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

BigIdeasinMathematics

forFutureElementaryTeachers

BigIdeasinGeometryandMeasurement

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

(UpdatedSummer2017)

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

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

John,Jason,Eric,Amy,Carol&Jen

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.

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.

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

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

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

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

ChapterOne

SeeingtheWorldGeometrically

ClassActivity1:TrianglePuzzleGeometryisthescienceofcorrectreasoningonincorrectfigures.

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

Whathappenedtothemissingsquare?

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

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.

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.

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.

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

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

b) Whatmightchildrenlearnabouttherelationshipsamongthepatternblocksbyworkingontheseproblems?

3) HereisapictureofallsevenTangrampiecesrearrangedtomakeatriangle.

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

a) Arguethatthevertexoftheyellowtrianglealongwiththeverticesofthelargeorangeandbluetrianglesreallydomeettomake180degrees(theyformastraightangle)atthebottomedgeofthepuzzle.

b) Arguethattheedgeoftheorangetrianglefitsperfectlywiththeedgesofthesquareandyellowtriangleandthatthebigbluetrianglereallyfitsperfectlyalongtheedgesoftheparallelogramandyellowtriangle.

ClassActivity2:DefiningMoments

Wherethereismatter,thereisgeometry.JohannesKepler(1571-1630)

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

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

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.

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.

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.

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?”

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.

TheseshapesfitMyRule TheseshapesdonotfitMyRule

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

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.

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?

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)

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.

Step2.Thisistheonlymissingpieceofourproof.Lookupthedefinitionsofatransversalandofalternateinteriorangles.CanyouseethatÐ1andÐ5arealternateinteriorangles?Whatisthecorrespondingtransversal?

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

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:

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.

4. Allrightanglesareequaltoeachother.

5a.Ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesameside

lessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthat

sideonwhicharetheangleslessthanthetworightangles.

5b.Throughapointnotonalinetherecanbedrawnexactlyonelineparalleltothegiven

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

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

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

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

Fig.4.28.Patternsonahundredboard

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

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

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.”

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.

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.

10) HerearesomemoreoftheCommonCoreStateStandardsrelatedtoreasoningaboutshapes.InwhatwaysdoHWproblems4)–7)addressthesestandards?

11) DrawaVenndiagramshowingtherelationshipbetweenallthevariousquadrilateralswehavestudied.HowdoesthisfitwiththeCommonCoreStateStandardsdescribedbelow?

CCSSGrade3:Reasonwithshapesandtheirattributes.

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

CCSSGrade5:Classifytwo-dimensionalfiguresintocategoriesbasedontheirproperties.

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

2. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.

ClassActivity4:AlltheAngles

DonotworryaboutyourdifficultiesinMathematics.Icanassureyouminearestillgreater.

AlbertEinstein

1) Hereisthedescription–fromtheCommonCoreStateStandardsforgradefour–ofhowtothinkaboutmeasuringanglesusingdegrees.Readitcarefullyandsketchapicturetohelpyourgroupmakesenseoftheirexplanation.

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

Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

2) Makesureeveryoneinyourgroupcanusetheirprotractortomeasure(indegrees)theangleindicatedbelow:

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

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.

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.

Whenyoutalkaboutangleswithchildren,wesuggestthatyoualwaysuseyourhandorarmtoshowthesweepoftheangleinadditiontoshowingthestaticpicture.

Childreningradefourlearntouseaprotractortomeasureanglesindegreesandtoaccuratelyestimatethemeasureofangles.BelowyouwillfindtheCCSSrelatedtoanglemeasureinthisgrade.Readthemcarefully.

CCSSGrade4:Geometricmeasurement:understandconceptsofangleandmeasureangles.

1. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:

a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommon

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

b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.

2. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesof

specifiedmeasure.

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

CCSSGrade4:Drawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.

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

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

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

Homework Energyandpersistenceconquerallthings. BenjaminFranklin

3) ChildreninyourclassmaysaythatÐABCissmallerthanÐDEF.Isthistrue?Astheirteacher,whatwouldyousaytothesechildren?

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

5) Astudentsclaimsthatthesumofthevertexanglesofahexagonis6×180becauseeachtrianglehas180degreesofanglesmeasureandshehasshownthat6trianglestomakeupthehexagon.Whatwillyousayasherteacher?

6) IntheReadandStudysectionweclaimthatthesumoftheexterioranglesofanypolygonisalways360°.Makeamathematicalargumenttosupportthisclaim.

ClassActivity5:ALogicalInterlude

Equationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsof geometry.

StephenHawkingInthepicturebelow,eachcardhasacolorononesideandashapeontheotherside.Whichcard(s)wouldyouhavetoturnovertobesurethatthefollowingstatementistrue?

Ifacardisredononeside,thenithasasquareontheotherside.

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.

