Big Ideas in Euclidean and Non-Euclidean Geometries€¦ · Geometry allows us to think spatially,...

BigIdeasinMathematicsforFutureMiddleGradesTeachersandElementaryMathSpecialists

BigIdeasinEuclideanandNon-EuclideanGeometries

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

(UpdatedSummer2017)

2

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

3

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

John,Jason,Eric,Amy,Carol&Jen

4

Hey!Readthis.Itwillhelpyouunderstandthebook. Theonlywaytolearnmathematicsistodomathematics. PaulHalmosThisbookwaswrittentopreparefuturemiddlegrades(Grades6-8)teachersandelementarymathematicsspecialistsforthemathematicalworkofteaching.Thefocusofthismoduleisgeometry,andmathematicsdoesn’tgetanybetterthanthat.Geometryallowsustothinkspatially,toseestructureinartandform,andtocreateandvisualizenew“worlds”withdifferentrules.Doestheword“geometry”calltomindthetwo-column-proofofyourhighschooldays?Longagomathematicseducatorsdecidedthatgeometryclasswouldbeagoodplacetoshowcasetheimportanceofdefinitions,reasoning,andproofinmathematicalthinking–really,thesethingsarevitalinallareasofmathematics–notjustgeometry–andifyouuseanyofourothermodules,you’llseethatthisisso.However,ifthetwo-columnproofhasruinedgeometryforyou,thenforgetaboutit.Youdon’tneedtodoanyhere.Youarefreetoreasoninanyformyouseefitaslongasyoucancommunicateyourargumenttoothers.Afterall,wemathematiciansrarelywriteaproofinsuchaform.We’dhatetobeconstrainedinthatway.Geometryisadomainforactionandactivities.TheNationalCouncilofTeachersofMathematics(NCTM)advocatesthatmiddlegradesstudentsdraw,measure,visualize,compare,transform,andclassifygeometricobjects(NCTM,2000).(Notealltheactionverbs!)Wewilldoallthesethingsinthismodule.Theideasinthisbookarefundamentallyimportantforyourstudentstounderstandandsotheyarefundamentallyimportantforyoutounderstand.Eachofthemodulesinthisserieswaswrittenbytwoormoreauthors.Toprepareourselvestowritethistext,westudiedfourStandards-basedcurriculumprojectsformiddleschoolstudents(thebooksyourfuturestudentsmightuse).ThoseprojectsareMathematicsinContext,ConnectedMathematics,MATHThematics,andMathScape.Alloftheseareactivity-basedandStandards-basedcurricula.Thismeansthatthemiddleschoolmaterialswerewrittensothatyourfuturestudentswillsolveproblemsandcreateunderstandingsbasedonconcreteexperiences.Incaseyouareskepticalaboutthesetypesofmaterialsforyourfuturestudents,letusassureyouthattheybetterencourageandsupportthetypesofbehaviorsandthinkingthatmathematiciansvaluethandotraditionalmaterials.Furthermore,theresearchsuggeststhatschoolsthatadoptStandards-basedmaterialsformorethantwoyearsshowsignificantlyhighertestscoresoneventraditionalmeasuresofmathematicalunderstandingthanmatchedschoolsthatadopttraditionalcurricula(Reys,Reys,Lapan,&Holliday,2003;Riordan&Noyce,2001;Briars,2001;Griffen,Evans,Timms,&Trowell,2000;Mullisetal.,2001).Weassureyouthattheideasyouwillmeetinthesepagesarevitallyconnectedtothemathematicscurriculumofyourfuturestudents,andwehopethatthetextiswritteninawaythatmakestheseconnectionsapparenttoyou.

In2000theNationalCouncilofTeachersofMathematics’(NCTM)wrotestandardsingeometryforGrades6-8.NCTMisthenationalorganizationforschoolmathematicsteachers.Readthese

5

standardscarefully,andasyouworktheproblemsinthistext,thinkabouthowtheyfitwithinthesecategories.NCTMGeometryStandardforGrades6–8 Instructionalprogramsfromprekindergartenthroughgrade12shouldenableallstudentsto—

Ingrades6–8allstudentsshould—

Analyzecharacteristicsandpropertiesoftwo-andthree-dimensionalgeometricshapesanddevelopmathematicalargumentsaboutgeometricrelationships

•preciselydescribe,classify,andunderstandrelationshipsamongtypesoftwo-andthree-dimensionalobjectsusingtheirdefiningproperties;

•understandrelationshipsamongtheangles,sidelengths,perimeters,areas,andvolumesofsimilarobjects;

•createandcritiqueinductiveanddeductiveargumentsconcerninggeometricideasandrelationships,suchascongruence,similarity,andthePythagoreanrelationship.

Specifylocationsanddescribespatialrelationshipsusingcoordinategeometryandotherrepresentationalsystems

•usecoordinategeometrytorepresentandexaminethepropertiesofgeometricshapes;

•usecoordinategeometrytoexaminespecialgeometricshapes,suchasregularpolygonsorthosewithpairsofparallelorperpendicularsides.

Applytransformationsandusesymmetrytoanalyzemathematicalsituations

•describesizes,positions,andorientationsofshapesunderinformaltransformationssuchasflips,turns,slides,andscaling;

•examinethecongruence,similarity,andlineorrotationalsymmetryofobjectsusingtransformations.

Usevisualization,spatialreasoning,andgeometricmodelingtosolveproblems

•drawgeometricobjectswithspecifiedproperties,suchassidelengthsoranglemeasures;

•usetwo-dimensionalrepresentationsofthree-dimensionalobjectstovisualizeandsolveproblemssuchasthoseinvolvingsurfaceareaandvolume;

•usevisualtoolssuchasnetworkstorepresentandsolveproblems;•usegeometricmodelstorepresentandexplainnumericalandalgebraicrelationships;

•recognizeandapplygeometricideasandrelationshipsinareasoutsidethemathematicsclassroom,suchasart,science,andeverydaylife.

ReprintedwithpermissionfromPrinciplesandStandardsforSchoolMathematics,©2000bytheNationalCouncilofTeachersofMathematics.Allrightsreserved.NCTMdoesnotendorsethecontentvalidityofthesealignments.BuildingontheworkofNCTM,morerecentlyagroupofleadersatthestate-levelhaveworkedtoarticulatestandardsinmathematicsthatprovidemorefocusedguidanceforteachersofeachgrade.ThiseffortresultedintheCommonCoreStateStandardsinMathematics,“astate-ledefforttoestablishasharedsetofcleareducationalstandardsforEnglishlanguageartsandmathematicsthatstatescanvoluntarilyadopt.Thestandardshavebeeninformedbythebestavailableevidenceandthehigheststatestandardsacrossthecountryandglobeanddesignedbyadiverse

6

groupofteachers,experts,parents,andschooladministrators…”(asfoundJanuary11,2012attheirwebsite:http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf).Asofthetimeofpublicationofthistext,moststateshadofficiallyadoptedthesestandards,andsoitisimportantforyoutoknowthemandthecontentandpracticesthattheyadvocate.BelowyouwillfindtheCommonCoreStateStandardsStandardsforMathematicalPracticeandtheContentStandardsforGeometryforgrades6–8asfoundathttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

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

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning

7

CommonCoreStateStandardsforGeometry:

GeometryGradeSix:Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.

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

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

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;use

coordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

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

GeometryGradeSeven:Draw,construct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem.

1. Solveproblemsinvolvingscaledrawingsofgeometricfigures,includingcomputingactuallengthsandareasfromascaledrawingandreproducingascaledrawingatadifferentscale.

2. Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.

3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

8

Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Mathematiciansviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjust

GeometryGradeEight:Understandcongruenceandsimilarityusingphysicalmodels,transparencies,orgeometrysoftware.

1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.b. Anglesaretakentoanglesofthesamemeasure.c. Parallellinesaretakentoparallellines.

2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbe

obtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.

4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbe

obtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

5. Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.Forexample,arrangethreecopiesofthesametrianglesothatthesumofthethreeanglesappearstoformaline,andgiveanargumentintermsoftransversalswhythisisso.

UnderstandandapplythePythagoreanTheorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

Solvereal-worldandmathematicalproblemsinvolvingvolumeofcylinders,cones,andspheres.

9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.

9

asimportantasanycontentyouteach.Whenyoustandbeforeyourclass,youarearepresentativeofthemathematicalcommunity;wewillhelpyoutobeagoodone.Noonecandothisthinkingforyou.Mathematicsisn’tasubjectyoucanmemorize;itisaboutwaysofthinkingandknowing.Youneedtodoexamples,gatherdata,lookforpatterns,experiment,drawpictures,think,tryagain,makearguments,andthinksomemore.Thebigideasofgeometryarenotalwayseasy.EachsectionofthisbookbeginswithaClassActivity.Theactivityisdesignedforsmall-groupworkinclass.Someactivitiesmaytakeyourclassaslittleas30minutestocompleteanddiscuss.Othersmaytakeyoutwoormoreclassperiods.Nosolutionsareprovidedtoactivities–youwillhavetosolvethemyourselves.TheReadandStudy,ConnectionstotheMiddleGrades,andHomeworksectionsarepresentedwithinthecontextoftheactivityideas.

10

TableofContentsLetnoonedestituteofgeometryentermydoors.

InscriptionovertheentrancetoPlato’sacademyCHAPTER1:ARGUINGFROMAXIOMSClassActivity1:TownRules…………..……………..……….…...…….………...……………………………………..p.14 AxiomaticSystemsandModels TheLanguageofMathematics NCTMReasoningandProofStandards ClassActivity2A:TwoFiniteGeometries………........…………..……………….……………………………….p.20

AffineandProjectiveFiniteGeometryAxioms Parallelism

Negation,Quantifiers,ConverseandContrapositiveformsCommonCoreStandardsforMathematicalPractice

ClassActivity2B:PointsofPappus….……….……….…………………….……………………………………………p.32 ClassActivity3:ReadingEuclid…………………………..……………………………………………………………….p.33 Euclid’sAxioms EuclideanLinesandAngles ClassActivity4:ConstructionZone……………………………………………………………………….……………..p.40 StraightEdgeandCompassConstruction SegmentandAngleBisectorsClassActivity5:IfYouBuildit….…………………………………………………………………………………….…….p.46 MoreConstructions Polygons TriangleCongruenceTheoremsSummaryofBigIdeasfromChapterOne………………………………………………………………………………p.52

CHAPTER2:LEARNINGANDTEACHINGEUCLIDEANGEOMETRYClassActivity6:CircularReasoning…..……………………….……….………………………………….....…….…p.54

EuclideanCircles Incenter,Orthocenter,CircumcenterandCentroid vanHielelevels

11

ClassActivity7:FindingFormulas………………..……………………………………………………..……………….p.62 LengthandArea MakingSenseofMeasurementFormulas ScalingClassActivity8:PlayingPythagoras…………..…………………………………………………………………….….p.67 ThePythagoreanTheorem ClassActivity9:NothingbutNet………………………………………………………………………………………….p.72 VolumeandSurfaceArea RightversusObliquePrisms PolyhedraClassActivity10:Slides,TurnsandFlips…………………….…………………….....................................p.78 RigidMotionsofthePlane

ClassActivity11:TransformativeThinking……………………………………..…………………………………..p.86

CompositionsofRigidMotionsClassActivity12:ExpandingandContracting……………………………………………………………………..…p.87

Dilations Similarity

ClassActivity13:StrictlyPlatonic(Solids)……………………………………………………………………………..p.91 SymmetriesinSpace

Congruence ClassActivity14:BuriedTreasure……………………….……………………………………………………………….p.98 TheCartesianPlane AnalyticGeometry

ClassActivity15:PlagueofLocus….……………………………………………………………………………………p.105 TheConicSections

EquationsofThingsGeometric ClassActivity16:ComparingStandards……………………...………………………………………………….....p.111 SummaryofBigIdeasfromChapterTwo…………….……………………………………………………………..p.112

12

CHAPTER3:EXPLORINGSTRANGENEWWORLDS:NON-EUCLIDEANGEOMETRIESClassActivity17:LifeonaOne-SidedWorld…………………………….…………………………………..……p.114 TheMöbiusStrip TheKleinBottle IdentificationSpacesClassActivity18:LifeinaTaxicabWorld…………………………………....……………………………………..p.119 MeasuringDistance CirclesandTriangles ClassActivity19:LifeonaSphericalWorld…………………………………….………………………………….p.124 LinesandDistance Parallelism TrianglesonaSphere ClassActivity20:LifeonaHyperbolicWorld……………………………………..................................p.132 ParallelLinesinHyperbolicSpace Triangles,Rectangles,andaRight-AngledPentagonClassActivity21:LifeinaFractalWorld……………………………………………………………………………..p.137 SelfSimilarity:NaturalandMathematical PerimeterandArea TheIterativeProcessandDimensionSummaryofBigIdeasfromChapterThree………………………………………………………………………….p.145

APPENDICES

References………………………………………………………………..……………………………………………………….p.147Euclid’sTheoremsandPostulates….………………………………………………………………….............…...p.148Glossary………………………………………………………………………….....................................................p.154PolygonCut-outs……………………………………………………………………………………………………………….p.165HyperbolicPaperTemplate………………………………………………………………………………………………..p.170

13

CHAPTERONE

ArguingfromAxioms

14

ClassActivity1:TownRules

Themathematicianstartswithafewpropositions,theproofofwhichissoobviousthattheyarecalledselfevident.Therestofhisworkconsistsofsubtledeductionsfromthem.

ThomasHenryHuxley(MSQ) WelcometothesuperfuntownofHilbert!Wehaveafewrulesherejusttobesureallourresidentshaveplentyoffriendsandhobbies.InHilbert,aclubisamembershiplist,andnotwodistinctclubshavethesamemembershiplist.Herearetheruleslegislatedforourclubs:

a) Everytwotownspeoplehaveaclubtowhichtheybothbelong,andthatclubisunique(meaningthatforeachpairofpeoplethereisonlyonesuchclub).

b) Everyclubhasatleasttwomembers.

c) Noclubcontainsallthetownspeople.

d) Ifyounameaclubandatownspersonwhoisnotamemberofthatclub,therewillbe

oneandonlyoneclubthatpersonbelongstothathasnomembersincommonwiththefirstclub.

WeareinterestedinhowmanypeoplecouldliveinHilbertandfollowtheserules.Checkfortownpopulationsofonethroughfive.Ineachcase,arguethatyouarecorrect.Anyconjectures(guesses)abouttownpopulationslargerthanfive?

15

ReadandStudy Geometryisthescienceofcorrectreasoningonincorrectfigures.

GeorgePolya(MQP)Whatisgeometryallabout?Duringthiscoursewewilltrytogiveyouseveraldifferentwaysofthinkingaboutgeometry.Thefirstisthis:Geometryisthestudyofidealshapesandspacesandtherelationshipsthatexistamongthem.Thewordidealisimportant.Geometry–andallmathematicsforthatmatter–isnotaboutrealobjects.Thinkaboutit,haveyoueverseenacircle?You’veseenplentyofrepresentationsofcircles,butanactualcircle(allpointsinaplaneequidistantfromagivenpoint)existsonlyinourminds.Andsodopointsandplanes.Allmathematicalobjectsarelikethis–theyareideas.Doinggeometry(andallmathematics)inaformalsensemeansstartingwithsomedefined(andsomeundefined)idealobjects,andsomerules(calledaxioms)andreasoningtoseewhatis“true”abouttheobjects.Weput“true”inquotes,becausewedon’tmeantrueinatheologicalsense,butratherwemean“true”withinthesystemwehavecreated.Hilbertisanexampleofanaxiomaticsystem.Wedescribedanobjectcalleda“club,”wegaveyousomerules(axioms)aboutthebehavioroftheseclubs,andthenweleftyoufigureoutwhatwas“true”aboutthetown.Hilbertisalsoageometry.Changetherulesabouttownspeopleandclubstorulesaboutpointsandlinesandyouwillseewhatwemean.TakeaminutetocomparethefollowingrulestotheonesintheTownRulesactivity.(Didyounoticetheitalics?Thatisyoursignaltodosomething.Mathematiciansreadmathbookswithpencilinhand.Weanswerquestionsandverifyanythingtheauthorsclaimtobetrue.Startdoingthistoo.Theitalicswillhelpyouremembertoslowdownandthinkwhileyouread.)

a) Everytwopointsareonauniqueline.b) Everylinecontainsatleasttwopoints.c) Nolinecontainsallthepoints.d) Ifyounamealineandapointnotontheline,therewillbeoneandonlyone

lineonthepointthatisparalleltothegivenline.Axiomaticsystemsincludefiveparts:undefinedterms(like‘member’),definedterms(like‘club’),axioms(like“Everyclubhasatleasttwomembers.”),theorems(thingsyoucandeducefromtheaxioms,like‘Hilbertcannothaveapopulationofexactlytwopeople.’)andproofsoftheorems(argumentsthatthetheoremsaretruebasedontheaxioms).

