Big Ideas Geometry and Data Analysis · We will be giving you other definitions of geometry as we...
Transcript of Big Ideas Geometry and Data Analysis · We will be giving you other definitions of geometry as we...
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BigIdeasinMathematicsforFutureTeachers
BigIdeasinGeometryandData
JohnBeam,JasonBelnap,EricKuennen,AmyParrott,CarolE.Seaman,andJenniferSzydlik
(UpdatedSummer2019)
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This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
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Hey!Readthis.Itwillhelpyouunderstandthebook.
Theonlywaytolearnmathematicsistodomathematics. PaulHalmos
Thisbookwaswrittentopreparefutureteachersforthemathematicalworkofteaching.Thefocusofthismoduleisgeometryanddata–andthisdomainencompassesmanydeepandwonderfulmathematicalideas.Thistextisnotintendedtohelpyourelearnyourelementarymathematics;itisaboutteachingyoutothinklikeamathematiciananditisabouthelpingyoutothinklikeamathematicsteacher.TheNationalCouncilofTeachersofMathematics(NCTM,2000)writes:
Teachersneedseveraldifferentkindsofmathematicalknowledge–knowledgeaboutthewholedomain;deep,flexibleknowledgeaboutcurriculumgoalsandabouttheimportantideasthatarecentraltotheirgradelevel;knowledgeaboutthechallengesstudentsarelikelytoencounterinlearningtheseideas;knowledgeabouthowtheideascanberepresentedtoteachthemeffectively;andknowledgeabouthowstudents’understandingcanbeassessed(p.17).
Wearegoingtoworktowardthesegoals.(Readthemagain.Thisisatallorder.Inwhichareasdoyouneedthemostwork?)Throughoutthisbook,wewillaskyoutoconsiderquestionsthatmayariseinyourelementaryclassroom.
Isasquarealwaysarectangle?
Whatdoesthisnumbercalledprepresent?Whatdoesitmeantomeasure“area”?
Cantworectangleswiththesameareahavedifferentperimeters?
Whatissospecialaboutrighttriangles? IfIbuya5-inchpizzaandmyfriendbuysa10-inchpizza,doessheget twiceasmuchtoeat?
HowmanypeopledoyouneedtosampleinordertoreasonablyfindouthowmanyhoursofTVsecondgraderswatchinaweek?
Whatdoesitmeantosaythattheprobabilityofaheadwhenflippingacoinis½?
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Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Weviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjustasimportantasanycontentyouteach.Whenyoustandbeforeyourclass,youarearepresentativeofthemathematicalcommunity;wewillhelpyoutobeagoodone.Noonecandothisthinkingforyou.Mathematicsisn’tasubjectyoucanmemorize;itisaboutwaysofthinkingandknowing.Youneedtodoexamples,gatherdata,lookforpatterns,experiment,drawpictures,think,tryagain,makearguments,andthinksomemore.Thebigideasofgeometryarenotalwayseasy–buttheyarefundamentallyimportantforyourstudentstounderstandandsotheyarefundamentallyimportantforyoutounderstand.EachsectionofthisbookbeginswithaClassActivity.Theactivityisdesignedforsmall-groupworkinclass.Someactivitiesmaytakeyourclassaslittleas20minutestocompleteanddiscuss.Othersmaytakeyoutwoormoreclassperiods.TheReadandStudy,ConnectionstoTeaching,andHomeworksectionsarepresentedwithinthecontextoftheactivityideas.Nosolutionsareprovidedtoactivitiesorhomeworkproblems–youwillhavetosolvethemyourselves.ThemathematicscontentinthisbookpreparesyoutoteachtheCommonCoreStateStandardsforMathematicsforgradesK-8.Thesearethestandardsthatyouwilllikelyfollowwhenyouareanelementaryteacher,sowewillhighlightaspectsofthemthroughoutthetext.Inorderforyoutoseehowthemathematicalworkyouaredoingappearsintheelementarygrades,wehavemadeexplicitconnectionstoBridgesinMathematicsfromTheMathLearningCenter.ThisistheonlineelementarygradesmathematicscurriculumadoptedbytheOshkoshAreaSchoolDistrict.Youwilloftenbeaskedtogotothesitebridges.mathlearningcenter.orgtoreadordoproblems.Yourinstructorwillprovideyouwithacodesothatyoucanaccessthesematerials. Allourbest,
John,Jason,Eric,Amy,Carol&Jen
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TableofContents
Tobeateacherrequiresextensiveandhighlyorganizedbodiesofknowledge.Shulman,1985,p.47
ChapterOne.......................................................................................................9ClassActivity1: TrianglePuzzle.......................................................................................10
WhatisGeometry................................................................................................................11
NatureofMathematicalObjects..........................................................................................11
MathematicalCommunication.............................................................................................13
ClassActivity2: DefiningMoments.................................................................................17
RoleofDefinitions................................................................................................................18
ParallelandPerpendicularLines..........................................................................................19
ClassActivity3: GetitStraight........................................................................................26
LanguageofMathematics....................................................................................................29
DeductiveversusInductiveReasoning.................................................................................29
IdeaofAxioms......................................................................................................................30
MakingConvincingMathematicalArguments.....................................................................32
ClassActivity4: AlltheAngles.........................................................................................39
MeasuringAnglesinDegrees...............................................................................................39
RegularPolygons..................................................................................................................41
ClassActivity5: ALogicalInterlude.................................................................................45
ConverseandContrapositive...............................................................................................46
SummaryofBigIdeas........................................................................................................49
ChapterTwo.....................................................................................................50ClassActivity6: MeasureforMeasure............................................................................51
StandardandNonstandardUnits.........................................................................................53
ApproximationandPrecision...............................................................................................54
ClassActivity7: Part1Triangulating................................................................................57
ClassActivity7: Part2AreaEstimation...........................................................................58
IdeaofArea..........................................................................................................................59
CoveringwithUnitSquares..................................................................................................60
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ClassActivity8: FindingFormulas....................................................................................64
MakingSenseofAreaFormulas...........................................................................................65
ClassActivity9: TheRoundUp........................................................................................70
Whatisπ?............................................................................................................................72
InscribedPolygons................................................................................................................74
ClassActivity10: PlayingPythagoras...............................................................................75
ProofsofthePythagoreanTheorem....................................................................................77
GeometricandAlgebraicRepresentations...........................................................................78
SummaryofBigIdeas........................................................................................................80
ChapterThree..................................................................................................81ClassActivity11: StrictlyPlatonic(Solids)........................................................................82
RegularPolyhedra................................................................................................................83
SpatialReasoning.................................................................................................................84
ClassActivity12: PyramidsandPrisms............................................................................87
CountingVertices,Edges,andFaces....................................................................................87
ClassActivity13: SurfaceArea........................................................................................90
IdeaofSurfaceArea.............................................................................................................91
ClassActivity14: NothingButNet...................................................................................94
NetsforCylindersandPyramids..........................................................................................94
ClassActivity15: BuildingBlocks.....................................................................................95
IdeasofVolume....................................................................................................................97
VolumesofPrismsandCylinders.........................................................................................97
ClassActivity16: VolumeDiscount..................................................................................99
Capacity..............................................................................................................................101
VolumesofPyramids,Cones,andSpheres........................................................................102
ClassActivity17: VolumeChallenge..............................................................................106
BuildingModelstoSpecification........................................................................................106
SummaryofBigIdeas......................................................................................................109
ChapterFour..................................................................................................110ClassActivity18: Slides.................................................................................................111
RigidMotions.....................................................................................................................112
Translations........................................................................................................................112
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ClassActivity19: Turn,Turn,Turn.................................................................................115
Rotations............................................................................................................................117
ClassActivity20: ReflectingonReflecting.....................................................................120
Reflections..........................................................................................................................122
ClassActivity21: Part1ASimilarTask..........................................................................125
ClassActivity21: Part2Zoom.......................................................................................126
SimilarPolygons.................................................................................................................127
ClassActivity22: SearchingforSymmetry.....................................................................130
SymmetriesinthePlane.....................................................................................................131
SummaryofBigIdeas......................................................................................................134
ChapterFive...................................................................................................135ClassActivity23: DiceSums..........................................................................................136
LanguageofProbability......................................................................................................137
ExperimentalandTheoreticalAnalysis..............................................................................141
ClassActivity24: RatMazes..........................................................................................144
TreeDiagrams....................................................................................................................145
CompoundEvents..............................................................................................................146
ClassActivity25: TheMaternityWard...........................................................................149
LawofLargeNumbers........................................................................................................150
IndependentEvents...........................................................................................................150
SummaryofBigIdeas......................................................................................................153
ChapterSix.....................................................................................................154ClassActivity26: StudentWeightsandRectangles........................................................155
TheLaguageofSampling....................................................................................................158
RandomSamples................................................................................................................159
ClassActivity27: SnowRemoval...................................................................................165
AnalyzingaSurvey..............................................................................................................166
ClassActivity28: Part1NameGames...........................................................................172
ClassActivity28: Part2IntheBalance..........................................................................174
RedistributionConceptoftheMean..................................................................................176
BalanceConceptoftheMean............................................................................................176
MedianandMode..............................................................................................................176
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FiveNumberSummary.......................................................................................................177
Percentiles..........................................................................................................................178
ClassActivity29: MeasuringtheSpread........................................................................183
Range,IQR,andMeanAbsoluteDeviation........................................................................185
ACriterionforFindingSuspectedOutliers.........................................................................185
Boxplots..............................................................................................................................186
ClassActivity30: TheMatchingGame...........................................................................191
Distributions,Skewness,andSymmetry............................................................................193
SummaryofBigIdeas......................................................................................................203
References.......................................................................................................................204
APPENDICES...................................................................................................206Euclid’sPostulatesandPropositions................................................................................207
Glossary..........................................................................................................................212
GraphPaper....................................................................................................................227
Tangrams.........................................................................................................................229
PatternBlocks.................................................................................................................230
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ChapterOne
SeeingtheWorldGeometrically
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ClassActivity1: TrianglePuzzle
Geometryisthescienceofcorrectreasoningonincorrectfigures. Originalauthorunknown,butquotedfromG.Polya,HowtoSolveIt.Princeton:
PrincetonUniversityPress.1945.Whathappenedtothemissingsquare?
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ReadandStudy
Geometricfiguresshouldhavethisdisclaimer:
“Noportrayalofthecharacteristicsofgeometricalfiguresorofthespatialpropertiesofrelationshipsofactualbodiesisintended,andanysimilaritiesbetweentheprimitiveconceptsandtheircustomarygeometricalconnotationsarepurely
coincidental.”
"GeometryandEmpiricalScience"inJ.R.Newman(ed.)TheWorldofMathematics,NewYork:SimonandSchuster,1956.
Mathematicalobjects–geometricobjectsincluded–arenotrealobjects.Theyareidealobjects.Thismightseemdisturbingbecausethismeansthatgeometricobjects–liketrianglesandcircles–donotexistinthephysicalworld.Youcandrawsomethingthatlookslikeatriangleoracircle,butitwon’thavethepreciseandperfectpropertiesthattheidealmathematicaltriangleorcirclehas.Thesketchwillsimplycalltomindtheidealobject.Whilewedrawlotsofpicturesingeometry,weneedtokeepinmindthatthepicturescanbemisleading.TaketheTrianglePuzzleasanexample.Thispuzzleiscompellingbecauseitreallylookslikethetopandbottomfiguresarebothtriangles.Theyarenot.Onlybyreasoningabouthowtheidealizedpiecesfittogethercanwediscoverthetruth.Thepuzzleisgovernedbyanunderlyingstructure–thepropertiesoftheshapesinvolveddeterminehowtheywillfittogether.Mathematicianslovetorevealhiddenstructureandtoexplainpatterns.Thisiswhatmathematicsisabout.Andthisiswhatgeometry,inparticular,isabout.Onedefinitionofgeometryisthatitisthestudyofidealshapesandtheirproperties,ofthepatternsthoseshapescanform,andoftheactionsonthoseshapesthatpreservetheirproperties.Wewillbegivingyouotherdefinitionsofgeometryaswegoalong.Watchforthevarietyofwaysofthinkingaboutgeometry.W H A T I S G E O M E T R Y N A T U R E O F M A T H E M A T I C A L O B J E C T S
Thisbookisdesignedsothatyougettodogeometry.Wegenerallydonotteachyoutechniquesandthenhaveyoupractice.Instead,weaskthatyouworkonproblemstohelpyouconstructimportantideas.Theproblemscanbedifficult–whichiswhywehopethatyouwillworkonthemingroupsandthendiscussthemasaclass.Wemadethemthatwayonpurpose.Webelievethataproblemisonlyaproblemifyoudon’tknowhowtosolveit.Ifyoudoknowhowtosolveit,thenitisjustanexercise.Wehopethisbookisfullofproblemsforyouandthatyouwill‘getstuck’alot.Trynottoletbeingstuckdiscourageyou.Itispartofdoingmathematics.Thereisnomagictechniqueforsolvingamathematicsproblem–whichisgood,becauseotherwisemathematicswouldn’tbeanyfun.Basically,youjusthavetowrestlewiththe
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problem.Theprocessmighttakeminutes,orhours,orevenyears.Therearestrategiesthatmayprovehelpful,andapurposeofthisfirstsectionistomakeyouawareofsomeofthethingspeopledotoworkonproblems.Fornow,wewouldlikeyoutoseparateyourthinkingaboutproblemsolvingintofourcategories:
1) Understandingtheproblem.Whatdoesitmeantosolvethisproblem?Doyouunderstandtheconditionsandinformationgiveninthestatementoftheproblem?(FortheTrianglePuzzleabove,thismeansunderstandingthatsinceareamustbepreserved,thepictureshavetobemisleadinginsomeway.Solvingthisproblemmeansshowingexactlyhowthepicturesaremisleading.)
2) Reflectingonyourproblemsolvingstrategies.Whatdidyoudotoworkonthe
problem?(Didyoustudythepiecestoseehowtheyfittogether?Randomlyorinsomesystematicmanner?Didyoukeeptrackofanything?Didyoudrawpictures?Didyoucomputesomething?)
3) Explainingthesolution.Whatistheanswertothequestion?(Exactlywhatiswrong
withthepictures?)
4) Justifyingthatyouarebothdoneandcorrect.Whydoesyoursolutionmakesense?Canyouprovethatyouarecorrectandthattheproblemiscompletelysolved?
Beforewegetintothisbookanyfurther,wemightaswelltellyouthatwe’rebossy.Throughoutthereadingsections(whichyoumustdo–youoweittothechildreninyourfutureclassrooms)wewillaskquestionsandissuecommandsinitalics.Dothethingswesuggestinitalics.Don’tworrythatitslowsthereadingdown.Mathematiciansreadvery,very,agonizinglyslowlyandcarefully,withpencilinhand.Wewriteonourbooks–alloverthem.Weverifyclaims;wedotheproblems;weasknewquestionsandtrytoanswerthem.Sowechallengeyoutodofourthingsthisterm.Firstandsecond,readeverywordofyourtextandworkhardoneachandeveryproblem.Third,makeacontributiontoeachdiscussionofaclassactivity.Andfinally,practicelisteningtoandmakingsenseofotherstudents’mathematicalideas.Asateacher,youwillneedtounderstandthemathematicalthinkingofothers;useyourclasstopracticethatskill.ConnectionstoTeaching
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Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoorganizeandconsolidatetheirmathematicalthinkingthrough
communication;tocommunicatetheirmathematicalthinkingcoherentlyandclearlytopeers,teachers,andothers;toanalyzeandevaluatethemathematicalthinking
andstrategiesofothers;andtousethelanguageofmathematicstoexpressmathematicalideasprecisely. NCTMPrinciplesandStandardsforSchoolMathematics,2000
Learningviaproblemsolvingandcommunicationofideasaretwomajorthreadsinelementarymathematicseducation.TheNationalCouncilofTeachersofMathematics(2000),inadoptingthePrinciplesandStandardsforSchoolMathematics,advocatedthatallstudentsofmathematicsengageinproblemsolvingandcommunication,bothoralandwritten,atallgradelevels.Theywrite:
“Solvingproblemsisnotonlyagoaloflearningmathematicsbutalsoamajormeansofdoingso”(p.52).“Communicationisanessentialpartofmathematicsandmathematicseducation.Itisawayofsharingideasandclarifyingunderstanding….Whenstudentsarechallengedtothinkandreasonaboutmathematicsandtocommunicatetheresultsoftheirthinkingtoothersorallyorinwriting,theylearntobeclearandconvincing”(p.60).
Mathematicseducatorsbelievethatproblemsolvingandwrittencommunicationarealsoessentialcomponentsofyourmathematicalpreparationtobecomeelementaryschoolteachers.Wewillbeaskingyoutowriteaboutmathematicsinthisclass.Wewillaskyoutowriteinterpretationsofproblems,descriptionsofstrategies,explanationsofsolutions,andjustificationsofsolutions.M A T H E M A T I C A L C O M M U N I C A T I O N
Howdowewriteaboutmathematics?Isn’tmathematicsallaboutnumberslike2andp?Andaboutsymbolslike║andÐ?Well,no,it’snot.Mathematiciansusesymbolslikethesetowritestatementsandtosolvesomeproblems,andyoucanusesymbolslikeÐand^and@whenyouwriteaboutgeometryproblems.Butmathematicsismuchmoreaboutexploringpatterns,makingconjectures,explainingresultsandjustifyingsolutions.Theseactivitiesrequireustowritewithwordsincompletesentencesthatusemathematicallanguageandlogicappropriately.Symbolsareatoolwewilluse.Butfornow,youshouldfocusonwritingwithwords–clearly,completely,correctly,andconvincingly.Asfutureteachersyoumustpracticecommunicatinginthelanguageofmathematics.Youwillhaveachancetopracticerightnowinthehomework.
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HomeworkYoualwayspassfailureonthewaytosuccess.
MickeyRooney(MQS)
1) TheTangrampuzzleiscomposedofsevenshapesincludingonesquare,oneparallelogram,twosmallisoscelesrighttriangles,onemedium-sizedisoscelesrighttriangle,andtwolargeisoscelesrighttriangles.Inthediagrambelow,thesevenpiecesarearrangedsothattheyfittogethertoformasquare.a) Tracethepieces,cutthemout,andthenidentifyeachone.Lookuptheterms
isosceles,righttriangleandparallelogramintheglossaryandlearnthosedefinitions.
b) Figureouthowtorearrangeallsevenpiecestoformatrapezoid.Noticethatyoufirstneedtounderstandtheproblem.Lookupthedefinitionofatrapezoidifyouneedtodoso.
c) Reflectonyourproblemsolvingstrategiesandwriteadescriptionofthestrategies
youusedtoworkontheaboveTangrampuzzle.
d) Explainthesolutionbygivingcarefulinstructions,usingwordsonly(nopictures),forarrangingthesevenpiecestoformatrapezoid.
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2) ChildrenintheearlyelementarygradescansolvepuzzlessimilartotheTangrampuzzleusingpatternblocks(asetofflatblocksinsixshapes:regularhexagon,isoscelestrapezoid,tworhombi,square,andequilateraltriangle).Lookupthetermsrhombus(thepluralformofrhombusisrhombi)andhexagonintheglossary,thendotheactivitydescribedbelow.
a) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide(yourprofessorshouldhaveacodeforyoutoviewthis).Spend10-15minuteslookingthroughUnit5Module1.ThenworkthroughPatternBlockPuzzle3.Howmanydifferentwaysarepossibletofillthisshape?Seeifyoucanfindatleast5.
b) Whatmightchildrenlearnabouttherelationshipsamongthepatternblocksbyworkingontheseproblems?
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3) HereisapictureofallsevenTangrampiecesrearrangedtomakeatriangle.
Youaregoingtobeginto“justifythatwearecorrect.”Thisinvolvesarguingthatthepiecesreallydofittogetherasshown.(Rememberthatjust“lookingliketheyfit”isn’tgoodenough.)Yougettoassumethattheoriginalpiecesreallyareallperfectshapesandthattheyoriginallyfitperfectlytoformasquarelikethis:
a) Arguethatthevertexoftheyellowtrianglealongwiththeverticesofthelargeorangeandbluetrianglesreallydomeettomake180degrees(theyformastraightangle)atthebottomedgeofthepuzzle.
b) Arguethattheedgeoftheorangetrianglefitsperfectlywiththeedgesofthesquareandyellowtriangleandthatthebigbluetrianglereallyfitsperfectlyalongtheedgesoftheparallelogramandyellowtriangle.
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ClassActivity2: DefiningMoments
Wherethereismatter,thereisgeometry.JohannesKepler(1571-1630)
Mathematicaldefinitionsareimportanttomathematiciansbecausetheygiveustheexactcriteriaweneedtoclassifyobjects.Usethedefinitionofapolygontodecidewhethereachoftheobjectsthesketchcallstomindarepolygons.Ifanobjectdoesnotmeetthedefinition,explainexactlyhowitfails.Visittheglossaryifyouneedtolookupterms.Apolygonisasimple,closedcurveintheplanecomposedonlyofafinitenumberoflinesegments.
1) 2)3) 4) 5)6) 7)
8) 9)
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ReadandStudy
Everythingyou’velearnedinschoolas“obvious”becomeslessandlessobviousasyoubegintostudytheuniverse.Forexample,therearenosolidsintheuniverse.There’snotevenasuggestionofasolid.Therearenosurfaces.Therearenostraightlines. R.BuckminsterFuller
Mathematicianscaredeeplyaboutthewordsweusetotalkaboutmathematics.Wehavemanyspecialwords,likeisosceles,thatdonotappearineverydaylanguage.Thesewordshaveprecisedefinitionsthatprovidepowerfulknowledgeabouttheobjectstowhichtheyrefer.Evencommonwordslikerighttakeonspecialmeaningwhentheyareusedinmathematicaltalk.R O L E O F D E F I N I T I O N S
Wewillsaylots,andwemeanlots,moreaboutthetermsweuseinmathematicsthroughoutthepagesofthisbook.Everytimeyouencounteratermyoudon’tknow,lookupthedefinitionintheglossaryandmakecertainyouunderstandjusthowthewordisusedinmathematicaltalk.Tohelpyoudothis,wewillcontinuetounderlineandboldfacemathematicaltermsthefirsttimeweusethem.Thiswillletyouknowthatthewordhasaparticularmeaninginmathematicstowhichyouneedtopayattention.Mathematicaldefinitionsaresoimportantbecause:
1) Definitionsprovideprecisecriteriafordescribingandclassifyingtheseidealobjects;
2) Definitionsdescriberelationshipsamongobjects;and
3) Definitionsgiveusthepowertomakemathematicalarguments.
Let’stalkabitmoreabouteachofthese.IntheClassActivityyouhadtheopportunitytomakesenseofadefinitionandtouseittoclassifypolygons.Diditsurpriseyouthatthedefinitionreliedonsomanyotherterms?Afterall,theideaofapolygondoesn’tseemthatcomplicated.Howeveryouwillfindthatyourstudentshavemanydifferentideasinmindaboutpolygons.Somewillthinkthatsolidshapesarepolygons.Somemightclassifyshapeswithcurvededges(likecircles)aspolygons.Inordertobesurethatweareallimaginingthesameidealobjects,wemustallhavethesamedefinition.Thatsaid,wehavetostartsomewherewhenwritingourdefinitions,andthatmeansthatnotalltermscanbepreciselydefined.Inparticular,ingeometry,weacceptthefollowingideasasundefined:point,line,plane,andspace.Thesetermscorrespondtothe0-dimensional,the1-dimensional,the2-dimensional,andthe3-dimensionalobjectstowhichthedefinitionsofgeometryapply.
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Maybeitseemsstrangethatsuchafundamentalobjectasalineisanundefinedterm,buteventhoughwedon’tdefineit,wecanunderstandalinetobeacollectionofpoints(alsoundefined)thatobeysasetofrules.Wehaveintuitiveideasaboutwhatapointorlineis,butwecanbestunderstandortalkaboutpointsorlinesintermsofamodel.Usefulmodelsofalineincludethecreaseinasheetofpaper,thestraightedgewherethewallofaroommeetsthefloor,atautpieceofthinstring,orthepicturebelow.Ofcourseeachofthesemodelsisonlyarepresentationofaline.A“true”linehasnowidthatall–onlylength–anditextendsindefinitelyinbothdirections.Likeallothermathematicalobjects,a“true”lineisanidealobject–itexistsonlyinourminds.Itmayalsoseemstrangethatwecan’tdefinealinebysayingthatitis“straight.”Thepropertyof“straightness”isanotherintuitiveideathatcarrieswithitthenotionof“shortestdistance.”Thatis,wesaythatalineis“straight”ifitismeasuringtheshortestdistancebetweenpoints(the“tautstring”idea).Theseintuitiveideasof“straight”workwellonflatsurfaces(andintheworldofEuclideangeometry),butarenotashelpfuloncurvedsurfacessuchasasphere.Whatistheshortestdistancebetweentwopointsonasphere?Findaball(oranorangeoraglobe)andastringandhavealook.Twocommonmodelsforaplaneareaflatsheetofpaperandthesurfaceofawhiteboard(providedwerememberthateachisonlyaportionoftheplanewhichactuallyextendsinfinitelyinalldirections).Asecondwaythatweusedefinitionsistocreaterelationshipsbetweenobjects.Forexample,wesaythattwolinesareparalleliftheylieinthesameplaneanddonotintersect.Thisdefinitionhelpsusunderstandtherelationshipbetweenlinesthatareparallelprovidedweunderstandwhataplaneisandwhatitmeansforlinestointersect.Alternatively,wecansaythattwolinesareparalleliftheylieinthesameplaneanddonothaveanypointsincommon.Thisseconddefinitionincorporatestheideaofnon-intersectionwithoutusingawordthatmaynotbeknown.P A R A L L E L A N D P E R P E N D I C U L A R L I N E S
Weusedefinitionsinathirdwaywhenwemakearguments.Wewilltalkaboutthisfurtherinthenextsection,butfornow,let’slookatasimpleexample:Supposewewanttoarguethatnotrianglecanalsobeasquare.Sincethedefinitionofatrianglestatesthatitisapolygonwithexactlythreesidesandthedefinitionofasquarestatesthatitisapolygonwithexactlyfourcongruentsidesandfourrightangles,andsincethreedoesnoteverequalfour;wecanconcludethatitisnoteverpossibleforatriangletohavefoursides.Soatrianglecanneveralsobeasquare.Nowthisargumentmayseemtrivial,butthepointhereisthatweusedefinitionstomakearguments.Usethedefinitionsof“parallel”and“perpendicular”toarguethattwolinesthatareparallelcanneveralsobeperpendicular.
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ConnectionstoTeachingInprekindergartenthroughgrade2allstudentsshouldrecognize,name,build,draw,
compare,andsorttwo-andthree-dimensionalshapes. NCTMPrinciplesandStandardsforSchoolMathematics,2000
Inrecentyearsmoststates(includingWisconsin)haveadoptedcommonstandardsforschoolmathematics.Thesestandards,calledtheCommonCoreStateStandards(CCSS),prescribethemathematicalcontentandpracticesthatteachersshouldaddressateachgradelevel.Asafutureteacher,youwillneedtoknowandunderstandthem.Thepracticestandardsdescribeexpectationsforstudentsacrossallgradelevels.Whichofthesestandardshaveyouexperiencedsofarinthisclass?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Inthisbook,wewillfocusonthecontentstandardsrelatedtogeometryandmeasurement,andwewillbeginnowwithgeometryforchildreninkindergartenandfirstgrade.Takeaminutetoreadthem.
CommonCoreStateStandardsforMathematicalPractice
Childrenshould…
1. Makesenseofproblemsandpersevereinsolvingthem.2. Reasonabstractlyandquantitatively.3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
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©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Inordertohelpchildrentodistinguishbetweendefiningandnon-definingattributes,youmightaskchildrentosortshapesintocategories.Forexample,youmightaskthattheyidentifyallthetrianglesinthefollowinggroupofshapes:
CCSSKindergarten:GeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres).
1. Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.
2. Correctlynameshapesregardlessoftheirorientationsoroverallsize.
3. Identifyshapesastwo-dimensional(lyinginaplane,“flat”)orthree-dimensional(“solid”).
Analyze,compare,create,andcomposeshapes.
4. Analyzeandcomparetwo-andthree-dimensionalshapes,indifferentsizesandorientations,usinginformallanguagetodescribetheirsimilarities,differences,parts(e.g.,numberofsidesandvertices/“corners”)andotherattributes(e.g.,havingsidesofequallength).
5. Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.
6. Composesimpleshapestoformlargershapes.Forexample,“Canyoujointhese
twotriangleswithfullsidestouchingtomakearectangle?”
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Whatconversationscouldyouhavewithchildrenregardingthisactivity?InwhatwaysmightyouuseittoaddresstheCCSSforkindergarten?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Anotherideaistoaskchildrentomakeuptherule,sorttheshapes,andthenhaveotherchildrenfigureoutarulethatwillgivethesame“sort.”Childrensometimesattendtoattributesinsolvingsortingproblemsthatwe,asmathematicians,wouldnotpayattentionto.Thusactivitiesliketheseprovideopportunitiestodrawchildren’sattentiontodifferentthings.Forexample,manychildrenwouldsaythatthis
figure▼is“upsidedown”orthattheseare“differentshapes”becausetheyareorienteddifferently.Mathematicianswouldsaythattheabovefiguresarethesameshape.Theydonottakeorientationoftwo-dimensionalshapesintoaccountwhendecidingifthoseshapesare“thesame.”Belowweshowa“studentsort”fromafirstgradeclassroom.CanyoufigureoutDevione’srule?Istheremorethanonerulethatcouldgivethesamesort?
CCSSGrade1:GeometryReasonwithshapesandtheirattributes.
1. Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthree-sided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.
2. Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.
3. Partitioncirclesandrectanglesintotwoandfourequalshares,describetheshares
usingthewordshalves,fourths,andquarters,andusethephraseshalfof,fourthof,andquarterof.Describethewholeastwoof,orfouroftheshares.Understandfortheseexamplesthatdecomposingintomoreequalsharescreatessmallershares.
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TheseshapesfitMyRule TheseshapesdonotfitMyRule TheCommonCoreStateStandardsaskthatyou,asateacher,alsohelpchildrentotalkaboutthepositionofobjects,toreasonabouthowobjectsarecomposedofotherobjects,andtorecognizewhetheranobjectistwo-dimensional(flat)orthree-dimensional.Whataresomeactivitiesthatyoumightdowithchildrentoaccomplishthesethings?
Homework
Theonlyplacesuccesscomesbeforeworkisinthedictionary. VinceLombardi
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
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A B
3) Thereareseveraltermsassociatedwithlinesthatyouneedtounderstandandusewithpropernotation.Takeafewminutestostudythese.
Supposewehavethethreepoints,A,B,andC.(Noticethatmathematicianscustomarilyusecapitallettersfromthebeginningofthealphabettodenotepoints.SometimeswearetalkingaboutenoughpointsthatwemakeitallthewaytoZ,butwealmostalwaysstartwithA.)ThelineABistheentiresetofpointsextendingforeverinbothdirections.We
commonlydenotealineas AB andrepresent AB asshownbelow(notethearrowsateachendindicatingthatthelinecontinues):
TherayABisthesetofpointsincludingAandallthepointsonthelineABthatareontheBsideofA.ThepointAiscalledthevertexoftheray.WecommonlydenotearayasAB andrepresent AB asshownbelow:
ThelinesegmentABisthesetofpointsbetweenAandB,includingbothAandB,whicharecalledtheendpointsofthelinesegment.WecommonlydenotealinesegmentasAB andrepresent AB asshownbelow:
4) Visittheglossaryandlearntheprecisedefinitionsforeachofthefollowingterms:square,
parallelogram,rectangle,rhombus,andtrapezoid.Makesureyoucanexplainthedefinitionsusinggoodmathematicallanguage.Sketchanexampleofeach.
5) Whichpropertiesaresufficienttodefinearectangle?Thatis,ifaquadrilateralhasaparticularproperty,doyouknowforcertainthatthequadrilateralmustbearectangle?Explainwhyyouransweris‘yes’or‘no’ineachcase.
a) Ifaquadrilateralhastwosetsofcongruentsides,thenitmustbearectangle.b) Ifaquadrilateralhasoppositeanglescongruent,thenitmustbearectangle.c) Ifaquadrilateralhasdiagonalsthatbisecteachother,thenitmustbearectangle.d) Ifaquadrilateralhastworightangles,thenitmustbearectangle.e) Ifaquadrilateralhascongruentdiagonals,thenitmustbearectangle.f) Ifaquadrilateralhasperpendiculardiagonals,thenitmustbearectangle.g) Ifaquadrilateralhastwosetsofparallelsidesandonerightangle,thenitmustbea
rectangle.h) Ifaquadrilateralhastwosetsofcongruentsidesandonerightangle,thenitmustbe
arectangle.
A B
A B
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Concave PolygonsConvex Polygons
6) Studythefollowingexamplesandformadefinitionofeachoftheseterms:convexandconcave,inyourownwords.Thenlookupthemathematicaldefinitionsintheglossary.Explainthemathematicaldefinitionsinyourownwords.
7) Athirdgradeclassweobservedwaslearningaboutparallellines.Theteacherexplained
thatparallellinesarelinesintheplanethathavenocommonpoints.Thenshedrewthepicturebelowandaskedthechildrenwhetherthelinesshownwereparallelornot.
Severalchildrenarguedthattheywereparallel.Whymighttheyhavesaidthat,andwhatwouldyousaytothemastheirteacher?
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ClassActivity3: GetitStraightGobackalittletoleapfurther. JohnClarke
1) Hereisanactivitytohelpyourupperelementarychildrenmaketheconjecturethattakentogether,theanglesofanytrianglecanformastraightangle.Eachgroupshouldcutoutalargeobtusetriangle,alargeacutetriangleandalargerighttriangle.Foreachtriangle,labelthevertexangles(inanyorder)#1,#2and#3.Then,foreachtriangle,tearoffthethreecornersandputthemtogethersothattheanglesareadjacent.Dothis,anddiscusswhatchildrenmightlearn.Didthisactivityprovethattheanglesofatrianglealwayscanformastraightangle?Whyorwhynot?
2) Again,asinpart1),eachgroupshouldcreatealargeobtusetriangle,alargeacutetriangleandalargerighttriangleand,foreachtriangle,labelthevertexangles#1,#2and#3.Butinsteadoftearingoffthecorners,thistimehaveonepersonuseaprotractortocarefullymeasureeachanglelabeled#1,anotherpersonmeasureeachanglelabeled#2,andanotherpersonmeasureeachanglelabeled#3(eachwithoutlookingattheothers’measurements).Wasthetotalanglemeasureforeachtriangle180degrees?Shouldithavealwaysbeen?Explainanydiscrepancies.
(continuedonthenextpage)
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3) Ourexplorationsinparts1)and2)mighthaveconvincedyouthatthesumofthevertexanglesofanytriangleisthesameasastraightangle,butmathematiciansdonotconsidereitherofthosedemonstrationstobeaproof.Whynot,doyouthink?Nowwearegoingtolearntomathematicallyprovethatsumofthevertexanglesofanytriangleisthesameasastraightangle.
Step1.Startwithanytriangle:Nowwe’llcreatealinethroughonevertexthatisparalleltotheoppositesideofthetriangleandlabelalltheanglessowecantalkaboutthem.Weknowwecanalwaysdothisbecauseitisanaxiomofplanegeometrythatthroughapointnotonalinetherecanbedrawnone(andonlyone)lineparalleltothegivenline.
Bytheway,itshouldn’tbeobvioustoyouwhywe’vedecidedtocreatethisparallelline;itjustturnsouttogiveusagreatwaytobeginourproof.InStep2,wearegoingtoprovethatÐ1iscongruenttoÐ5,andthatÐ2iscongruenttoÐ4.Sofornow,supposingthistobetrue,arguethatangles1,2,and3wouldcombinetoformastraightangle.
(continuedonthenextpage)
m
5
432
1
B
A
C
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Step2.Thisistheonlymissingpieceofourproof.Lookupthedefinitionsofatransversalandofalternateinteriorangles.CanyouseethatÐ1andÐ5arealternateinteriorangles?Whatisthecorrespondingtransversal?
CanyouseethatÐ2andÐ4arealternateinteriorangles?Whatisthecorrespondingtransversal?Nowarguethatiftwoparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.Youmayassumethatiftwoparallellinesarecutbyatransversal,thentheinterioranglesonthesamesideofthetransversalformastraightangle.(Note:Thisisabigthingtoassume.ItisoneofEuclid’sfundamentalassumptionsaboutgeometry.)
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ReadandStudy
Iargueverywell.Askanyofmyremainingfriends.Icanwinanargumentonanytopic,againstanyopponent.Peopleknowthisandsteerclearofmeatparties.
Often,asasignoftheirgreatrespect,theydon’teveninviteme. DaveBarry
Mathematicalthinkingalwaysinvolvesreasoningandmakingarguments,andwehaveawholevocabularyfordescribingthatprocess.Inthissection,wehighlightthetermswemathematiciansusetodescribefacetsofdoingmathematics.Theseareimportant.Makesureyouunderstandthem.L A N G U A G E O F M A T H E M A T I C S
1) Anaxiomisastatementthatweagreetoacceptwithoutproof.Itisanassumptionorstartingpoint.(Note:Anotherwordforaxiomispostulate.)