Itishelpfultoknowthatastatementanditscontrapositiveareequivalentbecausethatmeansthatyoucanproveastatementbyprovingitscontrapositive.ConnectionstotheElementaryGrades

Thebeginningofknowledgeisthediscoveryofsomethingwedonotunderstand. FrankHerbert

Nowthatwehavetalkedabitaboutdoingmathematics,wewanttoshowyouthattheCommonCoreStateStandardsrequirethatchildrenalsodomathematics.Giveanexampleofsomethingyouhavedoneinclasssofarthistermthatmeetseachofthesestandards.

CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

1. Makesenseofproblemsandpersevereinsolvingthem.

2. Reasonabstractlyandquantitatively.

3. Constructviableargumentsandcritiquethereasoningofothers.

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning.

Homework

Amultitudeofwordsisnoproofofaprudentmind. Thales

3) Decideifeachofthefollowingstatementsistrueorfalse.Iftrue,giveamathematicalexplanation.Iffalse,giveacounterexample.

a) Ifaquadrilateralisasquare,thenitisarectangle.b) Ifaquadrilateralhasapairofparallelsides,thenitmusthaveapairof

oppositesidesthatarecongruent.c) Ifthediagonalsofaquadrilateralareperpendiculartoeachother,thenthe

quadrilateralisarhombus.d) Ifaquadrilateralhasonerightangle,thenallofitsanglesmustberight

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

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

aretrue?

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

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°

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.

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:

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.

Homework

Homecomputersarebeingcalledupontoperformmanynewfunctions,includingthe consumptionofhomeworkformerlyeatenbythedog.

DougLarson

2) Studyeachboldandunderlinedtermusedinthissection.Thismeansyoushouldbeabletoexplainthedefinitionusinggoodmathematicallanguageandthatyoushouldbeablemakeexamplesandnon-examplesofeachterm.

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

4) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewearedescribing?Explainyouranswer.

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

5) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewearedescribing?Explainyouranswer.

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

6) Atwhichstepdoyouknowenoughtodrawatrianglethatiscongruenttotheonewe

aredescribing?Explainyouranswer.

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

7) Makeamathematicalargumentthatatrianglecanonlyhaveoneobtuseangle.

8) Makeamathematicalargumentthatthetwoacuteanglesofarighttrianglearecomplementary

9) MakeamathematicalargumentforTheorem7),thatinasingletriangle,theanglesthatareoppositethecongruentsidesmustbecongruent.

10) Makeamathematicalargumentthattwoacuteanglesofanisoscelesrighttriangleareeach45°.

11) Makeamathematicalargumentthateachangleinanequilateraltriangleis60°.

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.

ChapterTwo

Transformations,Tessellations,andSymmetries

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.

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

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

Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoapplytransformationsandusesymmetrytoanalyzemathematicalsituations.

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

Homework

Youmaybedisappointedifyoufail,butyouaredoomedifyoudon’ttry.

BeverlySills

2) Decideifeachofthestatementsabouttranslationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.

a) Correspondingsidesofanobjectanditstranslatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderatranslation,thenthelinesegment

joiningvertexAtovertex A isthetranslationvector.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe

parallelsidesofitsimagetobeverticalafteratranslation.d) IfAand A andBand B arecorrespondingpointsunderatranslation,itis

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

arenofixedpointsinatranslation.

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

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

ClassActivity8:Turn,Turn,Turn

Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple.

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

1) Rotatetheplane60degreescounterclockwiseaboutpointA. A

2) Rotatetheplane140degreesclockwiseaboutpointP.

(Thisactivityiscontinuedonthenextpage.)

3) Rotatetheplane90degreesclockwiseaboutapointQintheexactcenterofthesquare.

4) Studythethreerotationsinthisactivity.Whatrelationshipsdoyouseebetweenthe

originalfiguresandtheirrotatedimages?BetweencorrespondingpointsandthepointP?Makeasmanyconjecturesasyoucanaboutrotations.

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

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 ?

Homework

Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair. JohnQuincyAdams

2) Usethegridtorotatetheplane90degreescounterclockwiseaboutpointP.Showtheimageofthefiguresaftertherotation.

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.

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

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

1) Studythatdefinition.Makesureeveryoneinyourgroupunderstandshowitfitswiththeideaofareflection.SeeifyoucanusethedefinitiontohelpyoutosketchthereflectionoftriangleABCinlinel.

2) Carefullyusethedefinitionofareflectiontosketchthereflectionofthetrapezoidshown.Firstyouwillreflectitinlinemandthenyouwillreflectwhatyougetinlinen(thatisparalleltolinem).

Whatisthesinglerigidmotionthatwouldtaketheinitialfiguredirectlytothefinalfigure?Explain.

3) Whatwouldhappenifthelinesintersected?Tryitandthenuseyourobservationstomakeaconjecture.