16

Okay,beforewegoanyfurther,let’sclarifysomeofthelanguagethatmathematiciansusetotalkabouttheprocessofdoingmathematics.Herearesomeimportantdefinitions:

1) axiom:arulethatthemathematicalcommunityhasdecidetoacceptastruewithoutproof.Anaxiomisanassumption.

2) conjecture:aconjectureisahypothesisoraguessaboutwhatistruegiventheaxioms.Forexample,aftersomeexperiencewithHilbert,youmighthaveconjecturedthatthenumberofpeopleinHilbertmustbeaperfectsquare.

3) inductivereasoning:comingtoaconclusionbasedonexamples.Imightnoticethatthesun

rosethedaybeforeyesterday,itroseyesterday,anditrosetoday;soImightconcludethatthesunwillrisetomorrow.Thisisinductivereasoning.Thistypeofreasoningisoftenusedtogenerateaconjecture,butitisnotconsideredsufficientevidencebymathematicianstoproveageneralstatement.

4) deductivereasoning:comingtoconclusionbasedontheaxiomsandlogic.Thistypeof

reasoningisthehallmarkofmathematicalargument.

5) counterexample:acounterexampleisaspecificexamplethatshowsthataconjectureisfalse.Giveanexampleofacounterexample.

6) proof:amathematicalproofconsistsofadeductiveargumentthatestablishesthetruthof

aclaim.

7) theorem:atheoremisamathematicalstatementthathasbeenproventobetrue.Forexample,itisatheoremthatHilbertcannothaveapopulationofexactlythreepeople.Thisisnotstatedasaspecificaxiom,butyoucandeducethisbasedontheaxioms.Ourargumentgoessomethinglikethis:

SupposeAbe,Ben,andCalliveinHilbert.Then,becauserule(a)saysthateachpairmustbelongtoauniqueclubtogether,wemusthaveClub1consistingof,say,AbeandBen,Club2consistingofAbeandCal,andClub3consistingofBenandCal.Makecertainyouunderstandwhywemusthavethesethreeclubswhenwefollowrule(a).Wecannothave3peopleinanyclubbecauserule(c)statesthatallofthetownspeoplecannotbelongtooneclubtogether.Andwecannothaveanyclubsofonlyonepersonbecauserule(b)saysthateachclubmusthaveatleasttwomembers.SoClubs1,2,and3aretheonlypossibleclubswecanmakeandwemusthaveeachofthemtofollowrule(a).Makecertainyouunderstandwhyrules(b)and(c)forceustoconcludethatClubs1,2,and3aretheonlypossibleclubswecanmake.

17

Thuswehaveonlyonecase(Clubs1,2,and3)toinvestigatewithrespecttorule(d).Rule(d)saysthatifInameaclub(say,Club2)andapersonnotinthatclub(say,Ben),ImustfindoneandonlyoneotherclubtowhichBenbelongsthatdoesnotcontainAbeorCal.Butthatisnotpossiblesincethereareonlytwootherclubs,Club1whichcontainsAbeandClub3whichcontainsCal.Makecertainyouunderstandwhywecannotfollowrule(d).Therefore,sincewehavealreadyarguedthatwecannotformanyotherclubs,thereisnowayBencanbelongtoaclubwithoutAbeorCal.Thusrule(d)cannotbemetwithexactlythreepeoplelivinginHilbertandwehavemadeourargumentitisnotpossibleforexactlythreepeopletoliveinHilbert.

Takesometimehereandaskyourselfifyoureallyunderstandwhatyoujustread.Didyouanswerallofthequestions?Canyouexplaintheargumenttosomeoneelse?Moststudentsareinthehabitofreadingtextbookstoocasually.Theprevioussectionistough–itcouldeasilytakeacarefulreader20to30minutestoreadtheprecedingfourparagraphswithunderstanding.Rememberwhatwesaidaboutmathematiciansreadingslowlyandthoughtfullywithapencilinhand?Takingthetimetoreadthistextlikeamathematicianwouldisoneofthesurestwaystodeepenyourunderstandingofgeometry.(Anotherwayistodoallthehomeworkproblemswiththesametypeofcarefulthought.)Thereareacoupleofthingswestriveforwhencreatinganaxiomaticsystem.First,weneedthattheaxiomsbeconsistent.Inotherwords,theaxiomsshouldn’tcontradictoneanother.Second,wewantthesystemasleanaspossible–noredundancy.If(andonlyif)theaxiomsareconsistent,thenthereexistsamodelforthesystem.(KurtGödelprovedthistheoremin1930.)Inmakingtheargumentaboveweattemptedtocreateamodelofthesystemforthreepeople.Wegavespecificnamestothethreepeople(Abe,Ben,andCal)andnamesandmemberliststothethreeclubsweformed:Club1={Abe,Ben},Club2={Abe,Cal},andClub3={Ben,Cal}.Toformamodelwesimplyidentifyeachobjectinthesystemwithaconcreterepresentationinsuchawaythatthedefinitionsandrulesofthesystemmakesense.Amodelisakindof“superexample.”Itisaconcretewayto“see”amathematicalstructure.ThereareseveralpossiblemodelsforthefirstthreerulesofthePeopleandClubssystemforthreepeople.Wealreadysawthatthepeoplecouldbenamedandtheclubscouldbemodeledassetsofnames.Foranothermodel,wecouldleteachpersonbeoneofthelettersA,B,Candletaclubberepresentedbyalinesegment.Thismodelwouldlooklikethis:

B

A C

18

Eventhoughthesetwomodelslookdifferenttheyrepresentthesamesystem,thesamesetofinformationabouttherelationshipsbetweenthethreepeople(points)andtheclubs(lines)towhichtheybelong.Sometimesitisveryusefultohavemorethanonewaytolookat,orrepresent,thesamethinginmathematics.Itisimportanttorecognizewhendifferent-lookingobjectshavethesameunderlyingstructureorthesamesetofproperties.Youprobablyfoundafour-personmodelforthetownofHilbert.MaybeyouthoughtofasetofnamesandclubmembershipliststhatsatisfiedallHilbert’saxioms.Perhapsyoudrewasetofpointsandlinestoshowclubsthatmetalltheaxioms.Inanycase,youcreatedasetofobjectsandinterpretationsfortheundefinedtermsinsuchamannerthatalltheaxiomsweretrueatthesametimeusingyourinterpretations.Thatis,youcreatedamodelfortheaxiomsystem.Iftheaxiomsarenotconsistent,therecanbenomodel.Readthiswholesectionagain.Itisimportanttounderstandtheseideas.Andnowwecangiveyouaseconddefinitionofgeometry.Ageometryisanaxiomaticsystemaboutobjectscalled“points”andcollectionsofpointscalled“lines”andtherelationshipsbetweenpointsandlines,thatiswesayapointis“on”alineandalineis“on”(orcontains)apoint.Itmaysurpriseyoutolearnthattheremanygeometries.Euclideangeometryistheonetaughtinschool,anditisveryusefulforworkingonflatsurfaces.However,ifwewanttotalkaboutlinesandtrianglesonacurvedsurfacesuchasasphere(alsoapracticalconcern,sinceweliveonthesurfaceofasphere),weneedadifferentgeometry.TherulesofEuclidcan’tallbeobeyedonasphere.TheHilbertsystemforfourpeopleisalsoageometry.Beforewemoveontomakesomeconnectionstothemiddlegrades,thereareseveralimportantwordsusedintheTownRulesthatweneedtotalkabout.Thewordsareevery,unique,atleast,oneandonlyone,andall.Thesewordsareexamplesofwhatmathematicianscallquantifiers,wordsthattellussomethingimportantabouthowmanyobjectsareinvolvedinthestatement.Otherquantifyingwordsandphrasesaresome,exactly,atmost,each,thereis.Itiscrucialtounderstandthedistinctionsbetweenthesewordsandtousethemcarefullyinmakingarguments.Youwillbeaskedtodosointhehomeworkset. Homework: Childrenarenotvesselstobefilled,butlampstobelighted. HenrySteeleCommager