2) Inductivereasoningiscomingtoaconclusionbasedonexamples.Forexample,Iobservethat3,5and7areallprimenumbers.Now,basedontheseexamplesImightreason(incorrectly,bytheway)thatalloddnumbersareprime.OrImightnoticethatthesunrosedaybeforeyesterday,itroseyesterday,itrosetoday.SoImightconcludethatthesunwillrisetomorrow.Thisisinductivereasoning.
3) Deductivereasoningiscomingtoconclusionbasedonlogic.Forexample,Iwillargue
deductivelythatthesunwillcomeuptomorrow:Theearthiscaughtinthesun’sgravitysoitwon’tfloataway,andtheearthisspinning.Wearehereontheearthandsowhenourpartoftheearthturnstowardthesun,wesayit“comesup.”Aslongasnocatastropheoccurstochangethesefacts,thesunwillrisetomorrow.We’llgiveyouanother(moremathematical)exampleinaminute.
D E D U C T I V E V E R S U S I N D U C T I V E R E A S O N I N G
4) Aconjectureisahypothesisoraguessaboutwhatistrue.Forexample,aftersomeexperiencewithcircles,astudentmightconjecturethattwointersectingcirclesalwayssharetwodistinctpointsincommon.Conjecturesareoftenmadebasedoninductivereasoning.
5) Acounterexampleisaspecificexamplethatshowsaconjectureisfalse.Forexample,
thetwocirclesbelowaretangent(theyshareexactlyonepointincommon)andsotheaboveconjectureisshowntobefalsebythecounterexampleshownhere:
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6) proof:amathematicalproofconsistsofadeductiveargumentthatestablishesthetruth
ofaclaim.(Note:Bytruthwemeantruthinthecontextofthemathematicalworldthatiscreatedbytheaxioms.Somethingistrueifitisalogicalimplicationoftheaxioms.)
7) theorem:atheoremisamathematicalstatementthathasbeenprovedtobetrue.For
example,itisatheoremthatverticalanglesarecongruent.YouprovedthisintheClassActivity.(Note:Anotherwordfortheoremisproposition.)
WearegoingtousewhatyoudidintheClassActivitytohighlightthemeaningofsomeoftheseterms.First,youtoreapartavarietyoftrianglesandarrangedthemtoseethateachappearedtohaveastraightangle(180degrees)ofvertexangles.Thiswasinductivereasoning,becauseyouweretestingexamplesoftrianglestoseewhatseemedtrueabouttheirvertexanglemeasure.Atthispointitwouldhavebeenreasonabletoconjecturethatthesumofthevertexanglesofatriangleis180degrees.Thenyouwereaskedtomakeanargumentusingdefinitions,axioms,andlogicthatyouwerecorrect.Inotherwordsyougaveaproofthatthevertexanglesofatrianglesumto180degrees,andnowthatconjectureiscalledatheorem.Oneofthereasonsthatgeometryclass(remembertenthgrade?)hastraditionallyfocusedonproofisthattheaxiomsofgeometryareeasiertostatethantheaxiomsofarithmetic.Butproofispartofallmathematics.Don’tworry,wearenotplanningtofocusontwo-columnproofsoranaxiomaticdevelopmentofgeometry,butwewouldberemissifwedidn’tatleaststatetheoriginalaxioms(assumptions)ofplanegeometry.TheaxiomswerefirstmadeexplicitbyEuclid,aGreekmathematicianwholivedandworkedattheAcademyinAlexandria,Egypt.Heisbestknownforwritinga13-volumebookofmathematicscalledTheElements-thesecondmostpublishedbookintheworld.Ithasbeenusedasamathematicstextbookforover2000years.Inthefirsttwovolumesofthiswork,hedevelopedallofthe(then)knowntheoremsabouttwo-dimensional(plane)geometrystartingwithjustfiveaxioms:I D E A O F A X I O M S
TheAxiomsofEuclideanGeometry:1. Auniquestraightlinesegmentcanbedrawnfromanypointtoanyotherpoint.2. Alinesegmentcanbeextendedtoproduceauniquestraightline.3. Auniquecirclemaybedescribedwithagivencenterandradius.4. Allrightanglesareequaltoeachother.
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5a.Ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.
5b.Throughapointnotonalinetherecanbedrawnexactlyonelineparalleltothegivenline.
Uponassumingaxioms1-4,axioms5aand5bareequivalent.Axiom5aisEuclid’sversion,and5bisamoremoderninterpretationknownasPlayfair’sAxiom.Lookinthisbook’sappendixandyouwillfindtheaxioms(postulates)andtheorems(propositions)fromBook1ofEuclid’sElements,writtenmorethan2,000yearsago.Axiom5aishisfifthpostulate.Youmayfinditinterestingthatuntilthelate1800s,manymathematiciansthoughtEuclid’sfifthaxiomwasredundant–thatitalreadyhadtobetrueifaxioms1-4wereassumedtobetrue–butithassincebeenproventhatthosemathematicianswerewrong.Noticethateachoftheaxiomsdescribessomethingthatcanbeconstructedwiththeexceptionofthefourth.Axiom4saysthattheEuclideanplaneis,insomesense,uniform(nodistortions).Namely,itsaysthatwhereveryouconstructaperpendicularlinesontheplane(andsoformfourangles),thoseangleswillallhavethesamemeasure.ItturnsoutthatEuclidmadesomeimplicitassumptionsthatheshouldhavestatedasadditionalaxiomsinordertodogeometryrigorously–andsomodernmathematicianshaveextendedhislistofaxioms.Butyoudon’tneedtoworryaboutthosetechnicalitiesinthiscourse.Sketchapictureofaxioms1–3and5tohelpyoumakesenseofeach.Then,describehowyoucancreateanequilateraltrianglebyfollowingEuclid’saxioms.WealsowanttonotethatEuclidoftenusedtheword“equal”whenwewouldusetheword“congruent.”Today’smathematiciansuse“equal”whentheywanttocomparetwonumbers.Sowemightsaythat½isequalto0.5.Weusetheword“congruent”whenwewanttosaythattwoobjects(liketwotrianglesortwosegments)arethesamesizeandshape.Thebasicideahereisthattwoobjectsarecongruentinthecasewhereifoneobjectwasmovedtolieontopoftheotherobject,theywouldcorrespondexactly.Wewilldoamorecarefuljobofdefining“congruent”later.
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Whileprovingtheoremsisnotthefocusofthiscourse,wemayoccasionallyaskthatyoutrytoproveaEuclideantheorem.Whenwedo,youshouldturntotheAppendixwhereEuclid’spostulates(axioms)andpropositionsarelistedandfindit.Thenyouarefreetouse(assume)anypostulateandanypropositionlistedbeforetheoneyouaretryingtoprove.Forexample,sayyouwanttoprovethatinanisoscelestriangle,thebaseanglesarecongruent.GototheAppendix(reallydoit)andseeifyoucanfindthattheorem.Thencomerightbackhere.Let’sendthissectionwiththebigidea:Geometryintheplanearisesfromsomeintuitiveidealobjects,theirmathematicaldefinitions,thesefiveaxioms,andalotofdeductivereasoning.Infact,asecondpossibledefinitionofgeometryisthis:anaxiomaticsystemaboutidealobjectscalled“points,”collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.
ConnectionstoTeaching
Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstorecognizereasoningandproofasfundamentalaspectsofmathematics;tomakeandinvestigatemathematicalconjectures;todevelopandevaluatemathematicalargumentsandproofs;andtoselectandusevarioustypesofreasoningandmethodsofproof.
ReasoningandProofStandard,NCTMPrinciplesandStandardsforSchoolMathematics,2000
M A K I N G C O N V I N C I N G M A T H E M A T I C A L A R G U M E N T S
Itisimportantthatyouprovidechildreninyourfutureclasseswiththeopportunitiestoreallydomathematics.Theytooneedtouseinductivereasoning,makeconjectures,lookforcounterexamples,andmakedeductivearguments.Inordertounderstandbetterwhatyoushouldexpectfromchildren,readthefollowingfromthediscussionoftheReasoningandProofstandardforthePreK–2gradebandfoundinNCTMPrinciplesandStandardsforSchoolMathematics,2000,pages122–125.Noticetheiruseofmathematicallanguage.
Whatshouldreasoningandprooflooklikeinprekindergartenthroughgrade2?p.122
Theabilitytoreasonsystematicallyandcarefullydevelopswhenstudentsareencouragedtomakeconjectures,aregiventimetosearchforevidencetoproveordisprovethem,andareexpectedtoexplainandjustifytheirideas.Inthebeginning,perceptionmaybethepredominantmethodofdeterminingtruth:ninemarkersspreadfarapartmaybeseenas“more”thanelevenmarkersplacedclosetogether.Later,asstudentsdeveloptheirmathematicaltools,theyshoulduseempiricalapproachessuchasmatchingthecollections,whichleadstotheuseofmore-abstractmethodssuchascountingtocomparethecollections.Maturity,experiences,andincreasedmathematicalknowledgetogetherpromotethedevelopmentofreasoningthroughouttheearlyyears.»
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Creatinganddescribingpatternsofferimportantopportunitiesforstudentstomakeconjecturesandgivereasonsfortheirvalidity,asthefollowingepisodedrawnfromclassroomexperiencedemonstrates.
Thestudentwhocreatedthepatternshowninfigure4.27proudlyannouncedtoherteacherthatshehadmadefourpatternsinone.“Look,”shesaid,“there’striangle,triangle,circle,circle,square,square.That’sonepattern.Thenthere’ssmall,large,small,large,small,large.That’sthesecondpattern.Thenthere’sthin,thick,thin,thick,thin,thick.That’sthethirdpattern.Thefourthpatternisblue,blue,red,red,yellow,yellow.”Herfriendstudiedtherowofblocksandthensaid,“Ithinktherearejusttwopatterns.See,theshapesandcolorsareanAABBCCpattern.ThesizesareanABABABpattern.ThickandthinisanABABABpattern,too.Soyoureallyonlyhavetwodifferentpatterns.”Thefirststudentconsideredherfriend’sargumentandreplied,“Iguessyou’reright—butsoamI!”
Fig.4.27.Fourpatternsinone
Beingabletoexplainone’sthinkingbystatingreasonsisanimportantskillforformalreasoningthatbeginsatthislevel.
Findingpatternsonahundredboardallowsstudentstolinkvisualpatternswithnumberpatternsandtomakeandinvestigateconjectures.Teachersextendstudents’thinkingbyprobingbeyondtheirinitialobservations.Studentsfrequentlydescribethechangesinnumbersorthevisualpatternsastheymovedowncolumnsoracrossrows.Forexample,askedtocoloreverythirdnumberbeginningwith3(seefig.4.28),differentstudentsarelikelytoseedifferentpatterns:“Somerowshavethreeandsomehavefour,”or“Thepatterngoessidewaystotheleft.”Somestudents,seeingthediagonalsinthepattern,willnolongercountbythreesinordertocompletethepattern.Teachersneedtoaskthesestudentstoexplaintotheirclassmateshowtheyknowwhattocolorwithoutcounting.Teachersalsoextendstudents’mathematicalreasoningbyposingnewquestionsandaskingforargumentstosupporttheiranswers.“Youfoundpatternswhencountingbytwos,threes,fours,fives,andtensonthehundredboard.Doyouthinktherewillbepatternsifyoucountbysixes,sevens,eights,ornines?Whataboutcountingbyelevensorfifteensorbyanynumbers?”Withcalculators,studentscouldextendtheirexplorationsoftheseandothernumericalpatternsbeyond100.
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Fig.4.28.Patternsonahundredboard
p.123Students’reasoningaboutclassificationvariesduringtheearlyyears.Forinstance,whenkindergartenstudentssortshapes,onestudentmaypickupabigtriangularshapeandsay,“Thisoneisbig,”andthenputitwithotherlargeshapes.Afriendmaypickupanotherbigtriangularshape,traceitsedges,andsay,“Threesides—atriangle!”andthenput»itwithothertriangles.Bothofthesestudentsarefocusingononlyoneproperty,orattribute.Bysecondgrade,however,studentsareawarethatshapeshavemultiplepropertiesandshouldsuggestwaysofclassifyingthatwillincludemultipleproperties.Bytheendofsecondgrade,studentsalsoshouldusepropertiestoreasonaboutnumbers.Forexample,ateachermightask,“Whichnumberdoesnotbelongandwhy:3,12,16,30?”Confrontedwiththisquestion,astudentmightarguethat3doesnotbelongbecauseitistheonlysingle-digitnumberoristheonlyoddnumber.Anotherstudentmightsaythat16doesnotbelongbecause“youdonotsayitwhencountingbythrees.”Athirdstudentmighthaveyetanotherideaandstatethat30istheonlynumber“yousaywhencountingbytens.”Studentsmustexplaintheirchainsofreasoninginordertoseethemclearlyandusethemmoreeffectively;atthesametime,teachersshouldmodelmathematicallanguagethatthestudentsmaynotyethaveconnectedwiththeirideas.Considerthefollowingepisode,adaptedfromAndrews(1999,pp.322–23):
Onestudentreportedtotheteacherthathehaddiscovered“thatatriangleequalsasquare.”Whentheteacheraskedhimtoexplain,thestudentwenttotheblockcornerandtooktwohalf-unit(square)blocks,twohalf-unittriangular(triangle)blocks,andoneunit(rectangle)block(showninfig.4.29).Hesaid,“Ifthesetwo[squarehalf-units]arethesameasthisoneunitandthesetwo[triangularhalf-units]arethesameasthisoneunit,thenthissquarehastobethesameasthistriangle!”
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Fig.4.29.Astudent’sexplanationoftheequalareasofsquareandtriangularblockfaces
p.124Eventhoughthestudent’swording—thatshapeswere“equal”—wasnotcorrect,hewasdemonstratingpowerfulreasoningasheusedtheblockstojustifyhisidea.Insituationssuchasthis,teacherscouldpointtothefacesofthetwosmallerblocksandrespond,“Youdiscoveredthat»theareaofthissquareequalstheareaofthistrianglebecauseeachofthemishalftheareaofthesamelargerrectangle.”
Whatshouldbetheteacher’sroleindevelopingreasoningandproofinprekindergartenthroughgrade2?Teachersshouldcreatelearningenvironmentsthathelpstudentsrecognizethatallmathematicscanandshouldbeunderstoodandthattheyareexpectedtounderstandit.Classroomsatthislevelshouldbestockedwithphysicalmaterialssothatstudentshavemanyopportunitiestomanipulateobjects,identifyhowtheyarealikeordifferent,andstategeneralizationsaboutthem.Inthisenvironment,studentscandiscoveranddemonstrategeneralmathematicaltruthsusingspecificexamples.Dependingonthecontextinwhicheventssuchastheoneillustratedbyfigure4.29takeplace,teachersmightfocusondifferentaspectsofstudents’reasoningandcontinueconversationswithdifferentstudentsindifferentways.Ratherthanrestatethestudent’sdiscoveryinmore-preciselanguage,ateachermightposeseveralquestionstodeterminewhetherthestudentwasthinkingaboutequalareasofthefacesoftheblocks,oraboutequalvolumes.Oftenstudents’responsestoinquiriesthatfocustheirthinkinghelpthemphraseconclusionsinmore-precisetermsandhelptheteacherdecidewhichlineofmathematicalcontenttopursue.Teachersshouldpromptstudentstomakeandinvestigatemathematicalconjecturesbyaskingquestionsthatencouragethemtobuildonwhattheyalreadyknow.Intheexampleofinvestigatingpatternsonahundredboard,forinstance,teacherscouldchallengestudentstoconsiderotherideasandmakeargumentstosupporttheirstatements:“Ifweextendedthehundredboardbyaddingmorerowsuntilwehadathousandboard,howwouldtheskip-countingpatternslook?”or“Ifwemadechartswithrowsofsixsquaresorrowsoffifteen
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squarestocounttoahundred,wouldtherebepatternsifweskip-countedbytwosorfivesorbyanynumbers?”Throughdiscussion,teachershelpstudentsunderstandtheroleofnonexamplesaswellasexamplesininformalproof,asdemonstratedinastudyofyoungstudents(CarpenterandLevi1999,p.8).Thestudentsseemedtounderstandthatnumbersentenceslike0+5869=5869werealwaystrue.Theteacheraskedthemtostatearule.Annsaid,“Anythingwithazerocanbetherightanswer.”Mikeofferedacounterexample:“No.Becauseifitwas100+100that’s200.”Annunderstoodthatthisinvalidatedherrule,sosherephrasedit,“Isaid,umm,ifyouhaveazeroinit,itcan’tbelike100,becauseyouwantjustplainzerolike0+7=7.”Thestudentsinthestudycouldformrulesonthebasisofexamples.Manyofthemdemonstratedtheunderstandingthatasingleexamplewasnotenoughandthatcounterexamplescouldbeusedtodisproveaconjecture.However,moststudentsexperienceddifficultyingivingjustificationsotherthanexamples.
p.125Fromtheverybeginning,studentsshouldhaveexperiencesthathelpthemdevelopclearandprecisethoughtprocesses.Thisdevelopmentofreasoningiscloselyrelatedtostudents’languagedevelopmentandisdependentontheirabilitiestoexplaintheirreasoningratherthanjust»givetheanswer.Asstudentslearnlanguage,theyacquirebasiclogicwords,includingnot,and,or,all,some,if...then,andbecause.Teachersshouldhelpstudentsgainfamiliaritywiththelanguageoflogicbyusingsuchwordsfrequently.Forexample,ateachercouldsay,“Youmaychooseanappleorabananaforyoursnack”or“Ifyouhurryandputonyourjacket,thenyouwillhavetimetoswing.”Later,studentsshouldusethewordsmodeledforthemtodescribemathematicalsituations:“Ifsixgreenpatternblockscoverayellowhexagon,thenthreebluesalsowillcoverit,becausetwogreenscoveroneblue.”Sometimesstudentsreachconclusionsthatmayseemoddtoadults,notbecausetheirreasoningisfaulty,butbecausetheyhavedifferentunderlyingbeliefs.Teacherscanunderstandstudents’thinkingwhentheylistencarefullytostudents’explanations.Forexample,onhearingthathewouldbe“StaroftheWeek”inhalfaweek,Benprotested,“Youcan’thavehalfaweek.”Whenaskedwhy,Bensaid,“Sevencan’tgointoequalparts.”Benhadtheideathattodivide7by2,therecouldbetwogroupsof3,witharemainderof1,butatthatpointBenbelievedthatthenumber1couldnotbedivided.Teachersshouldencouragestudentstomakeconjecturesandtojustifytheirthinkingempiricallyorwithreasonablearguments.Mostimportant,teachersneedtofosterwaysofjustifyingthatarewithinthereachofstudents,thatdonotrelyonauthority,andthatgraduallyincorporatemathematicalpropertiesandrelationshipsasthebasisfortheargument.Whenstudentsmakeadiscoveryordetermineafact,ratherthantellthemwhetheritholdsforallnumbersorifitiscorrect,theteachershouldhelpthestudentsmakethatdeterminationthemselves.Teachersshouldasksuchquestionsas“Howdoyouknowitistrue?”andshouldalsomodelwaysthatstudentscanverifyordisprovetheirconjectures.Inthisway,studentsgraduallydeveloptheabilitiestodeterminewhetheranassertionistrue,ageneralizationvalid,orananswercorrectandtodoitontheirowninsteadofdependingontheauthorityoftheteacherorthebook.
NCTMPrinciplesandStandardsforSchoolMathematics
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Homework
Euclidtaughtmethatwithoutassumptionsthereisnoproof.Therefore,inanyargument,examinetheassumptions. EricTempleBell
1) DoalltheitalicizedthingsintheReadandStudysection.Writeadescriptionofthe
strategiesyouusedinsolvingtheproblemofcreatinganequilateraltriangleusingonlyEuclid’sfiveaxioms.
2) WhichofthechildrenfromtheConnectionsreadingareusingdeductivereasoningand
whichareusinginductivereasoning?Explain.
3) AccordingtotheNCTM,whatistheteacher’sroleinpromotingreasoningamongchildrenintheearlyelementarygrades?
4) Explainwhyverticalanglesformedbyintersectinglinesarethesame.
5) HavealookatProposition32oftheappendix.Doyouseethatitisthetheoremyou
provedintheClassActivity?Whichpropositionsthatcomebefore#32didyouuseintheproof?
6) DecideifeachofthefollowingstatementsaboutEuclideanlinesandanglesistrueorfalsebyexploringexamplesandlookingupdefinitions.Ifyoudecidethatastatementistrue,writeadeductiveargumentbasedonaxiomsanddefinitions.Ifyoudecidethestatementisfalse,giveacounterexampleoradeductiveargumentthatitisnotpossible.
a) Anytwodistinctlineswilleitherintersectinexactlyonepointortheywillbeparallel.
b) Thereexisttwoacuteangleswhicharesupplementary.c) Everytwolinesthatareeachparalleltoathirdlinemustbeparalleltoeach
other.d) Everytwolinesthatareeachperpendiculartoathirdlinewillbeperpendicular
toeachother.e) Everytwoacuteanglesmustbecomplementary.f) Thereexisttwooppositesidesinanytrapezoidwhichareparallel.g) Ifoneoftwosupplementaryanglesisacute,theotheranglemustbeobtuse.
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7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide(yourprofessorshouldhaveacodeforyoutoviewthis).Spend10-15minuteslookingthroughUnit6Module1.CarefullyreadthroughtheactivityGuessMyShape,thenmakeupyourownriddleforyoursecondgradestudentstosolve.
8) Wehaveaconjecture.Everyrectangleisaparallelogram.Giveaninductiveargumentthatthisconjectureistrue.Now,sincemathematiciansarenotsatisfieduntiltheyhaveadeductiveargument,giveoneofthose.
9) Wehaveanotherconjecture!Everyrectangleisasquare.Isthisconjecturetrueorfalse?
Ifitistrue,giveadeductiveargument.Ifitisfalse,giveacounterexample.
10) HerearesomemoreoftheCommonCoreStateStandardsrelatedtoreasoningaboutshapes.InwhatwaysdoHWproblems4)–7)addressthesestandards?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
11) DrawaVenndiagramshowingtherelationshipbetweenallthevariousquadrilateralswehavestudied.HowdoesthisfitwiththeCommonCoreStateStandardsdescribedbelow?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CCSSGrade3:Reasonwithshapesandtheirattributes.
1. Understandthatshapesindifferentcategories(e.g.,rhombuses,rectangles,andothers)mayshareattributes(e.g.,havingfoursides),andthatthesharedattributescandefinealargercategory(e.g.,quadrilaterals).Recognizerhombuses,rectangles,andsquaresasexamplesofquadrilaterals,anddrawexamplesofquadrilateralsthatdonotbelongtoanyofthesesubcategories.
CCSSGrade5:Classifytwo-dimensionalfiguresintocategoriesbasedontheirproperties.
1. Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.Forexample,allrectangleshavefourrightanglesandsquaresarerectangles,soallsquareshavefourrightangles.
2. Classifytwo-dimensionalfiguresinahierarchybasedonproperties.
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ClassActivity4: AlltheAngles
DonotworryaboutyourdifficultiesinMathematics.Icanassureyoumineare stillgreater.
AlbertEinstein
1) Hereisthedescription–fromtheCommonCoreStateStandardsforgradefour–ofhowtothinkaboutmeasuringanglesusingdegrees.Readitcarefullyandsketchapicturetohelpyourgroupmakesenseoftheirexplanation.
Anangleismeasuredwithreferencetoacirclewithitscenteratthecommonendpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
M E A S U R I N G A N G L E S I N D E G R E E S
2) Makesureeveryoneinyourgroupcanusetheirprotractortomeasure(indegrees)theangleindicatedbelow:
(continuedonthenextpage)
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3) Findaformulaforthesumofthemeasuresofthevertexanglesofann-gon(apolygonwithnsides).Youmayneedtocollectsomedata.Intheend,makeamathematicalargumentthattheformulayoufindwillworkforaconvexpolygonofanynumberofsides.(Theformulathatyoudevelopedwillworkforconcavepolygonsaswell,buttheargumentistrickier,sowearenotaskingyoutojustifyitatthistime.)
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ReadandStudy
Theknowledgeofwhichgeometryaimsistheknowledgeoftheeternal. Plato
Aregularpolygonisoneinwhichallofthelinesegmentsarecongruentandallofthevertexanglesarealsocongruent.Sketcharegulartriangleandaregularquadrilateral.Isthebelowrhombusaregularquadrilateral?Explain.
R E G U L A R P O L Y G O N S
Polygonsareoftennamedforthenumberofsidestheycontain.Infact,theprefix“poly”means“many”andtheroot“gon”means“side.”So“polygon”means“many-sided”figure.Inordertonamepolygons,you’llneedtoknowthefollowingprefixes: Five–“penta” Six–“hexa” Seven–“hepta” Eight–“octa” Nine–“nona” Ten–“deca” Twelve–“dodeca”Forexample,five-sidedpolygonsarecalledpentagons.Hereareacoupleexamplesofconvexpentagons.Drawanexampleofaconcavepentagon.Mathematiciansidentifythreetypesofanglesinapolygon:thevertexangles,thecentralangles,andtheexteriorangles.Studythehexagoninthefollowingdiagramandthenexplainthedifferencebetweenthethreetypesofanglesinyourownwords.Theboldarcsmarkthevertexangles;thethinsolidarcsmarkthecentralangles;andthedashedarcsmarktheexteriorangles.PointGisanyinteriorpoint.
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ThinkaboutthesumofthesixcentralanglesformedatpointG.Thissumwillalwaysbe360°regardlessofwhereintheinteriorpointGis,regardlessofhowmanysidesthepolygonhas,andregardlessofwhetherornotthepolygonisregular.Why?Wealsoclaimthatthesumoftheexterioranglesofanypolygonis360°.Wewillaskyoutomakeamathematicalargumenttosupportthisclaiminthehomeworksection.Thecaseofthesumofthevertexanglesofapolygonistheinterestingcase.IntheClassActivityyoufoundthatthissumdoesdependonthenumberofsidesinthepolygon.Doesitdependonwhetherthepolygonisregular?Explain.
ConnectionstoTeaching
Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.
CCSS,p.6
Anglesarenotoriouslydifficultidealobjectsforchildren.Ontheonehand,theyareoftendefinedasafigureformedbytworayswithacommonendpoint(and,infact,thatisexactlyhowwehavedefinedthem).Ontheotherhand,whatisimportantinmeasuringanangleisitsdegreeofturn.
A
B
C
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F
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Whenyoutalkaboutangleswithchildren,wesuggestthatyoualwaysuseyourhandorarmtoshowthesweepoftheangleinadditiontoshowingthestaticpicture.Childreningradefourlearntouseaprotractortomeasureanglesindegreesandtoaccuratelyestimatethemeasureofangles.BelowyouwillfindtheCCSSrelatedtoanglemeasureinthisgrade.Readthemcarefully.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CCSSGrade4:Geometricmeasurement:understandconceptsofangleandmeasureangles.
1. Recognizeanglesasgeometricshapesthatareformedwherevertworaysshareacommonendpoint,andunderstandconceptsofanglemeasurement:
a. Anangleismeasuredwithreferencetoacirclewithitscenteratthecommon
endpointoftherays,byconsideringthefractionofthecirculararcbetweenthepointswherethetworaysintersectthecircle.Ananglethatturnsthrough1/360ofacircleiscalleda“one-degreeangle,”andcanbeusedtomeasureangles.
b. Ananglethatturnsthroughnone-degreeanglesissaidtohaveananglemeasureofndegrees.
2. Measureanglesinwhole-numberdegreesusingaprotractor.Sketchanglesof
specifiedmeasure.
3. Recognizeanglemeasureasadditive.Whenanangleisdecomposedintonon-overlappingparts,theanglemeasureofthewholeisthesumoftheanglemeasuresoftheparts.Solveadditionandsubtractionproblemstofindunknownanglesonadiagraminrealworldandmathematicalproblems,e.g.,byusinganequationwithasymbolfortheunknownanglemeasure.
CCSSGrade4:Drawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.
1. Drawpoints,lines,linesegments,rays,angles(right,acute,obtuse),andperpendicularandparallellines.Identifytheseintwo-dimensionalfigures.
2. Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.
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Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module1.PrintoutandthenworkthroughtheMeasuringPatternBlockAnglesactivity.(Notethatinthisactivitystudentsarenotusingprotractorstomeasuretheangles.Youalsowillnotneedtouseaprotractor.).
Homework
Energyandpersistenceconquerallthings. BenjaminFranklin
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
3) ChildreninyourclassmaysaythatÐABCissmallerthanÐDEF.Isthistrue?Astheir
teacher,whatwouldyousaytothesechildren?
4) Usingtheresultsoftheclassactivity,findthemeasureofonevertexangleinan
equilateraltriangle,asquare,aregularpentagon,aregularhexagon,aregularoctagon,aregulardecagon,andaregulardodecagon.
5) Astudentclaimsthatthesumofthevertexanglesofahexagonis6×180becauseeachtrianglehas180degreesofanglesmeasureandshehasshownthat6trianglestomakeupthehexagon.Whatwillyousayasherteacher?
6) IntheReadandStudysectionweclaimthatthesumoftheexterioranglesofanypolygonisalways360°.Makeamathematicalargumenttosupportthisclaim.
B
A
CE
D
F
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ClassActivity5: ALogicalInterludeEquationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsof
geometry. StephenHawking
Inthepicturebelow,eachcardhasacolorononesideandashapeontheotherside.Whichcard(s)wouldyouhavetoturnovertobesurethatthefollowingstatementistrue?
Ifacardisredononeside,thenithasasquareontheotherside.
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ReadandStudy
Whenintroducedatthewrongtimeorplace,goodlogicmaybetheworstenemyofgoodteaching.
GeorgePolya TheAmericanMathematicalMonthly,v.100,no3.
HerearetwotheoremsaboutparallellinesthatcanbeprovedfromEuclid’saxioms.
1) Iftwolinesarecutbyatransversalandthealternateinterioranglesarecongruent,thenthelinesareparallel.
2) Iftwoparallellinesarecutbyatransversal,thenthealternateinterioranglesarecongruent.
Drawasketchtomakesurethatyouseewhateachofthesehastosay.
Noticethattheybothhavean‘if-then’statementform.Lotsofmathematicaltheoremsarelikethis.Whenatheoremisstatedin‘if-then’form,whateverfollowsthe‘then’isalwaystruewhenevertheconditionsstatedinthe‘if’partaremet.Youcanthinkofan‘if-then’statementasapromisethatiskeptunlessthe‘if’partistrueandthe‘then’partisnot.Wecallstatement2)theconverseofstatement1).Thesetwotheoremsmaysoundthesametoyou,buttheyarenotsayingthesamething.Tohelpyouseethis,let’schangethecontext.Hereisanotherpairofstatementsinwhichthesecondistheconverseofthefirst(andviceversa):
3) IfIliveinChicago,thenIliveinIllinois.
4) IfIliveinIllinois,thenIliveinChicago.Thinkabouteachofthestatements.Whichoftheseistrue?Sinceoneistrueandtheotherfalse,thesetwostatementscannotbesayingthesamething.Inotherwords,theconverseofastatementisalogicallydifferentstatementfromtheoriginalstatement.C O N V E R S E A N D C O N T R A P O S I T I V E
Nowhereisastatementthatislogicallyequivalenttotheoriginalstatementmadein3)above.Itiscalledthecontrapositiveofstatement3).
5) IfIdonotliveinIllinois,thenIdonotliveinChicago.
ThisisessentiallywhatyoufoundwhenyouworkedontheClassActivity.Writethecontrapositivetostatement4)above.
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Itishelpfultoknowthatastatementanditscontrapositiveareequivalentbecausethatmeansthatyoucanproveastatementbyprovingitscontrapositive.
ConnectionstoTeaching
Thebeginningofknowledgeisthediscoveryofsomethingwedonotunderstand. FrankHerbert
Nowthatwehavetalkedabitaboutdoingmathematics,wewanttoshowyouthattheCommonCoreStateStandardsrequirethatchildrenalsodomathematics.Giveanexampleofsomethingyouhavedoneinclasssofarthistermthatmeetseachofthesestandards.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CommonCoreStateStandardsforMathematicalPractice
Childrenshould…
1. Makesenseofproblemsandpersevereinsolvingthem.
2. Reasonabstractlyandquantitatively.
3. Constructviableargumentsandcritiquethereasoningofothers.
4. Modelwithmathematics.
5. Useappropriatetoolsstrategically.
6. Attendtoprecision.
7. Lookforandmakeuseofstructure.
8. Lookforandexpressregularityinrepeatedreasoning.
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Homework
Amultitudeofwordsisnoproofofaprudentmind. Thales
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
3) Decideifeachofthefollowingstatementsistrueorfalse.Iftrue,giveamathematical
explanation.Iffalse,giveacounterexample.a) Ifaquadrilateralisasquare,thenitisarectangle.b) Ifaquadrilateralhasapairofparallelsides,thenitmusthaveapairof
oppositesidesthatarecongruent.c) Ifthediagonalsofaquadrilateralareperpendiculartoeachother,thenthe
quadrilateralisarhombus.d) Ifaquadrilateralhasonerightangle,thenallofitsanglesmustberight
angles.e) Writetheconverseofeachofthestatementsina)–d)above.Whichare
true?f) Writethecontrapositiveofeachofthestatementsina)–d)above.Which
aretrue?
4) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module4.ThenworkthroughtheClockAngles&ShapeSketchesfromthehomelinksection.Howisdeductivereasoningusedtosolvetheseproblems?
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SummaryofBigIdeas
Hey!What’sthebigidea? Sylvester
• Geometryisastudyofidealobjects–notrealobjects.
• Definitionsallowustonameandcategorizeidealobjects,tocreaterelationships
betweenobjects,andtomakeargumentsaboutthepropertiesofobjects.• Onedefinitionofgeometryisthatitisthestudyofidealshapesandtheirproperties,of
thepatternsthoseshapescanform,andoftheactionsonthoseshapesthatpreservetheirproperties.
• Aseconddefinitionisthatgeometryisanaxiomaticsystemaboutobjectscalled
“points,”collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.
• Mathematicalthinkingalsoinvolvesdeductivereasoningandmakingarguments–this
meansusinglogicaswellasusingdefinitionsandaxioms.• Congruenceisanimportantrelationshipbetweengeometricobjects.Wesaytwo
objectsarecongruentiftheycoincidewhenplacedontopofeachother.
• InEuclideangeometrythesumoftheanglesinatriangleisalways180degreesandyoucanexplainwhythisisso.
• Youwillrepresentthemathematicalcommunityforyourstudents.Theywilllooktoyoutounderstandwhatwemathematiciansdoandhowwethink.Yourstudentswilltrytodiscernthemeaningsthatyou,yourcurriculummaterials,andotherstudentsgivetoideas,strategies,andsymbolsthroughtheirparticipationindoingmathematicsinyourclassroom.Youmustbeawarethattalkingaboutthemeaningsofwords,ideas,andsymbolsisanimportantpartofyourroleasateacher;andyoumustbetransparentandcarefulinyouruseofwords,symbols,andnotationwhenyouareteachingmathematics.
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ChapterTwo
MeasurementinthePlane
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ClassActivity6: MeasureforMeasureAndtherewentoutachampionoutofthecampofthePhilistines,namedGoliathof
Gath,whoseheightwassixcubitsandaspan. ISamuel17:4
1) Usingyourcubit(lengthfromelbowtofingertips)andhandspan,determinehowtall
GoliathwasbycuttingastringthatisaslongasGoliathwastall.Compareyourstringlengthwiththestringsofothersinyourgroup.Whatarethedifficultiesthatmightarisefromchoosingandusingunitsdeterminedbyeachperson’sownbody?Whydoyouthinkweusethe“foot”asacommonunitoflength?
2) Onthenextpage,arefourcopiesofthesamelinesegment.Usingeachoftherulersprovided,carefullymeasurethelinesegment.