ReadandStudy

WecoulduseuptwoEternitiesinlearningallthatistobelearnedaboutourownworldandthethousandsofnationsthathavearisenandflourishedandvanishedfromit.Mathematicsalonewouldoccupymeeightmillionyears.

MarkTwainThereareseveralwaystohelpchildrenpicturetheresultsofareflection.Thinkaboutthereflectionoftheparallelograminlineshownbelow.

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

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

Weallhaveafewfailuresunderourbelt.It’swhatmakesusreadyforthesuccesses. RandyK.Milholland,Webcomicpioneer

2) Onthesquaregridbelow,uselinenasthelineofreflectiontoreflectthegivenobjects.Labeleachimageappropriately.

3) Whatrelationshipsdoyouseebetweenoriginalfiguresandtheirreflectedimages?Betweentheobjects,theirimagesandthegivenlineofreflection?Makeasmanyconjecturesasyoucanaboutreflections.

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.

ClassActivity10:Zoom

Onecanstate,withoutexaggeration,thattheobservationofandthesearchforsimilaritiesanddifferencesarethebasisofallhumanknowledge.

AlfredNobel

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

ReadandStudy

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

1) Angle-AngleTriangleSimilarityTheorem(AA):Iftwoanglesofonetriangleare

congruenttotwoanglesofanothertriangle,thenthetrianglesaresimilar.ThistheoremappearedinhisBookVI(aboutsimilargeometricfiguresandproportionalreasoning)ratherthanhisBookIthatwehavestudiedpreviously.Homework

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

a) drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointCand

scalefactor2. b) instead,drawthesimilarshapethatistheresultofadilationoftheplanewithfixed

pointPandscalefactor2. c) Whatcommonalitiesanddifferencesdoyouobserveaboutthethreeshapesyouhave

drawn?

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.

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

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?

Onlyrepeatinginfinitepatternshavetranslationsymmetry.Theyaretheonlytypeofobjectthatcanbeslidandstillfallbackonthemselves.Imaginethatthispatternstripcontinuesforeverinbothdirectionssoifyouslideitonepatterntotheright(ortwoorthree…)itlooksjustthesame.

… …

Itouchthefuture.Iteach. ChristaMcAuliffeTheCommonCoreStateStandardsformathematicsmakeinformalideasofsymmetryatopicforgrade4.Whiletheymentiononlyreflectionsymmetry,childrenatthisage(andevenyounger)arecapableofexploringandrecognizing“turn”symmetryaswell.Readthisstandard.

Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2.CarefullyexaminetheMosaicGame.Playthegameonetimeandrecordyourresult.Explainhowyouknowthatyourarrangementproducesthemostlinesofsymmetry.Whatmightstudentslearnaboutsymmetryfromcompletingthisactivity?

Playthegameagain,butthistime,trytoproduceafigurewiththelargestorderofrotationalsymmetry.Again,explainyourreasoning.

CCSSGrade4:

1. Recognizealineofsymmetryforatwo-dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline-symmetricfiguresanddrawlinesofsymmetry.

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?

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

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

Traceablecopiesofmanyregularpolygons.Allofthemhavebeenscaledsothattheyhavethesamesidelength.

ReadandStudy

Everythinghasbeauty,butnoteveryoneseesit.Confucius

Atilingisanarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplanewithnogapsandnooverlaps.Anexampleusinga“T”shapepolygonisshownbelow.

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

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

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.

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.

7) Describesomereflectional,rotationalandtranslationalsymmetriesforthetessellationbelow.

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.

ChapterThreeMeasurementinthePlane

ClassActivity13:MeasureforMeasure

AndtherewentoutachampionoutofthecampofthePhilistines,namedGoliathofGath,whoseheightwassixcubitsandaspan.

ISamuel17:4

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

2) Onthenextpage,arefourcopiesofthesamelinesegment.Usingeachoftherulersprovided,carefullymeasurethelinesegment.