1) GobackanddoallthethingsinitalicsintheReadandStudysection.

2) Considerthefollowingsixstatements.Whichcarrythesamemeaning?Whichcanbetrueatthesametime,eventhoughtheydonotcarrythesamemeaning?Why?Whichcannotbetrueatthesametime?Why?

a) Thereisacatlivinginmyhouse.

19

b) Therearethreecatslivinginmyhouse.c) Ihaveexactlyonedoglivinginmyhouse.d) Thereisoneandonlyonedoglivinginmyhouse.e) Thereareatleastthreeanimalslivinginmyhouse.f) Someoftheanimalslivinginmyhousehavefourlegs.

3) Underlineeachofthequantifiersfoundinthestatementsintheprecedingproblem.

Explainwhateachonetellsyouaboutthenumberofanimalslivinginmyhouse

4) Provethat1+2+3+….+(n–1)+n=½[n×(n+1)].

5) Hereisanaxiomaticsystemwiththeundefinedtermscorner,square,andonandthefollowingaxioms:

I. Thereisasquare.II. Eachsquareisonexactlyfourdistinctcorners.III. Foreachsquare,thereareexactlyfourdistinctsquareswithexactlytwocorners

onthegivensquare.IV. Eachcornerisonexactlyfourdistinctsquares.

a) Createaninfinitemodelintheplaneforthissystem.b) Createafinitemodelforjustthefirstthreeaxioms(inotherwords,yoursetofobjects

willbefinite).c) Seeifyoucancreateafinitemodelforallfouraxioms.

6) HereareaxiomsforTriadGeometry:

I. Thereareexactlythreepoints.II. Eachpairofpointsisonexactlyoneline.III. Nolinecontainsallthepoints.

a) Makeamodelforthisfinitegeometryusingdotsaspointsandsegmentsaslines.Can

therebemorethanoneconfigurationthatsatisfiestheseaxioms?Explain.b) Nowmakeamodelusinglettersaspointsandpairsoflettersaslines.c) YourfirstmodelwiththedotsandsegmentswasaEuclideanmodel(onebasedon

geometryintheinfiniteflatplane)anditmighthavebeenmisleadingbecauselinesegmentsintheEuclideansensedonotexistinfinitegeometry.ListsomeotherfamiliarEuclideanobjectsthatdon’texistinanyfinitegeometry.

20

ClassActivity2A:TwoFiniteGeometries

Projectivegeometryisallgeometry. ArthurCayley(MQS)HereisanaxiomaticsystemforAffinePlaneFiniteGeometries:Wehaveafinitesetof‘points’and‘lines’sothatthefollowingaretrue(notethatagain‘point,’‘line’and‘on’areundefinedterms):

I. Everytwodistinctpointshaveexactlyonelineonthemboth.II. Givenalineandapointnotonthatline,thereisexactlyonelineonthe

pointthathasnopointsonthefirstline.III. Everylineisonatleasttwopoints.IV. Thereexistthreenon-collinearpoints.

Anaffineplanewithnpointsoneachlineissaidtohaveordern.

a) Sketchamodelforanaffineplaneoforder2.b) IsHilbertanaffinegeometry?Explain.c) Hereismodelforanaffineplaneoforder3.Checktoseethatitsatisfiesallthe

axioms.

Affineplaneoforder3

(Thisactivityiscontinuedonthenextpage.)

21

HereisanaxiomaticsystemforProjectivePlaneFiniteGeometries.

I. Everytwodistinctpointshaveexactlyonelineonthemboth.II. Everytwolineshaveexactlyonepointonthemboth.III. Everylineisonatleastthreepoints.IV. Thereexistthreenon-collinearpoints.

Aprojectiveplaneofordernhasn+1pointsoneachline.Wearegoingtodescribehowtosketchamodelofaprojectiveplaneofordertwobystartingwithamodeloftheaffineplaneofordertwo(seebelow)andaddingsomestructure.Hereistheplanforyourgrouptofollow:Collectthelinesparalleltoeachotherinaclass(inourpicture,eachsetofmutuallyparallellinesisthesamecolor).Foreachofthen+1classesofparallellines(inthiscasetherearethreeclasses),addanewpointthatwillbeoneachofthoselines(youmayneedtoextendthemandcurvethemaroundsothattheyintersect).Thenaddanothernewlinecontainingexactlyallofthesen+1newpoints.Tryit.

d) Howmanypointsareoneachline?Howmanylinesoneachpoint?Howmanypointstotalareinaprojectiveplaneoforder2?Howmanylines?

e) Givenalineandapointnotontheline,howmanylinesaretherethroughthegivenpointthatareparalleltothegivenline?

22

ReadandStudy

...tocharacterizetheimportofpuregeometry,wemightusethestandardformofamovie-disclaimer:Noportrayalofthecharacteristicsofgeometricalfiguresorofthespatialpropertiesofrelationshipsofactualbodiesisintended,andanysimilaritiesbetweentheprimitiveconceptsandtheircustomarygeometricalconnotationsarepurelycoincidental.

CarlG.Hempel,inTheWorldofMathematicsAfinitegeometryconsistsofafinitenumberofobjects(typicallycalledpoints)andtheirrelationships(typicallydescribedintermsofpointsbeing‘on’lines).Theterms‘point,’‘line,’andtherelationship‘on’areusuallyundefined.Thepropertiesareestablishedbyasetofaxiomsthatgoverntherelationships.Hilbertisanexampleofafinitegeometryifwethinkofpeopleaspointsandclubsaslinesandthetownrulesastheaxioms.TriadGeometryisanotherexampleofafinitegeometry. Therearetwotypesoffinitegeometriesthatareofparticularinteresttomathematicians:affineandprojective.(Infactsomemathematiciansdefineafinitegeometryinsuchawaythatthesearetheonlytwotypesoffinitegeometries.)Themodelyoumadeintheclassactivityisanexampleofourfirstprojectivegeometry.Whatistheprimarydistinctionbetweenthesetwoflavorsoffinitegeometry?Inaffineplanegeometry,throughapointnotonagivenline,wegetoneparallelline.Inprojectiveplanegeometry,wegetnone.Lookattheaxiomsforeachgeometrytoseewhichaxiomtellsyouaboutparallelism.Itmighthelptorecallthattwolineskandlareparallel,writtenk||l,iftheyareinthesameplaneandnopointisonbothkandl.Mathematicianshaveproventhataffineplanesofordernexistwhenevern=pk(wherepisprimeandkisawholenumber).Butwedon’tknowyetwhichothervaluesofngiveusaffineplanes.Wecallthatanopenquestion.Peopleareprobablyworkingonitrightnow. Intheclassactivity,youworkedtounderstandaxiomsandcreatemodelsfortheaxiomaticsystems.Weknowthatworkingwithaxiomsisn’teasy.Asyouworkonthistext,pleasekeepinmindthatouraimistohelpyouunderstandwhatageometryisfromtheperspectiveofmathematicians.Thespecificgeometriesweintroduceandthespecifictheoremsaboutthemarenotasimportantastheideathattherearemanygeometries,eachwithitsownaxiomsanditsownmodels,andthereisawayofthinkingandalanguagethatweusetostudythemall.ThisReadandStudyisdevotedtothelanguageofmathematicsthatyouwillneedtostudyaxiomsystems.Inparticular,belowwe’lldiscussseveraldistinctionsthatarenotalwaysimportantineverydayspeechbutarevitalforunderstandingmathematicalarguments.