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ReadandStudy
Itisnotenoughtohaveagoodmind.Themainthingistouseitwell. ReneDescartes
Thebigideaofmeasurementisthatofcomparinganattributeofanobjecttoanappropriateunit.Forexample,wemightcomparethelengthofourdesktoafoot-longrulerorwemightcomparetheareaofsheetofpapertotheareaofagreentriangle.Sothekeyquestionregardingmeasurementisthis:Howmanyoftheunitfitintotheobject?Whatmakesaunitappropriate?Well,firstitmusthaveadimensionthatmatchestheattributetobemeasured.Forexample,wemeasurelength(orwidthorheight)usingone-dimensionalunits.Hereisanexampleofaone-dimensionalunit:Wemeasureareausingtwo-dimensionalunitslikethis:Wemeasurevolumeusingthree-dimensionalunitslikethis:Wewilltalkmoreaboutvolumemeasurementinlatersections.Second,ifyouwanttobeabletocommunicatewithothers,ithelpsthattheunitbea‘standard’one.Astandardunitisonethattheculturehasagreedupon.Eachpersonhasamentalmodeloftheunitsoheorshecanpicturehowbigitis.Forexample,inourcultureafootisastandardunitformeasuringlength.InEurope(andmostoftherestoftheworld)ameterisastandardunitformeasuringlength.Ifwewanttotalkamongcultures,weneedtobeabletoconvertfromoneunittoanother.Thereareabout3.28feetinameter.Ifaroomis15feetlong,approximatelyhowmanymetersisthat?S T A N D A R D A N D N O N S T A N D A R D U N I T S
Third,itisusefulthattheunitbeofreasonablesizeinrelationtotheattributetobemeasured.Itwouldbeinconvenienttomeasurethelengthoffootballfieldusingmicrons(areallysmallone-dimensionalunit).Inthemetricsystemsizeisindicatedbytheprefix.Forexample,theprefixkilomeans1000times.Soakilometeris1000meters.Inthecurriculummaterialselementarystudentsareexpectedtousetheprefixesmilli(onethousandth),centi(onehundredth),andkilo.Youshouldmemorizetheseandbeabletomakeconversions.Approximatelyhowmanycentimetersarethereinafoot?Becausemeasurementalwaysinvolvescomparison,itisnecessarilyalwaysanapproximation.Weoftenindicateourdegreeofcertaintyaboutameasurementbasedonthewaywereportit.Forexample,ifIclaimadeskis1.23meterslong,thissuggeststhatIamconfidentintheaccuracytothehundredthofameter(tothecentimeter).Whenworkingwithchildrenyou
j ''
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shouldexplicitlydiscussthesefacetsofmeasurementandyoushouldbesuretoalwaysreporttheunitwithanyactualmeasurement.Sayingthatadeskis1.23longmeansnothing.Sayingthatitis1.23meterslongmakessense.Acommonmeasurementwemakeforaplaneobjectistomeasurethedistancearounditsboundary.Wecallthismeasurementtheperimeteroftheobject.(Whentheobjectisacircle,wecallthislengththecircumference.)Theperimeterofaplaneobjectisaone-dimensionalmeasurement–soweuselinearunitslikeinchesorcentimeters.Wecalculateaperimeterbysimplyaddingupthelengthsofthecurvesthatmakeuptheobject.Wecanmeasurethelengthsoflinesegmentswitharuler.Wewillfindaformulaforthecircumferenceofacircleinafutureactivity.A P P R O X I M A T I O N A N D P R E C I S I O N
ConnectionstoTeachingOnlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.
Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient. EugeneS.Wilson
TheCommonCoreStateStandardsformathematicsrequirethatchildrenbegintostudystandardmeasurementoflengthbeginningingrade2.ReadtheexcerptfromtheCCSSbelow.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
NoticethatchildrenaretobelearningbothmetricandEnglishunits.Comeupwithabenchmark(somethingtoimaginethatistherightlength)foreachoftheseunits:inches,feet,centimeters,andmeters.
CCSSGrade2:Measureandestimatelengthsinstandardunits.
1. Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.
2. Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.
3. Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.
4. Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthe
lengthdifferenceintermsofastandardlengthunit.
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TheCommonCoreStateStandardsformeasurementingrade1areshownbelow.Comparethemtothegrade2standards.Howdotheydiffer?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide.Spend10-15minuteslookingthroughUnit4Module1Session1TeacherFeet.Inwhatwaysdothechildrenneedtoanalyzetheprocessofmeasurementinthissession?
Homework
Manyoflife'sfailuresarepeoplewhodidnotrealizehowclosetheyweretosuccesswhentheygaveup.
ThomasA.Edison
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DotheConnectionsproblems.
3) Whataregoodbenchmarksforthemillimeter,centimeter,decimeter,andkilometer?Findsomethingthatisaboutthelengthofeach.
4) UseanappropriateEnglishunittoestimatea)thelengthofyourtable,b)thedistancefromChicagotoDenver,andc)thewidthofapencil.
5) Useanappropriatemetricunittoestimatea)thelengthofyourtable,b)thedistancefromChicagotoDenver,andc)thewidthofapencil.
CCSSGrade1:Measurelengthsindirectlyandbyiteratinglengthunits.
1. Orderthreeobjectsbylength;comparethelengthsoftwoobjectsindirectlybyusingathirdobject.
2. Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelengthunit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limittocontextswheretheobjectbeingmeasuredisspannedbyawholenumberoflengthunitswithnogapsoroverlaps.
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6) Findtheperimetersoftheplanefiguresshownbelowbymeasuringthem
a) UsinganappropriateEnglishunit.b) Usinganappropriatemetricunit.c) NowconvertyourmetricunitstoEnglishunitstocheckthatyourmeasurementsin
b)matchyoumeasurementsina).
7) Explainwhyitmakessensethattheperimeterofarectanglecanbefoundbycomputing2l+2wwherelisitslengthandwisitswidth.
8) Iwanttorunaroundthebelowlake.Iplantohugtheshoreline.HowfardoIhavetorun?Thinkaboutvariouswaysyoucoulduseamaptoanswerthisquestion.Howaccurateisyourestimate?Howcanyouimproveitsaccuracy?LakeJen
Scale1cm=3miles
9) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade2TeachersGuide.Spend10-15minuteslookingthroughUnit4Module1.Then,carefullyexaminetheworksheet4AEstimate&MeasureInchesRecordSheet.Whatconversationscouldyouhavewithyourstudentsaboutthebigideasoflinearmeasurementbasedonthisactivity?
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ClassActivity7: Part1Triangulating
Godowndeepenoughintoanythingandyouwillfindmathematics.DeanSchlicter
Defineonetriangleunittobetheareaofthegreentrianglepatternblock.Usingthisunit,determinetheareaofthissheetofpaper.Thenusethetriangleunittoestimatetheareaofeachofthepatternblockshapespicturesbelow.
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ClassActivity7: Part2AreaEstimationHereisamapofthegreatstateofWisconsin.Withoutlookinganythinguponline,yourgroupneedstomakeareasonableestimateofitsareainsquaremiles.Firstdiscussacoupleofdifferentwaysofdoingthisusingthemap.Thengoaheadandcarefullycomputeyourestimate.
http://www.nationsonline.org/oneworld/map/USA/wisconsin_map.htm
Wouldyouguaranteeyourestimatetothenearsest10squaremiles?Thenearest100squaremiles?Thenearest1000squaremiles?Somethingelse?Howdidyoudecide?
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ReadandStudy
Themanignorantofmathematicswillbeincreasinglylimitedinhisgraspofthemainforcesofcivilization.
JohnKemenyMathematiciansdefineareaasthequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigure.Wecommonlymeasurethisareaintermsofsquareunitssuchassquareinchesorsquarecentimeters.Sotofindtheareaofafigureyouneedtofindthenumberofsquareunitsitwouldtaketocoverthefigure.Wewantyoutoliterallydothisnow.Hereisaunitofarea(thesquarecentimeter):Traceitandthenseehowmanyofthoseunits(includingpartsofunits)ittakestocovertheshapebelow: Iftheshapeisarepresentationofalake,andacentimeteroflengthcorrespondsto3miles,thenwhatistheareaoftheactuallake?Explain.I D E A O F A R E A
Weoftenmeasuretheareaofirregularlyshapedobjectsjustbycounting(andestimating)thenumberofsquareunitswithintheboundaryoftheobject.Sometimeswesuperimposeagridtohelpwiththatestimate.Estimatetheareaofthisobjectnowassumingthateachsmallsquareisaunitofarea.Whenwemeasuretheareaofanobjectinsquareunits,noticethatwearetakingadvantageofthefactthatsquarestessellatetheplane–therearenogapsbetweenthesquares.Wehavealreadydiscoveredthatthereareothershapesthattessellatetheplaneaswell;twoofthem
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are,likethesquare,alsoregularpolygons.Whichones?Whydoyouthinkpeoplechosetousesquareunitsinsteadofsomeothershapethattessellatestheplane?
ConnectionstoTeaching
Teachingcreatesallotherprofessions.AuthorUnknown
ChildrenneedtohaveexperiencesfindingareasbycoveringfigureswithsquareunitslikeyoudidintheReadandStudysection.Ifyoushowthemformulastooearly,theywillsimplytrytorememberthoseformulasandtheymaynotfocusontheideaofarea.Besides,formulasworkonlywithalimitednumberofshapes,andsoestimatingareasusingthedefinitionisausefulskillinitsownright.Wesuggestthatchildreningrades2and3spendmanyweeksestimatingareasusingsquarestickersorsquaregridstocoverfigures.Someofthosefiguresshouldhavecurvedboundariesandsomeshouldhavespecialpolygonshapes.Childrenwillquicklybegintoproposeshortcutsontheirownthatwillleadnaturallytotheareaformulas.Forexample,ifchildrencomputeareasofthebelowrectangleshapesusingstickersoragrid,someofthemwillnoticethatashortcutforfindingareasofrectangleshapesistosimplymultiplythelengthoftherectanglebyitswidth.Thenyoucantalkwiththemaboutwhythisisso.Practicethatnowbydoingthebelowtasks.C O V E R I N G W I T H U N I T S Q U A R E S
Firstuseagridtoestimatehowmanyareaunitsittakestofilleachrectangleshape.Thenexplainwhyitmakessensethattheareaofarectanglecanbefoundbymultiplyingitslengthbyitswidth.Hereisaunitoflength:_____ Hereisaunitofarea:
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TheCommonCoreStateStandardsdescribethefollowingstandardsforchildreningrade3.Readthemtoseethatthethingschildrenshouldlearnaremanyofthethingswehavedescribedinthissection.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Wehavenottalkedexplicitlyabout2.c.and2.d.above.Sketchpicturestohelpyoutomakesenseofwhattheymeanbythose.
CCSSGrade3:Geometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.
1. Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.a. Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“one
squareunit”ofarea,andcanbeusedtomeasurearea.
b. Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.
c. Measureareasbycountingunitsquares(squarecm,squarem,squarein,square
ft,andimprovisedunits).
2. Relateareatotheoperationsofmultiplicationandaddition.a. Findtheareaofarectanglewithwhole-numbersidelengthsbytilingit,and
showthattheareaisthesameaswouldbefoundbymultiplyingthesidelengths.
b. Multiplysidelengthstofindareasofrectangleswithwholenumbersidelengthsinthecontextofsolvingrealworldandmathematicalproblems,andrepresentwhole-numberproductsasrectangularareasinmathematicalreasoning.
c. Usetilingtoshowinaconcretecasethattheareaofarectanglewithwhole-
numbersidelengthsaandb+cisthesumofa×banda×c.Useareamodelstorepresentthedistributivepropertyinmathematicalreasoning.
d. Recognizeareaasadditive.Findareasofrectilinearfiguresbydecomposing
themintonon-overlappingrectanglesandaddingtheareasofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
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Homework
ThemoreIpractice,theluckierIget.JerryBarber
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.
3) Useagridtoestimatecarefullytheareaofthebelowcircularfigureinsquare
centimeters.
4) Astudentcomestoyouandasks,"Whydoweusesquarecentimeterstomeasuretheareaofthecircle?Acircleisroundandnotsquare."Explaintoherwhywestillusesquarecentimeterstomeasuretheareaofacircle.
5) WewouldliketohavetheabovelakeclassifiedasoneoftheGreatLakes.AspartoftheapplicationtotheDepartmentoftheInterior,wehavetoreportitsarea.Estimatetheareaofthelakeinsquaremiles.
Scale1cm=3miles
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6) Hereisafloorplanforthefirstlevelofahouse.a) Whatisitsareainsquarefeetincludingthegarage?Assumethat1cm
represents6feetoflength.b) Howgoodisyourestimate?Areyouconfidenttothenearestsquarefoot?c) WhatCommonCoreStateStandardismetbythisproblem?Explain.
7) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade3TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3.
a. Howareareaandperimeterintroduced?b. PrintoutandthencarefullyworkthroughtheworksheetBayardOwl’sBorrowed
Tables.Whatconversationscouldyouhavewithyourstudentsaboutthebigideasofperimeterandareabasedonthisactivity?
8) AnNBAbasketballcourtmeasures50feetby96feet.Usethisinformationbelowtodeterminehowmanyacresitcovers.
1foot=12inches1yard=3feet1mile=1760yards1acre=4840squareyards1squaremile=640acres
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ClassActivity8: FindingFormulas
Onecannotescapethefeelingthatthesemathematicalformulashaveanindependentexistenceandanintelligenceoftheirown,thattheyarewiserthanwe
are,wisereventhantheirdiscoverers. HeinrichHertz
Wecommonlycomputeareasofsomespecialpolygonshapeswiththeuseofformulas.Theseformulascanbeexplained–theyarebasedongeometricdefinitionsandtheorems–andwewantyoutounderstandwhytheymakesense.Thatisthegoalofthisactivity.
1) Rectangle:Usetheideaofareaasthenumberofsquareunitsittakestocoveranobjecttoexplainwhyitmakessensethattheareaofarectangleissimplytheproductofitslengthandwidth.
2) Parallelogram:Showhowtocutandrearrangeaparallelogramtomakearectanglewith
thesamearea.Arguethattheresultingfigureisinfactarectangle.UsetheformulafortheareaofarectangletofindaformulafortheareaAofaparallelogramusingthebasebandtheheighthoftheparallelogram.
3) Triangle:Showhowtorearrangetwocopiesofthesametriangletomakeaparallelogram.Arguethattheresultingfigureisinfactaparallelogram.UsetheformulafortheareaofaparallelogramtofindaformulafortheareaAofatriangleusingthebasebandtheheighthofthetriangle.
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ReadandStudy
Theessenceofmathematicsisitsfreedom. GeorgCantor
Manyofyourstudents(andmanyoftheirparents)willthinkthatformulastellthewholestoryaboutarea.Infact,somepeoplemistakenlydefineareaas“lengthtimeswidth.”Everyclosedtwo-dimensionalshapehasarea,butaswehaveseeninanearliersection,onlyaveryfewoftheseshapeshaveformulaswecanusetocalculatethearea.Theareaofsomegeometricobjectsismoreeasilydeterminedthroughtheuseofareaformulas.IntheClassActivityyoudevelopedseveralusefulandwell-knownformulasthatarereadilyfoundthroughoutthecurriculum.Notethatalloftheseformulasarebasedonthechoiceofasquareastheunitofarea.Ifwehadusedadifferentshapeastheunit(aswedidintheprevioussection),alloftheformulaswouldhavechanged.Themostfundamentalareaformulaisfortheareaofarectangle:length´width.Alloftheotherformulasforareaarebuiltonthat.Asyouhaveseen,theseformulasarereallytheoremsthathavebeenproventowork.And,thesetheoremsareonlytruewhenweusetheunitsquareasourunit.Explaincarefullywhythatpreviousstatementissoimportant.Whathappensifwedonotusesquaresasourunit?YoushouldhaveyourupperelementarystudentsdoactivitieslikethoseintheClassActivitysothattheycanseethisforthemselves.IntheBridgesinMathematicscurriculumforgrade4,studentsdiscovertheformulasfortheareaandperimeterofrectangles.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module3Session3.Inwhatwaysdoesthisactivityhelpstudentstodevelopformulasfortheperimeterandareaofarectangle. M A K I N G S E N S E O F A R E A F O R M U L A S
ConnectionstoTeaching Knowingisnotenough;wemustapply.Willingisnotenough;wemustdo.
JohannWolfgangvonGoethe
Oncechildrenunderstandtheideasofperimeterandarea,theycansolvesomepracticalproblems.Wewillposesomenowthatspecificallyaddressthestandardsthatfollowthem.Dotheseproblemsandthenreadtoseewhichofthestandardswe’veaddressedwiththem.
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a) Ifarectangularroomhasalengthof5feetandanareaof60squarefeet,whatisitswidth?
b) Ifarectanglehasalengthof10feetandaperimeterof36feet,whatisitswidth?
c) Isittruethatrectangleswithbiggerperimetersalwayshavebiggerareastoo?Explain.
d) Isittruethatrectangleswithbiggerareasalwayshavebiggerperimeterstoo?Explain.
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©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Homework
Toliveacreativelife,wemustloseourfearofbeingwrong. JosephChiltonPearce
1) DotheproblemsandthendotheitalicizedstatementintheConnectionssection.
CCSSGrade3:Geometricmeasurement:recognizeperimeterasanattributeofplanefiguresanddistinguishbetweenlinearandareameasures.Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.
CCSSGrade4:Solveproblemsinvolvingmeasurementandconversionofmeasurementsfromalargerunittoasmallerunit.
1. Knowrelativesizesofmeasurementunitswithinonesystemofunitsincludingkm,m,cm;kg,g;lb,oz.;l,ml;hr,min,sec.Withinasinglesystemofmeasurement,expressmeasurementsinalargerunitintermsofasmallerunit.Recordmeasurementequivalentsinatwocolumntable.Forexample,knowthat1ftis12timesaslongas1in.Expressthelengthofa4ftsnakeas48in.Generateaconversiontableforfeetandincheslistingthenumberpairs(1,12),(2,24),(3,36),...
2. Usethefouroperationstosolvewordproblemsinvolvingdistances,intervalsoftime,liquidvolumes,massesofobjects,andmoney,includingproblemsinvolvingsimplefractionsordecimals,andproblemsthatrequireexpressingmeasurementsgiveninalargerunitintermsofasmallerunit.Representmeasurementquantitiesusingdiagramssuchasnumberlinediagramsthatfeatureameasurementscale.
3. Applytheareaandperimeterformulasforrectanglesinrealworldandmathematicalproblems.Forexample,findthewidthofarectangularroomgiventheareaoftheflooringandthelength,byviewingtheareaformulaasamultiplicationequationwithanunknownfactor.
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2) PracticeexplainingwhyeachoftheformulasfromtheClassActivitymakessense.
3) Oneofyourstudentsisconfusedaboutareacalculations.Adamwonderswhy,ifyoutakearectangleandmultiplythelengthofeachsideby2,theareaofthenewrectangleisn'ttwiceasbigastheareaoftheoldrectangle.Drawsomepicturestohelpyouseewhatisgoingonhere.Whatwillyousaytohim?
4) Explainwhyitmakessensethattheareaofatrapezoidisalways½h(b1+b2)whereb1andb2arethelengthsoftheparallelbasesandhistheheight.Youcandothisbypartitioningthetrapezoidregionintopieces,orbycuttingitapartandrearrangingit,orbyaddingonstructure.Youmayusewhatyouknowaboutfindingareasofrectangles,parallelograms,andtriangles.Seeifyoucanfindmorethanonewaytodothis.
5) Findtheareaofthetrapezoidshownbelowinasmanydifferentwaysasyoucan.
Assumethateachgridsquarerepresentsoneunitofarea.
* * * * * *
* * * * * *
* * * * * *
* * * * * *
6) Supposearectanglehasaperimeterof36units.Whatareallthepossiblewholenumberdimensionsoftherectangle?Makeagraphofwidthvs.area.Whichwidthgivesthegreatestarea?
7) Supposeyouhave100metersofflexiblefencingtomarkapastureoutontheplains.
Howwouldyousetitup(whatshape)toenclosethemostgrazingareaforyourcattle?Whatdimensionswouldyouuseifthepasturehadtobearectangle?
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8) Thetrianglebelowisconstructedona1cmgrid.Findtheareaofthetriangleusingatleastthreedifferentmethods.
9) Assumethatthetriangleareaaboverepresentspartofasignthatneedstobepainted.
Thescaleofthedrawingisthat1cmrepresents10feet.Theinstructionsonthepaintcansaythat1gallonofpaintwillcover100squarefeet.Howmanygallonsofpaintwillyouneedtobuyinordertopaintthetriangle?
1 cm
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ClassActivity9: TheRoundUp
Donotdisturbmycircles!Archimedes’finalwords
1) Acircleisdefinedasthesetofallpointsintheplanethatareequidistantfromagiven
pointcalledthecenter.Studythisdefinition.Iftheword“all”wasmissing,howwouldthatchangethings?Whatifthewords“intheplane”weremissing?
2) Howmanytimesdoesthediameterofacirclefitintoitscircumference?Gathersomedatatosee.
3) Exploretheideaoftheareaformulaforacircularregionbyrearrangingitintoaparallelogram-shapedfigure.Findtheareaofthe“parallelogram”intermsoftheradiusofthecircle.Whyisthisjusttheideaoftheargument?
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ReadandStudy
Itisnotoncenortwice,buttimeswithoutnumberthatthesameideasmaketheirappearanceintheworld.
Aristotle CirclesareasfundamentaltoEuclideangeometryasarepointsandlines.RecallthatEuclid’sthirdaxiomassuresusthatwecanalwaysmakeacircleofanysize(radius)wewant.Ofcourse,thecircleswedrawarestillonlyapproximationsofatruemathematicalcircle,justasthelinesegmentswemakearejustapproximationsofatruelinesegment.Acircleisthesetofallpointsintheplanethatareequidistantfromagivenpoint,calledthecenter(Ointhediagrambelow).Thediagramshowssomeoftheotherimportanttermsassociatedwithacircle.Becertainyouunderstandeachtermandcanexplainitsmathematicaldefinition,whichyouwillfindintheglossary.
Central Angle Ð AOC
Tangent
Secant
Chord DE Diameter AB Arc DE
Radius CO
Sector O
C
A
B
D
E
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Itisanamazingfactthat,foranysizecircle,theratioofthecircumferencetothediameteristhesamenumber.Thiswasknowninallearlycivilizations.Wecallthatratio“pi.”Soπ(pi)isthesymbolforthenumberoftimesthediameterofacirclefitsintoitscircumference.Readthatagain;itisimportant.Thisnumberpisanirrationalnumber.Thatmeansithasadecimalrepresentationthatneitherendsnorrepeats.Thiswasprovedin1761byamathematiciannamedJohannHeinrichLambert.Evenwhenweusethepkeyonacalculator,weareusinganapproximatevalue.Elementarystudentscommonlyuseeither3.14or
722 asanapproximatevalueforpwhen
carryingoutcalculationsinvolvingcircles.Alwaysbesuretomakethepointthatthisisjustanapproximation.W H A T I S Π ?
ConnectionstoTeaching
Attheageofeleven,IbeganEuclid,withmybrotherasmytutor.Thiswasoneofthegreateventsofmylife,asdazzlingasfirstlove.
BertrandRussellIntheBridgesinMathematicscurriculumforgrade4,studentsareintroducedtobasiccircleterminology.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module1Session5.Howarethepartsofacircleintroducedtothestudents?Withinthelesson,thestudentsareaskedtowritetheirowndefinitionofacircle.IssaandSuzigavethefollowingdefinitions.Aretheirdefinitionscorrect?Howasateacherdoyourespondtoeachofthesestudents? Issa’sdefinition:Acircleisashapethat’sroundandhas360°. Suzi’sdefinition:Acircleisashapethathasthesamewidthallthewayaround.
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Homework
Eureka!I’vegotit! Archimedes
1) DoalltheitalicizedthingsintheReadandStudysection.
2) SolvetheproblemsintheConnectionssection.
3) Studyeachboldandunderlinedtermusedinthissection.Thismeansyoushouldbe
abletoexplainthedefinitionusinggoodmathematicallanguageandthatyoushouldbeablesketchexamplesandnon-examplesofeachterm.
4) HowdoesthedefinitionofπleadtotheformulaC=2πr(whereCisthecircumferenceofacircleandrisitsradius)?
5) Ifacirclehasameasuredradiusof5inches,then,usingtheformulaforfindingtheareaofacircle,wewouldsaythattheareaofthecircleisapproximately78.5squareinches.Giveatleasttwodistinctreasonswhythecalculationisapproximate.
6) Explaintherelationshipbetweenasecantandachordofacircle,betweenaradiusandadiameter,andbetweenasecantandatangentofacircle.
7) Hereisanothersetofpicturesdesignedtogiveanintuitiveargumentthattheareaofacircularregionisπr2.Imaginethatthecircleshownismadeofcircularstringssittingoneinsidethenext.Thenyoutakeascissors,sniparadius,andflattenthestringstomakethetriangularshape.Whatistheareaofthetriangleintermsofr(theradiusofthecircle)?
8) IfIdoublethediameterofacircularregion,whathappenstoitscircumference?
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9) IfIdoublethediameterofacircularregion,whathappenstoitsarea?
10) Ifan8-inch(diameter)pizzacosts$5,howmuchshoulda16-inchpizzacost?Justifyyourresult.
11) Havealookatthecirclesbelow.
a) Carefullymeasuretheperimeterofthesquarethatisinscribedinsidethefirstcircle.
b) Carefullymeasuretheperimeteroftheregularpentagoninscribedinsidethesecondcircle.
c) Carefullymeasuretheperimeteroftheregularhexagoninscribedinsidethethirdcircle.
d) TrueorFalse?Asthenumberofsidesoftheinscribedpolygongrows,sodoestheperimeterofthepolygon.
e) TrueorFalse?Ifweinscribeapolygonwithinfinitymanysides,thentheperimeterofthatpolygonwillbeinfinitelylong.Explainyourthinking.
f) Howcouldyouusethesemeasurementstogetanestimateforπ?Explain.
Morethan2000yearsagoArchimedesfoundapproximateboundsforπusinginscribedandcircumscribed(outside)polygonswith,getthis,120sides.WestilluseArchimedesboundstoday( 7
2271223
<< ).Theaverageofthesetwovaluesis
roughly3.1419whichiscorrecttothreedecimalplaces.Notbadfor350BC.
I N S C R I B E D P O L Y G O N S
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ClassActivity10: PlayingPythagoras
Ihavehadmyresultsforalongtime,butIdonotknowyethowIamtoarriveatthem.
CarlFriedrichGaussThePythagoreanTheoremisthoughttobealmost4000yearsold.TheBabylonians,theEgyptians,andtheChineseallknewit.Thatis,theyknewthatthesumoftheareaofthesquaresonthelegsofarighttriangleequalstheareaofthesquareonthehypotenuse,andtheyusedthisfactnumericallyinconstructionandcommerceandsurveying.
1) Carefullymeasuretheareasofthesquaresinthebelowexampletoseeifthistheoremseemstrue.
2) Whathappensifthetriangleisn’tarighttriangle?Isthesumofthesquaresonthe“legs”(shortersides)ofanobtusetrianglemoreorlessthanthesquareonthelongestside?Whathappensinanacutetriangle?Drawsomediagramstosee.
(Thisactivityiscontinuedonthenextpage.)
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ThefirstproofofthetheoremisattributedtoPythagorasofSamos(it’sinGreece)around600B.C.Sincethen,hundredsofdifferentproofshavebeencreated.Youaregoingtoexploreoneofthem.a) Theproofbeginswithanyrighttriangle.Carefullydrawyourownandlabelthe
lengthofthehypotenusec,thelengthofthelongerlegb,andthelengthoftheshorterlega.
b) Nowmakefourcongruentcopiesofyourtriangle,cutthemout,andarrangethem
intoaquadrilateralasshownbelow.
c) Justifythattheboundaryoftheouterquadrilateralisasquare.
d) Justifythattheinnerquadrilateralisasquare.
e) Nowgiveanalgebraicproofthatc2=a2+b2byusingthefactthatthefivepolygonsformthelargefigure(sotheareaformulasforthefivemustsumtotheareaformulaofthelargesquare).
f) Whatthingsgowrongifthetrianglesarenotrighttriangles?
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ReadandStudy
I’mverywellacquaintedtoowithmattersmathematical,Iunderstandequations,boththesimpleandquadratical,AboutbinomialtheoremI’mteemingwithalotofnews--
Withmanycheerfulfactsaboutthesquareofthehypotenuse.Gilbert&Sullivan,“ThePiratesofPenzance”
ThePythagoreanTheoremisoneofthemostwell-knownandmostimportanttheoremsofallofelementarymathematics.ItisnamedaftertheGreekmathematician,Pythagoras,andEuclidincludeditasthefittingendtovolumeoneofTheElements.InEuclid’swordsthetheoremsaysthis:
Inright-angledtrianglesthesquareonthesideoppositetherightangleequalsthesumofthesquaresonthesidescontainingtherightangle.
Euclidintendedtheword“square”tomeanthephysicalsquaredrawnonthehypotenuseorlegoftherighttriangle.Sohistheoremsaysthattheareaoftheyellowsquare(inthefigureabove)isequaltothesumoftheareasoftheredandbluesquareswhenABCisarighttriangle.Todaywemorecommonlyuseanalgebraicstatement:
Ifarighttrianglehaslegsoflengthsaandbandahypotenuseoflengthc,thenc2=a2+b2.
Noticethathere,a,b,andcarenumbersrepresentingthelengthsofthevarioussidesofthetriangle.Sonowtheword“square”carriesthealgebraicmeaningdenotedbytheexponenttwo.P R O O F S O F T H E P Y T H A G O R E A N T H E O R E M
Therearemany(atonecount,atleast367)proofsofthePythagoreanTheorem.YourecreatedoneoftheproofsintheClassActivity.YouwillbeaskedtoexploretwoothersaspartoftheHomeworkforthissection.
C B
A
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ThePythagoreanTheoremisanexampleofatheoremwhoseconverseisalsoatheorem.StatetheconverseofthePythagoreanTheorem.Ifyouhavetolookup“converse,”visittheglossaryanddoso.TheconverseofthePythagoreanTheoremgivesusawaytodiscoverwhetherornotatriangleisarighttriangleevenwhenweknownothingabouttheanglemeasuresofthetriangle.Forexample,supposeweknowthatthesidesofatriangleareexactly3,4,and5incheslong.Isthisarighttriangle?Let’ssubstitutethevalues3fora,4forb,and5forc(Howdoweknowthatthehypotenusemustbethesideoflength5?)andcheckoutthePythagoreanrelationshipc2=a2+b2.Does52=32+42?Iftheansweris“yes,”thenthetriangleisarighttriangle.
ConnectionstoTeaching
Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstocreateanduserepresentationstoorganize,record,andcommunicate
mathematicalideas;toselect,apply,andtranslateamongmathematicalrepresentationstosolveproblems;andtouserepresentationstomodelandinterpret
physical,social,andmathematicalphenomena.NCTMPrinciplesandStandardsforSchoolMathematics,2000
Havingtwo(ormore)waystointerpretorrepresentamathematicalideaisanimportantcharacteristicofmathematicsforteaching.TheNCTMStandardsrecognizethisandcallforallelementarystudentsto“select,apply,andtranslateamongmathematicalrepresentationstosolveproblems.”Andso,asteachers,itisalsoimportantthatweunderstandamathematicalconceptfrommorethanonepointofviewinordertoassistourstudentstousedifferentrepresentationsofthatconcepttosolveproblems.G E O M E T R I C A N D A L G E B R A I C R E P R E S E N T A T I O N S
ThePythagoreanTheoremisonesuchideathatcanbeunderstoodfrommanyviewpoints:geometric,numeric,andalgebraic.Someofourstudentswillmoreeasilygraspthealgebraicapproachwhileotherswillpreferthemoreconcretegeometricornumericrepresentation.Asteachers,wemustmasterallrepresentationsinordertocarryoutourresponsibilitiestosupporteachstudent’slearning.
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Homework
Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackinwill.
VinceLombardi
1) DoalltheitalicizedthingsintheReadandStudysection.
2) JamesAbramGarfield(1831-1881),thecountry’stwentiethpresident,createdthisproofofthePythagoreanTheoremin1876,whilehewasamemberoftheHouseofRepresentatives.FindGarfield’sproofbyusingthediagrambelowbyfindingformulasfortheareaofthetrapezoidintwodifferentways.Howdoeshisargumentfailifthetrianglesarenotrighttriangles?
3) Whenthelengthsofthesidesofarighttriangleareallintegersthethreenumbers(a,b,
c)areknownasaPythagoreanTriple.Explainwhy(3,4,5)isaPythagoreanTriple.WhichofthefollowingtriplesofnumbersarePythagoreanTriples?a)(4,5,6) b)(4,6,8) c)(6,8,10)
4) Supposeatrianglehassidesoflength8,15,and17.Isitarighttriangle?
5) Whatistheexactheightofanequilateraltriangleifallsidesareoflength10?Oflength3?Oflength4?Oflengths?
6) Aclosetis3feetdeep4feetwideand12feethigh.Findthedistancefromonecornerat
thefloortothediagonallyoppositecornerattheceiling.Youmightwanttohavealookataboxtohelpyouseewhattodohere.
a
b
c
c b
a
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SummaryofBigIdeas Althoughthismayseemaparadox,allexactscienceisdominatedbytheidea
ofapproximation. BertrandRussell
• Measurementisthecomparisonofanattributeofanobjecttoaunit.Tomeasure
meanstoseehowmanyoftheunitfitintotheobjectyouaremeasuring.
• Lengthisaone-dimensionalmeasurement;areaistwo-dimensional;andvolumeisthree-dimensional.
• Allmeasurementisapproximate.• Thechoiceofaunitisafundamentalpartofthemeasuringprocess.
• Areaisthenumberofsquareunitsittakestocoveranobject.Itisnotdefinedas“length
timeswidth,”and,infact,thatformulaworksonlyinafewlimitedcases.
• Formulasforthecalculationofareacanbeexplainedbythegeometryoftheobjectsbeingmeasured.Asateacher,youwillneedtohelpyourstudentstoseewhereformulascomefromandwhytheymakesense.
• πisdefinedasthenumberoftimesthediameterofacirclefitsintoitscircumference.Itisanirrationalnumber.
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ChapterThree
TheThirdDimension
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ClassActivity11: StrictlyPlatonic(Solids)
IfIhaveevermadeanyvaluablediscoveries,ithasbeenowingmoretopatient attention,thantoanyothertalent. SirIssacNewton
Hereisadefinitionforyoutostudy:Apolyhedronisafinitesetofpolygon-shapesjoinedpairwisealongtheedgesofthepolygonstoencloseafiniteregionofspacewithinonechamber.Thepolygon-shapedsurfacesarecalledfaces.Itisrequiredthatthesurfacebesimple(notmorethanonechamberisenclosed)andclosed(youcan’tgetinfromtheoutsidewithouttearingit).Thesegmentswherethepolygonsmeetarecallededges.Thepointswhereedgesintersectarecalledvertices.
1) Usethedefinitiontodecidewhichofthebelowimagesof3-dimensionalobjectsrepresentpolyhedra(“polyhedra”isthepluralformofpolyhedron)
Apolyhedronisregularifallofthefacesarethesamecongruent,regularpolygonandalloftheverticeshaveexactlythesamenumberofpolygons.
2) UsethepiecesofaFrameworksÔset(manufacturedbyPolydron)tobuildalloftheregularpolyhedra.Besystematicsoyoucangiveanargumentthatyouhavefoundthemall.(Polyhedraarenamedforthenumberoffacestheycontain;forexample,apolyhedronwithtenfaceswouldbecalledadecahedron.)
3) Canyoubuildapolyhedronthatusesonlyonetypeofcongruentregularpolygonbutisnotregular?Explain.
4) Seeifyoucanfindshortcutwaysofcountingthenumbersofverticesoredgesofregularpolyhedra.
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ReadandStudy
Everythingshouldbemadeassimpleaspossible,butnotonebitsimpler. AlbertEinstein
IntheClassActivityyouwereaskedtobuildmodelsoftheregularpolyhedra.Thesespecialobjects,alsocalledthePlatonicsolids,havebeenknownsincebeforethedaysoftheGreekmathematics.Approximationsoftheregularpolyhedraevenoccurinnature.Inparticular,thetetrahedron,cube,andoctahedronshapesallappearascrystalstructures.Wealsofindpolyhedralshapesamonglivingthings,suchastheCircogoniaicosahedrashownbelow,aspeciesofRadiolaria,whichisshapedlikearegularicosahedron.
(http://en.wikipedia.org/wiki/Platonic_solids)Manyvirusesalsohavetheshapeofaregularicosahedron.Viralstructuresarebuiltofrepeatedidenticalproteinsubunitsandapparentlytheicosahedronistheeasiestshapetoassembleusingthesesubunits.R E G U L A R P O L Y H E D R A
Netsaretwo-dimensionalfiguresthatcanbefoldedintothree-dimensionalobjects.Belowarenetsfortheregulartetrahedronandforthecube.Imaginehoweachfoldsuptomakethe3-dimensionalobject.