ReadandStudy

Itisnotenoughtohaveagoodmind.Themainthingistouseitwell. ReneDescartes Thebigideaofmeasurementisthatofcomparinganattributeofanobjecttoanappropriateunit.Forexample,wemightcomparethelengthofourdesktoafoot-longrulerorwemightcomparetheareaofsheetofpapertotheareaofagreentriangle.Sothekeyquestionregardingmeasurementisthis:Howmanyoftheunitfitintotheobject?Whatmakesaunitappropriate?Well,firstitmusthaveadimensionthatmatchestheattributetobemeasured.Forexample,wemeasurelength(orwidthorheight)usingone-dimensionalunits.Hereisanexampleofaone-dimensionalunit:Wemeasureareausingtwo-dimensionalunitslikethis:Wemeasurevolumeusingthree-dimensionalunitslikethis:Wewilltalkmoreaboutvolumemeasurementinlatersections.Second,ifyouwanttobeabletocommunicatewithothers,ithelpsthattheunitbea‘standard’one.Astandardunitisonethattheculturehasagreedupon.Eachpersonhasamentalmodeloftheunitsoheorshecanpicturehowbigitis.Forexample,inourcultureafootisastandardunitformeasuringlength.InEurope(andmostoftherestoftheworld)ameterisastandardunitformeasuringlength.Ifwewanttotalkamongcultures,weneedtobeabletoconvertfromoneunittoanother.Thereareabout3.28feetinameter.Ifaroomis15feetlong,approximatelyhowmanymetersisthat?Third,itisusefulthattheunitbeofreasonablesizeinrelationtotheattributetobemeasured.Itwouldbeinconvenienttomeasurethelengthoffootballfieldusingmicrons(areallysmallone-dimensionalunit).Inthemetricsystemsizeisindicatedbytheprefix.Forexample,theprefixkilomeans1000times.Soakilometeris1000meters.Inthecurriculummaterialselementarystudentsareexpectedtousetheprefixesmilli(onethousandth),centi(onehundredth),andkilo.Youshouldmemorizetheseandbeabletomakeconversions.Approximatelyhowmanycentimetersarethereinafoot?Becausemeasurementalwaysinvolvescomparison,itisnecessarilyalwaysanapproximation.Weoftenindicateourdegreeofcertaintyaboutameasurementbasedonthewaywereportit.Forexample,ifIclaimadeskis1.23meterslong,thissuggeststhatIamconfidentinthe

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

Onlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient.

EugeneS.WilsonTheCommonCoreStateStandardsformathematicsrequirethatchildrenbegintostudystandardmeasurementoflengthbeginningingrade2.ReadtheexcerptfromtheCCSSbelow.

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

CCSSGrade2:Measureandestimatelengthsinstandardunits.

1. Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.

2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.

3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.

4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthe

lengthdifferenceintermsofastandardlengthunit.

TheCommonCoreStateStandardsformeasurementingrade1areshownbelow.Comparethemtothegrade2standards.Howdotheydiffer?

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

Homework

Manyoflife'sfailuresarepeoplewhodidnotrealizehowclosetheyweretosuccesswhentheygaveup.

ThomasA.Edison

2) DotheConnectionsproblems.

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

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

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

CCSSGrade1:Measurelengthsindirectlyandbyiteratinglengthunits.

1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.

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

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?

ClassActivity14A:Triangulating

Godowndeepenoughintoanythingandyouwillfindmathematics.DeanSchlicter

Defineonetriangleunittobetheareaofthegreentrianglepatternblock.Usingthisunit,determinetheareaofthissheetofpaper.Thenusethetriangleunittoestimatetheareaofeachofthepatternblockshapespicturesbelow.

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

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

Wouldyouguaranteeyourestimatetothenearsest10squaremiles?Thenearest100squaremiles?Thenearest1000squaremiles?Somethingelse?Howdidyoudecide?

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

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:

TheCommonCoreStateStandardsdescribethefollowingstandardsforchildreningrade3.Readthemtoseethatthethingschildrenshouldlearnaremanyofthethingswehavedescribedinthissection.

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

CCSSGrade3:Geometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.

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

squareunit”ofarea,andcanbeusedtomeasurearea.

b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.

c. Measureareasbycountingunitsquares(squarecm,squarem,squarein,square

ft,andimprovisedunits).

2. Relateareatotheoperationsofmultiplicationandaddition.a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,and

showthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.

b. Multiplysidelengthstofindareasofrectangleswithwholenumbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.

c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-

numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.

d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposing

themintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.

Homework

ThemoreIpractice,theluckierIget.JerryBarber

3) Useagridtoestimatecarefullytheareaofthebelowcircularfigureinsquare

centimeters.

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

5) WewouldliketohavetheabovelakeclassifiedasoneoftheGreatLakes.AspartoftheapplicationtotheDepartmentoftheInterior,wehavetoreportitsarea.Estimatetheareaofthelakeinsquaremiles.

Scale1cm=3miles

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

ClassActivity15:FindingFormulas

Onecannotescapethefeelingthatthesemathematicalformulashaveanindependentexistenceandanintelligenceoftheirown,thattheyarewiserthanweare,wisereventhantheirdiscoverers.

HeinrichHertzWecommonlycomputeareasofsomespecialpolygonshapeswiththeuseofformulas.Theseformulascanbeexplained–theyarebasedongeometricdefinitionsandtheorems–andwewantyoutounderstandwhytheymakesense.Thatisthegoalofthisactivity.

1) Rectangle:Usetheideaofareaasthenumberofsquareunitsittakestocoveranobjecttoexplainwhyitmakessensethattheareaofarectangleissimplytheproductofitslengthandwidth.

