediagramstosee.
(Thisactivityiscontinuedonthenextpage.)
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ThefirstproofofthetheoremisattributedtoPythagorasofSamos(it’sinGreece)around600B.C.Sincethen,hundredsofdifferentproofshavebeencreated.Youaregoingtoexploreoneofthem.a) Theproofbeginswithanyrighttriangle.Carefullydrawyourownandlabelthe
lengthofthehypotenusec,thelengthofthelongerlegb,andthelengthoftheshorterlega.
b) Nowmakefourcongruentcopiesofyourtriangle,cutthemout,andarrangethem
intoaquadrilateralasshownbelow.
c) Justifythattheboundaryoftheouterquadrilateralisasquare.
d) Justifythattheinnerquadrilateralisasquare.
e) Nowgiveanalgebraicproofthatc2=a2+b2byusingthefactthatthefivepolygonsformthelargefigure(sotheareaformulasforthefivemustsumtotheareaformulaofthelargesquare).
f) Whatthingsgowrongifthetrianglesarenotrighttriangles?
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ReadandStudy
I’mverywellacquaintedtoowithmattersmathematical,Iunderstandequations,boththesimpleandquadratical,AboutbinomialtheoremI’mteemingwithalotofnews--Withmanycheerfulfactsaboutthesquareofthehypotenuse.
Gilbert&Sullivan,“ThePiratesofPenzance”
ThePythagoreanTheoremisoneofthemostwell-knownandmostimportanttheoremsofallofelementarymathematics.ItisnamedaftertheGreekmathematician,Pythagoras,andEuclidincludeditasthefittingendtovolumeoneofTheElements.InEuclid’swordsthetheoremsaysthis:
Inright-angledtrianglesthesquareonthesideoppositetherightangleequalsthesumofthesquaresonthesidescontainingtherightangle.
Euclidintendedtheword“square”tomeanthephysicalsquaredrawnonthehypotenuseorlegoftherighttriangle.Sohistheoremsaysthattheareaoftheyellowsquare(inthefigureabove)isequaltothesumoftheareasoftheredandbluesquareswhenABCisarighttriangle.Todaywemorecommonlyuseanalgebraicstatement:
Ifarighttrianglehaslegsoflengthsaandbandahypotenuseoflengthc,thenc2=a2+b2.
Noticethathere,a,b,andcarenumbersrepresentingthelengthsofthevarioussidesofthetriangle.Sonowtheword“square”carriesthealgebraicmeaningdenotedbytheexponenttwo.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.
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Homework
Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackinwill.
VinceLombardi
1) DoalltheitalicizedthingsintheReadandStudysection.
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.)
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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.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
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)
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Homework
Gettingaheadinadifficultprofessionrequiresavidfaithinyourself.Thatiswhysomepeoplewithmediocretalent,butwithgreatinnerdrive,gomuchfurtherthanpeoplewithvastlysuperiortalent.
SophiaLoren
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
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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ConnectionstotheElementaryGrades
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
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Homework
Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.
JohnQuincyAdams
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
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?
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ReadandStudy
Mathematicsisnotacarefulmarchdownawell-clearedhighway,butajourneyintoastrangewilderness,wheretheexplorersoftengetlost.
W.S.AnglinIntheClassActivityyoutalkedaboutwhyitmadesensetocalculatethevolumeofarectangularprismbymultiplyingthelengthloftheprismbythewidthwoftheprismbytheheighthoftheprism(V=l´w´h).Theideahereisthatwecanthinkofaprismaslayersofthebasestackedoneuponthenext.Sovolumeistheareaofthebasemultipliedbytheheight(V=AreaofBase×h).Havealookatthepicturebelowtoseewhatwemean.
Willthatsameideaworkforacylinder?Isitsvolumetheareaofitsbasemultipliedbyitsheight?Makeasketchofacylinderanduseittohelpyoutoexplainyourthinking.ConnectionstotheElementaryGrades:
Thecureforboredomiscuriosity.Thereisnocureforcuriosity. DorothyParker
Childrenoftenbegintheirexplorationsofvolumebydeterminingthevolumeofvariousrectangularboxes.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit1Module1Sessions4and5.Howdoesthisactivityhelpstudentstounderstandvolume?
139
Homework
Whentheworldsays,"Giveup,"Hopewhispers,"Tryitonemoretime."
AuthorUnknown
1) DoalltheitalicizedthingsintheReadandStudysection.