23

Let’sdoit.Distinction1:(Astatementanditsnegation).IfwehavesomestatementP,thenthenegationofPisthestatement“NotP.”IfPistrue,then“notP”isfalse.AndifPisfalse,then“notP”istrue.Forexample,thenegationofthestatement,“Ilovegeometry,”isthestatement“Idonotlovegeometry.”Distinction2:(orversusand.)Hereisacasewhenmath-speakdiffersabitfromeverydayconversation.Whenamathematiciansayssomethinglike‘xisanelementofAorB,’(HereassumeAandBaresets–noticehowweusecapitalstodenotesetsandsmalllettersforelementsofsets–thisisprettytypical–butnotaruleoranything)shemeansthatxcouldbeinA,xcouldbeinBorxcouldbeinbothatthesametime.Whenshesays‘xisanelementofAandB”shemeansxisdefinitelyinboth.Decidewhethereachofthefollowingstatementsistrue:

1) Asquarehasfoursidesandatrianglehasfoursides.2) Asquarehasfoursidesoratrianglehasfoursides.3) Asquarehasfoursidesandatrianglehasthreesides.4) Asquarehasfoursidesoratrianglehasthreesides

Nowlet’sseehowwenegatean“and”sentencelike3)Asquarehasfoursidesandatrianglehasthreesides.Noticethatfortheabovesentencetobetrue(whichitis),bothpartsmustbetrue.Iftheabovesentenceisnottruetheneitherasquaredoesn’thavefoursidesoratriangledoesn’thavethreesides.Thatmeansthenegationisthesentence:

Asquaredoesnothavefoursidesoratriangledoesnothavethreesides.Inotherwords,“not(AandB)”means“(notA)or(notB).”Stopandthinkthisthrough.Writethenegationofthesentence:Iatepeasandpotatoes.Whataboutthenegationofan“or”sentence?Consider(true)sentence2):Asquarehasfoursidesoratrianglehasfoursides.Forthistobefalse,bothasquarecan’thavefoursidesandatrianglecan’thavefoursides.So“not(PorQ)”isequivalentto“(notP)and(notQ).”

Asquaredoesnothavefoursidesandatriangledoesnothavefoursides.

Writethenegationofthesentence:xisanelementofsetAorxisanelementofsetB.

24

Distinction3:(converseversuscontrapositive)Manymathematicalstatementsareconditionalstatements(“if-then”statements).Herearesomeexamples.Decidewhethereachistrueorfalse,andineachcaseexplainyourthinking.

1) Ifapolygonisasquare,thenitisarectangle.

2) Ifapolygonisarectangle,thenitisasquare.

3) IfyouliveinLosAngeles,thenyouliveinCalifornia.

4) Ifyoudon’tliveinCalifornia,thenyoudon’tliveinLosAngeles.

5) IfitisFriday,thentomorrowisSaturday.

Mathematicianscallthestatement‘IfQ,thenP’theconverseof‘IfP,thenQ’.SoStatement2istheconverseofStatement1above.Notethatthosetwostatementsarenotlogicallyequivalent(notalwaystrueatthesametime.).Astatementoftheform‘IfnotQ,thennotP’iscalledthecontrapositiveofthestatement‘IfP,thenQ.’Thecontrapositiveformislogicallyequivalentto‘IfP,thenQ”.SoStatement3andStatement4abovearelogicallyequivalentstatements.Thinkaboutittomakesurethatseemsright.WhatistheconverseofStatement5above?Howaboutthecontrapositive?Sometimeswhenyouwanttoprovethatanif/thenstatementistrue,itiseasiertoprovethatthecontrapositivestatementistruethanitistoprovetheoriginalstatementistrue.Andthepointhereisthatbyprovingthecontrapositiveyoualsoprovetheoriginal(becausetheyareequivalentstatements).Distinction4:(“Thereexists…”versus“Forall…”)Recallthatthesesentencestartersarecalledquantifiers,andtheytellyouwhetherthestatementisclaimingthatsomethingexists(thereisatleastone),orwhetherthestatementisageneralone(meaningthatitistrueforeverycase).Herearesomeexamples:

1) Everyclubhasatleastfourmembers.

2) Thereexistsaclubwithexactlyfourmembers.

3) Foreveryclubandforeachtownspersonwhoisnotamemberofthatclub,therewillexistoneandonlyoneclubthatpersonbelongstothathasnomembersincommonwiththefirstclub.

Noticethatthefirststatementmakesaclaimaboutallclubs,whereasthesecondstatementonlymakesaclaimaboutatleastoneclub.Thethirdstatementusesacombinationofquantifiers–whichbringsustoournextdistinction.

25

Distinction5:(‘Thereexistsanx,suchthatforally…’versus‘Forally,thereexistsanx,suchthat…’)Belowaretwostatementsthatmeanexactlythesamethinginreallifetalk,buthavequitedifferentmeaningsinmathematics.Canyoufigureouthowthefollowingstatementsmightbedifferent? Thereissomeoneforeveryone. Foreveryone,thereissomeone.Inmathematics-speak,thefirststatementsaysthatthereisonepersonfortheentiregroup--onepersonwhoisforallofus.Thesecondstatementsaysthateachofushasourownspecialperson.Foreachofus,thereissomeone,andmysomeonemaybedifferentfromyours(atleastIhopeso).Herearesomeexamplesofhowthislooksinamathematicalcontext.Decidewhethereachistrueorfalse.Makeanargumentineachcase.Fornow,assumebothxandymustbeintegers(elementsoftheset{…-3,-2,-1,0,1,2,3…}).

a) Forallx,thereexistsay,suchthatx+y=0.b) Thereexistsanx,suchthatforally,x+y=0.c) Thereexistsanx,suchthatforally,xy=0d) Forallxandforally,x+yisaninteger.e) Forallxandforally,x+y=7.f) Thereexistsanxandthereexistsay,suchthatx+y=7.g) Forallx,thereexistsaysuchthatx+y=7.

ConnectionstotheMiddleGrades:

Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.

Youmayhavenoticedthatwehavebeentalkingquiteabitaboutfacetsof“doingmathematics”ingeneralinthesefirstsections.Thisisbecauseyourmathematicalpractices–yourhabitsofmindandhabitsofbehaviorregardingmath–willprovideacontextforallofyourthinkingandworkingeometry,andtheywillalsogiveyourfuturestudentsamodelofwhatitmeanstodomathematics.BuildingontheworkoftheNationalCouncilofTeachersofMathematics(NCTM),theteamofeducatorsandmathematicianswhowrotetheCommonCoreStandardsforMathematicssingledouteightStandardsforMathematicalPracticethatstudentsshouldlearnduringthecourseof

26

theirschooling.Itwillbeyourjob,astheirteacher,tohelpthemtoestablishthesemathematicspractices.CarefullyreadthebelowCommonCoreStandardsforMathematicalPractices(CommonCoreStateStandardsasfoundathttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf).Wewillreturntothesethroughoutthetext.