2) Parallelogram:Showhowtocutandrearrangeaparallelogramtomakearectanglewiththesamearea.Arguethattheresultingfigureisinfactarectangle.UsetheformulafortheareaofarectangletofindaformulafortheareaAofaparallelogramusingthebasebandtheheighthoftheparallelogram.

3) Triangle:Showhowtorearrangetwocopiesofthesametriangletomakeaparallelogram.Arguethattheresultingfigureisinfactaparallelogram.UsetheformulafortheareaofaparallelogramtofindaformulafortheareaAofatriangleusingthebasebandtheheighthofthetriangle.

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.

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.

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.

* * * * * *

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

7) Supposeyouhave100metersofflexiblefencingtomarkapastureoutontheplains.

Howwouldyousetitup(whatshape)toenclosethemostgrazingareaforyourcattle?Whatdimensionswouldyouuseifthepasturehadtobearectangle?

8) Thetrianglebelowisconstructedona1cmgrid.Findtheareaofthetriangleusingatleastthreedifferentmethods.

9) Assumethatthetriangleareaaboverepresentspartofasignthatneedstobepainted.

Thescaleofthedrawingisthat1cmrepresents10feet.Theinstructionsonthepaintcansaythat1gallonofpaintwillcover100squarefeet.Howmanygallonsofpaintwillyouneedtobuyinordertopaintthetriangle?

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?

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

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.

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?

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.

ClassActivity17:PlayingPythagoras

Ihavehadmyresultsforalongtime,butIdonotknowyethowIamtoarriveat

them. CarlFriedrichGauss

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

1) Carefullymeasuretheareasofthesquaresinthebelowexampletoseeifthistheoremseemstrue.

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

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?

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.

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.

Homework

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackinwill.

VinceLombardi

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

3) Whenthelengthsofthesidesofarighttriangleareallintegersthethreenumbers(a,b,c)areknownasaPythagoreanTriple.Explainwhy(3,4,5)isaPythagoreanTriple.WhichofthefollowingtriplesofnumbersarePythagoreanTriples?

a)(4,5,6) b)(4,6,8) c)(6,8,10)

4) Supposeatrianglehassidesoflength8,15,and17.Isitarighttriangle?

5) Whatistheexactheightofanequilateraltriangleifallsidesareoflength10?Oflength3?Oflength4?Oflengths?

6) Aclosetis3feetdeep4feetwideand12feethigh.Findthedistancefromonecorneratthefloortothediagonallyoppositecornerattheceiling.Youmightwanttohavealookataboxtohelpyouseewhattodohere.

ClassActivity18:Coordination

Thoughtisonlyaflashbetweentwolongnights,butthisflashiseverything. HenriPoincare

Whichofthetrianglespicturedabovearesimilartoeachother?Justifyyouranswerinasmanywaysasyoucan.(Youmayassumethatthedotsareequallyspacedonthegridinboththehorizontalandverticaldirections,andthattheverticesofthetriangleslieexactlyonthedotsastheyappearto.)

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

-10 -5 5 10

D: (2, - 3)

C: (4, 0)

B (-2, 5)

A (- 3, -1)

origin

y-axis

x-axis

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:

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)?

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

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

Coordinategeometryisatopictobeintroducedingrade5accordingtotheCommonCoreStateStandardsformathematics.Readthedescriptionbelowtobesurethatitfitswithwhatwehavediscussedinthissection.

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

CCSSGrade5:Graphpointsonthecoordinateplanetosolvereal-worldandmathematicalproblems.

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

2. Representrealworldandmathematicalproblemsbygraphingpointsinthefirstquadrantofthecoordinateplane,andinterpretcoordinatevaluesofpointsinthecontextofthesituation.

Homework

Energyandpersistenceconquerallthings.

BenjaminFranklin

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

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

Whichregularpolygonscanbemadeonthegeoboard?Ifitisnotpossibletomakeaparticularregularpolygon,explainwhyitisnotpossible.

3) Findtheperimetersandareasofthefiguresonthegeoboardsquaregridsbelow.YoumightfindthePythagoreanTheoremhelpful.

4) Thereare14differentsegmentlengthsthatarepossibletomakeona5by5geoboard.Findthemall.

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.

ChapterFour

TheThirdDimension

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.

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.

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.

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)

Homework

Gettingaheadinadifficultprofessionrequiresavidfaithinyourself.Thatiswhysomepeoplewithmediocretalent,butwithgreatinnerdrive,gomuchfurtherthanpeoplewithvastlysuperiortalent.

SophiaLoren

2) Writeoutthedetailsofamathematicalargumentthatthereareexactlyfiveregularpolyhedra.

3) Thefollowingpictureisoftengivenasanexampleofaregularicosahedron.Examinethepicturecarefullyanddeterminewhythispictureisclaimingtobesomethingthatitisnot.