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Beforeyoureadfurther,gotothissiteandbuildyourselfoneofeachoftheregularpolyhedra.Youwillneedthemforthehomework.http://www.mathsisfun.com/platonic_solids.html
ConnectionstoTeaching
Studentsingrades3–5shouldexaminethepropertiesoftwo-andthree-dimensionalshapesandtherelationshipsamongshapes.Theyshouldbeencouragedtoreason
aboutthesepropertiesbyusingspatialrelationships. NCTMPrinciplesandStandards,2000
ThefollowingexcerptfromtheNCTMStandardsforGrades3–5Geometry(p.168)describestheimportanceofvisualizationandspatialreasoningastoolselementarystudentscanusetounderstandthepropertiesofgeometricobjectsandtherelationshipbetweenthesepropertiesandtheshapes.Readtheseparagraphsandstudytheexamples.Thenbuildthebuildingtheydescribe.S P A T I A L R E A S O N I N G
Usevisualization,spatialreasoning,andgeometricmodelingtosolveproblemsStudentsingrades3–5shouldexaminethepropertiesoftwo-andthree-dimensionalshapesandtherelationshipsamongshapes.Theyshouldbeencouragedtoreasonaboutthesepropertiesbyusingspatialrelationships.Forinstance,theymightreasonabouttheareaofatrianglebyvisualizingitsrelationshiptoacorrespondingrectangleorothercorrespondingparallelogram.Inadditiontostudyingphysicalmodelsofthesegeometricshapes,theyshouldalsodevelopandusementalimages.Studentsatthisagearereadytomentallymanipulateshapes,andtheycanbenefitfromexperiencesthatchallengethemandthatcanalsobeverifiedphysically.Forexample,“Drawastarintheupperright-handcornerofapieceofpaper.Ifyouflipthepaperhorizontallyandthenturnit180°,wherewillthestarbe?”Muchoftheworkstudentsdowiththree-dimensionalshapesinvolvesvisualization.Byrepresentingthree-dimensionalshapesintwodimensionsandconstructingthree-dimensionalshapesfromtwo-dimensional»representations,studentslearnaboutthecharacteristicsofshapes.Forexample,inordertodetermineifthetwo-dimensionalshapeinfigure5.15isanetthatcanbefoldedintoacube,studentsneedtopayattentiontothenumber,shape,andrelativepositionsofitsfaces.
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Fig.5.15.Ataskrelatingatwo-dimensionalshapetoathree-dimensionalshape
Studentsshouldbecomeexperiencedinusingavarietyofrepresentationsforthree-dimensionalshapes,forexample,makingafreehanddrawingofacylinderorconeorconstructingabuildingoutofcubesfromasetofviews(i.e.,front,top,andside)likethoseshowninfigure5.16.
Fig.5.16.Viewsofathree-dimensionalobject(AdaptedfromBattistaandClements1995,p.61)
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Homework
Gettingaheadinadifficultprofessionrequiresavidfaithinyourself.Thatiswhysomepeoplewithmediocretalent,butwithgreatinnerdrive,gomuchfurtherthan
peoplewithvastlysuperiortalent. SophiaLoren
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
2) Writeoutthedetailsofamathematicalargumentthatthereareexactlyfiveregular
polyhedra.
3) Thefollowingpictureisoftengivenasanexampleofaregularicosahedron.Examinethepicturecarefullyanddeterminewhythispictureisclaimingtobesomethingthatitisnot.
4) Isitpossibletobuildapolyhedronwhereallofthefacesarecongruentregularpolygonsbutthepolyhedronisnotregular?Explain.
5) Usethefivemodelsyoubuilttofindaformulathatrelatesnumbersofvertices,edges,andfacesinregularpolyhedra.
6) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade
1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Session2MysteryBagSorting.Makeupyourownmysteryforyourstudentstosolvesimilartothosegiveninthelesson.
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ClassActivity12: PyramidsandPrisms
Itisbettertoknowsomeofthequestionsthanalloftheanswers. JamesThurber
Therearetwospecialcategoriesofpolyhedrawewillexploreinthisactivity:pyramidsandprisms.Apyramidisapolyhedroninwhichonefaceiscalledthebaseandalloftheremainingfacesaretrianglesthatshareacommonvertex(calledtheapex).Apyramidisnamedfortheshapeofitsbase.Forexample,asquarepyramidisonewhosebaseisasquare.(Noticethatthebaseofapyramidneednothavetheshapeofaregularpolygon.Itcouldlooklikethefigurebelow,forexample.)
1) Youhavealreadybuiltapyramidwithanequilateraltrianglebase(thetetrahedron).In
yourgroup,sketchapyramidwithasquarebaseandanotherwithahexagonalbase.
2) Useyourpicturestohelpyoufindformulasforthenumberoffaces,thenumberofvertices,andthenumberofedgesinapyramidwhosebaseisann-gon.Thenprovethatyourformulaswillworkforallpyramids.
Aprismisapolyhedroninwhichtwoofthefaces(calledthebases)arecongruentandlie on parallel (non-intersecting) planes andtheremainingfacesareparallelograms.Theprismisalsonamedafteritsbase.Iftheparallelogramfacesarerectangular,theprismisarightprism.Iftheparallelogramfacesarenon-rectangular,theprismisanobliqueprism.
3) Which(ifany)oftheregularpolyhedraareprisms?Explain.
4) Sketchanobliqueprism.C O U N T I N G V E R T I C E S , E D G E S , A N D F A C E S
5) Sketchaprismwithatriangularbaseandonewithahexagonalbase.Thenuseyour
picturestofindformulasforthenumberoffaces,thenumberofvertices,andthenumberofedgesinaprismwhosebasesarecongruentn-gons.Provethatyourformulaworksinthecaseofallprisms.
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ReadandStudy
Themediocreteachertells.Thegoodteacherexplains.Thesuperiorteacherdemonstrates.Thegreatteacherinspires.
WilliamArthurWard
ThemostfamouspyramidsaretheEgyptianpyramids.Thesehugestonestructuresareamongthelargestman-madeconstructions.InAncientEgypt,apyramidwasreferredtoasthe"placeofascendance."TheGreatPyramidofGizaisthelargestinEgyptandoneofthelargestintheworldwithabasethatisover13acresinarea.ItisoneoftheSevenWondersoftheWorld,andtheonlyoneoftheseventosurviveintomoderntimes.TheMesopotamiansalsobuiltpyramids,calledziggurats.Inancienttimesthesewerebrightlypainted.Sincetheywereconstructedofmud-brick,littleremainsofthem.TheBiblicalTowerofBabelisbelievedtohavebeenaBabylonianziggurat.AnumberofMesoamericanculturesalsobuiltpyramid-shapedstructures.Mesoamericanpyramidswereusuallystepped,withtemplesontop,moresimilartotheMesopotamianzigguratthantheEgyptianpyramid.ThelargestpyramidbyvolumeistheGreatPyramidofCholula,intheMexicanstateofPuebla.Thispyramidisconsideredthelargestmonumenteverconstructedanywhereintheworld,andisstillbeingexcavated.Modernarchitectsalsousethepyramidshapeforbuilding.AnexampleistheLouvrePyramidinParis,France,inthecourtoftheLouvreMuseum.DesignedbytheAmericanarchitectI.M.Peiandcompletedin1989,itisa20.6meter(about70foot)glassstructurewhichactsasanentrancetothemuseum.Themostcommonexampleofaprisminthe“realworld”isitsoccurrenceinoptics,whereaprismisatransparentopticalelementwithflat,polishedsurfacesthatrefractlight.Theexactanglesbetweenthesurfacesdependontheapplication.Thetraditionalgeometricalshapeisthatofatriangularprismwithatriangularbaseandrectangularsides,andincolloquialuse"prism"usuallyreferstothistype.Prismsaretypicallymadeoutofglass,butcanbemadefromanymaterialthatistransparenttothewavelengthsforwhichtheyaredesigned.Aprismcanbeusedtobreaklightupintoitsconstituentspectralcolors(thecolorsoftherainbow).Theycanalsobeusedtoreflectlight,ortosplitlightintocomponentswithdifferentpolarizations.
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ConnectionstoTeaching
Itisthesupremeartoftheteachertoawakenjoyincreativeexpressionandknowledge.
AlbertEinsteinElementarystudentsbegintheirstudyof3-dimensionalshapesinthefirstandsecondgrades.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade1TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2Sessions4and5.Whatideasaboutprismsandpyramidsareemphasizedinthesesections?NowlookattheHomelinksectionfromSession5.Examinequestions4and5carefully.Determineseveralreasonswhyyourchosenfiguredoesnotbelong.Inquestion5,determineatleasttwodifferentfiguresonwhichtoplacethe“X”.
Homework
Iattributemysuccesstothis:Inevergaveortookanyexcuse. FlorenceNightingale
1) DotheproblemsintheConnectionssection.
2) Makeamathematicalargumentfortheformulasyoufoundforthenumberoffaces,the
numberofvertices,andthenumberofedgesinapyramidwhosebaseisann-gon.
3) Makeamathematicalargumentfortheformulasyoufoundforthenumberoffaces,thenumberofverticesandthenumberofedgesinaprismwhosebasesarecongruentn-gons.
4) Provethattheformulayoufoundrelatingthevertices,edges,andfacesofaregularpolyhedraalsoholdsforthen-gonalpyramidandforthen-gonalprism.
5) Thefollowingaredescriptionsofpyramidsandprisms.Identifytheprismorpyramidandsketchanet.a) Apolyhedronwith5facesand9edges.b) Apolyhedronwithtwohexagonalfacesandtheremainingfacesarerectangles.c) Apolyhedronwith12edgesand8verticies.d) Apolyhedronwith7facesand7verticies.
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ClassActivity13: SurfaceArea
Mathematicsmaybedefinedastheeconomyofcounting.Thereisnoprobleminthewholeofmathematicswhichcannotbesolvedbydirectcounting.
ErnstMach
1) Use14interlockingunitcubestobuildthe3-dimensionalfigurepictured.
a) Whatisthesurfaceareaofthisobject?
b) Whatisitsvolume?
c) Isthisapolyhedron?Explain.
2) Whatareallthepossiblevaluesforthesurfaceareaoffiguresmadewith14interlocked
cubes?Explain.
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ReadandStudy
Numberrulestheuniverse. Pythagoras
Surfaceareaisthemeasureoftheboundaryofathree-dimensionalobjectinthesamewaythatperimeteristhemeasureoftheboundaryofatwo-dimensionalobject.Andjustlikewemeasureperimeterbyaddingupthelengthsofeachsectionoftheboundaryoftheobject,wemeasuresurfaceareabyaddinguptheareasofeachfaceoftheboundaryoftheobject.Forexample,supposewehavearectangularprismthatis3cmlongby5cmwideby8cmtall.Takeaminutetosketchthatfigure.I D E A O F S U R F A C E A R E A
Thismeansthatwehavetwofacesthatarerectanglesthatare3cmby5cm(andsohaveanareaof15squarecmor15cm2),twofacesthatarerectanglesthatare5cmby8cm(andsohaveanareaof40squarecmeach),andtwofacesthatare3cmby8cm(andsohaveanareaof24squarecmeach).Thenthesurfaceareaoftheprismis15+15+40+40+24+24=158squarecm.(Checkourwork.)Noticethatweusesquareunitstomeasuresurfaceareasinceitisameasureoftwo-dimensional(flat)space.Thereisnoneedtodevelopcompletelynewformulasforsurfacearea.Wecanusewhicheverareaformulasareappropriategiventheshapesthatmakeupthefacesoftheobject.
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ConnectionstoTeaching
Theimportantthingisnottostopquestioning.Curiosityhasitsownreasonforexisting.
AlbertEinsteinElementarystudentscanuseinterlockingcubes(suchastheonesweusedintheClassActivity)tocreate“buildings”andthenusethosebuildingstobuildanunderstandingofsurfaceareathroughdrawingthebuildingsfromvariousview-points.Forexample,thebuildingbelowcanbeviewedfromthetop,front,andside.
Buildyourownbuildingoutof5cubes.Thensketchthefigureanditscorrespondingtop,front,andsideviews.Herearethetop,front,andsideviewsofanotherbuilding.Sketchapossiblebuildingwiththeseviews.
Howdotheseactivitiesrelatetothesurfaceareaofthebuilding?Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit1Module2Session4.Theendofthelessonlistsseveralquestionsforyoutoaskyourstudents.Thinkuptwomorequestionsyoucouldaskyourstudentsabouttheactivity.
top front right side
top front right side
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Homework
Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.
JohnQuincyAdams
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
2) Supposeyouareaskedtobuildanobjectusinginterlockingunitcubesthathasasurfaceareaof36squareunits.a) Howmanyunitcubesataminimumwouldyouneed?b) Whatisthemaximumnumberofunitcubesyoucoulduse?c) Forwhatthree-dimensionalshapewillthevolumebegreatestforafixedsurface
area?Makeaconjecture.
3) Belowisanetforatetrahedron.Ifeachequilateraltrianglehassidesoflength5cm,whatisthesurfaceareaofthetetrahedron?(Thoselittleextrapartsaretabsthathelpyoutotapeittogether–don’tincludethose.)
4) Whatamountofpaper(area)wouldyouneedtomakeanetforthecubewithanedgelengthof7cm?(Ignoreanypaperneededfortabs.)
5) Sketchanetforarectangularprismthatis4cmby5cmby7cm.Whatisthesurfaceareaofthatprism?
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ClassActivity14: NothingButNet
Puremathematicsis,initsway,thepoetryoflogicalideas. AlbertEinstein
1) Figureouthowtobuildapapermodelofarightcylinderwitharadiusof3cmanda
heightof10cm.Useyourmodeltohelpyoufindaformulaforthesurfaceareaofacylinder.
N E T S F O R C Y L I N D E R S A N D P Y R A M I D S
2) Buildapapermodelofapyramidthathasa6cmby6cmsquarebaseandaheightof4
cm.Whatisitssurfacearea?
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ClassActivity15: BuildingBlocks
Measurewhatismeasurable,andmakemeasurablewhatisnotso. Galileo
Volumeisameasureoftheamountofspaceenclosedbyathree-dimensionalobject.Onewaytodefinevolumeisasthenumberofcubes(cubicunits)thatittakestofilltheenclosedspace.Usingthisdefinition,wecanmeasurevolumebycarefullyestimatingthenumberofcubesthatfitwithintheobject.Someobjectshaveformulasthatwillhelpustocomputevolume.Whatisaformulaforthevolumeofarectangularprismwithlengthl,widthw,andheighth?Whydoesitmakesense?Next,youaregoingtobuildthreeobjectsoutofsmallwoodencubes(withhalf-inchedges)andlargewoodencubes(withone-inchedges).Followthestepsbelow.StepA:Buildsomethingoutof10smallwoodencubes.We’llcallthisfigure,“ObjectA.”StepB:Using10largewoodencubes,buildalargerversionofObjectA.We’llcallthisfigure,“ObjectB.”StepC:Usingasmanysmallcubesasnecessary,reproduceObjectB.(ItshouldbethesamesizeandshapeasObjectB,butbuildoutofsmallcubesinsteadoflargeones.)We’llcallthisfigure,“ObjectC.”
1) HowmuchtallerisObjectBthanObjectA?
2) HowmuchbiggeristhesurfaceareaofObjectBcomparedtoObjectA?
(Thisproblemcontinuesonthenextpage.)
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3) HowmuchbiggeristhevolumeofObjectBcomparedtoObjectA?
4) WhatistherelevanceofObjectCtoansweringthesequestions?
5) WhatifIgaveyoualargerblockthatis3timesthelengthofthesmallcube;howwouldyouranswersto1-3change?Whatabout4timesthelength?
6) WhatifIgaveyouasmallerblockthatis½timesthelengthofthesmallcube;howwouldyouranswersto1-3change?
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ReadandStudy
Mathematicsisnotacarefulmarchdownawell-clearedhighway,butajourneyintoastrangewilderness,wheretheexplorersoftengetlost.
W.S.AnglinIntheClassActivityyoutalkedaboutwhyitmadesensetocalculatethevolumeofarectangularprismbymultiplyingthelengthloftheprismbythewidthwoftheprismbytheheighthoftheprism(V=l´w´h).Theideahereisthatwecanthinkofaprismaslayersofthebasestackedoneuponthenext.Sovolumeistheareaofthebasemultipliedbytheheight(V=AreaofBase×h).Havealookatthepicturebelowtoseewhatwemean.
Willthatsameideaworkforacylinder?Isitsvolumetheareaofitsbasemultipliedbyitsheight?Makeasketchofacylinderanduseittohelpyoutoexplainyourthinking.I D E A S O F V O L U M E
V O L U M E S O F P R I S M S A N D C Y L I N D E R S
ConnectionstoTeaching:
Thecureforboredomiscuriosity.Thereisnocureforcuriosity. DorothyParker
Childrenoftenbegintheirexplorationsofvolumebydeterminingthevolumeofvariousrectangularboxes.Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade5TeachersGuide.Spend10-15minuteslookingthroughUnit1Module1Sessions4and5.Howdoesthisactivityhelpstudentstounderstandvolume?
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Homework
Whentheworldsays,"Giveup,"Hopewhispers,"Tryitonemoretime."
AuthorUnknown
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoallthoseproblemsintheConnectionssection.
3) Whenyoudoublethelengthofallthesidesofacube,whathappenstoitsvolume?Whydoesthishappen?Whathappenswhenyoutriplethelength?
4) Belowwehavesketchedanetforacube.a) Buildapapermodelofacubethatistwiceaslongineachlineardimension.b) Buildapapermodelofacubethathastwicethesurfaceareaofthecubesuggested
bythenet.c) Buildapapermodelofacubethathastwicethevolumeofthecubesuggestedby
thenet.
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ClassActivity16: VolumeDiscount
ThelawsofnaturearebutthemathematicalthoughtsofGod.Euclid
1)Reasonwithunitcubestocreatevolumeformulasforthefollowingobjects:
a) arectangularprism
b) atriangularprism
c) acylinder1) Userice--notformulas--toanswerthenextfourquestions:
a) Howdoesthevolumeofthesquarepyramidcomparetothevolumeofthesquareprism?Usethisinformationtocreateavolumeformulaforasquarepyramid.
(Thisactivitycontinuesonthenextpage.)
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b) Howdoesthevolumeofthetriangularpyramidcomparetothevolumeofthetriangularprism?Usethisinformationtocreateavolumeformulaforatriangularpyramid.
c) Howdoesthevolumeoftheconecomparewiththevolumeofthecylinder? Usethisinformationtocreateavolumeformulaforacone.
d) Howdoesthevolumeoftheconecomparewiththevolumeofthesphere?Usethisinformationtocreateavolumeformulaforasphere.
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ReadandStudy
Ifyouwouldbearealseekeraftertruth,itisnecessarythatatleastonceinyourlifeyoudoubt,asfaraspossible,allthings. ReneDescartes
IntheClassActivityyouobservedthatthevolumeofaprismisaboutthreetimesthevolumeofapyramidwiththesameheightandbase,andthatthevolumeofacylinderisthreetimesthevolumeofaconewiththesameheightandradius.Itturnsoutthatfortheidealobjects,thefactorofthreeisexactlycorrect.Thesenicerelationshipsmakeformulasforvolumerelativelystraightforwardifwebuildonformulaswealreadyknow.C A P A C I T Y
InanearlierClassActivity,youexplainedwhyitmakessensethatthevolumeofarectangularprismisV=(l´w´h).Now,sinceyouhaveseenthatthisvolumeisthreetimesthevolumeofthepyramidwiththesameheightandrectangularbase,thevolumeofthepyramidshouldbegivenbytheformulaV= 13 (l´w´h).
Wecangeneralizethesetwoformulassothattheyapplytoallprismsandallpyramids(andtoallcylindersandallcones).Noticeinthevolumeformulafortherectangularprismthatl´wisjusttheareaoftherectangularbase.Ifthebasehasadifferentshape,wejustneedtousetheappropriateareaformulatofindtheareaofthebaseandthenmultiplybytheheighttofindthevolumeoftheprism:V=(AreaofBase)´hforallprismsandcylinders.
FigurefromG.S.Rehill’sInteractiveMathsSeries.
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Likewise,wecanuseV= 13 (Areaofbase´h)forallpyramidsandcones.
Thesphereisanotherthree-dimensionalshapethathasawell-knownvolumeformula, 34
3V r=
,whereristheradiusofthesphere.Thisformulacomesfromcalculus–sowedon’thavethemachinerytoproveitworks–howeveryou(andyourstudents)canobservethattheformulaseemsplausibleusingthewaterexperiment.Here’stheidea.Sinceacylinderhasvolumeπr2×h,andthecylinderyouusedintheClassActivityhasaheightof2r,thismeansthatthecylinderyouusedtopourwaterhasavolumeof2πr3.Takeaminutetocarefullythinkthisthrough.V O L U M E S O F P Y R A M I D S , C O N E S , A N D S P H E R E S
Now,sinceaconehasonethirdthevolumeofacylinderwiththesamebase,theconeyouusedmusthaveavolumeof1/3×(2πr3).Soifittakestwoconestofillasphereitthewaterpouringexperiment,thenitmakessensetoconjecturethatthevolumeofaspheremustbegivenbytheformula, 34
3V r= .Makesurethat
youunderstandwhatwe’resayinghere.Thecorrespondingformulaforthesurfaceareaofasphereis 24SA r= .
Ofcourse,therearemanythree-dimensionalobjectsforwhichwedonothaveformulastocalculatetheirvolume.Forallofthese,wecanusethe“capacity”definitionthatwasdiscussedintheClassActivity.Whenwemeasurevolumeusingcapacitywecommonlyuseunitslikethecup,thequart,thegallon,theliter,etc.Whenwefindvolumeusingaformulawecommonlyuseunitslikecubicinches,cubicyards,andcubiccentimeters.Volumeisathree-dimensionalmeasuresotheunitsusedwillallbecubicunits.
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ConnectionstoTeachingPuremathematicsis,initsway,thepoetryoflogicalideas.
AlbertEinsteinChildreningradethreeshouldhaveexperiencesmeasuringvolumesusingwaterandweighingphysicalobjects.HerearetherelevantCommonCoreStateStandards.Readthem.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Ingrade5,studentsshouldlearntodomanyofthethingswehavetalkedaboutinthelasttwosections.Theyshouldthinkofvolumemeasurementasboththenumberofcubicunitsrequiredtofilla3-dimensionalobject,andasthecapacityoftheobject.Theyshouldunderstandhowtothinkaboutthevolumeofaprismandmakesenseofsomevolumeformulas.YouwillfindtherelevantCommonCoreStateStandardsforgrade5below.Readthem.Whatdotheymeanwhentheysaythatstudentsshouldrecognizevolumeas“additive?”
CCSSGrade3:Solveproblemsinvolvingmeasurementandestimationofintervalsoftime,liquidvolumes,andmassesofobjects.
1. Tellandwritetimetothenearestminuteandmeasuretimeintervalsinminutes.Solvewordproblemsinvolvingadditionandsubtractionoftimeintervalsinminutes,e.g.,byrepresentingtheproblemonanumberlinediagram.
2. Measureandestimateliquidvolumesandmassesofobjectsusingstandardunitsofgrams(g),kilograms(kg),andliters(l).Add,subtract,multiply,ordividetosolveone-stepwordproblemsinvolvingmassesorvolumesthataregiveninthesameunits,e.g.,byusingdrawings(suchasabeakerwithameasurementscale)torepresenttheproblem.
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©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CCSSGrade5:Geometricmeasurement:understandconceptsofvolumeandrelatevolumetomultiplicationandtoaddition.
1. Recognizevolumeasanattributeofsolidfiguresandunderstandconceptsofvolumemeasurement.
a. Acubewithsidelength1unit,calleda“unitcube,”issaidtohave“onecubic
unit”ofvolume,andcanbeusedtomeasurevolume.
b. Asolidfigurewhichcanbepackedwithoutgapsoroverlapsusingnunitcubesissaidtohaveavolumeofncubicunits.
2. Measurevolumesbycountingunitcubes,usingcubiccm,cubicin,cubicft,and
improvisedunits.
3. Relatevolumetotheoperationsofmultiplicationandadditionandsolverealworldandmathematicalproblemsinvolvingvolume.
a. Findthevolumeofarightrectangularprismwithwhole-numbersidelengths
bypackingitwithunitcubes,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengths,equivalentlybymultiplyingtheheightbytheareaofthebase.Representthreefoldwhole-numberproductsasvolumes,e.g.,torepresenttheassociativepropertyofmultiplication.
b. ApplytheformulasV=l×w×handV=b×hforrectangularprismstofind
volumesofrightrectangularprismswithwholenumberedgelengthsinthecontextofsolvingrealworldandmathematicalproblems.
c. Recognizevolumeasadditive.Findvolumesofsolidfigurescomposedoftwo
non-overlappingrightrectangularprismsbyaddingthevolumesofthenon-overlappingparts,applyingthistechniquetosolverealworldproblems.
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Homework
Toclimbsteephillsrequiresaslowpaceatfirst. WilliamShakespeare
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DotheitalicizedthingsintheConnectionssection.
3) Supposeyouhavearightcircularcylinderwithheighthandradiusrandanoblique
circularcylinderwithheighthandradiusr.Dothesetwocylindershavethesamevolume?Compareastraightstackofpenniestoa“slanted”stacktosee.(Reallydoit.)Ineachcase,howisheightmeasured?
4) Theclosetinmylivingroomhasanoddshapebecausemyapartmentisonthetopfloorofahousewithaslantedroof.Theclosetis6feettallinfrontbutonly4feettallinback.Itis3feetdeepand12feetwide.
a) Buildascaleddownpapermodelofmycloset.Reallydothis.Itwillhelpyouwiththerestofthisproblem.
b) Howmanycubicfeetofstoragedoesithold?c) Iwanttopainttheinsidewallsandceilingofmycloset.Howmanysquarefeet
willIneedtopaint?
5) Howmanycubicfeetofwaterdoesasemi-cylindrical(halfacylinder)troughholdthatis10feetlongby1footdeep?Howmanycubicinchesisthat?
6) Iused1500cubicinchesofheliumtofillmyballoon.Assumingmyballoonisasphere,tothenearesttenthofaninch,whatisitsdiameter?Whatisitssurfacearea?
7) Amovietheatersellspopcorninaboxfor$2.75andpopcorninaconefor$2.00.Thedimensionsoftheboxandtheconearegiven.Whichisthebetterbuy?Explain.
8) Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade
5TeachersGuide.Spend10-15minuteslookingthroughUnit6Module3Session4.Howdoesthisactivityfurtherstudents’understandingofvolume?
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ClassActivity17: VolumeChallengeIfpeopledonotbelievethatmathematicsissimple,itisonlybecausetheydonot
realizehowcomplicatedlifeis.JohnLouisvonNeumann
1) Yourgroupshouldworktogethertobuildpapermodelsofeachofthefollowingobjects
insuchawaythatthevolumeofeachis60cm3.(Otherthanthat,youmayuseanydimensionsyoulike.)
a) Cylinderb) Rectangularprismthatisnotacubec) Pyramidwithasquarebased) Prismwithanequilateraltrianglebase
2) Findthesurfaceareaofeachoftheobjectsyoubuiltin1)above.
B U I L D I N G M O D E L S T O S P E C I F I C A T I O N
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ReadandStudyandConnectionstoTeaching
It'snotthatI'msosmart;it'sjustthatIstaywithproblemslonger. AlbertEinstein
Bythetimetheyreachupperelementaryschool,studentscansolvemanypracticalproblemsingeometry.Howeverthesestudentsarenotreadytosimplyapplyformulastosolveproblems;insteadtheyneedtousemodelsinordertomakesenseofproblems.Astheirteacher,yourjobwillbetomakesurethatchildreninyourclassesbuildandhandleappropriatemodels.NoticethattheCommonCoreStateStandardsforgrade6explicitlyrequirethatstudentsusehands-onmodelstoexploreideas.Studentsareaskedtocuttrianglesandothershapesapart,todrawpictures,topackspaceswithunitcubes,tousethecoordinateplane,andtobuildandusenets.Readthesestandards.Havewedoneofthesethingsinthisbooksofarthisterm?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CCSSGrade6:Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.
1. Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
2. Findthevolumeofarightrectangularprismwithfractionaledgelengthsbypackingitwithunitcubesoftheappropriateunitfractionedgelengths,andshowthatthevolumeisthesameaswouldbefoundbymultiplyingtheedgelengthsoftheprism.ApplytheformulasV=lwhandV=bhtofindvolumesofrightrectangularprismswithfractionaledgelengthsinthecontextofsolvingreal-worldandmathematicalproblems.
3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
4. Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
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Homework
Theelevatortosuccessisnotrunning;youmustclimbthestairs.ZigZiglar
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
2) Atriangleonacoordinateplanehasverticesat(2,0),(7,0)and(7,8).
a) Sketchthepolygononacoordinateplane.b) Whatisthelengthofthehypotenuse?c) Whatistheareaofthepolygon?d) WhatCommonCoreStateStandardisaddressedbythisproblem?
3) AreplicaoftheGreatPyramidstands2feettallandis3.15feetlongonaside(ithasa
squarebase).a) Approximatelyhowmuchvolumedoesthisreplicatakeup?Usethemodelyoubuilt
intheClassActivitytohelpyoutothinkaboutthisproblem.b) Whatisthesurfaceareaofthereplica?c) Supposethescaleofthereplicatotherealthingisinis1footto240feet.Whatis
thevolumeofGreatPyramid?Whatisitssurfacearea?d) WhichoftheCommonCoreStateStandardsaremetbythisseriesofproblems?
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SummaryofBigIdeas
Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes
• Thereareexactlyfiveregularpolyhedra–andchildrencanunderstandwhythisisso.
• Surfaceareaisameasureofthesumofareasofthefacesofathree-dimensionobject.
Itisthenumberofsquareunitsittakestocoverthefacesoftheobject.Aunitofsurfaceareaisflatlikethis:
• Volumeisameasureofthenumberofcubicunitsittakestofillathreedimensionalobject.Aunitofvolumeisthree-dimensionalandlookslikethis:
• Volumecanalsobemeasuredbytheamountofliquid(orsand)ittakestofillathreedimensionalobject.
• Thevolumeofaprismorcylindercanbefoundbycomputingtheareaofthebaseandmultiplyingthatbyitsheight(thenumberoflayersofthebase).Thisideaworksforbothrightandobliqueobjects.
• Ittakesthevolumeofthreepyramidstofillaprismwiththesamebaseandheight,andittakesthevolumeofthreeconestofillacylinderwiththesamebaseandheight.
• Childrenneedtohavemanyyearsofexperiencesbuildingandusingavarietyofmodels.
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ChapterFour
Transformations,Tessellations,andSymmetries
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ClassActivity18: Slides
Forthethingsofthisworldcannotbemadeknownwithoutaknowledgeofmathematics.
RogerBacon
Informally,thinkofatranslationasamotionwhich“slides”theentireplaneinonedirectionaparticulardistance.Inordertotranslateanobjectwemustknowhowfartoslideitandwemustknowthedirectiontouse.Thesetwopiecesofinformationareusuallygiventousintheformofatranslationvector(alsocalledatranslationarrow).
1) OnthesquaregridbelowyouaregiventranslationvectorRSandseveralgeometricobjectsontheplane.ShowwhereeachobjectendsupaftertheplaneistranslatedbyvectorRS.
2) Hereisadefinitiontostudy:AtranslationAA’isarigidmotionoftheplanethattakesAtoA’,andforallotherpointsPontheplane,PgoestoP’wherevectorPP’andvectorAA’havethesamelengthanddirection.Discuss,inyourgroups,howthisdefinitionfitswiththeaboveidea.
3) Whatrelationshipsdoyouseebetweentheoriginalfiguresandtheirtranslatedimages?
Betweentheobjects,theirimages,andthetranslationvector?Makeasmanyconjecturesasyoucanabouttranslations.
4) IfwetranslatetheplaneusingRSandthenperformasecondtranslation,say,ST,whatistheresultingrigidmotion?Explain.
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ReadandStudy
Onlythecuriouswilllearnandonlytheresoluteovercometheobstaclestolearning.Thequestquotienthasalwaysexcitedmemorethantheintelligencequotient.
EugeneS.WilsonTransformationsareabigcategoryofmotionswecanapplytothepointsofaplanewhichcauseobjectsintheplanetochangetheirposition,ortheirsize,oreventheirshape.Sometransformations,likescaling,onlychangethesizeofanobject.Othertransformations,likeashearingcanchangeboththesizeandshapeofanobject.Theobjectthatistheresultofatransformationappliedtoanobjectiscalledtheimageoftheobjectunderthattransformation.Forexample,therectanglebelowisenlarged1½timestoproducethescaledimageandisshearedhorizontallytoproducetheshearedimage.
Inthisandthenexttwosections,wewillstudythreetransformationsthatchangethepositionofanobjectbutdonotchangeitssizeoritsshape.Thesetypesoftransformationsarecalledrigidmotions.R I G I D M O T I O N S
T R A N S L A T I O N S
Rigidmotionsarethetransformationsoftheplaneforwhichthedistancebetweenpointsispreserved.Inotherwords,iftwopointswereacertaindistanceapartbeforethemotion,thentheyarestillthatsamedistanceapartafterthemotion.(Whydoesthename“rigidmotion”makesense?)Usingtheideaofrigidmotions,wecanmorepreciselydefinecongruence:twoobjectsarecongruentifthereexistsaseriesofrigidmotionswhereoneobjectistheimageoftheother.Rigidmotionsincludetranslations,rotations,andreflections.Eachofthesetransformationswillmovetheplaneinauniqueway.Translationsslidetheplaneaparticulardistanceinaparticulardirection.Rotationsturntheplaneeitherclockwiseorcounterclockwisearoundafixedcenter.Reflectionsmovetheplanebyflippingitacrossaline.Itturnsoutthatallrigidmotionsoftheplanearecombinationsofjustthesemoves.Asyoudiscoveredintheclassactivity,atranslationvectorisusedtodescribethedistanceanddirectioneachpointismovedinatranslation.Iftheendpointsofthevectoraregivenascoordinatesonasquaregrid,wecandescribethedistanceanddirectioneachpointismovedassomanyunitsupordownandsomanyunitsrightorleft.UsethislanguagetodescribethetranslationvectorRSonthepreviouspage.
sheared image
scaled imageoriginalrectangle
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Weuseastandardnotationtolabeltheverticesoftheimageofanobjectunderatranslation.Forexample,iftheoriginalobjectistherectangleABCD,thenitsimageislabeled DCBA .Herevertex A oftheimagecorrespondingtovertexAoftheoriginalrectangle,vertex B tovertexB,etc.ThepointsAand A arecalledcorrespondingpoints.ThesidesABand BA arecalledcorrespondingsides.
ConnectionstoTeaching
Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentstoapplytransformationsandusesymmetrytoanalyzemathematical
situations. NCTMPrinciplesandStandardsforSchoolMathematics,2000
Studentsintheelementarygradescanpredictanddescribetheresultsofsliding,flipping,andturningtwo-dimensionalshapes.Variouselementarycurriculausedifferentapproaches.Somemakeuseofmanipulativesandothershavestudentscutoutshapesinordertophysicallyperformtheslide,flip,orturntheshapes.
Homework
Youmaybedisappointedifyoufail,butyouaredoomedifyoudon’ttry.BeverlySills
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Decideifeachofthestatementsabouttranslationsistrueorfalse.Iftrue,givea
mathematicalexplanation;iffalse,explainwhyorgiveacounterexample.a) Correspondingsidesofanobjectanditstranslatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderatranslation,thenthelinesegment
joiningvertexAtovertex A isthetranslationvector.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe
parallelsidesofitsimagetobeverticalafteratranslation.d) IfAand A andBand B arecorrespondingpointsunderatranslation,itis
possibleforthelines AA and BB tointersect.e) Everypointintheplanemovestoanewpositionunderatranslation,i.e.,there
arenofixedpointsinatranslation.f) Underarotation,linesegmentsthatjoinpointstotheirimagesareallcongruent.
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3) Belowyouwillfindacoordinategrid.Applythefollowingthreetranslationstoatrianglewithverticesinitiallylocatedat(0,0),(-2,-3),and(3,-3).Whatisashortcutwayofdoingpartc)?
a) up5,left3 b)down2,right4 c)up5,left3followedbydown2,right4
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ClassActivity19: Turn,Turn,Turn
Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple.
S.GudderInformally,arotationinvolves“turning”theplaneaboutafixedpoint.Inordertospecifyarotation,weneedanangle(withdirection,clockwiseorcounterclockwise)andafixedpoint(calledthecenteroftherotation).Ineachcaseyourgroupshouldusetracingpaperandacompassandprotractortofigureoutwhereeachshapeendsupafterthegivenrotation.(Therearequestionsonthenextpagetoo.)
1) Rotatetheplane60degreescounterclockwiseaboutpointA. A
2) Rotatetheplane140degreesclockwiseaboutpointP.
P.
(Thisactivityiscontinuedonthenextpage.)
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3) Rotatetheplane90degreesclockwiseaboutapointQintheexactcenterofthesquare.
4) Studythethreerotationsinthisactivity.Whatrelationshipsdoyouseebetweenthe
originalfiguresandtheirrotatedimages?BetweencorrespondingpointsandthepointP?Makeasmanyconjecturesasyoucanaboutrotations.
5) Hereisthedefinition.Studyittoseehowitfitswiththeideaofrotation.ArotationaboutapointPthroughanangleqisatransformationoftheplaneinwhichtheimageofPisPand,iftheimageofAis 'A ,then PA @ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation.
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ReadandStudy
Wedon’tseethingsastheyare.Weseethingsasweare. AnaisNin
Arotationisa“turn”aboutagivenpointcalledthecenterthroughagivenangleofturn.(Theturncanbemadeclockwiseorcounterclockwise.Thisistypicallyindicatedintheproblem.)R O T A T I O N S
Formally,arotation(aboutapointPthroughanangleq)isatransformationoftheplaneinwhichtheimageofPisPand,iftheimageofAis 'A ,thenPA @ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation.HavealookatthefollowingillustrationofthemotionofturningtriangleABCclockwise240°aroundpointP.