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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ConnectionstotheElementaryGrades
Puremathematicsis,initsway,thepoetryoflogicalideas.AlbertEinstein
Childreningradethreeshouldhaveexperiencesmeasuringvolumesusingwaterandweighingphysicalobjects.HerearetherelevantCommonCoreStateStandards.Readthem.
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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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http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
CCSSGrade5:Geometricmeasurement:understandconceptsofvolumeandrelatevolumetomultiplicationandtoaddition.
1. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.
a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubic
unit”ofvolume,andcanbeusedtomeasurevolume.
b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.
2. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,and
improvisedunits.
3. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.
a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengths
bypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.
b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofind
volumesofrightrectangularprismswithwholenumberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.
c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwo
non-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
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Homework
Toclimbsteephillsrequiresaslowpaceatfirst. WilliamShakespeare
1) DoalltheitalicizedthingsintheReadandStudysection.
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?
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ClassActivity25:VolumeChallenge
Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeis.
JohnLouisvonNeumann
1) Yourgroupshouldworktogethertobuildpapermodelsofeachofthefollowingobjectsinsuchawaythatthevolumeofeachis60cm3.(Otherthanthat,youmayuseanydimensionsyoulike.)
a) Cylinderb) Rectangularprismthatisnotacubec) Pyramidwithasquarebased) Prismwithanequilateraltrianglebase
2) Findthesurfaceareaofeachoftheobjectsyoubuiltin1)above.
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ReadandStudyandConnections It'snotthatI'msosmart;it'sjustthatIstaywithproblemslonger. AlbertEinsteinBythetimetheyreachupperelementaryschool,studentscansolvemanypracticalproblemsingeometry.Howeverthesestudentsarenotreadytosimplyapplyformulastosolveproblems;insteadtheyneedtousemodelsinordertomakesenseofproblems.Astheirteacher,yourjobwillbetomakesurethatchildreninyourclassesbuildandhandleappropriatemodels.NoticethattheCommonCoreStateStandardsforgrade6explicitlyrequirethatstudentsusehands-onmodelstoexploreideas.Studentsareaskedtocuttrianglesandothershapesapart,todrawpictures,topackspaceswithunitcubes,tousethecoordinateplane,andtobuildandusenets.Readthesestandards.Havewedoneofthesethingsinthisbooksofarthisterm?
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
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.
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Homework
Theelevatortosuccessisnotrunning;youmustclimbthestairs.ZigZiglar
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
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?
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SummaryofBigIdeasfromChapterFour
Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes
• Thereareexactlyfiveregularpolyhedra–andchildrencanunderstandwhythisisso.
• Surfaceareaisameasureofthesumofareasofthefacesofathree-dimensionobject.Itisthenumberofsquareunitsittakestocoverthefacesoftheobject.Aunitofsurfaceareaisflatlikethis:
• Volumeisameasureofthenumberofcubicunitsittakestofillathreedimensionalobject.Aunitofvolumeisthree-dimensionalandlookslikethis:
• Volumecanalsobemeasuredbytheamountofliquid(orsand)ittakestofillathreedimensionalobject.
• Thevolumeofaprismorcylindercanbefoundbycomputingtheareaofthebaseandmultiplyingthatbyitsheight(thenumberoflayersofthebase).Thisideaworksforbothrightandobliqueobjects.
• Ittakesthevolumeofthreepyramidstofillaprismwiththesamebaseandheight,andittakesthevolumeofthreeconestofillacylinderwiththesamebaseandheight.
• Childrenneedtohavemanyyearsofexperiencesbuildingandusingavarietyofmodels.
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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.
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BOOKIPROPOSITIONSProposition1.
Onagivenfinitestraightlinetoconstructanequilateraltriangle.
Proposition2.Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.
Proposition3.Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.
Proposition4.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhaveanglescontainedbytheequalstraightlinesequal,theywillalsohavethebaseequaltothebase,thetrianglewillbeequaltothetriangle,andtheremainingangleswillbeequaltotheremaininganglesrespectively,namelythosewhichtheequalsidessubtend.
Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.
Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.
Proposition7.Giventwostraightlinesconstructedonastraightline(fromitsextremities)andmeetinginapoint,therecannotbeconstructedonthesamestraightline(fromitsextremities),andonthesamesideofit,twootherstraightlinesmeetinginanotherpointandequaltotheformertworespectively,namelyeachtothatwhichhasthesameextremitywithit.
Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.
Proposition9.Tobisectagivenrectilinealangle.
Proposition10.Tobisectagivenfinitestraightline.
Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.
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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.
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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.
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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.
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Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.
Proposition46.Onagivenstraightlinetodescribeasquare.
Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.
Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.
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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
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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
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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
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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
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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
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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
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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
169
170
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