4) Isitpossibletobuildapolyhedronwhereallofthefacesarecongruentregularpolygonsbutthepolyhedronisnotregular?Explain.

5) Usethefivemodelsyoubuilttofindaformulathatrelatesnumbersofvertices,edges,andfacesinregularpolyhedra.

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

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.

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.

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.

ClassActivity21:SurfaceArea

Mathematicsmaybedefinedastheeconomyofcounting.Thereisnoprobleminthewholeofmathematicswhichcannotbesolvedbydirectcounting.

ErnstMach

1) Use14interlockingunitcubestobuildthe3-dimensionalfigurepictured.

a) Whatisthesurfaceareaofthisobject?

b) Whatisitsvolume?

c) Isthisapolyhedron?Explain.

2) Whatareallthepossiblevaluesforthesurfaceareaoffiguresmadewith14interlocked

cubes?Explain.

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.

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

Homework

Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.

JohnQuincyAdams

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

area?Makeaconjecture.

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

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

5) Sketchanetforarectangularprismthatis4cmby5cmby7cm.Whatisthesurfaceareaofthatprism?

ClassActivity22:NothingButNet Puremathematicsis,initsway,thepoetryoflogicalideas. AlbertEinstein

1) Figureouthowtobuildapapermodelofarightcylinderwitharadiusof3cmandaheightof10cm.Useyourmodeltohelpyoufindaformulaforthesurfaceareaofacylinder.

2) Buildapapermodelofapyramidthathasa6cmby6cmsquarebaseandaheightof4cm.Whatisitssurfacearea?

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.)

3) HowmuchbiggeristhevolumeofObjectBcomparedtoObjectA?

4) WhatistherelevanceofObjectCtoansweringthesequestions?

5) WhatifIgaveyoualargerblockthatis3timesthelengthofthesmallcube;howwouldyouranswersto1-3change?Whatabout4timesthelength?

6) WhatifIgaveyouasmallerblockthatis½timesthelengthofthesmallcube;howwouldyouranswersto1-3change?

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?

Homework

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

AuthorUnknown

2) DoallthoseproblemsintheConnectionssection.

3) Whenyoudoublethelengthofallthesidesofacube,whathappenstoitsvolume?Whydoesthishappen?Whathappenswhenyoutriplethelength?

4) Belowwehavesketchedanetforacube.a) Buildapapermodelofacubethatistwiceaslongineachlineardimension.b) Buildapapermodelofacubethathastwicethesurfaceareaofthecubesuggested

bythenet.c) Buildapapermodelofacubethathastwicethevolumeofthecubesuggestedby

thenet.

ClassActivity24:VolumeDiscount ThelawsofnaturearebutthemathematicalthoughtsofGod.

Euclid1)Reasonwithunitcubestocreatevolumeformulasforthefollowingobjects:

a) arectangularprism

b) atriangularprism

c) acylinder2) Userice--notformulas--toanswerthenextfourquestions:

a) Howdoesthevolumeofthesquarepyramidcomparetothevolumeofthesquareprism?Usethisinformationtocreateavolumeformulaforasquarepyramid.

(Thisactivitycontinuesonthenextpage.)

b) Howdoesthevolumeofthetriangularpyramidcomparetothevolumeofthetriangularprism?Usethisinformationtocreateavolumeformulaforatriangularpyramid.

c) Howdoesthevolumeoftheconecomparewiththevolumeofthecylinder? Usethisinformationtocreateavolumeformulaforacone.

d) Howdoesthevolumeoftheconecomparewiththevolumeofthesphere?Usethisinformationtocreateavolumeformulaforasphere.

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.

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

Thesphereisanotherthree-dimensionalshapethathasawell-knownvolumeformula, 34

,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.

Puremathematicsis,initsway,thepoetryoflogicalideas.AlbertEinstein

Childreningradethreeshouldhaveexperiencesmeasuringvolumesusingwaterandweighingphysicalobjects.HerearetherelevantCommonCoreStateStandards.Readthem.

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

CCSSGrade3:Solveproblemsinvolvingmeasurementandestimationofintervalsoftime,liquidvolumes,andmassesofobjects.

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

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

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.

Homework

Toclimbsteephillsrequiresaslowpaceatfirst. WilliamShakespeare

2) DotheitalicizedthingsintheConnectionssection.

3) Supposeyouhavearightcircularcylinderwithheighthandradiusrandanoblique

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

4) Theclosetinmylivingroomhasanoddshapebecausemyapartmentisonthetopfloorofahousewithaslantedroof.Theclosetis6feettallinfrontbutonly4feettallinback.Itis3feetdeepand12feetwide.

a) Buildascaleddownpapermodelofmycloset.Reallydothis.Itwillhelpyouwiththerestofthisproblem.

b) Howmanycubicfeetofstoragedoesithold?c) Iwanttopainttheinsidewallsandceilingofmycloset.Howmanysquarefeet

willIneedtopaint?