NoticehoweachvertexofthetrianglemovesalongacirclewhosecenterisP.Whatpartofthedefinitionsaysthatthismusthappen?Howcanweseethe240°angleinthepictureabove?HowarethesegmentsAPand PA related?ThesegmentsBPand PB ?ThesegmentsCPandPC ?
C'
A'B'
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Homework
Courageandperseverancehaveamagicaltalisman,beforewhichdifficultiesdisappearandobstaclesvanishintoair.
JohnQuincyAdams
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Usethegridtorotatetheplane90degreescounterclockwiseaboutpointP.Showtheimageofthefiguresaftertherotation.
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3) Decideifeachofthestatementsaboutrotationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.
a) Correspondingsidesofanobjectanditsrotatedimagearealwaysparallel.b) If DCBA istheimageofABCDunderarotation,thenthelinesegmentjoining
vertexAtovertex A goesthroughthecenteroftherotation.c) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe
parallelsidesofitsimagetobeverticalafterarotation.d) IfAand A andBand B arecorrespondingpointsunderarotation,itispossible
forthelines AA and BB tointersect.e) Everypointintheplanemovestoanewpositionunderarotation,i.e.,thereare
nofixedpointsinarotation.f) Underarotation,linesegmentsthatjoinpointstotheirimagesareallcongruent.
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ClassActivity20: ReflectingonReflecting
Mathematics,rightlyviewed,possessesnotonlytruth,butsupremebeauty–abeautycoldandaustere,likethatofsculpture.
BertrandRussellInformally,areflectionflipstheentireplaneaboutagivenlineresultinginitsmirrorimage.OfficiallyareflectioninalinelisarigidmotionoftheplaneinwhichtheimageofapointPonlisP,andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof
'AA .
1) Studythatdefinition.Makesureeveryoneinyourgroupunderstandshowitfitswiththeideaofareflection.SeeifyoucanusethedefinitiontohelpyoutosketchthereflectionoftriangleABCinlinel.
l
(Thisactivityiscontinuedonthenextpage.)
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2) Carefullyusethedefinitionofareflectiontosketchthereflectionofthetrapezoidshown.Firstyouwillreflectitinlinemandthenyouwillreflectwhatyougetinlinen(thatisparalleltolinem).
m n
Whatisthesinglerigidmotionthatwouldtaketheinitialfiguredirectlytothefinalfigure?Explain.
3) Whatwouldhappenifthelinesintersected?Tryitandthenuseyourobservationstomakeaconjecture.
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ReadandStudy
WecoulduseuptwoEternitiesinlearningallthatistobelearnedaboutourownworldandthethousandsofnationsthathavearisenandflourishedandvanished
fromit.Mathematicsalonewouldoccupymeeightmillionyears. MarkTwain
Thereareseveralwaystohelpchildrenpicturetheresultsofareflection.Thinkaboutthereflectionoftheparallelograminlineshownbelow.
Onewaytoseewheretheimageshouldbelocatedistotracetheparallelogramandthelineofreflectiononasheetofthinpaperandthenphysicallyflipthepaperoverandplaceitbackontopoftheoriginalpapersothatthetwolinescoincide.Thecopyoftheparallelogramonthetracingpaperisnowpositionedastheimageofthereflection.Useasheetofpaperandtrythismethod.R E F L E C T I O N S
Anotherwaytovisualizeareflectionimageistophysicallyfoldtheoriginalsheetofpaperalongthelineofreflection.Theoriginalobjectanditsimageunderreflectionshouldnowcoincide,asinan“ink-blot”drawing.Trythis.Reallydoit.RecallthatareflectioninalinelisatransformationoftheplaneinwhichtheimageofapointPonlisP,andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA .Thisdefinitionofareflectionprovidesinsightintohowwecansketchtheimageofareflection.Eachpointmusttravelalongalineperpendiculartothelineofreflectionsothatthatlineisthemidpointbetweenofthelinesegmentconnectingcorrespondingpoints.Usethatmethodtosketchtheimageoftheparallelogramabove.
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Therearemanyinstancesofreflectioninphysicalphenomena.Commonexamplesincludethereflectionoflight,sound,andwaterwaves.Weareallfamiliarwiththephenomenaoflightreflection–takealookinamirror.
Homework
Weallhaveafewfailuresunderourbelt.It’swhatmakesusreadyforthesuccesses. RandyK.Milholland,Webcomicpioneer
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Onthesquaregridbelow,uselinenasthelineofreflectiontoreflectthegivenobjects.
Labeleachimageappropriately.
3) Whatrelationshipsdoyouseebetweenoriginalfiguresandtheirreflectedimages?Betweentheobjects,theirimagesandthegivenlineofreflection?Makeasmanyconjecturesasyoucanaboutreflections.
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4) Usethedefinitionofareflectiontoreflectthebelowobjectinline.
5) Decideifeachofthestatementsaboutreflectionsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.
a) Correspondingsidesofanobjectanditsreflectedimagearealwaysparallel.b) If DCBA istheimageofABCDunderareflection,thenthelinesegment
joiningvertexAtovertex A isperpendiculartothelineofreflection.c) If DCBA istheimageofABCDunderareflection,thenthelinesegment
joiningvertexAtovertex A isbisectedbythelineofreflection.d) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe
parallelsidesofitsimagetobeverticalafterareflection.e) IfAand A andBand B arecorrespondingpointsunderareflection,itispossible
forthelines AA and BB tointersect.f) Everypointintheplanemovestoanewpositionunderareflection,i.e.,there
arenofixedpointsinareflection.g) Underareflection,linesegmentsthatjoinpointstotheirimagesareall
congruent.
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ClassActivity21: Part1ASimilarTask
Onecanstate,withoutexaggeration,thattheobservationofandthesearchforsimilaritiesanddifferencesarethebasisofallhumanknowledge.
AlfredNobel
Hereisthedefinitionofanewtypeoftransformation:
Adilation(withcenterPandscalefactork>0)isamotionoftheplaneinwhichtheimageofPisPandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.
Studythisdefinitionanduseittodothefollowing:
1. DrawtheimageoftheseobjectsunderadilationwithcenterPandscalefactor2
2. DrawtheimageoftheseobjectsunderadilationwithcenterCandscalefactor½
3. Writedownsomeconjecturesthatyouhaveaboutdilationsandtheeffecttheyhaveonobjects.
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ClassActivity21: Part2Zoom
Inthepictureabove,westartedbydrawingthesmallertriangleonacomputerscreen,andthenwezoomedin.Thetrianglegotbiggerandmovedtothenewposition.Onepointonourscreenremainedfixedwhenwedidthiszoom.Findthatpoint.Whatwasthescalefactorofthezoom?Inasmanywaysasyoucan,findevidenceforyouranswer.
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ReadandStudy
Oursimilaritiesbringustoacommonground;ourdifferencesallowustobefascinatedbyeachother.
TomRobbinsWehavedifferentnotionsof“sameness”ingeometry.Inthestrongestsense,ifwesaytwoobjectsarethesame,wemeantheyarecongruent.Butwemightalsorefertotwoobjectshavingthesameshapeeveniftheyaren’tcongruent.Forinstance,thetwotrianglesinourClassActivityhavethesameshape,buttheyarenotthesamesize.Observethattheircorrespondinganglesarecongruent,andthattheircorrespondingsidesareproportional.Makesurethatyouunderstandwhatthismeans.Thesetwotrianglesaren’tcongruent,buttheyarewhatwecall“similar”.S I M I L A R P O L Y G O N S
Formally,wesaythattwoobjectsintheplanearesimilarifonecanbeobtainedfromtheotherbycomposingarigidmotion(tochangetheobject’spositionifnecessary)witha“dilation”.Adilation(withcenterPandscalefactork>0)isamotionoftheplaneinwhichtheimageofPisPandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.Inotherwords,PisafixedpointandallotherpointsarepushedradiallyoutwardfromPorpulledradiallyinwardtowardP,sothateachpoint’sdistancefromPhasbeenmultipliedbythescalefactor.Itcanbeshownthattwopolygonsaresimilarifandonlyiftheircorrespondingvertexanglesarecongruent,andtheircorrespondingsidesareproportional.Euclidprovedatheoremaboutsimilartriangles:
2) Angle-AngleTriangleSimilarityTheorem(AA):Iftwoanglesofonetriangleare
congruenttotwoanglesofanothertriangle,thenthetrianglesaresimilar.ThistheoremappearedinhisBookVI(aboutsimilargeometricfiguresandproportionalreasoning)ratherthanhisBookIthatwehavestudiedpreviously.
Homework
Therearenosecretstosuccess.Itistheresultofpreparation,hardwork,andlearningfromfailure.
ColinPowel
1) Usingyourcompass,drawacircle.PlaceapointCattheexactcenterofyourcircle,andapointPsomewhereoutsideofthecircle.Thenbeginningwiththatcircle,
a) drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointCandscalefactor2.
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b) Instead,drawthesimilarshapethatistheresultofadilationoftheplanewithfixedpointPandscalefactor2.
c) Whatcommonalitiesanddifferencesdoyouobserveaboutthethreeshapesyouhavedrawn?
2) Beginningwiththetrianglepicturedbelow,drawthesimilartrianglethatistheresultofadilationoftheplanewithfixedpointPandscalefactor1/2.
3) Decideifeachofthesestatementsaboutdilationsistrueorfalse.Iftrue,giveamathematicalexplanation;iffalse,explainwhyorgiveacounterexample.
a) Correspondingsidesofanobjectanditsdilatedimagearealwaysparallel.b) Dilationspreserveangles(i.e.anyangleanditsdilatedimagearealways
congruent).c) IfA’andB’aretheimagesoftwopointsAandBunderadilationwithscale
factork,thenthedistanceA’B’isktimesthedistanceAB.d) Iftheparallelsidesofatrapezoidarehorizontal,thenitispossibleforthe
parallelsidesofitsimagetobeverticalafteradilation.e) IfAand A andBand B arecorrespondingpointsunderadilation,itispossible
forthelines AA and BB tointersect.f) Everypointintheplanemovestoanewpositionunderadilation,i.e.,thereare
nofixedpointsinadilation.g) IfA’istheimageofAunderadilationwithfixedpointPandscalefactork,then
thedistanceAA’isktimesthedistancePA.
4) Onenight,a6-foottallmanstood10feetfromalamppost.Thelightfromthelamppostcasta12footshadowoftheman.Howtallwasthelamppost?
P
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5) Supposetwoobjectsaresimilarandthescalefactorofthedilationis1.Whatelsecanyousayabouttherelationshipbetweenthosetwoobjects?
6) LookagainatEuclid’strianglesimilaritytheorem(AA).Giventheotherthingsyouhavealreadylearnedabouttriangles,itisequivalenttosaying:ifallthreeanglesofonetrianglearecongruenttothecorrespondinganglesofanothertriangle,thenthetrianglesaresimilar.Considerthefollowingconjectureaboutquadrilaterals:ifallfouranglesofonequadrilateralarecongruenttothecorrespondinganglesofanotherquadrilateral,thenthequadrilateralsaresimilar.Isthisconjecturetrueorfalse?Giveanargumenttosupportyouranswer.
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ClassActivity22: SearchingforSymmetry
Themathematicalsciencesparticularlyexhibitorder,symmetry,andlimitation;andthesearethegreatestformsofthebeautiful.
AristotleAsymmetryofanobjectisarigidmotionoftheplaneinwhichtheimagecoincideswiththeoriginalobject.Therearetwoprimarytypesofsymmetry.Intuitively,anobjecthasreflectionsymmetryifitcanbecutbyalineofreflectionintotwopartsthataremirrorimagesofeachother.Thisbutterflyhasreflectionsymmetryandsodoesthisarrow.Sketchthelineofreflectionineachcase.
Anobjecthasrotationsymmetryifitcanberotatedaroundacenterpointthroughacertainangleandendupwiththeimagecoincidingwiththeoriginal.Wewouldsaythattherecyclingsignbelowhas120,240and360degreerotationalsymmetry.
1) Findallthesymmetriesofthecapitallettersinthefollowingtypeface:
ABCDEFGHIJKLMNOPQRSTUVWXYZ
2) Ineachcase,sketchapolygonwiththegivensymmetries,orexplainwhysuchapolygoncannotexist.
a) nolinesofreflectionsymmetry,but180°(and360°)rotationsymmetriesb) 90°(and360°)rotationsymmetriesandnoothersymmetries.c) 2linesofreflectionsymmetry,360°rotationsymmetry,andnoother
symmetriesd) 6linesofreflectionsymmetrye) nolinesofreflectionsymmetry,but90°,180°,270°(and360°)rotational
symmetryf) anytranslationsymmetry
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ReadandStudy
Mathematicsisthesciencewhichuseseasywordsforhardideas. E.KasnerandJ.Newman
Afigure,picture,orpatternissaidtobesymmetricifthereisatleastonerigidmotionoftheplanethatleavesthefigureunchanged.Forexample,thisleafis(prettymuch)symmetricbecausethereisalineofreflectionsymmetry.
Manyobjectsinnaturedisplaythiskindofbilateralsymmetry.ThelettersinATOYOTAalsoformasymmetricpattern:ifyoudrawaverticallinethroughthecenterofthe“Y”andthenreflecttheentirephraseacrosstheline,theleftsidebecomestherightsideandviceversa.Thepicturedoesn’tchange.S Y M M E T R I E S I N T H E P L A N E
Theorderofarotationsymmetryisdeterminedbycountingthenumberofturnstheobjectcanmakeandcoincidewithitselfbeforereturningtoitsoriginalposition.Theanglemeasureofthesmallestturnisdeterminedbydividing360°bythatnumberofturns.Whydoesthismakesense?Forexample,anequilateraltrianglehas“order3rotationsymmetry.”B A CTheturnof120°takesvertexAtovertexB;theturnof240°takesvertexAtovertexC;andtheturnof360°takesvertexAbacktovertexA.Whataretherotationsymmetriesofthesquare?Oftheregularhexagon?Oftheregularoctagon?Oftheregularn-gon?
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Onlyrepeatinginfinitepatternshavetranslationsymmetry.Theyaretheonlytypeofobjectthatcanbeslidandstillfallbackonthemselves.Imaginethatthispatternstripcontinuesforeverinbothdirectionssoifyouslideitonepatterntotheright(ortwoorthree…)itlooksjustthesame.
… …
ConnectionstoTeaching
Itouchthefuture.Iteach. ChristaMcAuliffe
TheCommonCoreStateStandardsformathematicsmakeinformalideasofsymmetryatopicforgrade4.Whiletheymentiononlyreflectionsymmetry,childrenatthisage(andevenyounger)arecapableofexploringandrecognizing“turn”symmetryaswell.Readthisstandard.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Goonlinetobridges.mathlearningcenter.organdfindtheBridgesinMathematicsGrade4TeachersGuide.Spend10-15minuteslookingthroughUnit5Module2.CarefullyexaminetheMosaicGame.Playthegameonetimeandrecordyourresult.Explainhowyouknowthatyourarrangementproducesthemostlinesofsymmetry.Whatmightstudentslearnaboutsymmetryfromcompletingthisactivity?Playthegameagain,butthistime,trytoproduceafigurewiththelargestorderofrotationalsymmetry.Again,explainyourreasoning.
CCSSGrade4:
1. Recognizealineofsymmetryforatwo-dimensionalfigureasalineacrossthefiguresuchthatthefigurecanbefoldedalongthelineintomatchingparts.Identifyline-symmetricfiguresanddrawlinesofsymmetry.
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Homework
Byperseverancethesnailreachedtheark. CharlesHaddonSpurgeon
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
2) Classifythesymmetriesforthefollowingtrafficsigns–(considertheentiresign–not
justtheinteriordesign).
3) Isitpossibleforanobjecttohaverotationsymmetrieswithouthavingreflection
symmetries?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.
4) Isitpossibleforanobjecttohavetworeflectionsymmetrieswithouthaving180°rotationsymmetry?Ifitis,giveanexampleofsuchanobject.Ifitisnot,giveanargumenttosupportthatconclusion.
5) Onlineatbridges.mathlearningcenter.org,findtheBridgesinMathematicsGrade1
TeachersGuide.Unit5Module3Session1introducesNinePatchInventions,whichSession2usestocreateNinePatchMini-quilts.Spend10-15minutesstudyingthesesessions.Whatideasaboutrotationalsymmetryareemphasizedinthesesections?
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SummaryofBigIdeas
Ifanidea’sworthhavingonce,it’sworthhavingtwice. TomStoppard
• Therearethreedistinctwaysto“move”theplanewithoutchangingtheshapeorsizeof
objects:thetranslation,therotation,andthereflection.• Atranslationslidestheplaneagivendistanceinagivendirection.
• Arotationturnstheplanearoundagivenpointthroughagivenamountofrotation
(usuallygivenindegrees).• Areflectionflipsanobjecttoitsmirrorimageacrossalineofreflection.
• Oneofthegoalsofelementaryschoolgeometryinstructionisthatstudentslearnto
visualizeandapplytransformations.
• Twoobjectsaresimilarifonecanbeobtainedfromtheotherbyarigidmotionandadilation.
• Symmetryisaphenomenonofthenaturalandartisticworldsthatcanbeexplained
withthelanguageofrigidmotions.
• Mathematiciansmostoftentalkabouttwotypesofsymmetry:reflectionsymmetry,inwhichanobjectisdividedbyalineofreflectionintotwopartsthataremirrorimagesofeachother,androtationsymmetry,whereanobjectisrotatedaroundacenterpointthroughacertainangleandendsupoccupyingthesamepositionintheplane.
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ChapterFive
Probability
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ClassActivity23: DiceSums
Theexcitementthatagamblerfeelswhenmakingabetisequaltotheamounthemightwintimestheprobabilityofwinningit.
BlaisePascalRolltwodiceatonceandifthesumis2,3,4,5,10,11or12,yourgroupwins.Ifthesumis6,7,8,or9,theinstructorwins.Isthisafairgame?Dothisproblemtwoways.First,actuallyplaythegamelotsoftimesandcollectdatatodecide.Then,doatheoreticalanalysisoftheproblemtodecide.(Youwillneedtobeginbydecidingwhatitmeansthatagameofchanceis“fair.”)
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ReadandStudy
Areasonableprobabilityistheonlycertainty.E.W.Howe
Gamesofchancehavebeenaroundforatleastaslongaspeoplehaverecordedhistory,butitwasn’tuntilthe16thcenturythatmathematiciansfirstbegantostudytherulesthatmighthelpanswerquestionssuchas“Isthisgamefair?”or“HowlikelyamItodrawaroyalflushinagameofpoker?”Wecalltheareaofmathematicsthatcontainstheserulesprobabilitytheory.Simplyput,probabilityisthestudyofchanceorrandomevents.Supposewerollastandardsix-sideddieonce.Wedon’tknowtheresultoftherollinadvance,sowesaythattheoutcomeisrandom.However,theoutcomeisnotcompletelyunpredictable.Ourdiehassixsidesandsoweknowtheoutcomewillbeoneofsixnumbers:1,2,3,4,5,or6.Wealsoknow(assumingwehaveafairdie)thatanyoneoftheseoutcomesisequallylikely;thatis,ifwerollthediemany,many,manytimesandrecordalltheoutcomes,wewillgetapproximatelythesamepercentageof‘1’saswedo‘2’saswedo‘3’sandsoforth.Supposewewouldliketobeabletoassignanumbertoexpresshowlikelyitisthatwegeta5ononeroll.Inotherwords,wewanttoknowtheprobabilityofgettinga5ononerollofastandard6-sideddie.Bytheway,theuseofgoodmathematicallanguageisahabit.Ifyouwanttousemathematicallanguagecorrectlywithyourelementarystudents,youmustpracticedoingsonow.Youwillfindunderlinedvocabularyandideasdefinedforyouintheglossary.Gothereandhavealookatthedefinitionof“probability”now.L A N G U A G E O F P R O B A B I L I T Y
Inordertotalkabouttheanswertothisquestion,wemustagreeonawaytomeasureprobability.Fourmainruleshavebeenadopted:
1) Theprobabilityofanoutcomeisanumberfrom0to1.Ifanoutcomeiscertaintooccur(suchas,attheequatorthesunwillrisetomorrow),thenitsprobabilityis1.Ifanoutcomecannothappen(suchas,themoonwillfallintothePacificOceantonight),thenitsprobabilityis0.Ifanoutcomeisneithercertainnorimpossible,thentheprobabilitywillbesomenumberbetween0and1.
2) Thesumoftheprobabilitiesofallpossibleoutcomesofarandomexperimentis1.
3) Theprobabilitythataneventwilloccuris1minustheprobabilitythatitwillnotoccur.Thatis,aneventwilleitheroccuroritwillnotoccur,andthesumofthosetwoprobabilitiesmustbe1.
4) Iftwoeventsaredisjoint(theyhavenooutcomesincommon),theprobabilitythatoneortheotherwilloccuristhesumoftheprobabilitiesthateachwilloccurbyitself.
Sowemustthinkabouthowwecanassignanumberfrom0to1totheoutcome“obtaininga5whenrollingonedie”thatwewillcallitsprobability.Todosomathematiciansusethelanguageofsets.Herecometheofficialdefinitionsofrandomexperiment,outcome,event,andsample
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space.Arandomexperimentisanactivity(suchasrollingour6-sideddie)wheretheoutcome(1,2,3,4,5,or6)cannotbeknowninadvance.Asetofallpossibleoutcomes,{1,2,3,4,5,6},isasamplespaceforthatexperimentandaneventisanysubsetofthesamplespace(suchas“rollinga5”or“rollinganevennumber”).Noteasubtledistinctionherebetweenanoutcomeandanevent:anoutcomeisanelementofthesamplespace,whereasaneventisasubsetofthesamplespace.Soifthesamplespaceis{1,2,3,4,5,6}then‘3’iscalledanoutcome,whereas{3}iscalledanevent.{1,3,5}isanotherevent.Listafewmoreeventsforthissamplespaceusinggoodcurly-bracketsetnotation.Doestheemptysetmeetthedefinitionofan“event?”Whyorwhynot?Okay,nowbyrulenumber1,theprobabilityofeachoutcomeinthesamplespacewillbeanumberbetweenzeroandone,andbyrulenumber2,thesumofalltheprobabilitiesoftheoutcomesinthesamplespacemustequalone.Sohowdowedeterminetheprobabilityof“rollinga5?”Wellthatturnsouttobeprettyeasyinthecasewherealltheoutcomesareequallylikelytooccur.(Wewillassumethatourdieisevenlybalancedandaperfectcube-wecallsuchadiefair–andsowearejustaslikelytoobtainanyoneofthenumbersfrom1to6.)Whenoutcomesareequallylikely,theprobabilitythatanyoneoccursisjusttheratioof1tothetotalnumberofoutcomes.Inthecaseofrollingonefairdie,theprobabilityofrollinga‘5’is .Whenaneventismorecomplicated(suchasobtainingasumof3whenrollingtwodice),wecanusethesameapproachbutweneedtochooseoursamplespacewisely.Supposeweareinterestedinthesumobtainedwhenrollingtwofairdice.Ifwelistthepossibleoutcomesforthesum,wegetthesamplespaceof{2,3,4,5,6,7,8,9,10,11,12},whichcontainselevenoutcomes.Unliketheearlierexampleofrollingonediehowever,theseoutcomesarenotequallylikely.Forexample,thereareseveralwayswecouldobtainasumof7(1+6,2+5,etc.),butonlyonewaywecouldobtainasumof12(6+6).Whentheoutcomesarenotequallylikely,weneedadifferentwaytodeterminetheprobabilityofeachoutcome.Inthisexample,theprobabilityofeachsumisnot .Onewaytodeterminetheprobabilitiesofeachsumistoconsiderasamplespaceoftherandomexperimentofrollingtwodicewheretheoutcomesareequallylikely.Forexample,wecanthinkoftheoutcomesinthissamplespaceasorderedpairs(x,y),wherexisthenumberon
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thefirstdieandyisthenumberontheseconddielikeyoumayhavedonewhenyouworkedontheclassactivity.Thissamplespaceisshowninthetablebelow.Theadvantageofusingthissamplespaceisthatalloutcomeshereareequallylikely.
1 2 3 4 5 6
1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
Apossiblesamplespaceforarolloftwodice
Noticethatthereare36equallylikelyoutcomesinthissamplespaceandsotheprobabilityof
eachoutcomeis .Wecanusethissamplespacetocomputetheprobabilityoftheevent“obtainingasumof3”(that’stheevent{(1,2),(2,1)})bycountingthenumberofpossibleoutcomesthathaveasumof3.Wemustbecarefultomakecertainthatweactuallydocountallpossibleoutcomesthathaveasumof3.Forexample,inthesamplespace,theoutcome(1,2)whichgives1+2=3andtheoutcome(2,1)whichgives2+1=3aretwoseparateelements.Thatis,therearetwowaystogeneratethesumof3:1)wecanrolla1onthefirstdieandthena2ontheseconddieor2)wecanrolla2onthefirstdieandthena1ontheseconddie.Tohelpyourstudentsthinkaboutthiswesuggesttwothings.First,havethemdoanexperimentwhereeveryonerollsapairofdiceforfiveminutesandtalliesthenumberoftimestheyrollasumof2andseparatelythenumberoftimestheyrollasumof3.Inthisway,theywillseethatthesumof3istwiceaslikelytooccur.Second,havethemeachholdonereddieandonegreendie.Askthattheymakeasumof3.Thenaskthattheymakeasumofthreeadifferentwaysotheycanseethattheoutcome(1,2)giving1+2=3andtheoutcome(2,1)giving2+1=3aretwoseparateelementsinthesamplespace.Trythisyourself.Nowmakeasumof2andnoticethatthereisnootherwaytodothis.
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Rollinga1onthereddieanda2onthegreendieisadifferentoutcomefromrollinga2onthereddieanda1onthegreendie,butthereisonlyonewaytohavea1oneachdie.Okay,backtooursamplespacewith36outcomes.Usingprobabilityrule4,theprobabilityofrollingtwodicewithasumof3,writtenP(Sum=3),isthesumoftheprobabilitiesofeach
outcomewhichequals + = = .WecanalsocomputeP(Sum=3)astheratioofthenumberofoutcomesthathaveasumof3tothenumberoftotaloutcomessinceallthe
outcomesareequallylikely.SoP(Sum=3)= .
Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.056),asapercent(5.6%)orasodds(2:34,thechancesofhavingasumofthreetothechancesofhavingasumotherthanthree).Checkallthistobesureitmakessensetoyou.Let’sconsideranotherexample:Whataremychancesofdrawingafacecardfromastandarddeckof52playingcards?Beforeyoureadfurther,takeaminutetoseeifyoucanfigureitout.
Hereisourthinking.Payattentiontothelanguageweuseaswellastothewaywesolvetheproblem.Astandarddeckofcardscontains52cards,13ofeachoffoursuits.Eachsuitcontains3facecards,thejack,thequeen,andtheking,soIhave4×3=12waysofdrawingafacecardand52waysofdrawinganycard.Sotheprobabilityofdrawingafacecardfromadeckofcards
is .Thisprobabilitycanbeexpressedasafraction( ),asadecimal(0.231),asapercent(23.1%)orasodds(12:40,thechancesofdrawingafacecardtothechancesofnotdrawingafacecard).Inthisexample,theexperimentis“drawingacardfromadeckofcards,”theeventis“drawingafacecard,”andthesamplespacecontains52equallylikelyoutcomes(the52cardsinthedeck).Let’slookatonemoreexample:Whatistheprobabilityofspinningredoneachofthespinnersbelow?Takeamomenttothinkthisthrough.
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362
181
5212
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Whatisthesamplespaceforeachspinner?Arealltheeventsinthesamplespaceequallylikely?Inbothofthesepictures,wehaveasamplespaceof{red,blue,green,yellow}.However,inthespinnerontheright,theeventsarenotequallylikely.Tocalculatetheprobabilityofspinningredoneitherspinner,wearereallylookingattheareaofthecirclethatisreddividedbytheentireareaofthecircle.Inotherwords,thefractionofthecirclethatisshadedredortheratiooftheareaofredtotheareaoftheentirecircle.Areaisonewaywecanthinkaboutprobabilitiesgeometrically.Dependinguponthesituation,wemightwanttothinkaboutafractionallength,area,orvolume.
ConnectionstoTeaching
Ingrades3–5,allstudentsshouldunderstandthatthemeasureofthelikelihoodofaneventcanberepresentedbyanumberfrom0to1.
NCTMPrinciplesandStandards,p.176InbothoftheReadandStudyscenarioswehavebeendiscussingwhatwecalltheoreticalprobabilities,thatisprobabilitiesthatareassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieoradeckofcards).Thismayhaveleftyouwiththeimpressionthatallrandomexperimentscanbeanalyzedinatheoreticalmanner.Notso.Considertheexperimentof“flipping”alargemarshmallow.Nowitcouldlandonaflatcircularendoronitsside(thecurvedpartofthecylinder),sowe’llidentifythesamplespaceastheset:{end,side}.Childrenmaythinkthatthesetwooutcomeseachhaveaprobabilityof½sincetherearetwochoices,butsomeexperimentingwithactualmarshmallowswillshowthatthesearenotequallylikelyoutcomes.Furthermore,atheoreticalanalysisoftheprobabilitiesofeachoutcomewouldrequireknowledgeofphysicsandgeometry,andmightdependonthesurface,thetemperature,orotherfactors.Incaseslikethat,itisbettertoperformanexperimenttodeterminetheprobability.Inotherwords,itisbesttohavethechildrenflipthemarshmallowalargenumberoftimesandtousethedatatoestimatetheprobabilityofeachoutcome.Probabilitiesthatarebasedondatacollectedbyconductingexperiments,playinggames,orresearchingstatisticsarecalledexperimentalprobabilities.Asateacher,itisimportantthatyougiveyourstudentsplentyofexperiencecalculatingnotjusttheoreticalprobabilities,butalsoexperimentalprobabilities.E X P E R I M E N T A L A N D T H E O R E T I C A L A N A L Y S I S
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Homework
You'llalwaysmiss100%oftheshotsyoudon'ttake.WayneGretzky
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.
2) Ifyoudecidedthatthegame“DiceSums”wasunfair,changetherulessothatthegame
isfair.Istheremorethanonesetofrulesthatwouldgiveafairgame?Ifso,howmanydifferentsetsofrulescanyoumake?
3) Consideranewgamecalled“DiceDifferences.”Twodicearetossedandthesmallernumberissubtractedfromthelargernumber.PlayerIscoresonepointifthedifferenceisodd.PlayerIIscoresonepointifthedifferenceiseven.(Note:Zeroisanevennumber.)Isthisafairgame?Ifso,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.
4) Readthe“TheseBeansHaveGottoGo!”activityfromUnit2Module1Session1oftheBridgesinMathematicsGrade2HomeConnections,onpage29-34.
a. Howwouldyouplaceyourbeansifyouwereplayingthisgame(andwantedtowin).
b. Howwouldyouexpectatypical2ndgradertoplacetheirbeans?c. Whatanswerswouldyouexpectthatyourfuture2ndgradestudentsmightwrite
toquestions3,4and5onpage34.
5) Abagcontainstwowhitemarbles,fourgreenmarbles,andsixredmarbles.Therandomexperimentistodrawamarblefromthebag.
a) Writeareasonablesamplespaceforthisexperiment.Usesetnotation.b) Whatistheprobabilityofdrawingawhitemarble?Aredmarble?Agreen
marble?Explainyouranswers.c) Whatistheprobabilityofdrawingablackmarble?Explain.d) Whatistheprobabilityofnotdrawingagreenmarble?Explain.e) Howmanymarblesofwhatcolorsmustbeaddedtothebagtomakethe
probabilityofdrawingagreenmarbleequalto0.5?
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6) Abagcontainsseveralmarbles.Somearered,somearewhite,andtherestaregreen.
Theprobabilityofdrawingaredmarbleis andtheprobabilityofdrawingawhite
marbleis .
a) Whatistheprobabilityofdrawingagreenmarble?Explain.b) Whatisthesmallestnumberofmarblesthatcouldbeinthebag?Explain.c) Couldthebagcontain48marbles?Ifso,howmanyofeachcolor?Explain.d) Ifthebagcontainsfourredmarblesandeightwhitemarbles,howmany
greenmarblesdoesitcontain?Explain.
7) Decidewhetherthefollowinggameisfairorunfair:Playershavethreeyellowchips,eachwithanAsideandaBside,andonegreenchipwithanAsideandaBside.Flipallfourchips.PlayerIwinsifallthreeyellowchipsshowA,ifthegreenchipshowsA,orifallchipsshowA.Otherwise,PlayerIIwins.Ifthegameisfair,makeanargumentthatitis.Ifthegameisunfair,changetherulessothatthegameisfair.
8) The6outcomesofrollingafairdieareequallylikely;soarethe52outcomesofa
randomdrawofonecardfromadeckofcards.Nameatleastthreemorerandomexperimentsthatgenerateequallylikelyoutcomes.Nameatleastthreethatproduceoutcomesthatarenotequallylikely.
9) Supposeyoubreaka25cmlongspaghettinoodlerandomly.Whatistheprobabilitythatoneofthetwopiecesislessthan5cmlong?
10) Supposeapointwasrandomlychosenwithintheinteriorofthelargestsquarebelow,whatistheprobabilitythatthepointisonwhite?Now,supposeapointwasrandomlychosenwithintheinteriorofthelargestcircle.Whatistheprobabilitythatthepointisonblack?
11) Considertherandomexperimentoftossingthreecoinsatonce.Writeasamplespaceforthisexperimentthathasequallylikelyoutcomes.Nowwriteanothersamplespaceforthissameexperimentthatdoesnothaveequallylikelyoutcomes.
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ClassActivity24: RatMazesThetroublewiththeratraceisthatevenifyouwin,you'restillarat.
LilyTomlin
1) Supposethataratissentintoeachofthebelowmazesatthe‘start.’Iftheratcannotgobackwards,butotherwisemakesallthedecisionsatrandom,whatistheprobabilitythatshefindsacheeseineachcase?Explainyouranswersasyouwouldtoanupperelementaryschoolstudent.
2) Makearatmazetoillustrateeachoftheseproblems:a) Youflipacoinandthenrolladie.WhatistheprobabilitythatyougetaHead
andthenamultipleofthree?b) Youflipacointhreetimes,whatistheprobabilitythatyougetexactlytwo
Heads?
Start
Start
Start
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ReadandStudy
Chancefavorsonlythepreparedmind. LouisPasteur
Manysituationsrequiresuccessivechoices–likeintheratmazeproblemswheretherathadtomakeatleasttwodecisionsaboutwhichpathtotaketogettothecheese.Hereisanotherexample:gettingdressedforschoolinthemorningrequiresseveraldecisions.Imustchooseoneoftwopairsofjeans(onlytwoareclean,abluepairandablackone),oneofthreesweaters(red,purple,blue),andoneoffourpairsofshoes(let’sjustcallthemA,B,C,&D)Let’ssaythatshoepairsBandDdon’tcoordinatewiththeblackjeans,butshoepairsA&Ccanbewornwitheitherpairofjeans.1)WhatistheprobabilitythatIwearbluejeans,aredsweater,andCshoes?2)Whatistheprobabilitythatmyoutfitcontainsthecolorblue?3)WhatistheprobabilitythatIweartheBpairsofshoes?T R E E D I A G R A M S
Treediagrams(orratmazes)areusefulinfindinganswerstoquestionslikethese.Belowwehavemadeatreethatcountsallthepossibleoutfits(orpathsinthemaze).Inotherwords,thechoicesforgettingdressedlooklikethis.
Nowassumingthatallchoicesaremadeatrandom,thereareacouplewaystocomputetheprobabilitiesinquestions1-3above.Takeaminutetodosobeforereadingfurther.Mypersonalfavoriteisthesending-rats-into-the-mazetechnique.I’llsend,say,24ratsintothetopofthemaze.ThenIexpect12togoleftdownthebluejeanchoice,4ofthosetogodowneachsweatercolor,and1tocomeoutofeachoftheshoechoicesA,B,CandD.Ofthe12thatwillgorightdowntheblackjeanchoice,4willgodowneachsweaterrouteand2willemergefromeachshoetypeAandC.Nowwecanusethisinformationtoanswertheabovequestions.1)Since1outof24comeoutofthebluejeans,redsweater,Cshoespath,theprobabilityof
C
C
A
A
A
D
C B
D
C B
C
B A
A
D
A
blue sweater
blue sweater
red sweater red sweater
black jeans blue jeans
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thatis .2)Sincealltheblue-jeansratsandalsotheblue-sweaterratscontainblue(that’s12+
4), or istheprobabilitythatmyoutfitcontainsblue.3)Since3ratsendupwithBshoes,
theprobabilitythatIwearthoseshoesis or .C O M P O U N D E V E N T S
Okay,nowyoumightbethinking,buthowdidsheknowtosend24ratsintothemaze?Doesthatnumbermatter?Tosee,youtryit.Firstuse48rats.Thentry100toseewhathappens.Reallydoit.Whatdidyoulearn?Anothersolutionistousetheprobabilityequivalentofthemultiplicationcountingprincipletofindtheprobabilitythatwereachtheendofeachbranch.Forexample,theprobabilitythatIwearbluejeans,aredsweater,andshoepairAis!