Mathematics|StandardsforMathematicalPracticeTheStandardsforMathematicalPracticedescribevarietiesofexpertisethatmathematicseducatorsatalllevelsshouldseektodevelopintheirstudents.Thesepracticesrestonimportant“processesandproficiencies”withlongstandingimportanceinmathematicseducation.ThefirstofthesearetheNCTMprocessstandardsofproblemsolving,reasoningandproof,communication,representation,andconnections.ThesecondarethestrandsofmathematicalproficiencyspecifiedintheNationalResearchCouncil’sreportAddingItUp:adaptivereasoning,strategiccompetence,conceptualunderstanding(comprehensionofmathematicalconcepts,operationsandrelations),proceduralfluency(skillincarryingoutproceduresflexibly,accurately,efficientlyandappropriately),andproductivedisposition(habitualinclinationtoseemathematicsassensible,useful,andworthwhile,coupledwithabeliefindiligenceandone’sownefficacy).

1. Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.

27

2. Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.

3. Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.

4. Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

28

5. Useappropriatetoolsstrategically.

Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvingamathematicalproblem.Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.

6. Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.

7. Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthewellremembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

29

Homework:

Discoveryconsistsofseeingwhateverybodyhasseenandthinkingwhatnobodyhasthought.

AlbertSzent-Gyorgyi1) Ifyouhaven’talreadydoneso,gobackanddoallthethingsinitalicsintheReadandStudy

section.

2) ThefirstCommonCoreStandardforMathematicalPracticeisaboutmakingsenseofproblemsandperseveringinsolvingthem.Readthatstandardagain.Towhatextentdoyoumonitoryourownthinkingasyousolveaproblem?DidyoudoanyofthethingsdescribedasyouworkedontheClassActivity?Explain.

3) Explain,asyouwouldtoyourmiddlegradesstudents,whyastatementoftheform‘not(A

orB)’isequivalentto‘(notA)and(notB).’Anexamplemayhelp.

4) HereisatheoremaboutthetownofHilbert:IfaclubinHilberthasexactlynmembers,thenalloftheclubshaveexactlynmembers.

a) Statetheconverseofthistheorem.Isittrue?

b) Statethecontrapositiveofthistheorem.Isittrue?

8. Lookforandexpressregularityinrepeatedreasoning.

Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlookbothforgeneralmethodsandforshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthemtothegeneralformulaforthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.

30

5) Eric’sGeometryhasthefollowingundefinedterms:book,library,on;andthissetofaxioms:

AxiomI:Thereisatleastonebook.AxiomII:Eachlibraryhasexactlyfourbooksonit.AxiomIII:Eachbookhasexactlytwolibrariesonit.

a) Makeamodelthatsatisfiestheaxioms.b) UseyourmodeltomakesomeconjecturesaboutEric’sGeometry.c) Seeifyoucanprovethatoneofyourconjecturesistrue.d) IsEric’sGeometryafinitegeometry?Explain.e) WritethenegationofAxiomIII.

6) ThethirdCommonCoreStandardforMathematicalPracticesisaboutmakingarguments.

Readthatparagraphagain.DescribesomespecificthingsthatyoudidwhenyouworkedontheTownRulesClassActivitythatwouldfitthatstandard.

7) Spend15minutesonthisquestion:Isitpossibletohaveafinitegeometrywhereifyouaregivenalineandapointnotontheline,youcanhavemorethanonelinethroughthepointthatisparalleltothegivenline?Recallthatparallellinesinfinitegeometryneednot‘lookparallel.’Relyonthedefinitionof“parallel”tohelpyouthinkaboutthis.

8) Itturnsoutthatyoucanalwaysturnaffineplanemodelsintoprojectiveplanemodelsbydoingthemodificationyoudidintheclassactivity:Collectthelinesparalleltoeachotherinaclass.Foreachofthen+1classesofparallellines,addanewpointthatwillbeoneachofthoselines.Thendefineallofthesen+1newpointstoallbeonthesameline.Seeifyoucancreateamodelforaprojectiveplaneoforder3bymodifyingtheaffineplaneoforder3belowasdescribed.Thenchecktobesureyourmodelfulfillsalltheprojectiveplaneaxioms.

Affineplaneoforder3

31

9) Herearetwotheoremsaboutaffineplanes:AffineTheorem1:Ifsomelineofanaffineplanehasnpointsonit,theneachlinehasnpointsonitandeachpointhasn+1linesonit.AffineTheorem2:Inanaffineplaneifsomelinehasnpointsonit,thentherearen2pointsandn(n+1)lines,andeachlinehasnlinesparalleltoit(includingitself).Herearetwotheoremsaboutprojectiveplanes:ProjectiveTheorem1:Ifonelineofaprojectiveplanehasn+1pointsonit,thenalllineshaven+1pointsonthemandallpointshaven+1linesonthem.ProjectiveTheorem2:Inaprojectiveplaneifsomelinehasn+1pointsonit,thentherearen2+n+1pointsandn2+n+1lines.

a) Checktoseethatourmodelforanaffineplaneoforder3satisfiesTheorems1and2above.

b) Checkyourprojectiveplaneoforder2fromtheclassactivityandseeifitsatisfiesboththeorems.

c) StatethecontrapositiveofAffineTheorem1.d) StatetheconverseofProjectiveTheorem1.e) Comparethetheoremsaboutaffineplanestotheprojectiveplanestheorems.f) Usetheaffineplaneaxiomstoprovethattheminimumnumberofpointsinany

affineplaneisfourandtheminimumnumberoflinesissix.

32

ClassActivity2B:PointsofPappus

Geometryenlightenstheintellectandsetsone’smindright. IbnKhaldun(MQS)

InthefirsthalfofthefourthcenturyPappusofAlexandriawroteaguidetoGreekgeometrytitledTheMathematicalCollection.InthatguidehediscussedtheworkofEuclid,ArchimedesandPtolemy,presentingtheirtheorems,constructionsandarguments.CarefullyreadthefollowingtheoremofPappus:

IfA,BandCarethreedistinctpointsononelineandA’,B’andC’arethreedifferentdistinctpointsonasecondline,thentheintersectionoflineAC’andlineCA’,lineAB’andBA’,andlineBC’andCB’arecollinear(thethreeintersectionpointsalllieonthesameline).

Undertherequirementthatthespecifiedlinesintersect,thisbecomesaEuclideanTheorem,meaningthatitistrueinthefamiliarflatinfiniteplaneofyourhighschooldays.

1) Drawsomecarefulsketches,usingdifferentconfigurationsofA,BandCandA’,B’andC’andseeifthistheoremseemstobetrue.DoesitstillholdifB’isn’tbetweenA’andC’?Justtomakeitsowecanalltalkaboutthisasaclass,labeltheintersectionpointofAB’andBA’asD,theintersectionofAC’andCA’asE,andtheintersectionofBC’andCB’asF.Whatsituationsmustbeavoidedtoensurethatallninepointsexist?

2) Now,let’sleavetheEuclideanworldandconsiderjusttheninepointsofPappusalong

withtheir“lines”asafinitegeometry.(Inotherwords,now,nootherpointsexistexceptthenineandlinesarejustsetsofpoints.)

a) Howmanylinesappearonyoursketches?Howmanypointsoneachline?How

manylinesoneachpoint?

b) Givenalineandapointnotontheline,howmanyotherlinescontainthegivenpointandintersectthegivenline?Stateaconjecturebasedonyourobservations.Whatsortofcounterexamplewouldberequiredinordertoproveyourconjecturefalse?

c) Givenalineandapointnotontheline,howmanylinesonthegivenpointare

notonthegivenline?Stateaconjecturebasedonyourobservations.Whatsortofcounterexamplewouldberequiredinordertoproveyourconjecturefalse?

d) Seeifyoucancreatetheaxiomsforwhichthissystemisamodel.

33

ClassActivity3:ReadingEuclid

Euclidtaughtmethatwithoutassumptionsthereisnoproof.Therefore,inanyargumentexaminetheassumptions.

EricTempleBellinH.EvesReturntoMathematicalCirclesInyourgroup,carefullystudythepostulates(anotherwordforaxioms)ofEuclid’sGeometry.ThesearebasicallytheoriginalformulationsfromEuclid’stext–butEuclidwroteinGreekandnotinEnglish,sotheyhavebeentranslatedforyou.Takeoutyourcompass(circlemaker)andstraightedge(linemaker)andseehowthepostulatescorrespondwiththesetools.