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

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

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

8) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3Session4.Howdoesthisactivityfurtherstudents’understandingofvolume?

ClassActivity25:VolumeChallenge

Ifpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonotrealizehowcomplicatedlifeis.

JohnLouisvonNeumann

1) Yourgroupshouldworktogethertobuildpapermodelsofeachofthefollowingobjectsinsuchawaythatthevolumeofeachis60cm3.(Otherthanthat,youmayuseanydimensionsyoulike.)

a) Cylinderb) Rectangularprismthatisnotacubec) Pyramidwithasquarebased) Prismwithanequilateraltrianglebase

2) Findthesurfaceareaofeachoftheobjectsyoubuiltin1)above.

ReadandStudyandConnections It'snotthatI'msosmart;it'sjustthatIstaywithproblemslonger. AlbertEinsteinBythetimetheyreachupperelementaryschool,studentscansolvemanypracticalproblemsingeometry.Howeverthesestudentsarenotreadytosimplyapplyformulastosolveproblems;insteadtheyneedtousemodelsinordertomakesenseofproblems.Astheirteacher,yourjobwillbetomakesurethatchildreninyourclassesbuildandhandleappropriatemodels.NoticethattheCommonCoreStateStandardsforgrade6explicitlyrequirethatstudentsusehands-onmodelstoexploreideas.Studentsareaskedtocuttrianglesandothershapesapart,todrawpictures,topackspaceswithunitcubes,tousethecoordinateplane,andtobuildandusenets.Readthesestandards.Havewedoneofthesethingsinthisbooksofarthisterm?

CCSSGrade6:Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.

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

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

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

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

Homework

Theelevatortosuccessisnotrunning;youmustclimbthestairs.ZigZiglar

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

3) AreplicaoftheGreatPyramidstands2feettallandis3.15feetlongonaside(ithasasquarebase).

a) Approximatelyhowmuchvolumedoesthisreplicatakeup?UsethemodelyoubuiltintheClassActivitytohelpyoutothinkaboutthisproblem.

b) Whatisthesurfaceareaofthereplica?c) Supposethescaleofthereplicatotherealthingisinis1footto240feet.Whatis

thevolumeofGreatPyramid?Whatisitssurfacearea?d) WhichoftheCommonCoreStateStandardsaremetbythisseriesofproblems?

SummaryofBigIdeasfromChapterFour

Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes

• Thereareexactlyfiveregularpolyhedra–andchildrencanunderstandwhythisisso.

• Surfaceareaisameasureofthesumofareasofthefacesofathree-dimensionobject.Itisthenumberofsquareunitsittakestocoverthefacesoftheobject.Aunitofsurfaceareaisflatlikethis:

• Volumeisameasureofthenumberofcubicunitsittakestofillathreedimensionalobject.Aunitofvolumeisthree-dimensionalandlookslikethis:

• Volumecanalsobemeasuredbytheamountofliquid(orsand)ittakestofillathreedimensionalobject.

• Thevolumeofaprismorcylindercanbefoundbycomputingtheareaofthebaseandmultiplyingthatbyitsheight(thenumberoflayersofthebase).Thisideaworksforbothrightandobliqueobjects.

• Ittakesthevolumeofthreepyramidstofillaprismwiththesamebaseandheight,andittakesthevolumeofthreeconestofillacylinderwiththesamebaseandheight.

• Childrenneedtohavemanyyearsofexperiencesbuildingandusingavarietyofmodels.

APPENDICES

Euclid’sPostulatesandPropositions

Euclid'sElementsThispresentationofElementsistheworkofJ.T.Poole,

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.

BOOKIPROPOSITIONSProposition1.

Onagivenfinitestraightlinetoconstructanequilateraltriangle.

Proposition2.Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.

Proposition3.Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.

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

Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.

Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.

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

Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.

Proposition9.Tobisectagivenrectilinealangle.

Proposition10.Tobisectagivenfinitestraightline.

Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.

Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.

Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.

Proposition14.Ifwithanystraightline,andatapointonit,twostraightlinesnotlyingonthesamesidemaketheadjacentanglesequaltotworightangles,thetwostraightlineswillbeinastraightlinewithoneanother.

Proposition15.Iftwostraightlinescutoneanother,theymaketheverticalanglesequaltooneanother.

Proposition16.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisgreaterthaneitheroftheinteriorandoppositeangles.

Proposition17.Inatriangletwoanglestakentogetherinanymannerarelessthantworightangles.

Proposition18.Inanytrianglethegreatersidesubtendsthegreaterangle.

Proposition19.Inanytrianglethegreaterangleissubtendedbythegreaterside.