"×!
%× !&= !
"&.Whydowesayweareusing
themultiplicationcountingprinciplehere?FindtheprobabilitiesofeachoftheoutfitsIcouldchoose.Doyouneedtocalculateeachprobabilityindividually?Orcanyoumakeanargumentthatalloftheoutfitswithbluejeansareequallylikely?Whatabouttheoutfitswithblackjeans?Finally,let’sthinkaboutthelasttwoquestionsweposedinthefirstparagraphusingthissecondapproach:Whatistheprobabilitythatmyoutfitcontainsthecolorblue?MyoutfitwillcontainblueifIwearbluejeansorifIwearthebluesweater.Halfofthepossibleoutfitsincludethebluejeansandonethirdoftheremaininghalfoftheoutfitsusethebluesweater.Soatotalof!
"+ !
%× !"= !
"+ !
*= "
%oftheoutfitscontainblue;theprobabilitythatIwearblueis66 "
%%.
UsethisideatofindtheprobabilitythatIweartheBpairofshoes.
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ConnectionstoTeaching
Ingrades6–8allstudentsshouldcomputeprobabilitiesforsimplecompoundevents,usingsuchmethodsasorganizedlists,treediagrams,andareamodels.
NCTMPrinciplesandStandards,p.248Treediagramsarepartoftheelementarygradescurriculumbecausetheygivechildrenawayto“see”allofthepossibleoutcomesandtokeeptrackoftheprobabilitiesofeachchoicethatmightbemadeinacompoundevent.Bytheway,itisnotenoughtoknowjustonewaytodothingsanymore.Ifyouaregoingtobeateacher,youwillneedtomakesenseofthethinkingofyourstudentseveniftheirmethodsaredifferentfromyours.Practicingdifferentwaysofthinkingaboutandexplainingproblemswillmakeyouabetterandmoreflexibleteacher.Homework
Onceyoulearntoquit,itbecomesahabit.VinceLombardi
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Decideifeachofthefollowingstatementsistrueorfalse.Ifitistrue,explainwhy.Ifitis
false,rewritethestatementsothatitistrue.
a) Iftheprobabilityofaneventhappeningis ,thentheprobabilitythatitdoesnothappenis .
b) Theprobabilitythatanoutcomeinthesamplespaceoccursis1.c) Ifaneventcontainsmorethanoneoutcome,thentheprobabilityofthe
eventisthesumoftheprobabilitiesofeachoutcome.d) IfIhavea60%chanceofmakingafirstfreethrowanda75%chanceof
makingasecond,thentheprobabilitythatImissbothshotsis10%.
3) Makearatmazetomodeleachoftherandomexperiments.a) Ihaveabagcontainingthreechips.TheredchiphasasideAandasideB.
TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.Idrawonechipatrandomfromthebagandtossit.
b) Threechipsaretossed.TheredchiphasasideAandasideB.TheyellowchiphasasideBandasideC.ThebluechiphasbothsidesmarkedA.
4) Supposethechipsin#3abovearealltossed.Useyoursecondratmazetodetermine
theprobabilityofgettingexactlya)oneC b)oneB c)oneAd)twoCs e)twoBs f)twoAs
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5) Supposethatthreechipsaretossedasin#3.PlayerIscoresapointifapairofAsturnup,andplayerIIscoresapointifapairofBsoraCturnsup.Isthisafairgame?Ifso,explainwhy.Ifnot,changetherulestomakeitfair.
6) Abasketballplayerhasa70%freethrowshootingaverage.Theplayergoesupforaone-and-onefreethrowsituation(thismeansthattheplayershootsonefreethrow,andonlyifshemakesit,shegetstoattemptasecondshot).Whatistheprobabilitytheplayerwillmake0shots?1shot?2shots?
7) SupposethatItossafaircoinuptofourtimesoruntilIgetaHead(whichevercomesfirst).
a) Makeamazetomodelthisrandomexperiment.b) Writeoutthesamplespaceusinggoodnotation.c) WhatistheprobabilitythatyougetTTH?
8) Place2blackcountersand1whitecounterintoapaperbagandshakethebag.Without
looking,draw1counter.Thendrawasecondcounterwithoutputtingthefirstcounterbackintothebag.Ifthe2countersdrawnarethesamecolor,youwin.Otherwise,youlose.Whatistheprobabilityofwinning?Doestheprobabilityofwinningchangeifyoureplacethefirstcounterbeforedrawingthesecondcounter?Whyorwhynot?Whatistheprobabilityofwinningifyoureplace?
9) Supposeyouspinbothspinnersbelow.Whatistheprobabilityyouspinred-red?Whataboutayellowononeandgreenontheother?
10) Here’showyouplayourlottery:youwritedownanysixnumbersfromtheset{1,2,3,…35,36}.Thensixwinningnumbersaredrawnfromthatsetatrandom(withoutreplacement–soallofthemwillbedifferentnumbers).
a) Ifyoumatchallthewinnersinorder,youwin$5,000,000.00.Whatistheprobabilityofthat?
b) Ifyoumatchallthewinnersbutnotnecessarilyinorder,youwin$1,000,000.00.Whatistheprobabilityofthat?
c) HowaretheseproblemsrelatedtoourPhotographandCommitteeproblems?Explain.Bespecific.Howwouldyouexplainthistosomeone?
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ClassActivity25: TheMaternityWard
WhenIwasakid,Iwassurroundedbygirls:oldersisters,oldergirlcousinsjustdownthestreet….EachyearIwouldwishforababybrother.Itneverhappened.
WallyLamb
Isitmorelikelythat70%ormoreofthebabiesbornonagivendayareboysinasmallhospitalorinalargehospital(ordoesn’titmatter)?Yourclassshoulddecideonasimulationtodoinordertohelpyoutoanswerthisquestion.
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ReadandStudy
“Ithinkyou’rebeggingthequestion,”saidHaydock,“andIcanseeloomingaheadoneofthoseterribleexercisesinprobabilitywheresixmenhavewhitehatsandsixmenhaveblackhatsandyouhavetoworkitoutbymathematicshowlikelyitisthatthehatswillgetmixedupandinwhatproportion.Ifyoustartthinkingaboutthings
likethat,youwouldgoroundthebend.Letmeassureyouofthat!”AgathaChristieinTheMirrorCrack’d
Ourintuitionoftenmisleadsuswhenwethinkaboutprobabilities.Initially,youmayhavethoughtthatthechancesofhaving70%ormoreofthebabiesbornbeboyswouldbebetterinalargehospitalbecauseitismorelikelythatmorebabiesareborninalargehospitalononedayandthereforemorelikelytohavemoreboys.Itismorelikelythatthenumberofbabiesbornonagivendayislargerinalargehospitalthanitisinasmallone,butthisdoesnotmeanthatitismorelikelythatthepercentageofboybabieswillbe70%ormoreinthelargehospital.L A W O F L A R G E N U M B E R S
Infact,thelargerthehospital(andthereforethelargerthenumberofbabiesbornonagivenday),themorelikelyitisthatthepercentageofboysbornwillbe50%(theapproximateprobabilityofhavingababyboyinasinglebirth).Mathematically,wecallthisresultthelawoflargenumbers,whichstatesthatinrepeated,independenttrialsofarandomexperiment,(suchasbabiesbeingborninahospital),asnumberoftrials(births)increases,theexperimentalprobabilitiesobserved(theprobabilitythatababybornisaboy)willconvergetothetheoreticalprobability(inthiscase,50%).Inotherwords,thelargerthesample,themorelikelyyouaretogetclosetothetheoreticalresult.Sothemorebirthsthereareonagivenday(thelargerthehospital),themorelikelyitisthatthepercentageofboysbornisnear50%.Thefactthatthetrialsshouldbeindependentisimportant.(Besuretousetheglossaryfornewmathematicalterms;don’tassumethatthemathematicalmeaningofawordisthesameasthemeaningthewordmighthaveincommonusage.)Forourpurposesconsidertwoeventstobeindependentiftheoccurrenceornon-occurrenceofoneeventhasnoeffectontheprobabilityoftheothereventhappening.Inthehospitalproblem,thebirthofababyboytoonemotherhasnoeffectontheboy/girloutcomeofthenextbirthinthehospital.Theoutcomesofthetwobirthsareindependentofoneanother.Whatabouttherolloftwodice?Aretheoutcomesoneachdieindependent?Whataboutthreeflipsofacoin?Isthehead/tailoutcomeoneachflipindependent?I N D E P E N D E N T E V E N T S
Notalleventsareindependent.Considertheevent“itwillraintomorrow”andtheevent“theweddingwillbeheldoutsidetomorrow.”Theprobabilitythattheweddingwillbeheldoutsideisaffectedbywhetheritrains.Writedownanotherexampleoftwoeventsthatarenotindependent.Onewaytoteachchildrenaboutthelawoflargenumbersistorunlotsofsimulationslikeyoudidinclasswiththehospitalproblem.Thatis,wemodeledthenumberofboyandgirlbirthsonagivendayinvarioussizehospitals,calculatedtheaveragepercentageofboybirthsforeach
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sizehospitalandthencomparedtheresults.Inordertohavethissimulationworkweneededamodel(acoinperhaps)thathastheapproximatelythesameprobabilityofsuccessasdoestheeventweareinvestigating.Andweneedtobecertainthatthenumberof‘births’weconsiderlargeislargeenoughforthelawoflargenumberstoapply.Apracticalwaytodesigncollectingdataonalargenumberoftrialsinyourclassroomistopooleveryone’sdata.Youwereaskedtorunasimulationintheclassactivity.Howdidyoudesignyoursimulation?Howmightyouhaveusedtherollofonefairdie?Howdidyoudecidethenumberofbirthsthatwouldrepresentasmallhospitalandthenumberofbirthsthatwouldrepresentalargehospital?Whatwastheeffectofpoolingalltheclasses’data?
Homework
IamalwaysdoingthatwhichIcannotdo,inorderthatImaylearnhowtodoit.PabloPicasso
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Acoinhasbeentossedfivetimesandhascomeupheadseachtime.Whichofthe
followingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.
3) Acoinhasbeentossedfivehundredtimesandhascomeupheadseachtime.Whichof
thefollowingstatementsaretrue?a) Thereisanequalchanceofcomingupheadsortailsonthenexttoss.b) Thecoinismorelikelytocomeupheadsonthenexttoss.c) Thecoinismorelikelytocomeuptailsonthenexttoss.d) Iquestionwhetherthecoinisfair.
4) Explainhowthelawoflargenumbersisrelatedtoyouranswerstothequestioninproblems2)and3)above.
5) Explainhowthelawoflargenumbersisrelatedtothedeterminationofexperimentalprobabilities.Whatdoesthislawtellyouaboutsimulationsyouwilldoinyourelementaryclassrooms?
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6) Afaircoinhasbeentossedathousandtimesandthenumberofheadsandtailsrecorded.Whichofthefollowingstatementsismostlikelytrue?
a) Thenumberoftailsis601andthenumberofheadsis399.b) 47%ofthetosseswereheads.c) Therewere20moreheadstossedthantails.
7) Supposeweflipacoin10,000,000,000times(justsuppose)andnoticethat
5,801,595,601areheads.Whatcanyousayaboutthecoin?Explain.
8) Designasimulationtodeterminetheexperimentalprobabilityofdrawingaspadefromastandarddeckofcards.Howcanyoubereasonablycertainyourresultisclosetothetheoreticalprobability?Carryoutyoursimulationandcompareyourresulttothetheoreticalprobabilityofdrawingaspadefromadeckofcards.
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SummaryofBigIdeas
Anideaisalwaysageneralization,andgeneralizationisapropertyofthinking.Togeneralizemeanstothink.
GeorgHegel
• Arandomexperimentisanactivitywheretheoutcomecannotbeknownin
advance.
• Theprobabilityofaneventhappeningistheproportionoftimesthateventwouldoccuriftherandomexperimentwasperformedaverylargenumberoftimes.
• Itisimportantforyoutogiveyourstudentsexperiencescalculatingprobabilities
experimentallyaswellastheoretically.• Treediagramshelpstudentsthinkthroughprobabilityideas.• Thelawoflargenumberstellsusthatthelargerthesample,themorelikelyweare
togetclosetothetheoreticalresult.
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ChapterSix StatisticsandDealingwithData
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ClassActivity26: StudentWeightsandRectangles
Themathematicsisnotthere‘tilweputitthere. SirArthurEddington(MQS)
Whatistheaverageweightofallthestudentsregisteredforclassesatyourschoolthissemester?Inyourgroups,makeaplanforsomewaytogatherinformationtoanswerthisquestion.Bespecificaboutwhatyouwilldoandwhenyouwilldoit.Atthispoint,timeandcostisnoobject.(Waittodiscussthisquestionasaclassbeforeyouturnthepage.)
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Now,havealookatthepopulationof100rectanglesonthenextpage.Noticethateachonehasanarea(asize).Forexample,rectanglenumber74hassize6squareunitsbecauseitiscomposedof6smallsquares.Carefullyselectasampleoftenrectanglesthatyouthinkwillhaveanaveragesizethatisclosetoaveragesizeofalltherectanglesonthepage.Makesuretopersonallychooseeachoftheten.Thenfindtheaveragesizeofyourself-selectedten.Yourclasswillthenmakeafrequencygraphshowingeveryone’saverages.Next,eachpersonshouldselect10rectanglesatrandom.(Yourinstructorwillhaveaplanfordoingthis.)Thenfindtheaveragesizeofyourrandomly-selectedten.Yourclassshouldmakeanotherfrequencygraphshowingeveryone’saverages.Finally,estimate,basedonthefrequencygraphs,whatistheaveragesizeofall100rectanglesonthepage?Explainyouranswer.Whatdoesallthishavetodowiththestudent-weightproblem?
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A Population of Rectangles
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ReadandStudy
Heusesstatisticsasadrunkenmanuseslampposts–forsupportratherthanillumination.
AndrewLang Thisrectanglesizeproblemcontainsahugelessoninstatistics.Huge.Itisthis:humansarenogoodatselectingsamples.Ifyouwantunbiasedinformation,randomsamplesarethewaytogo.Inordertotalkmoreaboutthis,wearegoingtointroducethelanguageofsampling.Rememberthatwewantyoutolearnthesedefinitionsandtousethem.First,asamplereferstoasubsetofapopulation.Inourstudentweightproblem,thepopulationofinterestwasallstudentsregisteredforclassesatyourschoolthissemester.Inanidealworld,wejustwouldhaveeverysinglememberofthepopulationcometogetweighedandwewouldcalculatetheaverageweight.Unfortunately,thiswouldlikelyproveimpossible.Lots(most?)ofthestudentswouldnotshowuptobeweighed,andforcingthemtodosowouldbeunethical.Worse,thosewhowouldbewillingtoshowupmightbedifferentthanthosewhowouldrefuse.Peoplewhowereweight-consciousoroverweightmightbemorelikelytoavoidsteppingonyourscale.Sointheend,youwouldendupwithabiasedsample.Anothersolutionmightbetosendoutanonymoussurveystotheentirepopulationaskingeachpersontorecordhisorherweight.Butwouldyoureturnthatsurvey?Mostpeoplewouldnot(masssurveysmailedoutlikethattendtohaveverylowresponserates)andthosepeoplewhoreturnthemareagaindifferentfromthepopulation.Additionally,whenasurveyreliesonparticipantstoself-reporttheirowndata(likeweight),sometimesthatdataisn’taccurate.Okay,wellwemighttrytocalleveryoneorgodoor-to-doortocollectinformationonstudentweights.Thattypeofsamplingyieldsmuchhigherresponseratesbutcouldproveveryexpensiveandtakealotoftime(particularlyifyouhappentohavethousandsofstudentsinthepopulation). T H E L A G U A G E O F S A M P L I N G
So,whatarewetodo?Hereisthestatistician’ssolution.Forgetaboutgatheringdata(information)fromeachandeverymemberofthepopulationandfocusyourtime,energyandresourcesinsteadongettinginformationfromjustarandomsampleofthepopulation.Thatwayyoucanaffordtogodoor-to-doorandgetdatafrommostmembersofthesample.Itstillwon’tbeperfect.Butitturnsoutthatthisistheverybestyoucandoinafreesociety.Whyshouldthesampleberandom?Whynotsimplycollectdataontheweightsofyourfriends,ormaybeeveryoneinyourdorm,orwhynotstandinfrontofthecampuslibraryandaskeveryonewhowalksbyfortheirweights?Becauseofthethreatof…samplingbias.Asamplingprocedureisbiasedifittendstoover-sample(orunder-sample)populationmemberswithcertaincharacteristics.Thiswillmostsurelyhappenifyouself-selectyour
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sample–afterall,thestudentsyouknowhavedifferentcharacteristicsthanthepopulationofyourcampus.Whataresomeofthosedifferences?Stopandwritedownatleastthree.Butyoursamplewillalsobebiased,inamoresubtleway,ifyoustandinfrontofthelibraryandaskpeople“atrandom.”Thisisbecauseyouarenotreallyaskingrandompeople.Youareaskingthepeoplewhowalkbythelibraryatthattimeofday(maybeoversamplingpeoplewhostudyoftenorunder-samplingthosewhohaveaclassatthattime).Youareaskingpeoplewhohavealookatyouanddecidetheywanttoansweryourquestion(unwittinglyoversamplingpeoplelikeyourself).Giveanotherreasonwhythelibrarysamplemightbebiased.Samplingbiasissneaky.Itcreepsinbasedonwhenyousampleandwhereyousampleandhowyousample(aswellaswhoyousample).Watchforitasyouanalyzestudies.Sothesolutionistogeneratearandomsampleandthentouseyourresourcestogetthebestpossibleresponseratefromthem.Herearetwotypesofrandomsamplingthatareconsideredacceptable.Thefirstisthebest.Itiscalledsimplerandomsamplinganditismathematicallyequivalenttopullingnamesoutofahat.Everymemberofthepopulationhasthesamechanceofbeinginthesampleasanyothermemberofthepopulationandanysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesize.Stop.Readthatlastsentenceagainandthinkaboutit.Ifyouhadyourregistrargeneratealistofallstudentsenrolledinclassesthissemesterandyouassignedeachstudentanumberandthenhadyourcalculatorgenerate100randomnumbersandusedthosenumberstogetnames,thenthatwouldbesimplerandomsampling.IfIlinedupyourclassinarowandthentookeverythirdpersontomakemysample,wouldthatbesimplerandomsampling?Explain.R A N D O M S A M P L E S
Thereareotherformsofacceptablesampling.Forexample,clustersamplingisrandomsamplinginstages.Forexample,ifyourandomlychosetenpagenumbersfromtheregistrar’slistofstudentsandthenrandomlychose10peoplefromeachofthosepages,thenthatwouldbeclustersampling.TheU.S.Censushasmadeuseofclustersamplingwhenithasattemptedtofindthosepeoplewhodidnotreturntheircensusforms.We’lltellthatstoryinalatersection,fornowletussaythatacensusmeansthatyougatherinformationfromeachandeverymemberofapopulation.TheU.S.Censusisbutoneexampleofanattemptatacensus.Ifyoutriedtogetweightinformationfromeverystudentregisteredatyourcampus,thenyoutoowouldbeattemptingacensus.Itisprettymuchimpossibletoconductacensuswithalargepopulationbecauseyounevergetdatafromeveryone.Ifyoucollectdatafromasubsetofthepopulation,youarereallyconductingasurvey.Okay,herearetwomoretermstoknow:thenumericinformationthatyouwanttoknowaboutapopulation(ifyoucouldgetit)iscalledaparameter;thenumericinformationyougetfromasampleiscalledastatistic.Inourexample,theparameterwewereafterwastheaverage
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weightofallthestudentsregisteredforclassesatyourschool.Ifwecoulddoacensusofthatpopulation,wecouldfindthatparameter.However,wehavediscussedsomeofreasonswhywe’renotlikelytogetthatparameter.Instead,wewillsettleforperformingasurveyofasampleofthepopulation,andwhatwewillgetisastatistic.Hereisourilluminatingpicture:
Population Sample(fromthepopulation)
Askeveryone,it’scalledacensus. Askasubset,it’scalledasurvey.#computedfromthepopulation↔parameter. #computedfromthesample↔statistic.Thedifferencebetweentheparameterandthestatisticiscalledsamplingerror.Samplingerrorisanidea.Inpractice,whenconductingasurvey,youaren’tgoingtoknowwhatitisbecauseallyouwillhaveisastatistic.You’llprobablyneverknowtheexactparameter.Butwestillusethesewordstotalkabouttheideas.Onesourceofsamplingerrorwehavealreadydiscussed:samplebias.Anothersourceofsamplingerrorischanceerror.Chanceerroriserrorduetothediversityofthemembersofthepopulationandthefactthatyouarejustcollectingdatafromasubsetofthem.Ifthepopulationisdiverse(different)intheirweights,forexample,thenifyoupickarandomsampleofthem,youmayendupwithanaverageweightthatisabitdifferentfromthepopulationweight.Youmayeven,duetochancealone,getsomepeoplewhoareverythinorveryheavyinthesample.ThinkbacktoyourrandomsampleoftenrectanglesfromtheClassActivity.Theaveragesizeofyourrandomrectangleswaslikelycloseto,butnotexactly,theaveragesizeofall100rectanglesonthesheet.Thedifferencewasduetochanceerror.Nowthinkbacktotheaveragesizeofyourtenself-selectedrectangles.Thedifferencebetweenthatnumberandtheaveragesizeofall100rectanglesonthesheetisduetobothchanceerrorandsamplebias.Howdoyoureducesamplebias?Makeyoursamplerandom.
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Howdoyoureducechanceerror?Makeyoursamplebigger.Sojusthowbigdoesasamplehavetobe?Itseemslikeyouwouldneedsomesignificantsamplingrate(likemaybe25%ofthepopulation),butitturnsoutthatgoodsurveyscanbedonewithquitesmallsamples.Forexample,nationalpollingtopredicttheoutcomeofthepresidentialelectionisoftendonewithonlyafewthousandpeople.That’safewthousandoutofmorethan100millionlikelyvoters.Samplingrate[(#inthesample)÷(#inthepopulation)]canbesosmallbecausepeoplearenotallthatdiversefromasamplingperspective.Thinkofitthisway:supposeyouaregoingtosampleabigpotofsouptoseehowittastes.Aslongasthatsoupisstirredupreallywell,andthechunksofmeatsandvegetablesintherearen’ttoobigortoodifferent,allyouneedisasmallbitetoknowtheflavor.Bytheway,whatwasthesamplingrateofyourrandomsamplefromtherectanglesheet?Figureitout.
ConnectionstoTeachingIngrades3-5allstudentsshould–proposeandjustifyconclusionsandpredictionsthatarebasedondataanddesignstudiestofurtherinvestigatetheconclusionsand
predictions. NationalCouncilofTeachersofMathematics
PrinciplesandStandardsforSchoolMathematic,p.176Ideasofsamplingaresurprisinglycomplex–butgroundworkforthemcanbelaidintheelementaryschool.TheNCTMStandardsadvocatesthatstudentsingrades3-5should
“…begintounderstandthatmanydatasetsaresamplesoflargerpopulations.Theycanlookatseveralsamplesdrawnfromthesamepopulation,suchasdifferentclassroomsintheirschool,orcomparestatisticsabouttheirownsampletoknownparametersforalargerpopulation…theycanthinkabouttheissuesthataffecttherepresentativenessofasample–howwellitrepresentsthepopulationfromwhichitisdrawn–andbegintonoticehowsamplesfromthepopulationcanvary(p.181).”
Noticetheuseofthetechnicallanguage.Whatdotheymeanby“knownparametersforalargerpopulation?”Givesomeexamples.Supposethatyourthirdgradershavecollecteddatafromtheirclassonthenumberoftimeslastweekthattheyeathotlunchatschoolandfoundthefollowingdatadisplayedbelowasafrequencyplot(eachXrepresents1childintheclass):
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X X X X X X X X X X X X X X X X X X X X 0 1 2 3 4 5 #ofHotLunchDayslastweekOfcourse,therearemanyquestionsyoumightaskyourstudentsaboutthesedata–buttherearesomequestionsparticularlyrelatedtosamplingthatshouldbeaskedanddiscussed.Questions1:Wasthereanythingspecialaboutlastweekthatmighthavemadethesedatadifferentthanifwe’daskedthesamequestionnextweekordoyouthinklastweekwasatypicalweek?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Question2:Howdoyouthinkthedatamighthavebeendifferentifwe’daskedthefifthgradersthesamequestion?
Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Questions3:Woulditbeokaytolabelthisgraph“NumberofDaysEachWeekthatChildrenEatHotLunchatOurSchool?”Whyorwhynot?Inwhatwaysmighttheyrespondtothatquestion?Whatpointsaboutsamplingcouldyoumakeineachcase?Noticethatthesequestionshelpchildrenthinkaboutwhetherthedataisrepresentativeofapopulationlargerthanthesampleof20third-gradersshownonthegraph,andithelpsthemto
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begintothinkcriticallyaboutwhatthedatasaysandwhatitdoesnotsay.Samplingbecomesanexplicittopicinthemiddlegrades.
Homework
Weknowthatpollsarejustacollectionofstatisticsthatreflectwhatpeoplearethinkingin‘reality.’Andrealityhasawellknownliberalbias.
StephenColbert
1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.2) Explainthedifferencebetweenacensusandasurvey.
3) Explainthedifferencebetweenaparameterandastatistic.
4) TrueorFalse?Thebiggerthesamplethesmallerthesamplebias.Explainyourthinking.
5) Sometimesawebsiteorpublicationwilladvertiseapollaskingitsviewersto‘callin’or
‘clicktorespond’toquestionsonatopic.Inthiscasewesaythatthesampleisself-selected.Whattypesofbiasesarelikelyinthistypeofsurvey?Explain.
6) Itispossibletobiasresultsofasurveybyaskingquestionsusing“charged”language.ConsiderthesequestionsfromtheRepublicanNationalCommittee’s2010ObamaAgendaSurvey,andfromaDemocraticStateSenator’sdistrictsurvey.(Canyoutellwhichiswhich?)Explainhoweachquestionmightbeaskedinalessleadingmanner.
a) “Doyoubelievethatthebestwaytoincreasethequalityandeffectivenessof
publiceducationintheU.S.istorapidlyexpandfederalfundingwhileeliminatingperformancestandardsandaccountability?”
b) “DoyousupportthecreationofanationalhealthinsuranceplanthatwouldbeadministeredbybureaucratsinWashington,D.C.?”
c) “DoyoubelievethatBarackObama’snomineesforfederalcourtsshouldbeimmediatelyandunquestionablyapprovedfortheirlifetimeappointmentsbytheU.S.Senate?”
d) “Doyousupportoropposelegislationtorestoreindexingofthehomesteadtaxcredit,whichwouldboostoureconomyandhelppeopletostayintheirhomes?”
e) “DoyousupportoropposelimitinglocalcontrolandweakeningenvironmentalstandardstoencourageironoremininginWisconsin?”
7) TheColumbusCityCouncilwantedtoknowwhetherthe65,000citizensofthecityapprovedofaproposaltospend$1,000,000torevitalizethedowntown.Thefollowingstudywasconductedforthispurpose.Asurveywasmailedtoevery10thpersononthe
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listofregisteredvotersinthecity(itwasmailedto3500people)askingabouttheproposal.CitizenswereaskedtocompletethesurveyandreturnittotheCouncilbymailinaself-addressed,stampedenvelope.3100peopleresponded;ofthese,75%supportedtheexpenditureand25%didnot.
a) Thepopulationforthisstudyis
A)allregisteredvotersinthecity B)allcitizensofthecityofColumbus C)the3500peoplewhoweresentasurvey D)the3100peoplewhoreturnedasurvey E)noneoftheabove
b) Whatwasthesamplingrateforthissurvey?
c) The"75%"reportedaboveisa A)parameter B)population C)sample D)statistic E)noneoftheabove
d) Thissurveysuffersprimarilyfrom A)samplingbias B)chanceerror C)havingasamplesizethatistoosmall D)non-responsebias
e) TrueorFalse?Theresultsofthissurveymaybeunreliablebecauseregisteredvotersmaynotberepresentativeofallthecitizens.Explainyourthinking.
f) TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribedas
simplerandomsampling.Explainyourthinking.
g) TrueorFalse?Samplebiascouldbereducedbytakingalargersampleofregisteredvoters.Explainyourthinking.
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ClassActivity27: SnowRemoval
Errorsusinginadequatedataaremuchlessthanthoseusingnodataatall.CharlesBabbage
Inordertostudyhowsatisfiedthe120,000citizensofGreenBay,WIarewithsnowremovalfromcitystreets,Iconductedthefollowingstudy.IstoodinfrontoftheGreenBayWalmartonaMondaymorninginFebruaryfrom9:00untilnoonandIaskedeverythirdpersonwhowalkedinwhethertheyweresatisfiedwithsnowremovalfromstreetsinGreenBay.Onehundredandeightpeoplesaidtheyweresatisfied.Twenty-fivepeoplesaidtheywerenot,and18refusedtoanswermyquestion.Answerthebelowquestioninyourgroups.
1) Describethepopulationforthestudy.
2) DidIconductacensusorasurvey?Explain.
3) Whatwasthesamplingrate?(Writethisinseveraldifferentways.)
4) Whatwastheresponserate?(Writethisinseveraldifferentways.)
5) TrueorFalse?Thisstudysuffersprimarilyfromnon-responsebias.Explain.
6) TrueorFalse?Thisstudysuffersprimarilyfromsamplingbias.Explain.
7) TrueorFalse?Theresultsofmystudymaynotbevalidbecausecityemployeesmayhavebeenpartofthesample.Explain.
8) TrueorFalse?Amainproblemwithmystudyisthatmysamplesizeistoosmall.Explain.
9) TrueorFalse?PeopleshouldconcludefrommystudythatGreenBaycitizensare
generallyhappywithsnowremovalfromcitystreets.
10) Describeatleastthreewaystoimprovemystudy.
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ReadandStudy
Youcanuseallthequantitativedatayoucanget,butyoustillhavetodistrustitanduseyourownintelligenceandjudgment.
AlvinToffler
“Fouroutoffivedentistssurveyedrecommendsugarlessgumfortheirpatientswhochewgum.”You’veheardthatclaim.Itisperhapsthemostwellrememberedstatistical‘fact’inthehistoryofmarketingresearch.Butwhatdoesittellyou?Themakersofsugarlessgumhopethisstatisticconvincesyoutopurchasetheirproduct.Butbeforeyourunoutforapackofsugarlessgum,youshouldaskafewquestionsaboutthis‘fact.’Howwerethedentistsinthesamplechosen?Whatwastheresponserate?Whatwasthesamplesize?Whatwastheexactquestiontowhichthedentistswereaskedtorespond?Andevenifitturnsoutthatthesurveyusedexemplarymethodology,notethattheresultdoesnotsuggestyouthatyoushouldtakeupchewingsugarlessgum.Itonlysuggeststhatifyoualreadychewgum,thesedentistssuggestyouchewsugarlessgum.Allsurveysarenotcreatedequal.Infact,somearedownrightmisleading.Itisimportantforyoutothinkcriticallyaboutinformationbasedonsurveydata.A N A L Y Z I N G A S U R V E Y
Surveysaremeanttoprovideinformationaboutapopulation.Theycanhelpcompaniesunderstandhowtomarketproducts;theycanhelpgovernmentalagenciesunderstandconstituents;theycanhelpcampaignspredictpublicreactionstoideasorevents;theycanhelpsecondgradeteachersunderstandwhichmoviestheirstudentsprefer.Hereisanexampleofonerealsurvey.TheUnitedStateCensusBureauusessurveystounderstandthepopulationofourcountry.(Bytheway,asteachersyoucangetgooddataforyourstudentstoanalyzefromtheU.S.CensusBureauwebsite.Keepthatinmind.)Forexample,theyconductasurveyofU.S.housingunitseveryotheryearinordertomakedecisionsaboutfederalhousingprojects.Intheirwords,theiraimisto“[p]rovideacurrentandongoingseriesofdataonthesize,composition,andstateofhousingintheUnitedStatesandchangesinthehousingstockovertime.”Hereisafigureshowingsomeoftheresultsofthe2007AmericanHousingSurvey(AHS).Takeafewminutestostudyit.
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U.S.CensusBureau,CurrentHousingReports,SeriesH150/07AmericanHousingSurveyfortheUnitedStates:2007.U.S.GovernmentPrintingOffice,Washington,DC.
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Theseresultsdrivenationalpolicy,sowehopethattheAHSiswelldesigned.Let’shavealookatthemethodology.ReadthefollowingparagraphbasedontheinformationfromtheUSCensusBureauwebsite:TheAHShassampledthesame53,000addressesineachodd-numberedyearsince1985.Thoseaddresseswerechoseninthefollowingmanner.
“FirsttheUnitedStateswasdividedintoareasmadeupofcountiesorgroupsofcountiesandindependentcitiesknownasprimarysamplingunits(PSUs).AsampleofthesePSUswasselected.ThenasampleofhousingunitswasselectedwithinthesePSUs”(U.S.CensusBureau,2007,p.B-1).
TheAHSincludesbothoccupiedandvacanthousingunitsandallpeopleinthosehousingunits.Dataiscollectedusingcomputer-assistedtelephoneandpersonal-visitinterviews.Whenahousingunitwasvacant,informationwasgatheredfromneighbors,rentalagents,orlandlords.TheAHSisavoluntarysurveymeaningthathouseholdscandeclinetoanswerquestions.Interviewsforthedatapresentedabovewereconductedfrommid-ApriltoSeptember2007.
Okay,shouldwebelievetheAHSresults?Let’stalkthroughananalysisofthesurveydesign.First,whatisthepopulationforthissurveyandwhatisthesample?Accordingtothe2001census,thereareabout200,000,000housingunitstotalintheUnitedStates–thatistheapproximatepopulation.Thesamplecontainedabout53,000ofthesehomes.Thatgivesasamplingrateofabout0.0027%.Thismightseemsmall,butthesamplewascarefullyselectedusingclustersamplingandsothesamplingerrorwillbefairlysmalltoo.(Whywasthisclustersampling?Explainwhybasedonthedefinition.)Inotherwords,thereisreasontobelievethatthehouseholdsinthesamplearerepresentativeofhouseholdsintheUnitedStates.Let’sgoonestepfurtherinthinkingaboutthequalityofthedata.Itcouldbethatthesamplewaswell-chosen,butthatmanyhouseholdsgavefaultyinformation(orsimplyrefusedtogiveinformationatall).LuckilytheCensusBureauusedthebestpossiblemethodforcollectingdata–theycalledfirst,andiftheycouldnotreachahouseholdbyphone,theysentsomeonetotheaddresstoconductaninterviewwitheithertheoccupants,orinthecaseofanemptyunit,aneighbororlandlordorrealestateagent.Thislikelyproducedafairlyhighresponserate.(Infact,theBureaureportsan89%responserate.Intheothercaseseithernoonewasathomeevenafterrepeatedvisits,theoccupantsrefusedtobeinterviewed,ortheaddresscouldnotbelocated.)So,ourtakeisthatthemethodologyfortheAHSissoundandtheresultsarebelievable.Wewantyoutolearntothinklikethiswheneveryouneedtojudgethebelievabilityofaclaimbasedondata.
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ConnectionstoTeaching
Mathematicsisasmuchasaspectofcultureasitisacollectionofalgorithms.CarlBoyer
The Common Core standards in grades 6 and 7 for surveys are listed below. Read them carefully and then determine which standards we have addressed so far.
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
CCSSGrade6:Developunderstandingofstatisticalvariability.1. Recognizeastatisticalquestionasonethatanticipatesvariabilityinthedata
relatedtothequestionandaccountsforitintheanswers.Forexample,"HowoldamI?"isnotastatisticalquestion,but"Howoldarethestudentsinmyschool?"isastatisticalquestionbecauseoneanticipatesvariabilityinstudents'ages.
CCSSGrade6:Summarizeanddescribedistributions.
5.Summarizenumericaldatasetsinrelationtotheircontext,suchasby:A.Reportingthenumberofobservations.B.Describingthenatureoftheattributeunderinvestigation,includinghowitwasmeasuredanditsunitsofmeasurement.
CCSSGrade7:Userandomsamplingtodrawinferencesaboutapopulation.
1. Understandthatstatisticscanbeusedtogaininformationaboutapopulationbyexaminingasampleofthepopulation;generalizationsaboutapopulationfromasamplearevalidonlyifthesampleisrepresentativeofthatpopulation.Understandthatrandomsamplingtendstoproducerepresentativesamplesandsupportvalidinferences.
2. Usedatafromarandomsampletodrawinferencesaboutapopulationwithanunknowncharacteristicofinterest.Generatemultiplesamples(orsimulatedsamples)ofthesamesizetogaugethevariationinestimatesorpredictions.Forexample,estimatethemeanwordlengthinabookbyrandomlysamplingwordsfromthebook;predictthewinnerofaschoolelectionbasedonrandomlysampledsurveydata.Gaugehowfarofftheestimateorpredictionmightbe.