Euclid’sPostulates(Axioms)(quotedfromThomasL.Heath’stranslationofEuclid’sElements,2002)

Letthefollowingbepostulated:1. Todrawastraightlinefromanypointtoanypoint.

2. Toproduceafinitestraightlinecontinuouslyinastraightline.

3. Todescribeacirclewithanycenteranddistance.

4. Thatallrightanglesareequaltooneanother.

5. That,ifastraightlinefallingontwostraightlinesmakestheinterioranglesonthesame

sidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.

Onthenextpage,youwillfindthefirstproofthatappearsinEuclid’stext.Studyit.

34

UsedwithpermissionfromHeath,T.L.(2002)translationofEuclid’sElements.

D.Densmore(Ed.)GreenLionPress,SantaFe,NewMexico.pp.3.WhatexactlyisEuclidprovinghere?Whatthingsdoyounoticeabouttheformofhisargument?EuclidcitesDef.15:Acircleisaplanefigurecontainedononelinesuchthatallthestraightlinesfallinguponitfromonepointamongthoselyingwithinthefigureareequaltooneanother(Heath,2002,p.1).Doesthiscorrespondwithdefinitionofacircleweprovidedintheglossary?Explain.

35

ReadandStudy:

Apointisthatwhichhasnopart.Alineisabreadthlesslength. Euclid,Elements

InthischapterwewillexploretheworldofgeometrycreatedbyEuclid’spostulates(axioms).EuclidlivedafterPlatoinGreecearound300BC.HeisknownprimarilyforhisworkonElements,atextthatlaidaxiomaticfoundationsforgeometryintheplane.Thistexthashadatremendousinfluenceonmathematicsbecauseofthesystematicwayitpresentsgeometrypropositions(theorems)logicallyderivedfromoneanother.Euclid’sgeometryisaworldofflatplanescoveredwithinfinitelymanypointsandnoholes,endlessstraightlines,andcirclesthatlookjustlikethecircleofyourelementaryschooldays.Hisistheworldwhereifyouseealineandapointnotonthatline,youwillfindexactlyonelineparalleltothegivenline.Hisistheworldinwhichyoudidyourhighschoolgeometry.Infact,inhighschoolhisgeometrywasyouronlygeometry;whatwewantyoutoknownowisthatEuclideangeometryisbutoneofmanygeometries.Wehavealreadyseensomeothergeometriesthatarefinite.Laterwewillstudysomenon-Euclideangeometriesthatcontaininfinitelymanypoints.HerearetheEuclideanPostulates(Axioms)perhapswritteninamoreuser-friendlyformthanthatwhichyousawintheClassActivity:

1) Auniquestraightlinesegmentcanbedrawnfromanypointtoanyotherpoint.

2) Astraightlinesegmentcanbeextendedtoproduceauniquestraightline.3) Acirclemaybedescribedwithanycenteranddistance.4) Allrightanglesareequaltoeachother.5) VersionA:Iftwolinesarecutbyatransversalandtheinterioranglesonthesameside

arelessthantworightangles,thenthelineswillmeetonthatside.

VersionB:Throughagivenpointnotonaline,therecanbedrawnonlyonelineparalleltothegivenline.(ThesetwoversionsoftheFifthPostulateareequivalent–andforthepurposesofthiscourse,youcanusewhicheveroneismostconvenientforyouinanygivenargument.VersionBisalsoknownasPlayfair’sAxiom.)

AsyousawintheClassActivity,withtheexceptionofnumberfour,theseaxiomsareallconstructive.Bythiswemeanthattheyareaboutwhatcanbeconstructedusingonlya

36

straightedge(a‘linemaker’)andacompass(a‘circlemaker’).Axiomfourisalittlebitdifferent.Ittellsusthatnomatterwhereweareontheplane,allrightanglesarecongruent.SoitprovidesuswiththeideathatEuclid’splaneisuniforminsomeway–thatis,nomatterwhereyouraiseaperpendicularordropaperpendicular,theanglesyouconstructwillallbethesame.EuclidalsolistedintheElementssomeadditionalaxioms(likethebelow)thathecalledCommonNotions.

1) Thingsequaltothesamethingarealsoequaltooneanother.

2) Ifequalsareaddedtoequals,thewholesareequal.3) Ifequalsaresubtractedfromequals,theremaindersareequal.4) Thingswhichcoincidewithoneanotherareequal.

Fromthisleansetoftools,Euclidthencarefullybegantobuildthetheorems(hecalledthempropositions)ofhisgeometry.Itisworthnotingherethattoday’smathematicianshavefoundEuclid’ssetofaxiomsabittoolean,andtheyhaveaddedmanymoreaxiomstoEuclideangeometry.Forexample,inProposition1,whenEuclidgaveaproofthathecouldconstructanequilateraltriangle,hemadetwocircleseachhavingtheradiusofthegivensegmentandusedapointwherethosecirclesintersectedtoidentifyavertexofthetriangle.Modernmathematicianswouldnotethathewasimplicitlyassumingthatthosetwocircleswouldintersectinapoint(thatpointwouldn’tbemissingfromthegeometryoranything),andhavedecidedthatthereshouldbeanaxiomtothataffect.However,Eucliddidaprettygoodjoboverall–and2300yearslaterhisbookElementsisstillthe“bible”ofgeometry.WealsowanttonotethatEuclidoftenusedtheword“equal”whenwewouldusetheword“congruent.”Today’smathematiciansuse“equal”whentheywanttocomparetwonumbers.Sowemightsaythat½isequalto0.5.Weusetheword“congruent”whenwewanttosaythattwoobjects(liketwotrianglesortwosegments)arethesamesizeandshape.Thebasicideahereisthattwoobjectsarecongruentinthecasewhereifoneobjectwasmovedtolieontopoftheotherobject,theywouldcorrespondexactly.Wewilldoamorecarefuljobofdefining“congruent”later.Justlikewedidinourfinitegeometryworlds,wewillnowtrytoseewhattheoremswecanproveusingEuclid’sassumptions.Infactthegamewewillplayfortherestofthechapterandthenextisthis:wheneveryouareaskedtoproveaEuclideantheorem,youshouldturntotheAppendixwhereEuclid’spostulatesandpropositionsarelistedandfindit.Thenyouarefreetouse(assume)anypostulateandanypropositionlistedbeforetheoneyouaretryingtoprove.Forexample,sayyouwanttoprovethatinanisoscelestriangle,thebaseanglesarecongruent.GototheAppendix(reallydoit)andseeifyoucanfindthattheorem.Thencomerightbackhere.

37

SincethatparticulartheoremispartofProposition5,thatmeansyoumayuseanyofthePropositions1–4aswellasanyofthepostulatesinyourargument,andasyouusethem,youshouldcitethem.Beforewedoanexampletoshowyouwhatanargumentmightlooklike,youwillneedtoreviewsomerelevantEuclideanGeometrydefinitionsaboutparallellinesandangles.(Inthisgeometry,point,line,planeandangleareundefinedterms.)Spendsometimereviewingthefollowingterms.

• Twoanglesaresupplementsiftogethertheymaketworightangles.• Twoanglesarecomplementsiftogethertheymakearightangle.

• Verticalanglesareanglesoppositeeachotherwhentwolinesintersectinapoint.

• Twolinesareparalleliftheylieinthesameplaneandsharenocommonpoint.

• Twolinesareperpendiculariftheyformrightverticalanglesatapointofintersection.

• Twoobjectsarecongruentiftheycanbemadetocoincidewithoneanother.(Ifyoumovedoneontopoftheother,itwouldfitexactly.)

• Atransversalcouldbeanylinethatintersectstwoormorelines.Checkoutthisdiagramshowingtwolines(landm)cutbyatransversal(n).AlsonotethatwhileLineslandmlookparallelinourpicturetheydon’talwayshavetobeso.

• Angles1and5arecorrespondingangles.Angles2and6arealsocorrespondingangles.

Whichotherpairsofanglesarecorrespondingangles?

l

m

n

8 765

3421

38

• Angles4and6arealternateinteriorangles.SoareAngles3and5.