Proposition20.Inanytriangletwosidestakentogetherinanymanneraregreaterthantheremainingone.

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

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

Proposition23.Onagivenstraightlineandatapointonittoconstructarectilinealangleequaltoagivenrectilinealangle.

Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.

Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.

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

Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.

Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.

Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.

Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.

Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.

Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.

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

Proposition34.Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.

Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.

Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.

Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.

Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.

Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.

Proposition46.Onagivenstraightlinetodescribeasquare.

Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.

Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.

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

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

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

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

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

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

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

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

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

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

GridPaper

References

Givecreditwherecreditisdue. Authorunknown

• Bauersfeld,H.(1995).Thestructuringofstructures:Developmentandfunctionofmathematizingasasocialpractice.InL.Steffe&J.Gale(Eds.),ConstructivisminEducation(pp.137-158).Hillsdale,NJ:Erlbaum.

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

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

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

Publishers.

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

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

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

• MathematicalQuotationsServer(MQS)atmath.furman.edu.

• NationalCouncilofTeachersofMathematics.(2000).PrinciplesandStandardsfor

SchoolMathematics.Reston,VA:NCTM.

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

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

Birkhïser.

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

Documents

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

National Harbor Stops - Old Town Trolley Tours · Circle Sheridan Circle Thomas Circle Scott Circle v ve ve ... GOLDEN TRIANGLE DUPONT CIRCLE CRYSTAL CITY ADAMS MORGAN THEODORE ROOSEVELT

Answers and Teachers’ Notes · 1. cylinder 2. pyramid 3. cube: the square hole and the circle (if the circle is big enough) cone: the triangle hole and the circle hole Activity

The Versatile Circle - AMSIamsi.org.au/wp-content/uploads/sites/15/2014/04/The-Versatile-Circle.key1.pdf · The Versatile Circle!!!!! right-angled triangle!!!!! A conversation about

AREAS AND PERIMETERS TRIANGLE SQUARE RECTANGLE RHOMBUS TRAPEZIUM CIRCLE.

Geometry 6 Level 1. Parts of a circle Why is this triangle isosceles?

Area of a Parallelogram Area of a Triangle Circumference & Area of a Circle.

Geometry. Circumcircle of a Triangle For any triangle, there is a unique circle that is tangent to all three vertices of the triangle This circle is the.

Y5 vocab bkmk complete - Wanborough Bookmarks/Y5... · add, addition more, plus, increase ... measure, measurement size compare ... circle, circular, semi-circle triangle, ...

square circle triangle rectangle hexagonhawkerresources.weebly.com/uploads/2/8/3/0/... · Circle corners . Count them. triangle pentagon square rhombus hexagon pentagon rectangle

square, triangle, circle, diamond, rectangle pentagon ...familiesnews.com/wp-content/uploads/2014/06/Shape-Hunt-coverpa… · square oval cylinder Cross cube cone on triangle star

N R C H R VISION E S - Great Maths Teaching · PDF fileSecondary Data Tally Chart Frequency Table ... Square Circle Equilateral Triangle Isosceles Triangle Rhombus Kite Parallelogram

SQUARE RECTANGLE TRIANGLE CIRCLE 1 ) COUNT · PDF fileSQUARE RECTANGLE TRIANGLE CIRCLE 1 ) COUNT SHAPES AND WRITE BELOW:-Circles:- _____? Triangles:-- _____? Squares:- _____? ... ACTIVITY

Exploring Pascal's Triangle - Berkeley Math Circle

circle triangle square diamond øcom · PDF filecircle triangle square diamond øcom Images (c) Jupiter Co. star oval rectangle heart øcom Images (c) Jupiter Co

circle triangle crafts rug clay signs eggs pottery pyramid sphere blueprint architect discussion

1st period by Warming Up Chant and Do Circle, circle, go, go, go! Triangle, triangle, win, win, win! Rectangle, rectangle, huh, huh, huh! Square, square,

circle square triangle · 2020. 5. 19. · cone cylinder heart rhombus circle square triangle . Title: document Author: csgarbac Created Date: 3/24/2020 12:42:00 PM

Trigonometry for Any Angle. SineCosineTangentCosecantSecantCotangent Abbreviation Reciprocal Function Co-function Right Triangle Definition Unit Circle.

Geo Circle Triangle

gceguide.xyz CHAPTER 1 THE TRIANGLE Medians. Circumcentre and circumcircle. Orthocentre and pedal triangle. Nine-point circle. Euler line. Incentre and excentres. Concurrence and collinearity.

SQUARE RECTANGLE TRIANGLE CIRCLE 1 ) COUNT · PDF fileSQUARE RECTANGLE TRIANGLE CIRCLE 1 ) COUNT SHAPES AND WRITE BELOW:-Circles:- ___? Triangles:-- _? Squares:- ___? ... ACTIVITY