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Homework
USATodayhascomeoutwithanewsurvey--apparently3outofevery4peoplemakeup75%ofthepopulation.
DavidLetterman
1) DoalltheitalicizedthingsintheReadandStudysection.
2) DoalltheitalicizedthingsintheConnectionssection.3) Spendafewminuteslookingatthefigureonoccupiedhomesfromthe2007AHS.Write
aparagraphhighlightingthethingsthatyoufindmostinterestingornotableabouttheseresults.
4) Inordertofindouthowitsreadersfeltaboutitscoverageofthe2008Presidential
Election,alocalnewspaper(withacirculationofabout10,000)ranafront-pagerequestintheSundaypaperthatreadersgototheirwebsiteandcompleteanonlineanonymoussurveythatcontainedthefollowingtwoquestions:
I. Whatisyourpoliticalaffiliation?____Democrat____Republican____Independent____Other
II. Howwouldyourateourcoverageofthe2008presidentialelection?____slantedleft____slantedright____unbiased____noopinion
Sixhundredreadersresponded.Ofthose,70%ofDemocrats,40%ofRepublicans,66%ofIndependents,and20%of“Other”saidthatthecoveragewasunbiased.
a) Analyzethesamplingmethod.Isthesampleofrespondentslikelytoberepresentativeofthepopulationthepaperwantstorepresent?Givespecificreasonswhyorwhynot.
b) Whatdoesthedatatellusaboutthepaper’scoverageofthe2008PresidentialElection?Explain.
c) Describesomewaystoimprovethissurvey.
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5) GototheUnitedStatesCensusBureauwebsite(http://www.census.gov/schools/)andfindtheTeachersandStudentslink.Spend15minutesbrowsingtheactivitiesthere.ThendesignamathematicslessonforGrade3usingtheStateFactsforStudentslink.
6) AresearcherwantstopredicttheoutcomeoftheFrostbiteFallsmayoralelection.In
ordertodothis,shemailedasurveytoeverytenthpersononalistofall40,980registeredvotersinthecity.Onethousandfifty-sixpeoplereturnedthesurvey.34%ofrespondentssaidtheysupportedSnidelyWhiplash;48%saidtheysupportedDudleyDoRight;theremaining18%wereundecided.
I. Thepopulationforthisstudyis
A)allmayoral-racevotersinthecityofFrostbiteFalls B)allcitizensofthecityofFrostbiteFalls C)thepeoplewhoweresentasurvey D)thepeoplewhoreturnedasurvey E)noneoftheabove
II. Theresponseratewas___________.III. Thesamplingratewas___________.
IV. The"48%"reportedaboveisa
A)parameter B)population C)sample D)statistic E)noneoftheabove
V. Thissurveysuffersprimarilyfrom
A)selectionbias B)nonresponsebias C)chanceerror D)havingasamplesizethatistoosmall
VI. TrueorFalse?Theresultsofthissurveymaybeunreliablebecausemembersofthecitygovernmentmayhavebeenincludedinthesample.Explainyourthinking.
VII. TrueorFalse?Themethodofsamplinginthissurveycouldbestbedescribed
assimplerandomsampling.Explainyourthinking.
VIII. TrueorFalse?Biascouldbereducedbytakingevery5thpersononthelistofregisteredvotersratherthanevery10th.Explainyourthinking.
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ClassActivity28: Part1NameGames
Ifeellikeafugitivefromthelawofaverages. WilliamH.Mauldin
1) Eachpersonintheclassshouldwritehisorherfirstnameonstickynotessothatthere
isoneletteroneachoftheperson’snotes.(Soforexample,IwouldwriteJononenote,Eonanother,andNonthethird.)Putyournamesinstacksontheboardlikewedidintheexamplebelow.Nowyourclassneedstorearrangeyourclasses’stickynotes(orevenpartsofstickynotes)sothateverystackendsupwiththesamenumberofletters.Dothatnow.
AmiJen Mo Omar…
Themean(oraverage)ofasetofnumericalobservationsisthesumofalltheirvaluesdividedbythenumberofobservations.Explainwhyitmakessensethatyourmethodjustcomputedthemeannamelengthforyourclass.
2) Nowwewillmakeadifferenttypeofgraph,abargraph.Eachpersonshouldgetonestickynotetoplaceonthegraphbasedonthelengthofhisorherfirstname.ForexampleiftherewereeightpeopleintheclasswithnamesMo,Ami,Jen,Omar,Karen,Steve,Sienna,andJuanitathebargraphwouldlooksomethinglikethis:
frequencies
____________________________________ 234567 NameLengths
Figureoutawaytorearrangethenotestohelpyoutofindthemeannamelengthonagraphlikethis.Whydoesitmakesensethatyourmethodjustcomputedthemeannamelengthforyourclass?Howisthismethoddifferentfromthemethodyouusedinquestion1?Discussthis.
(Thisactivitycontinuesonthenextpage.)
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3) Themedianofasetofnumericalobservationsisthemiddleobservationwhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”observations,themedianistheaverageofthem.Decidehowyoucoulduseeachtypeofgraphabovetohelpyoufindthemediannamelength.
4) Whichislikelytobelargerforyourclass?Themeannumberofsiblingsorthemediannumber?Explain.Then,collectthedataandseeifyouarecorrect.
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ClassActivity28: Part2IntheBalance
Sodivinelyistheworldorganizedthateveryoneofus,inourplaceandtime,isinbalancewitheverythingelse.
JohannWolfgangvonGoethe
Yourtaskistoputweightsonthenumberedpegsofabalancescaletokeepthesystembalanced.
1) Findseveraldifferentarrangementsof3weightssothatthesystemwillbalance.Makeageneralconjectureaboutallthearrangementsthatwillbalancethesystem.
2) Ifyouhadoneweight1totherightandtwoweights2totheleft,predictwhereyouwouldneedtoplaceafourthweighttobalancethesystem.
3) Findseveralwaystobalancefourweightswhereonlyoneweightisontheleftside.Makeageneralconjectureaboutallthearrangementsofthistypethatwillbalancethesystem.
4) Predictsomearrangementsthatwillbalancefor5weights.Makeageneralconjecture
aboutallthearrangementsof5weightsthatwillbalancethesystem.
5) Supposethereare4catswithmeanweightof10pounds.Youknowtheweightsof3ofthecats.Thoseweightsare:4pounds,7pounds,and15pounds.Howcanyouusethebalancetofigureouttheweightofthe4thcat?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)
6) Usethebalancetofigureoutthisproblem:Youhavetakenfourmathquizzes.Your
scoreswere82,67,85,and72.Whatdoyouneedtoscoreonthe5thquizforyouraveragequizscoretobe75?(Don’tdoacalculation.Figureitoutwiththebalanceonly.)
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ReadandStudy
Theaverageadultlaughs15timesaday;theaveragechild,morethan400times. MarthaBeck
Therearethreestatisticsthatarecommonlyusedtodescribethecenterofadistributionofnumericaldata:mean,medianandmode.Themeaniscomputedbysummingthevaluesanddividingbythenumberofthem.Butabetterwayforchildrentothinkaboutmeanisusingtheideaof“eveningoff.”Themeanistheaveragevalue,thevalueeachpersonwouldgetifallthedatawassharedevenly.Forexample,supposethepicturebelowshowsthecookiesthatbelongtoeachchild.
Keesha’scookies:
Andi’scookies:
Tim’scookies:
Myosia’scookie:
Carlos’cookie: Asyousawintheclassactivity,inordertofindthemeannumberofcookies,thechildrencouldimaginesharingcookies(orevenpartsofcookies)sothattheyeachchildhasexactlythesamenumber.Bytheway,isthisdatadisplayliketheoneinproblem1)ortheoneinproblem2)fromtheClassActivity?Explain.
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Keesha’scookies:
Andi’scookies:
Tim’scookies:
Myosia’scookies:
Carlos’cookies: Sothechildrenseethatthemeannumberofcookiesis3.Inthiscasewedidn’tneedtobreakupcookiesinordertosharethem,butyoucertainlyshoulddoexampleslikethatwithyourstudents.R E D I S T R I B U T I O N C O N C E P T O F T H E M E A N
Whatwehaveillustratedhereistheredistributionconceptofthemean:thatthemeanvalueofasetofnumbersiswhateachobservationwouldgetifthedatawasredistributedtothateachobservationhadthesameamount.Thisconceptisessentiallythesameasthepartitiveconceptofdivision:takingatotalamountandsharingitequallyintoanumberofgroups,andseeinghowmucheachgroupgets.B A L A N C E C O N C E P T O F T H E M E A N
Inclassactivity12b,wealsodiscoveredthebalancepointconceptofthemean,namely,thatthemeanisthepointinadistributionwherethesumofthedistancesthatthedataisabovethemeanbalanceswiththesumofthedistancesthatthedataisbelowthemean.Themodeofasetofdataisthevaluethatoccursmostfrequently.Whatisthemodeforthecookiedata?Sometimesadatasetwillhavemanymodesbecauselotsofvalueswilloccurwiththesamemaximumfrequency.M E D I A N A N D M O D E
Theideaofmedianistheideaofmiddleorcentervalue.Sowesawthatthenumberofcookiesheldbythefivepeopleis4,3,7,1,and1.Ifweputthesevaluesinnumericalorder,weget: 1 1 3 4 8
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Whydoweneedtoputtheminnumericalorderfirst?Howwouldyouexplainthistoachild?Weseethatthemiddlevalueis3(we’llwriteM=3.)Inthecaseofanevennumberofvalues,wehavenoonemiddlevalue,andsothemedianistheaverageofthetwomiddlevalues.Findthemean,mode,andmedianofmybowlingscores:
112 114 120 123 127 127 130 167 220Whydoesitmakesensethatthemeanislargerthanthemedianforthesedata?Explain.F I V E N U M B E R S U M M A R Y
WewilldefineQ1(thefirstquartile)asthemedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorder.Forexampleusingmybowlingscores,127isthemedian.
112 114 120 123|127| 127 130 167 220Sothescores112,114,120and123arepositionedbeforethemedian.Themedianofthosescoresis(114+120)/2=117.SoQ1=117.Q3(thethirdquartile)isthemedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder:127,130,167and220.Q3=148.5.Nowwecanreportsomethingwecallafive-numbersummaryforthesescores:Minimum=112, Q1=117, M=127, Q3=148.5, andMaximum=220Bythewaychildrenmaywanttoknowifduplicatesinthedatacanbethrownout.Whatwouldyousaytothem?Why?Whataboutvaluesofzero–couldtheybediscarded?Whatwouldyousaytothechildrenaboutthat?
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Weusedasmalldatasetforthepurposeofillustratingtheseideas.Wenotethatforlargedatasets,thesestatistics(mean,median,mode,quartiles,andpercentiles)aretypicallycomputedusingacalculatororcomputer.We’llgiveyoutheopportunitytousetechnology-ifyouhaveit-andworkwithalargerdatasetinalaterclassactivity.Asateacher,youwillneedtodiscusswithparentsthemeaningoftheirchild’sscoreonastandardizedexam,solet’sspendafewminutesontheideasthatyoumightfindusefulinthatcontext.Typicallythesescoresarereportedaspercentiles.Ifachildinyourclassscoresinthenthpercentilethismeansthat“npercent”ofthechildrenwhotookthetestscoredatorbelowthatscore.Forexample,ifachildscoresinthe80thpercentileonatest,thatmeansthat80%ofthechildrenwhotookthetestscoredatorbelowherscore.Anotherwaytothinkaboutthisistoimagineliningup100childrenaccordingtotheirtestscores,thischildwouldbenumber80fromthebottomintheline(ornumber20fromthetop).Inthiscontextthemedianisthe50thpercentile.WhatpercentileisQ3?P E R C E N T I L E S
ConnectionstoTeaching
Ingrades3-5allstudentsshouldusemeasuresofcenter,focusingonthemedian,andunderstandwhateachdoesanddoesnotindicateaboutthedataset.
NationalCouncilofTeachersofMathematics
PrinciplesandStandardsforSchoolMathematics,p.176Childreningrades3-5caneasilyunderstandandcomputemedianandmode,buttypicallytheideaofmeanisamoredifficulttopic.Thisisnotbecauseitishardtocalculateamean;ratheritisbecauseitismoredifficulttounderstandtheideaofamean.Belowisanexampleofanactivityappropriateforchildreningrade4thatrepresentsanopportunityfortheteachertodiscussmeanasaredistributionprocessratherthanasacomputation.Explainhowyoucouldusestickynotestohavechildrenuseevening-offtofindthemeanofthebelowdatawithouteverdoingthestandardcomputation.Reallydoit.
Thefamilysizesforchildreninourclassareshownbelow.Makeabargraphforthesedataandthenfindthemeannumberoffamilymembers.
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Family sizeJenson 8Lewis 3McDuff 7EarlesSeaman
45
Ortiz 4PetersToddMiene
653
Weknowthatthestandardcomputation(addthevalues,thendividebythenumberofvalues)isquickerand‘easier’todo.Butasanelementarymathematicsteacher,yourjobistohelpyourchildrenunderstandbigideas–andnottogivethemshortcutsthatallowthemtogetrightanswerswithoutunderstandingtheideas.Childrenmustlearnthatmathematicsmakessense,andthatthinking(insteadofmemorizing)isthewaywedomathematics. Infifthgrade,studentsarefirstexposedtotheideaofthemean.Whatconceptofthemeanisusedhere?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Homework
Wecannotteachpeopleanything;wecanonlyhelpthemdiscoveritwithinthemselves.
GalileoGalilei
1) DoalltheitalicizedthingsintheReadandStudysection.2) DoalltheitalicizedthingsintheConnectionssection.
CCSSGrade5:Representandinterpretdata.1. Makealineplottodisplayadatasetofmeasurementsinfractionsofaunit(1/2,
1/4,1/8).Useoperationsonfractionsforthisgradetosolveproblemsinvolvinginformationpresentedinlineplots.Forexample,givendifferentmeasurementsofliquidinidenticalbeakers,findtheamountofliquideachbeakerwouldcontainifthetotalamountinallthebeakerswereredistributedequally.
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3) Lookatthe“FulcrumInvestigation”inUnit8,Module1,Session3ofthe4thgradeBridgesinMathematicscurriculum.Inthisactivity,studentsaregivenapencil,a12-inchrulerandseveral“tiles”(weightsorpennies)allofthesameweight.Byplacing6tileson“0”ontheruler,theycansimulatea60-poundfourthgradestudent(Sarah)sittingontheendofaseesaw.Anotherweightisplacedontheotherendoftheseesaw(at12inches).Ifthefulcrumisinthemiddleofthepencil(at6”),thenitwouldtake6tiles(representing60pounds)attheotherendto“lift”orbalanceSara.Usinganactualpencil,ruler,andweights(suchaspennies),predictthenexperimenthowmuchweightittakestobalanceSarahifthefulcrumisplacedatthe4-inchmark,the5-inchmark,the7-inchmark,andthe8-inchmark.
a. Answerquestions1-6inthe“FulcrumInvestigation”.b. HowcanyouusetheconceptoftheMeantoanswerquestions5and6inthis
investigation?c. A“ChallengeQuestion”inthenextsessionasks“IfSarahcan’tmovethefulcrum
fromthemiddleoftheseesaw,whatcanshedotobalancewiththe40-poundfirstgrader?Whydoesthismethodwork?”Answerthisquestion.Howdoyouthinkfourthgraderswouldbeabletothisout?Howcanyouusetheconceptofthemeantofigurethisout?
d. Repeatpartc,butnowsupposeshewastobalancewiththe100-pound7thgraderwithoutmovingthefulcrum.
4) Hereisadatasetofthenumberofyearsmembersofthemathdepartmenthavebeen
employedatmyschool:7,25,7,15,8,23,27,6,1,3,28,22,24,9,15,14,18,8.Computethemean,medianandmodeforthesedata.Whatdothesedatatellyouaboutthegroupofmathematicsfacultyatthisschool?
5) DecidewhethereachofthefollowingisTrueorFalse.Ineachcase,arguethatyouare
right.
a) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemean.
b) T F Iftenpeopletookanexamthenitispossiblethatallbut oneofthemscoredlessthanthemedian.c) T F Iftenpeopletooka100-pointexam,itispossiblethatthe meanandmedianare90pointsapart.d) T F Iftenpeopletookathousand-pointexam,itispossiblethe themeanandthemedianare90pointsapart.
e) T F Themeanincomeofyourtownislargerthanthemedian income.
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6) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyourschool
andnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.
7) Supposethattenpeoplegraduatedwithadegreeincomputersciencefromyourschool
andnowtheirmedianannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Isitmoreconvincingorlessconvincingthanthescenariointhepreviousproblem?Explain.
8) Supposethatonehundredpeoplegraduatedwithadegreeincomputersciencefrom
yourschoolandnowtheirmeanannualincomeis$75,000.Ifmoneyisyouronlyconcern,doesthisconvinceyoutomajorincomputerscience?Explain.Isthismoreorlessconvincingthanthescenariosintheprevioustwoproblems?Explain.
9) Annneedsameanof80onherfiveexamsinordertoearnaBinherclass.Herexam
scoressofarare78,90,64and83.WhatdoessheneedtogetonherfifthexamtoearntheB?Dothisprobleminatleasttwodifferentways.Onewayshouldusetheredistributionconceptofthemean.Anothershouldusethebalancepointconceptofthemean.
10) Awaytothinkaboutthemedianofasetofnumbersisthatitisthevaluethatsplitsthe
distributionintotwoequal‘counts.’Considerthisfrequencygraphof18scoresonaquiz:
X X
X X X X X X X
X X X X X X X X X 1 2 3 4 5 6 7 8 9 10
ScoreonaQuiz(outoftenpossiblepoints)
a) Findthemedianscoreandexplain,asyouwouldtoastudent,whyitmakessense.b) Wesaythatthisdistributionisskewedleftbecauseithasanasymmetrictail(or
evenoutliers)totheleft.Inskewedleftdata,doyouexpectthemeantobehigherorlowerthanthemedian?Explainyourthinking.
c) Nowimaginethatthedistributionaboveshows18equalweightsarrangedonaseesaw.Ifyouwantedtheseesawtobalance,youwouldneedtoplacethefulcrumatthemean.Inotherwords,youcanthinkaboutmeanasthebalancingpointofthedistribution.Findthemeanandseeifitlookslikethebalancingpoint.Whydoesitmakesensethatthemeanissmallerthanthemedianforthesedata?
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11) Achildinyourfourth-gradeclassscoreda200outof500onastandardizedmathematicsassessmentinwhichtheaveragescorewas240.Thisstudentisreportedasscoringinthe35thpercentile.Youneedtoexplaintotheparentsexactlywhatthismeans.Whatdoyoutellthem?Areyouconcernedaboutthischild’sperformance?Explainyourthinking.
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ClassActivity29: MeasuringtheSpreadIt’snotthatGooddoesn’ttriumphoverEvil,it’sthatthepointspreadistoosmall.
BobThavesHerearetheexamscoresfortwostudents,AndreasandBelle:Andreas:45,57,69,69,80,94 Belle: 54,70,71,72,72,77Whichofthesestudentshadbetterscores?Explainhowyoudecidedbeforeyoureadfurther.Noticethatthesestudents’scoresareverysimilarbasedonthemeasuresofcenter(meanandmedian)–buttheyarenotsosimilarbasedonthespreadoftheirscores.Spreadisanideathatiscapturedbydifferentstatistics.Herearethreecommonmeasuresofspread:
1) Range=maximum-minimum.Computetherangeforeachstudent’sscores.Whatisitthatrangemeasures?Beasspecificasyoucan.
2) Inter-QuartileRange(IQR)=Q3–Q1.ComputetheIQRofeachstudent’sscores.What
isitthattheIQRmeasures?Beasspecificasyoucan.
3) MeanAbsoluteDeviation(MAD)=!-
𝑥! − 𝑥 + 𝑥" − 𝑥 + 𝑥% − 𝑥 +⋯+ 𝑥- − 𝑥 Inthisformulanisthenumberofvalues,𝑥isthemean,andeach“x”isaspecificvalue.Recallthat,forexample|-3|means“theabsolutevalueof-3.”ComputethemeanabsolutedeviationofAndreas’scoresandthenBelle’sasagroup.Whatisitthatthemeanabsolutedeviationmeasures?Beasspecificasyoucan.
(Thisactivitycontinuesonthenextpage.)
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Nowwehavethemachineryweneedtoidentifyanoutlierinthedata.First,adefinition:Anoutlierisaspecificobservationthatlieswelloutsidetheoverallpatternofthedata.Youmightask,whatdoesitmeantobe“welloutside”theoverallpatternofthedata?Gladyouasked.Wehaveacriterionforthat.Itiscalledthe1.5×IQRcriterion:avalueisconsideredanoutlierifitfallsmorethan1.5×IQRaboveQ3orifitfallsmorethan1.5×IQRbelowQ1.Let’susethiscriteriontotestAndreas’scoresforoutliers:45,57,69|69,80,94M=(69+69)/2=69andsitsinthepositionheldbythebarintheabovedataset,sothescores45,57and69arepositionedbelowthemedian(M).Q1=57andlikewise,69,80and94arepositionedabovethemediansoQ3=80.
IQR=Q3-Q1=80–57=23 1.5×IQR=34.5 Soavalueisanoutlierifitismorethan34.5pointsaboveQ3ormorethan34.5pointsbelowQ1.Inotherwords,outliersinAndreas’dataarebiggerthan80+34.5=114.5orsmallerthan57–34.5=22.5.Sincetherearenoscoresthatbigorthatsmallinhisdata,hehasnooutliersbasedonthecriterion.CheckBelle’sscoresforoutliersusingourcriterion.HowcanitbethatBellehasanoutlieramongherscoreswhenAndreasdoesnot?Explain.
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ReadandStudy
Americaisanoutlierintheworldofdemocracieswhenitcomestothestructureandconductofelections.
ThomasMann
IntheClassActivityyouworkedwiththreedifferentstatisticsthatdescribethespreadofthedata.Therangeshowsyoutheentirespreadofthedataset.TheIQRisameasureofthespreadofthemiddlehalfofthedata.Themeanabsolutedeviationmeasureshowspreadoutthedatais(onaverage)arounditsmean.Wewillusetwoprimarystatisticsfordeterminingthecenterofadistributionofdata:meanandmedian,andtwoprimarymeasuresofspreadofthedata:meanabsolutedeviationandinterquartilerange(IQR).So,howdotheyfittogetherandwhywouldwechoosetoreportoneoveranother?Ourfirstconsiderationisthis:doesthedatacontainoutliers?See,weknowthatanoutliercanhaveabigeffectonthemean(butnotonthemedian).Forexample,considerthesetestscoreswithmean75.44andmedian78:
67,68,68,71,78,79,80,83,85Beforeyougoanyfarther,checkthesescoresforoutliersusingthe1.5×IQRcriterion.R A N G E , I Q R , A N D M E A N A B S O L U T E D E V I A T I O N
A C R I T E R I O N F O R F I N D I N G S U S P E C T E D O U T L I E R S
Now,supposethatthescoreof67wasreplacedbyascoreof12.(Noticethat’sanoutlier.)Thischangehasnoeffectonthemedian(whichisstill78)butitdoesaffectthevalueofthemean.Computethenewmeantocheck.Wesaythatthemedianis“resistanttooutliers”andthatthemeanis“affectedbyoutliers.”Ifyourdatacontainsanoutlier(orseveraloutliers)thataffectthemean,thenthemedianisoftenthebetter(morehonest?)measureofcenterforyoutoreportforthosedata.Whichofthemeasuresofspreadisresistanttooutliersandwhichisaffectedbythem?Takeafewminutestofigureitout.
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Nowwearegoingtointroduceadisplayofthefive-number-summaryofadataset:theboxplot.Recallthatthefive-number-summaryconsistsoftheminimum,Q1,themedian,Q3,andthemaximumvaluewhenthedataisarrangedinnumericorder.Hereissomefictitiousdataonthenumberofyearsmathematicsfacultymembershavebeeninourdepartment.Takeaminutetofindthemedianandthequartilesforthesedata. 1,3,6,7,7,8,8,9,14,15,15,18,22,23,24,25,27,28B O X P L O T S
Hereisanexampleofaboxplotdisplayofthedataonnumberofyearsmathematicsfacultymembershavebeeninourdepartment.
NumberofYearsofService
Noticethataboxplotisnothingmorethanapictureofthefive-numbersummary.The“box”partshowsQ1,M,andQ3,andthe“arms”stretchoutthereachtheminandmaxoneitherside.Whenmakingaboxplot,besurethatyournumberlinehasaconsistentscale,thatyoulabelyourdisplay,andthatyouprovideinformationaboutthenumberofvalues(inthiscaseN=18)shownontheplot.
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ConnectionstoTeaching
Realistsdonotfeartheresultsoftheirstudy.FyodorDostoevsky
The Common Core expects students to analyze and interpret data. Read the standards below. Which questions or activities have you done so far address each of these standards?
©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
Homework
Ohwhocantelltherangeofjoyorsettheboundsofbeauty? SaraTeasdale
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Answerallofthequestionsinthe“Mean,Mode&Range”probleminUnit4,Module4,
Session2intheBridgesinMathematicsGrade4StudentBook.Then,asafutureteacher,considerthefollowingquestions:Whymightachildanswer17inchesonquestion6?Whyachildanswer36inches?Whymightachildanswer42inches?Whymightachildanswer52inches?
CCSSGrade6:Developunderstandingofstatisticalvariability.2.Understandthatasetofdatacollectedtoanswerastatisticalquestionhasadistributionwhichcanbedescribedbyitscenter,spread,andoverallshape.3.Recognizethatameasureofcenterforanumericaldatasetsummarizesallofitsvalueswithasinglenumber,whileameasureofvariationdescribeshowitsvaluesvarywithasinglenumber.
CCSSGrade6:Summarizeanddescribedistributions.
4.Displaynumericaldatainplotsonanumberline,includingdotplots,histograms,andboxplots.5.Summarizenumericaldatasetsinrelationtotheircontext,suchasby:
C.Givingquantitativemeasuresofcenter(medianand/ormean)andvariability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.D.Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.
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3) Consideragainthislinegraphof18scoresonaquiz:
X X
X X X X X X X
X X X X X X X X X 1 2 3 4 5 6 7 8 9 10
ScoreonaQuiz(outoften)
a) Whatistherangeofthesedata?b) WhatistheIQR?c) TrueorFalse?(Dothiswithoutperformingacalculation.)Themeanabsolute
deviationofthesedataisabout5.Explainyourthinking.d) Areeitherofthescores1or2anoutlier?Useourcriteriontocheck.e) Whatmeasureofcenterandwhatmeasureofspreadwouldyoureportforthese
dataandwhy?f) Makeaboxplotofthesedata.g) Howcouldyouusetheboxplottoquicklyspotoutliersinthedata?Explain.
4) Herearethosebowlingscoresonceagain:
112 114 120 123 127 127 130 167 220
a) Which(ifany)ofthesescoresareoutliersbasedonourcriterion?b) NowgototheintroductiontothistextandreadtheCommonCoreState
StandardsforGrade6.Noticethattheyaskthatchildren“summarizenumericaldatasetsinrelationtotheircontext.”Dothethingstheysuggestforthesetofbowlingscores.
i. Reportingthenumberofobservations.ii. Describingthenatureoftheattributeunderinvestigation,includinghow
itwasmeasuredanditsunitsofmeasurement.iii. Givingquantitativemeasuresofcenter(medianand/ormean)and
variability(interquartilerangeand/ormeanabsolutedeviation),aswellasdescribinganyoverallpatternandanystrikingdeviationsfromtheoverallpatternwithreferencetothecontextinwhichthedataweregathered.
iv. Relatingthechoiceofmeasuresofcenterandvariabilitytotheshapeofthedatadistributionandthecontextinwhichthedataweregathered.”
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5) EditorsofEntertainmentWeeklyrankedeverysingleepisodeevermadeofStarTrek:TheNextGenerationfrombest(ranked#1)toworst(ranked#178).Thentheycompiledtherankingsbasedontheseasoninwhichtheepisodeaired.Belowareboxplotsshowingtherankingsofepisodesineachofthesevenseasons(fromWorkshopStatistics,p.89):
StarTrekRankingsplottedbySeason
a) Whichwasthebestoverallseasonandwhichwastheworstbasedontheserankings?Makeanargumentineachcase.
b) Doanyoftheseseasonshaverankingsthatareoutliers(forthatseason)?Explain.
c) Whichseason(s)hasadistributionofrankingsthatisskewedleft?Explain.d) Createaseriesofgoodquestionsyoucouldaskupperelementarystudents
abouttheseboxplots;thenanswerthemyourself.
6) Thisisameanabsolutedeviationcontest.Youmayuseonlynumbersintheset{1,2,3,4,5}(Youcanusethesamenumberasmanytimesasyoulike).
a) Selectfournumberswiththelowestmeanabsolutedeviation.(Actuallycompute
it).Arethosetheonlyfournumbersthatgivethelowestmeanabsolutedeviation?
b) Selectfournumberswiththehighestmeanabsolutedeviation.(Actuallycomputeit).Arethosetheonlyfournumbersthatgivethehighestmeanabsolutedeviation?
7) Whichislikelybigger?
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a) Themeannew-housecostinSeattle,WAorthemediannew-housecost?Explain.
b) Themeanorthemedianinaskewed-leftdistribution?Explain.c) Therangeofasetofnumbersorthemeanabsolutedeviationoftheset?
Explain.d) Thesecondquartileorthemedian?Explain.e) Now,giveanexampleofasetof8numberswhosemeanis1000biggerthanits
medianorexplainwhyitisimpossibletodoso.
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ClassActivity30: TheMatchingGame
Letthepunishmentmatchtheoffense. Cicero
Onthefollowingpageyouwillfindbargraphsdisplayingdistributionsforeightdifferentsetsofexamscores.Thenumericalsummariesforthosedatasetsarelistedbelow.
DataSet Mean Median Mean
AbsoluteDeviation
A
44.5 44.0 7.7
B
37.0 36.0 4.0
C
32.5 31.5 12.0
D
40.0 41.5 25.5
E
11.5 3.0 20.0
F
40.0 38.5 11.0
G
33.5 34.5 10.0
H
20.5 20.5 10.0
1) Thesamenumberofstudentstookeachexam.Howmanywasthat?Explainhowyouknow.
2) Matcheachsummarywithadistribution.Makesuretowriteexplanationsforyourchoices.
3) Oneofthebargraphsismissing.Figureoutwhichdatasethasthemissingbargraph,thencreateapossiblebargraphforthatdataset.
4) Whatarethingsthatanupperelementarystudentcouldlearnbyworkingonthisactivity?Bespecific.
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ReadandStudy
Anapprenticecarpentermaywantonlyahammerandsaw,butamastercraftsmanemploysmanyprecisiontools.
RobertL.KruseInthissectionwearegoingtotalkspecificallyaboutsomewaystodisplaydataandaboutfeaturesofdatadistributions.First,adistributionofdataissimplyadisplaythatshowsthevaluesofthevaluesandtheirfrequencies(orrelativefrequencies).Oftenwhenwerefertoadistribution,wemeanthepatternofthedatawhenitismadeintoabargraphoralinegraphshowingvaluesofthedatasetonthehorizontalaxis,andeitherfrequencyofvaluesorrelativefrequency(percentsofvalues)ontheverticalaxis.D I S T R I B U T I O N S , S K E W N E S S , A N D S Y M M E T R Y
AllthreeofthedistributionsyouareabouttoseewerefoundattheQuantitativeEnvironmentalLearningProjectwebsite(http://www.seattlecentral.edu/qelp/Data_MathTopics.html).ForexamplethebargraphbelowshowsthedistributionofozoneconcentrationinSanDiegointhesummerof1998asmeasuredbytheCaliforniaAirResourcesBoard.TheEPAhasestablishedthestandardthatozonelevelsover0.12partspermillimeter(ppm)areunsafe.
Approximatelyhowmanydaysaredisplayedalltogether?InhowmanydayswastheozonelevelinSanDiegoclassifiedasunsafebytheEPA?Istheozonedistributionskewedrightorskewedleftorneither?Explain.
0
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Nowtakeaminutetomakesenseofthereservoirdatabelow.
Abouthowmanyreservoirsareshownonthisbargraph?Whatpercentofreservoirsarefilledtoatleast90%capacity?Doyouexpectthemeanofthereservoirdatatobehigherthanthemedianortheotherwayaround?Explain.ThegraphsincludesreservoirsinbothOregonandArizona.WheredoyouthinkArizona’sreservoirsarelikelyappearingonthebargraph?Why?
0510152025303540
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
>100
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Finally,belowwehaveonemoregraphshowingafairlysymmetricdistributionofthenumberofhurricanesoccurringineachtwo-weekperiodbeginningJune1.
Onwhichdatesarehurricanesmostlikelybasedonthesedata?Inabargraphtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxiscouldrepresentvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights).Ifyouhavecollecteddataonthenumberofseatswonbyvariouspartiesinparliamentaryelections,youcouldmakeabargraphofthatdatabylistingpartiesalongthehorizontalaxisandmakingthebar-heightsrepresenteitherfrequencyorrelativefrequencyofyoursamples’responses.Thisbargraph(datafromwikipedia.org)showsboththeresultsofa2004election,andthoseof1999election.Takeamomenttomakesenseofthesedata.
020406080100120140160
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ConnectionstoTeaching
Instructionalprogramsfromprekindergartenthroughgrade12shouldenablestudentstoformulatequestionsthatcanbeaddressedwithdataandcollect,organize,anddisplayrelevantdatatoanswerthem.
NationalCouncilofTeachersofMathematicsPrinciplesandStandardsforSchoolMathematics,p.176
Childreninelementaryschoollearntomakeandinterpretpictograms,frequencyplots(sometimescalledlinegraphs),bargraphs,andpiecharts.Onordertohelpyoudistinguishamongthesetypesofdisplays,let’sbeginwithanexampleofeach.ThedatabelowshowhowchildreninMr.Jensen’sclasstraveltoschool.
Student Transportation Student Transportation Student TransportationAbby Bus Sasha Foot Xia FootWinton Bicycle Lucy Bus Kim FootDexter Car Joe Car Ben CarTrevor Car Aiden Bus Leah CarAudrey Car Justin Foot Jeffery FootMavis Foot Campbell Car Queztal CarKate Foot Sonja Car Cheyenne BicycleLim Bicycle Jen Bus Steve Bicycle
First,hereisapictogramofthedata. TransportationPictogramforMr.Jensen’sClass
Bus
Bike
Car
Foot
=1childonfoot =1childdriventoschool =1biker =1busrider
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Apictogramrepresentseachvalueasameaningfulicon.Thedisplaycanbeorientedeitherverticallyorhorizontally(ascanlineplotsandbargraphs).Noticehowthegraphiscarefullylabeledandthatakeyisprovidedfortheicons.Makesuretohelpchildrengetinthehabitoflabelinganddefiningtheirsymbols.Alsostressthatitisimportantthatalltheiconsbethesamesizeorthegraphcouldbemisleading.Apictogramcanbeunderstoodbychildrenasyoungasthoseinkindergarten.Afrequencyplot(sometimescalledalineplotifthehorizontalaxisisanumberline)ofthesamedataisshownbelow.Notethatthisplotisaprecursortothebargraphinthesensethatitdisplaysfrequencyofresponseinamoreabstractmannerthandoesthepictogram. TransportationFrequencyPlotforMr.Jensen’sClass X X X X X X X X X X X X X X X X X X X X X X X X Bus Bike Car FootCreateabargraphforthedatafromMr.Jensen’sclass.CreateapiegraphforthedatafromMr.Jensen’sclass.Explainhowyoudecidedhowbigtomakeeachsector.Itisimportantnotonlythatchildrenlearntodisplaydatabutalsothattheylearntomakesenseofchartsandgraphstheyfindinbooksandothermedia,andtobecriticalofmisleadingdisplays.Considertheexamplesbelow:
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Amyclaimsthatthechildreninherkindergartenclasslikecatsmorethandogs.Shehasmadethefollowinggraphofherclassmates’petpreferencestosupportherclaim.Whyisthispictogrammisleading?Explain.Whatwouldyousaytothischild?
Childrenwholikecats
Childrenwholikedogs TuffTruckCompanyclaimsthattheirtrucksarethemostreliableandshowsthebelowgraphtomaketheircase.Whyisthisbargraphmisleading?Explain.(Bytheway,thisisareal-lifeexample.Onlythenameshavebeenchanged.)
Infact,criticalanalysisofdatadisplaysispartofthecurriculumforchildreninupperelementaryschool.
Perc entoftruc ks s tillontheroad,fiveyears afterpurc has e
93%
94%
95%
96%
97%
98%
99%
TuffTruck B randB B randC
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Homework
Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butalackofwill. VinceLombardi
1) DoalloftheitalicizedthingsintheReadandStudysection.2) DoalltheproblemsintheConnectionssection.Whataretheexactpercentagesforthe
piechart?Howcanyoucomputetheanglestothenearestdegree?Doitandthenexplainyourwork.
3) HereisatableofdatashowingthepetsownedbychildreninMs.Green’sthirdgrade
class.
Student PetsintheHouseAnna NoneJames 1catand1dogCleo 3dogsViolet 1catAndre 6fishJose 7fish,1dog,2catsAidos NoneYvonne NoneTyrise 1hamsterEllen 2dogsTomas 1dogYani 3fishand2catsOwen NoneAudrey 1cat
a) Makeafrequencyplotofthesedatawithchildren’snamesonthehorizontalaxis.b) Makealineplotwith‘numberofpets’onthehorizontalaxis.c) Makeabargraphofthesedatawith‘numberofpets’onthehorizontalaxis.d) Makeapiechartofthesedata.e) Whatproblemsmightchildrenencounterintryingtorepresentthesedatainthe
aboveways?4) Answerthequestionsinthe“FavoriteBooks”probleminUnit2Module4Session1in
theGrade3BridgesinMathematicscurriculum.Howcanyouusethisproblemtodiscussthesimilaritiesanddifferencesbetweenpicturegraphsandbargraphs?
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5) Answerthequestionsinthe“GiftWrapFundraiser”probleminUnit2Module4Session2intheGrade3BridgesinMathematicscurriculum.Here’ssomeadditionalquestionstothinkaboutasafutureteacher:
a) Explainwhyachildmightgivetheanswerof“5”toquestion2abouthowmanystudentssold7rollsofgiftwrap.
b) Explainwhyachildmightgivetheanswerof“7”toquestion4abouthowmanyrollsofgiftwrapSarahsold.
c) Whyistheanswertoquestion5NOTthetotalnumberofXsonthelineplot?d) MakeafrequencyplotforthesamedatawhereeachXrepresentsarollofgift
wrap.Thenexplaintousethegraphtoanswerquestions1-5.
6) ThisgraphshowslengthsofthewingsofhousefliesfromtheQuantitativeEnvironmentalLearningProject.(OriginaldatafromSokal,R.R.andP.E.Hunter.1955.AmorphometricanalysisofDDT-resistantandnon-resistanthouseflystrainsAnn.Entomol.Soc.Amer.48:499-507)
a) Howwouldyoudescribetheshapeofthisdistributionintermsofskewnessand
symmetry?b) Approximatelyhowmanyfliesweremeasuredforthisstudy?Explain.c) TrueorFalse?Themeanofthesedataisequaltothemedian.Explain.d) TrueorFalse?About90%offlieshavewingssmallerthan51.5(×0.1mm).
Explain.e) Makeupagoodquestiontoaskanelementaryschoolchildaboutthisdataset.
7) Hereisagraphshowing100yearsofdataonWorldwideEarthquakesbiggerthan7.0on
theRichterScalefromtheQuantitativeEnvironmentalLearningProject(datafromtheUSGeologicalSurvey)Soforexample,therewere4yearswhenthenumberofglobalquakes(above7.0)wasabout35.
02468101214161820
37.5 39.5 41.5 43.5 45.5 47.5 49.5 51.5 53.5 55.5
Freq
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Length(x.1mm)
HouseflyWingLengths
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a) Whatdoesthisgraphtellyou?Bespecific.b) Estimatethemedianannualnumberofquakesabove7.0.c) Howwouldyoudescribethisshapeofthedistributionintermsofskewnessor
symmetry?Whatdoesthatshapetellyouaboutearthquakes?
8) Considerthisgraphmakingthecasethatwearehavinganational‘crimewave!’Whatdoyouthink?Doesthegraphprovidecompellingevidence?Explain.
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S ource: S tatis ticalAbs tractoftheUnitedS tates ,2008
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9) Astemandleafplot(orsimplystemplot)isalistingofallthedatatypicallyarrangedsothetensplacemakesthestemandtheonesarethe‘leaves.’Forexample,herearescoresonaGeometryFinalExamarrangedasastemplot:
ExamsScoresinGeometry
2|03|2,74|5|6,8,9,96|2,4,5,5,7,87|0,0,2,4,5,6,78|0,1,1,2,4,49|2,4,510|0
Wewouldreadthescoresas20,32,37,56,58,59,59,62etc.
a) Describetheshapeofthedistributionofscoresintermsofskewnessandsymmetry.b) Whatisthemedianscore?c) Makeaboxplotofthesedata.d) Givesomeexamplesofdatasetsforwhichastemplotwouldbeimpractical.
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SummaryofBigIdeas
Hey!What’sthebigidea?BugsBunny
• Random sampling helps reduce sample bias.
• There are two ways we can conceptualize the mean – as a balancing point and as a
redistribution.
• The mean, median, and mode are statistics that describe the center of data.
• A five-number summary, the IQR, and the mean absolute deviation all describe the spread of a data set.
• There are many different representations for data displays such as pictograms, line plots, and pie charts. You should choose the chart that best displays the data without misrepresenting the data.
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References
Givecreditwherecreditisdue. Authorunknown
• Bauersfeld,H.(1995).Thestructuringofstructures:Developmentandfunctionof
mathematizingasasocialpractice.InL.Steffe&J.Gale(Eds.),ConstructivisminEducation(pp.137-158).Hillsdale,NJ:Erlbaum.
• BridgesinMathematicsusedthroughoutwithpermissionfromtheMathLearningCenter,http://www.mathlearningcenter.org/
• CommonCoreStateStandardsformathematicsasfoundin2012at©Copyright2010NationalGovernorsAssociationCenterforBestPracticesandCouncilofChiefStateSchoolOfficers.Allrightsreserved.
• Dawson,L.(2004).TheSalkpoliovaccinetrialof1954:risks,randomizationandpublicinvolvement,ClinicalTrials,1,pp.122-130.
• EducationalDevelopmentCenter.(2007)ThinkMath!Orlando,FL.HarcourtSchool
Publishers.
• Halmos,P.(1985).IWanttobeaMathematician,Washington:MAASpectrum,1985.
• Gottfried,W.L.,NewEssaysConcerningHumanUnderstanding,IV,XII.
• Ma,L.(1999).KnowingandTeachingElementaryMathematics:Teachers’UnderstandingofFundamentalMathematicsinChinaandtheUnitedStates.Mahwah,N.J.:LawrenceErlbaum.
• MathematicalQuotationsServer(MQS)atmath.furman.edu.
• NationalCouncilofTeachersofMathematics.(2000).PrinciplesandStandardsfor
SchoolMathematics.Reston,VA:NCTM.
• Shulman,L.S.(1985).Onteachingproblemsolvingandsolvingtheproblemsofteaching.InE.A.Silver(Ed.),TeachingandLearningMathematicalProblemSolving:multipleresearchperspectives(pp.439-450).Hillsdale,NJ:Erlbaum.
• Steen,L.A.andD.J.Albers(eds.).(1981).TeachingTeachers,TeachingStudents,Boston:
Birkhïser.
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• Theearlydevelopmentofmathematicalprobability.p.1293-1302ofCompanionEncyclopediaoftheHistoryandPhilosophyoftheMathematicalSciences,editedbyI.Grattan-Guinness.Routledge,London,1993.
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APPENDICES
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Euclid’sPostulatesandPropositions
Euclid'sElementsThispresentationofElementsistheworkofJ.T.Poole,
DepartmentofMathematics,FurmanUniversity,Greenville,SC.©2002J.T.Poole.Allrightsreserved.
BookI
POSTULATES
Letthefollowingbepostulated:1.Todrawastraightlinefromanypointtoanypoint.2.Toproduceafinitestraightlinecontinuouslyinastraightline.3.Todescribeacirclewithanycenteranddistance.4.Thatallrightanglesareequaltooneanother.5.That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.
COMMONNOTIONS1.Thingswhichareequaltothesamethingarealsoequaltooneanother.2.Ifequalsbeaddedtoequals,thewholesareequal.3.Ifequalsbesubtractedfromequals,theremaindersareequal.4.Thingswhichcoincidewithoneanotherareequaltooneanother.5.Thewholeisgreaterthanthepart.
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BOOKIPROPOSITIONSProposition1.
Onagivenfinitestraightlinetoconstructanequilateraltriangle.Proposition2.
Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.Proposition3.
Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.
Proposition4.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhaveanglescontainedbytheequalstraightlinesequal,theywillalsohavethebaseequaltothebase,thetrianglewillbeequaltothetriangle,andtheremainingangleswillbeequaltotheremaininganglesrespectively,namelythosewhichtheequalsidessubtend.
Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.
Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.
Proposition7.Giventwostraightlinesconstructedonastraightline(fromitsextremities)andmeetinginapoint,therecannotbeconstructedonthesamestraightline(fromitsextremities),andonthesamesideofit,twootherstraightlinesmeetinginanotherpointandequaltotheformertworespectively,namelyeachtothatwhichhasthesameextremitywithit.
Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.
Proposition9.Tobisectagivenrectilinealangle.
Proposition10.Tobisectagivenfinitestraightline.
Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.
Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.
Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.
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Proposition14.Ifwithanystraightline,andatapointonit,twostraightlinesnotlyingonthesamesidemaketheadjacentanglesequaltotworightangles,thetwostraightlineswillbeinastraightlinewithoneanother.
Proposition15.Iftwostraightlinescutoneanother,theymaketheverticalanglesequaltooneanother.
Proposition16.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisgreaterthaneitheroftheinteriorandoppositeangles.
Proposition17.Inatriangletwoanglestakentogetherinanymannerarelessthantworightangles.
Proposition18.Inanytrianglethegreatersidesubtendsthegreaterangle.
Proposition19.Inanytrianglethegreaterangleissubtendedbythegreaterside.
Proposition20.Inanytriangletwosidestakentogetherinanymanneraregreaterthantheremainingone.
Proposition21.Ifononeofthesidesofatriangle,fromitsextremities,therebeconstructedtwostraightlinesmeetingwithinthetriangle,thestraightlinessoconstructedwillbelessthantheremainingtwosidesofthetriangle,butwillcontainagreaterangle.
Proposition22.Outofthreestraightlines,whichareequaltothreegivenstraightlines,toconstructatriangle:thusitisnecessarythattwoofthestraightlinestakentogetherinanymannershouldbegreaterthantheremainingone.[I.20]
Proposition23.Onagivenstraightlineandatapointonittoconstructarectilinealangleequaltoagivenrectilinealangle.
Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.
Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.
Proposition26.Iftwotriangleshavethetwoanglesequaltotwoanglesrespectively,andonesideequaltooneside,namely,eitherthesideadjoiningtheequalangles,ofthatsubtendingoneoftheequalangles,theywillalsohavetheremainingsidesequaltotheremainingsidesandtheremainingangletotheremainingangle.
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Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.
Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.
Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.
Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.
Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.
Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.
Proposition33.Thestraightlinesjoiningequalandparallelstraightlines(attheextremitieswhichare)inthesamedirections(respectively)arethemselvesalsoequalandparallel.
Proposition34.
Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.
Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.
Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.
Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.
Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.
Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.
Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.
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Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.
Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.
Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.
Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.
Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.
Proposition46.Onagivenstraightlinetodescribeasquare.
Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.
Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.
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Glossary"WhenIuseaword,"HumptyDumptysaid,inaratherscornfultone,"itmeansjust
whatIchooseittomean-neithermorenorless.""Thequestionis,"saidAlice,"whetheryoucanmakewordsmeansomanydifferent
things.""Thequestionis,"saidHumptyDumpty,"whichistobemaster-that'sall."
LewisCarroll,ThroughtheLookingGlass Acuteangle–ananglethatmeasureslessthan90degreesAcutetriangle–atrianglewiththreeacuteanglesAdjacentangles–twonon-overlappinganglesthatshareavertexandacommonrayAlternateexteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines
thatlieoutsidethelinesandonoppositesidesofthetransversalAlternateinteriorangles–twonon-adjacentanglesformedbyatransversalofapairoflines
thatliebetweenthelinesandonoppositesidesofthetransversalAngle–thefigureformedbytworayswithacommonendpointAnglebisector–thelinethroughthevertexofananglethatdividestheangleintotwo
congruentanglesApex(ofapyramid)–thecommonpointofthenon-basefacesofapyramidApex(ofacone)–thecommonpointofthelinesegmentsthatcreateaconeArc–thesetofpointsonacirclebetweentwogivenpointsofthecircle(Thereareactuallytwo
arcsbetweenanytwogivenpoints;theshorteroneiscalledtheminorarcandthelongeroneiscalledthemajorarc.)
Area–thequantityoftwo-dimensionalspaceenclosedbyaclosedplanefigureAttribute–apropertyofageometricobjectthatcanbemeasured(suchaslength)or
categorized(suchascolor)Axiom–astatementthatweagreetoacceptastruewithoutproofAxiomaticsystem–asetofundefinedterms,definitions,axioms,andtheoremsthatcreatea
mathematicalstructure
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Axis(ofacone)–thelinejoiningtheapextothecenterofthe(circle)baseAxisofsymmetry–alineinspacearoundwhichathree-dimensionalobjectisrotatedBargraph:adatadisplayinwhichtheverticalaxisrepresentsacount(frequency)orapercent(relativefrequency)andthehorizontalaxisrepresentsvaluesofacategoricalvariable(likecolors),valuesofanumericdiscretevariable(likeshoesize),orvaluesofacontinuousvariable(likeheights),groupedintointervals.Baseangles(ofanisoscelestriangle)–theanglesthatareoppositethecongruentsidesofan
isoscelestriangleBilateralsymmetry–anobjecthasbilateralsymmetrywhenithasexactlyonelineof
reflectionalsymmetryBisect–todivideageometricobjectsuchasalinesegmentoranangleintotwocongruent
piecesBoundary–thesetofpointsthatseparatetheinsideofaclosedplanarobjectfromtheoutsideCategoricalData:Dataforwhichitmakessensetoplaceanindividualobservationintooneofseveralgroups(orcategories).Census:Anydatacollectionmethodinwhichdataiscollectedfromeachandeverymemberofthepopulation.Center(ofacircle)–thepointthatisequidistantfromallpointsonthecircleCentralangle–ananglewhosevertexisacenterofageometricobjectChord–alinesegmentwhoseendpointsaredistinctpointsonagivencircleChanceError:Errorinsamplingduesimplytothefactthatasampleisnotexactlythesameasapopulationduetochancealone.Circle–thesetofallpointsintheplanethatarethesamedistancefromagivenpoint,called
thecenterCircumference–thecircumferenceofacircleisitperimeterCircumscribedcircle–thecirclethatcontainsalltheverticesofapolygonClosedcurve–acurvethatstartsandstopsatthesamepoint
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Clustersampling:Randomsamplinginstages.Coincide–twoobjectsaresaidtocoincideiftheycorrespondexactly(areidentical)Collinearpoints–pointsthatlieonthesamelineCompass–aninstrumentusedtoconstructacircleComplementaryangles–twoangleswhosemeasuressumto90degreesConcavepolygon–apolygonforwhichatleastonediagonalliesoutsidethepolygonConcurrentlines–threeormorelinesthatintersectinthesamepointCone(circular)-athree-dimensionalgeometricobjectconsistingofalllinesegmentsjoininga
singlepoint(calledtheapex)toeverypointofacircle(calledthebase)Congruentobjects–twogeometricobjectsthatcoincidewhensuperimposedConjecture–aguessorahypothesisContrapositive(of“IfA,thenB.”)–“IfnotB,thennotA,”whereAandBarestatementsConverse(of“IfA,thenB.”)–“IfB,thenA,”whereAandBarestatementsConvexpolygon–apolygonallofwhosediagonalslieinsidethepolygonCoordinateplane–aplaneonwhichpointsaredescribedbasedontheirhorizontalandvertical
distancesfromapointcalledtheoriginCoplanarlines–linesthatlieinthesameplaneCorrespondingangles-twoanglesformedbyatransversalofapairoflinesthatlieonthe
samesideofthetransversalandalsolieonthesamesideofthepairoflinesCorrespondingpoints–apairofpoints,oneofwhichistheoriginalpointandtheotherof
whichistheimageofthatpointunderatransformationCounterexample–anexamplethatdemonstratesthatastatement(conjecture)isfalseCurve–asetofpointsdrawnwithasinglecontinuousmotion
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Cylinder(circular)–athree-dimensionalgeometricobjectconsistingoftwoparallelandcongruentcircles(andtheirinteriors)andtheparallellinesegmentsthatjoincorrespondingpointsonthecircles
Data:AnyinformationcollectedfromasampleDecagon–apolygonwithexactlytensidesDeductivereasoning–theprocessofcomingtoaconclusionbasedonlogicDegree–aunitofanglemeasureforwhichafullturnaboutapointequals360degreesDiagonal–thelinesegmentjoiningtwonon-adjacentverticesofapolygonDiameter–alinesegmentthroughthecenterofacirclewhoseendpointslieonthecircleDilation(withcenterPandscalefactork>0)–amotionoftheplaneinwhichtheimageofPis
PandtheimageA’ofanyotherpointAisontherayPAsothatthedistancePA’isktimesthedistancePA.
Dimension(ofarealspace)–thenumberofmutuallyperpendiculardirectionsneededto
describethelocationofthesetofpointsinthatspaceDistance(onacoordinateplane)–thesizeoftheportionofastraightlinethatliesbetweenthe
twopointsonthecoordinateplaneasmeasuredbythedistanceformula:22 bad += ,whereaisthehorizontaldistancebetweenthepoints(asmeasuredon
thex-axis)andbistheverticaldistance(asmeasuredonthey-axis)Distinct(geometricobjects)-twoobjectsthatdonotsharealltheirpointsincommonDistributionofdata:adisplaythatshowsthevaluesthedatatakesalongwiththefrequency(orrelativefrequency)ofthosevaluesDisjoint(events):TwoeventsaredisjointiftheyhavenooutcomesincommonDodecagon–apolygonwithexactlytwelvesidesEdge–thelinesegment(side)thatissharedbytwofacesofapolyhedronEndpoint(ofalinesegment)–oneoftwopointsthatdeterminesalinesegmentEquiangularpolygon–apolygonallofwhosevertexanglesarecongruent
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Equallylikelyoutcomes:Alltheoutcomesofarandomexperimentaresaidtobeequallylikelyif,inthelongrun,theyalloccurwiththesamefrequency(theyallhavethesameprobabilityofoccurring)
Equilateralpolygon–apolygonallofwhosesidesarecongruentEvent:anysubsetofthesamplespaceofarandomexperiment Example(ofadefinition)–ageometricobjectthatsatisfiestheconditionsofthedefinitionExperimentalprobabilitiesareprobabilitiesthatarebasedondatacollectedbyconducting
experiments,playinggames,orresearchingstatisticsExteriorangle–theangleformedbyasideofapolygonandtheextensionofanadjacentsideFace–apolygon(withinterior)thatformsaportionofthetwo-dimensionalsurfaceofa
polyhedronFirstquartile(Q1):themedianofthedatapositionedstrictlybeforethemedianwhenthedataisplacedinnumericalorderFiveNumberSummary:ofnumericaldataconsistsofthefollowingfivestatistics:Minimumvalue,Q1,Median,Q3,MaximumvalueFixedpoint–apointwhoselocationremainsthesameunderatransformationGeneralization–theextensionofastatement(aboutapattern)thatistrueforspecificvalues
ofn(anaturalnumber)toastatement(aboutthatpattern)thatistrueforallvaluesofnGeoboard–amanipulativetypicallycomposedofaboardwith25pegsarrangedina5x5
squarearrayGeoboardpolygon–apolygonwhoseverticesareallpoints(pegs)onageoboardHeight(ofatriangle)–lengthofthelinesegmentfromavertexperpendiculartotheopposite
side.ThislinesegmentisoftencalledthealtitudeofthetriangleHeptagon–apolygonwithexactlysevensidesHexagon–apolygonwithexactlysixsidesHistogram:Adatadisplayinwhichtheverticalaxisrepresentsarateandthehorizontalaxisisactuallythex-axis-inotherwords,itisacontinuouspieceoftherealnumberlineandassuchitcontainsallrealnumbersinaninterval.Valuesofthevariablearebrokenintointerval
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‘classes.’(Note:manytextsusetermshistogramandbargraphinterchangeably.Wewillnotdosointhistext.)Homogeneous(verticesinatessellation)–verticesthathaveexactlythesamepolygons
meetinginexactlythesamearrangementHomogeneous(verticesinapolyhedron)–verticesthathaveexactlythesamepolygonfaces
meetinginexactlythesamearrangementHypotenuse–thesideofarighttriangleoppositetherightangleImage(ofatransformation)–thesetofpointsthatresultfromthemotionofanobjectbya
translation,arotation,orareflectionIndependent(events):Twoeventsaresaidtobeindependentifknowingthatoneoccurs(orknowingitdoesnotoccur)doesnotchangetheprobabilityoftheotheroccurringInductivereasoning–theinformalprocessofcomingtoaconclusionbasedonexamplesInscribedpolygon–thepolygoninsideacirclewhoseverticesalllieonthecircleInteriorangle–anyoneofthealternateinterioranglesformedbyatransversaltotwolinesInter-QuartileRange(IQR)=Q3–Q1.ItisthespreadofthemiddlehalfofthedataIntersection(oftwolines)–thepointthelineshaveincommonIntersection(oftwosets)–thesetofelementsthatarecommontobothsetsIsosceles–havingatleastonepairofcongruentsidesJustification–anargumentbasedonaxioms,definitions,andpreviouslyprovenresultstoshow
thataconjectureistrueLawoflargenumbers:Inrepeated,independenttrialsofarandomexperiment,asnumberoftrialsincreases,theexperimentalprobabilitiesobservedwillconvergetothetheoreticalprobability. Inotherwords,theaverageoftheresultsobtainedfromalargenumberoftrialsshouldbeclosetotheexpectedvalue,andwilltendtobecomecloserasmoretrialsareperformed.Leg–asideofarighttriangleoppositeanacuteangleLength–thedistancebetweentwopointsonaone-dimensionalcurve
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Line–anundefinedone-dimensionalsetofpointsunderstoodcovertheshortestdistanceandtoextendinoppositedirectionsindefinitely
Lineofreflection–thelineaboutwhichanobjectisreflectedtoformitsmirrorimageLinesegment–thesetofallpointsonalinebetweentwogivenpointscalledtheendpointsMass–aconceptofphysicsthatcorrespondstotheintuitiveideaof“howmuchmatterthereis
inanobject;”unlikeweight,themassofanobjectdoesnotdependupontheobject’slocationintheuniverse
Mean:Themean(oraverage)valueofasetofnumericdataisthesumofallthevaluesdividedbythenumberofvaluesMeanAbsoluteDeviation(MAD):MAD=!
-𝑥! − 𝑥 + 𝑥" − 𝑥 + 𝑥% − 𝑥 +⋯+ 𝑥- − 𝑥 ,
whichgivestheaveragedistancebetweenthemeanandeachofthendatapoints.Measure–todeterminethequantityofanattribute(orofafundamentalconceptsuchastime)
usingagivenunitMedian:Themedianofasetofnumericvaluesisthemiddlevaluewhenthedataisplacedinnumericalorder.Inthecasewheretherearetwo“middle”values,themedianistheaverageofthetwoMetricsystemofmeasurement–thesystemofmeasurementunitsinwhichthereisone
fundamentalunitdefinedforeachquantity(attribute)withmultiplesandfractionsoftheseunitsestablishedbyprefixesbasedonpowersoften
Midpoint–thepointonalinesegmentthatdividesitintotwocongruentlinesegmentsMode:thevaluethatoccursmostfrequentlyModel–arepresentationofanaxiomsysteminwhicheachundefinedtermisgivenaconcrete
interpretationinsuchawaythattheaxiomsallholdMultiplicationprinciple:IfyouhaveAdifferentwaysofdoingonethingandBdifferentwaysofdoinganother(andonechoicedoesnotaffecttheother)thenthetotalnumberofwaystodobothAandBisA×BNet–atwo-dimensionalfigurethatcanbefoldedintoathree-dimensionalobjectNonagon–apolygonwithexactlyninesides
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Noncollinear(points)–asetofpointsnotallofwhichlieonthesamelineNon-example(ofadefinition)–anexamplethatdemonstratemostoftheconditionsofa
definitionbutthatfailstosatisfyatleastoneconditionNonresponsebias:BiasedinformationthatresultsfromthefactthatsomegroupsinthesampleweremorelikelytorespondthanothersNon-standardunitofmeasure–aunitofmeasurewhosevalueisnotestablishedbyreference
toanacceptedstandard;forexample,ablockNumericalData:numberdataforwhichitmakessensetoperformarithmeticoperations(suchasaveraging).Obliqueprism(pyramid,cylinder)–aprism(pyramid,cylinder)thatisnotrightObtuseangle–ananglewithmeasuregreaterthan90degreesObtusetriangle–atrianglewithoneobtuseangleOctagon–apolygonwithexactlyeightsidesOrder(ofarotationalsymmetry)–thenumberofdifferentrotationsthatareasymmetryofan
objectOrientation–thedirection,clockwiseorcounterclockwise,ofthereadingoftheverticesofa
polygoninalphabeticalorderOutcome:Anyonethingthatcouldhappeninarandomexperiment.Outlier:aspecificvaluethatlieswelloutsidetheoverallpatternofthedata.Parallellines–coplanarlineswithnopointsincommonParallelogram–aquadrilateralinwhichbothpairsofoppositesidesareparallelParameter:Somenumericalinformationthatisdesiredfromanentirepopulation.Usuallyaparameterisanunknownvaluethatisestimatedbyastatistic.Partition–adivisionofageometricobjectintoasetofnon-overlappingobjectswhoseunionis
theoriginalobjectPentagon–apolygonwithexactlyfivesides
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Percentile:Ascoreinthenthpercentilemeansthat“npercent”ofthepeoplewhotookthetestscoredatorbelowthatscorePerimeter(ofaplaneobject)–thelengthoftheboundaryoftheobjectPerpendicularbisector–thelinethroughthemidpointofalinesegmentthatisalso
perpendiculartothelinesegmentPerpendicularlines–twolinesthatintersecttoformfourrightanglesPi(p)–theexactnumberoftimesthediameterofacirclefitsintoitscircumference(orthe
ratioofthecircumferenceofacircletoitsdiameter);thisratioisanirrationalnumberthatisconstantforallsizecirclesandisapproximatelyequalto3.14159
Planarcurve–acurvethatliesentirelywithinaplanePlane–anundefinedtwo-dimensionalsetofpointsunderstoodtobe“flat”andtoextendinall
directionsindefinitelyPlaneofsymmetry–aplaneinspaceaboutwhichathree-dimensionalobjectisreflectedPlatonicsolid–aregularpolyhedronplusitsinteriorPoint–anundefinedzero-dimensionalobject;alocationwithnosizePolygon–afinitesetoflinesegmentsthatformasimpleclosedplanarcurvePolyhedron(plural:polyhedra)–afinitesetofpolygon-shapedfacesjoinedpairwisealongthe
edgesofthepolygonstoencloseafiniteregionofspacewithinonechamberPopulation:thelargestgroupaboutwhichtheresearcherorsurveyorwouldlikeinformationPrism–apolyhedroninwhichtwoofthefaces(calledthebases)arecongruentandlie on parallel
(non-intersecting) planes andtheremainingfacesareparallelogramsProbability:referstotheproportionoftimesaneventwouldoccuriftherandomexperimentwasperformedaverylargenumberoftimesProof–adeductiveargumentthatestablishesthetruthofaclaimProtractor–aninstrumentusedtomeasureanglesPyramid–apolyhedroninwhichonefaceiscalledthebaseandalloftheremainingfacesare
trianglesthatshareacommonvertex(calledtheapex)
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Pythagoreantriple–threepositiveintegersthatsatisfythePythagoreantheoremQuadrilateral–apolygonwithexactlyfoursidesQuantifier(inlogic)–awordorphrase(suchas“all”or“atleastone”)thatindicatesthesizeof
thesettowhichthestatementappliesRadius(plural:radii)–thelinesegmentjoiningapointonacircletothecenterofthecircleRandomExperiment:anyexperimentwheretheoutcomedependsonchanceandcannotbeknownbeforehand.Whileinarandomexperiment,thespecificoutcomecannotbeknown,thereisnonethelessaregulardistributionofoutcomesafteraverylargenumberoftrials.Range=maximumvalue–minimumvalueRay–thesetofpointsonalinebeginningatagivenpoint(calledtheendpoint)andextending
inonedirectiononthelinefromthatpointRectangle–aquadrilateralwithfourrightanglesReflection–(inalinel)isatransformationoftheplaneinwhichtheimageofapointPonlisP,
andifAisapointnotonlandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA
Reflectionsymmetry(2-dimensional)–areflectioninalineinwhichtheimageoftheobject
coincideswiththeoriginalobjectReflectionsymmetry(3-dimensional)–areflectioninaplaneinwhichtheimageoftheobject
coincideswiththeoriginalobjectRegularpolygon–apolygonwithallsidescongruentandallvertexanglescongruentRegularpolyhedron–apolyhedronwhosefacesareeachthesameregularpolygonwiththe
samenumberoffacesmeetingateachvertexRegulartessellation–atessellationthatcontainsonlyoneregularpolygonResponseRate:Theratioofthenumberofrespondents(peoplewhoactuallytookpartinthestudy)tothenumberwhowereinvitedtoparticipate.Responsevariable:avariablewhosevariationisexplainedbyanothervariable(calledanexplanatoryvariable).
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Rhombus(plural:rhombi)–aquadrilateralwithfourcongruentsidesRightangle–ananglethatisexactlyonefourthofacompleteturnaboutapointRightprism(pyramid,cylinder)–aprism(pyramid,cylinder)inwhichthelinejoiningthe
centersofthebases(theapexofthepyramidtothecenterofitsbase)isperpendiculartothebase
Righttriangle–atrianglewithonerightangleRigidmotions-transformationsoftheplanethatpreservedistancesbetweenpoints(theydo
notdistorttheshapeorsizeofobjects)Rotation(aboutapointPthroughanangleq)–atransformationoftheplaneinwhichthe
imageofPisPand,iftheimageofAis 'A ,then PA@ 'PA and 'm APA =q.PointPiscalledthecenteroftherotation
Rotationsymmetry(2-dimensional)–arotationaboutapointinwhichtheimagecoincides
withtheoriginalobjectRotationsymmetry(3-dimensional)–arotationaboutanaxisofsymmetryinwhichtheimage
coincideswiththeoriginalobjectSample:asubsetofapopulationfromwhichdataiscollectedSamplebias:Biased(inaccurate)informationthatresultsfromapoorlychosensample.Samplingerror:thedifferencebetweenthevalueofaparameterandthevalueofthestatisticthatrepresentsit.Samplingerrorcanincludechanceerror,samplingbias,andnonresponsebias.Samplingrate:[(#inthesample)÷(#inthepopulation)]Samplespace:ThesamplespaceofarandomexperimentisthesetofallpossibleoutcomesScalefactor,avaluebywhichwemultiplyeachoriginaldimensiontofindthenewlengthsScalenetriangle–atrianglenoneofwhosesidesarecongruentScaling–atransformationoftheplanethatcauseseitheramagnificationorashrinkingofan
objectinwhichtheimageremainssimilartotheoriginalobjectScatterplot:Adisplaythatshowstherelationshipbetweentwonumericalvariablesmeasured
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onthesamesampleofindividuals.Thevaluesofonevariableareshownonthehorizontalaxis,andvaluesoftheothervariableareshownontheverticalaxis.Eachindividualisplottedasapointrepresentinganorderedpair(variable1,variable2).
Secant–alinethatintersectsacircleintwodistinctpointsSector–theportionofacircleanditsinteriorbetweentworadiiSelf-selectedsample:asampleinwhichrespondentsvolunteeredtoparticipate.Semiregularpolyhedron–apolyhedronthatcontainstwoormoreregularpolygonsasfaces
whicharearrangedsothatallverticesarehomogeneousSemiregulartessellation–atessellationthatcontainstwoormoreregularpolygonsarranged
sothattheverticesarehomogeneousShearing–atransformationoftheplanethatchangestheshapeofanobjectSide–oneofthelinesegmentsthatmakeupapolygonSimilarobjects–objectswhereonecanbeobtainedfromtheotherbycomposingarigid
motionwithadilationSimplecurve–acurvethatdoesnotintersectitselfSimpleRandomSample(SRS):AsampleinwhicheverysubsetofthepopulationhasthesamechanceofbeinginthesampleasanyothersubsetofthesamesizeSkewedleft:AdistributionofdataisskewedleftifithasanasymmetrictailtotheleftSkewedright:AdistributionofdataisskewedrightifithasanasymmetrictailtotherightSlope(ofaline)–theverticaldistancerequiredtostayonalineforaoneunitchangein
horizontaldistanceSpace–anundefinedtermthatdenotesthesetofpointsthatextendsindefinitelyinthree
dimensionsSphere–thesetofpointsin(three-dimensional)spacethatareequidistantfromagivenpoint,
calledthecenterSquare–aquadrilateralwithfourrightanglesandfourcongruentsides
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Standardunitofmeasure–aunitofmeasurewhosevalueisestablishedbyreferencetoanacceptedstandard;forexample,themeterisdefinedtobeoneten-millionthofthedistancefromtheequatortothenorthpole
Straightangle–ananglethatmeasureshalfaturn(ortworightangles)Straightedge–aninstrumentusedtoconstructlinesegmentsStatistic:Anynumericalinformationcomputedfromasample.(Forexample,ameancalculatedfromasampleisastatistic,whichcanbeusedasanestimateforthepopulationmean,whichwouldbeaparameterStemandleafplot(orsimplystemplot):alistingofallthedatatypicallyarrangedsothetensplacemakesthestemandtheonesarethe‘leaves.’Supplementaryangles–twoangleswhosemeasuressumto180degreesSurface–thesetofpointsthatformtheboundaryofasolidthree-dimensionalobjectSurfacearea–thesumoftheareasofthefacesofaclosed3-dimensionalobjectSurvey:Anydatacollectionmethodinwhichdataiscollectedfromjustasubsetofthe
populationSymmetricDistribution:Adistributionthatissymmetricaboutthemedian.Symmetry(ofanobject)–atransformationoftheobjectinwhichtheimagecoincideswiththe
originalTangent(toacircle)–alinethatintersectsacircleinexactlyonepoint Tessellation–anarrangementofpolygonsthatcanbeextendedinalldirectionstocoverthe
planewithnogapsandnooverlapsinsuchawaythatverticesonlymeetotherverticesTheorem–astatementthathasbeenproventrueTheoreticalProbabilities:probabilitiesassignedbasedonassumptionsaboutthephysicaluniformityandsymmetryofanobject(suchasadieorcoinorbagofcandies).Thirdquartile(Q3):themedianofthedatapositionedstrictlyafterthemedianwhenthedataisplacedinnumericalorder.Tiling–anarrangementofpolygonsthatcanbeextendedinalldirectionstocovertheplane
withnogapsandnooverlaps
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Time–ameasurablepartofthefundamentalstructureoftheuniverse,adimensioninwhicheventsoccurinsequence
Transformation–amovementofthepointsofaplanethatmaychangethepositionorthesize
andshapeofobjectsTranslation(byavectorAA’)–arigidmotionoftheplanethattakesAtoA’,andforallother
pointsPontheplane,PgoestoP’wherevectorPP’andvectorAA’havethesamelengthanddirection
Translationsymmetry–atranslationoftheplanesuchthattheimagecorrespondstothe
originalobjectTranslationvector–anarrowthatgivesthedirectionanddistance(itslength)thatapointis
movedduringatranslationTransversal–alinewhichintersectstwoormorelinesTrapezoid–aquadrilateralwithexactlyonepairofparallelsidesTriangle–apolygonwiththreesidesTrivialrotation–therotationof360°;itisarotationalsymmetryofeveryobjectUndefinedterm–atermwhichhasanintuitivemeaning,butnoformaldefinitionUnion(ofsets)–thesetcontainingeveryelementofeachsetUnitofmeasure–anobjectisusedforcomparisonwithanattributeUnitsquare–asquarethatisoneunitbyoneunitandthushasanareaofonesquareunitUnitcube–acubethatisoneunitbyoneunitbyoneunitandhasavolumeofonecubicunitVenndiagram–apictureinwhichtheobjectsbeingstudiedarerepresentedaspointsona
planeandsimpleclosedcurvesaredrawntogroupthepointsintodifferentclassifications.Venndiagramsareusedtovisualizerelationshipsamongsetsofobjects.
Vertex(plural:vertices)–thecommonendpointoftwoadjacentsidesofapolygonVertexangle–theangleformedbyadjacentsidesofapolygonVertex(ofapolyhedron)–theintersectionoftwoormoreedgesofapolyhedron
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Verticalangles–anonadjacentpairofanglesformedbytwointersectinglinesVolume–ameasureofthecapacityofa3-dimensionalobjector,alternatively,thequantityof
spaceenclosedbya3-dimensionalobjectWeight–ameasureoftheforceofgravityonanobject;oftenusedinterchangeablywithmass;
differencesinthemeasuresofweightandmassarenegligibleatsealevelonearth
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GraphPaper
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Tangrams
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PatternBlocks
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