• Angles1and7arealternateexteriorangles.SoareAngles2and8.

Whichpairsofanglesontheabovepictureareverticalangles?

Euclidprovedmanytheoremsaboutlinesandangles.Let’shavealookattheformofsuchanargumentnow.Theideafortheproofisslick–andithadtobe:Eucliddidn’thavemuchmachinerybuiltuptouse.Theorem(Postulate5):Inanisoscelestriangle,thebaseanglesarecongruent.Supposethat∆ABCisanisoscelestriangle.

[Noticethatwebeganbystatingwhatisassumedandwedrewapicturewithlabelstohelpothersfollowalong.Thisisagoodpracticethatyoushoulddoalso.]

Now,weknowsegmentACiscongruenttoBC. [Becausethetriangleisisosceles.]

WealsoknowthatsegmentCBiscongruenttoCA. [Strange.Weknow.Justbearwithus.]

Also,ÐACBiscongruenttoÐBCA(becausetheyarethesameangle,)andABiscongruenttoBA.So,∆ABCiscongruentto∆BACbyProposition4.Therefore,ÐCABiscongruenttoÐCBA,andwearedone.

[Noticehowwesetituptocomparethetriangletoitself-butbackwards–sowecoulduseProposition4.ThiswasEuclid’sslickidea.]

Don’tworry.Wearen’toftenthiscleverandwedon’texpectyoutobeeither.Butwewillaskthatyoutrytomakesomeofthemorestraightforwardarguments.

39

Homework: Youalwayspassfailureonthewaytosuccess. MickeyRooney

1) GobackanddoallthethingsinitalicsintheReadandStudyandtheConnectionssections.

2) ThesixthStandardforMathematicalPracticefromtheCommonCoreStateStandardsarguesinpartthat“mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning…Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.”Inotherwords,knowingandunderstandingprecisedefinitionsisaveryimportantmathematicalpractice.Makeyourselfadefinitionsquizandlearntheboldedandunderlinedtermsinthissection.

3) ReadthefollowingselectionofEuclid’sPropositionsfromtheappendixanddrawanotated

sketchforeachtohelpyouunderstandwhatitissaying.Identifywhatisgiven(assumed)inthestatementandwhatisconcludedbythestatement.

a. Proposition13b. Proposition14c. Proposition15d. Proposition27e. Proposition28f. Proposition29g. Proposition30

4) ProveProposition15:Iftwostraightlinescutoneanother,thentheymakeverticalangles

equaltooneanother.

5) ProveProposition30,thatstraightlinesparalleltothesamestraightlineareparalleltoeachother.

40

ClassActivity4:ConstructionZone

Thehumanmindhasfirsttoconstructforms,independently,beforewecanfindtheminthings.

AlbertEinstein

1) ShowthatitispossibletoconstructaraywhichbisectsangleABC.Whatpropositionisthis?CheckAppendixAtosee.Thenprovethatyourconstructionworks.

2) Showthatitispossibletoconstructalinewhichbothbisectsandisperpendiculartolinesegment AB .(Wecallsuchalinetheperpendicularbisectorof AB .)Whichpropositionisthis?Checktosee.Thenprovethatyourconstructionworks.

B

A

C

A B

41

ReadandStudy:

Themathematicianisentirelyfree,withinthelimitsofhisimagination,toconstructwhatworldshepleases.

JohnWilliamNavinSullivan MathematicalQuotationsServerToconstructageometricobjectistocreateitusingonlystraightlinesegmentsandcircles(Euclid’sfirst,second,andthirdaxioms).Thetoolsweusearethestraightedge,tomakelinesegments,andthecompass,tomakecircles(orarcsofcircles).Infact,aswementionedearlier,youcanthinkofyourstraightedgeasyourline-makerandyourcompassasyourcircle-maker.Youcannotmeasureanythingwitharuleroraprotractoraspartofyourconstruction.Togiveyouanexampleofhowmathematiciansdescribeandjustifyconstructions,wewillshowitispossibletodropaperpendiculartoagivenlinethroughagivenpointnotontheline.Supposewehaveline(n)andapointnotontheline(P).ItispossibletoconstructalinethroughPthatisperpendiculartolinen(Proposition12).Takeoutyourstraightedgeandcompassandfollowalong.FirstweusethecompasstoconstructacirclecenteredatPthatintersectslinenintwopointswecancallAandB(wejustneedtodrawthearccontainingAandB).

NoticethatAPandBParebothradiiofthiscircle(andthus𝐴𝑃 ≅ 𝐵𝑃).Nowwewillusethissameradiusandconstructtwocircles,onecenteredatAandonecenteredatB.(Again,weonlyneedtodrawthearcsofthesecirclesthatintersectbelowlinen.)LabelthepointofintersectionofthesetwocirclesC.

nBA

P

42

ThefinalstepinourconstructionistodrawthelineconnectingpointsPandCwithastraightedge.Thislinewillbeperpendiculartolinen.Wejustdescribedthe“howto”oftheconstructionofaperpendicularline.It’simportanttobeabletocarryoutthisprocedureastherewillbemanyoccasionsonwhichyouwillneedtoconstructperpendicularlinesinthisclass.Itisevenmoreimportanttounderstandwhyweclaimthatthisprocedureproducesperpendicularlines.Wecall“explainingwhyitworks”justifying(orproving)theconstruction.RecallthatanypostulateaswellasanypropositionnumberedbelowProposition12isfairgameforouruse.

Lookagainatourconstructiondiagram.WeclaimthatthelineCPisperpendiculartolinen.Howcanwejustifythisclaim?Well,weknowthatperpendicularlinesarelinesthatintersectatright

n

C

BA

P

nD

C

BA

P

43

angles.So,ifwecanshowthatÐADP,ÐPDB,ÐBDC,andÐCDAareeachright,thenwecanconcludethatlinesPCandnareperpendicular.Furthermore,weclaimthatitissufficienttoshowthatjustoneoftheseanglesisright.(Makeanargumentforthisclaimrightnow.Whyareallfouranglesrightanglesifjustoneangleisknowntobearightangle?)WewillshowthatÐADPisarightangle.Noticethedashedlinesegmentsinthediagram.Theywillbeveryusefulinmakingourargument.“Adding”extralinesorlinesegments(whichwerenotpartoftheconstructionprocess)toadiagramisoftenahelpfulstrategyindesigningageometricproof.WeobservedbeforethatAPandBParebothradiiofthesamecircle(andso𝐴𝑃 ≅ 𝐵𝑃bythewaywedidtheconstruction).Now,wecanalsorealizethatACandCBareradiiofcirclescongruenttothefirstcircle.Thus,allfourdashedlinesegmentsarecongruentbyconstructionandwemadeeachofthemusingPostulate3.Makecertainyoucanexplainthispartoftheproofinyourownwords.Thesefourcongruentlinesegmentsformtwotriangles,∆CAPand∆CBP,thatshareacommonsideCP.SonowwecansaythatthesetrianglesarecongruentbyProposition8(SomeofyoumayknowthisastheSide-Side-Sidetrianglecongruencetheorem).Youmightbeaskingwhywewanttotalkabout∆CAPand∆CBP.WesaidwewantedtoproveÐADPisarightangleandÐADPisn’tevenapartof∆CAPor∆CBP.Well,let’stakealookat∆DAPand∆DBPwhichdocontainÐADP(andÐBDP).Ifwecouldshowthesetwotriangleswerecongruent,wewouldbemakingprogresstowardourgoal.Why?Wehavealreadynotedthat𝐴𝑃 ≅ 𝐵𝑃.DPisacommonside.WecouldusetheProposition4ifweknewthattheincludedangles,ÐAPDandÐBPD,werecongruent.Aha,nowitmakessensetowanttoknow∆CAP@∆CBP.ÐAPDandÐBPDarecorrespondingpartsofcongruenttriangles∆CAPand∆CBP,andsotheyarecongruent.Makesureyouunderstandwhywesaythis.Okay,nowwecansay∆DAP@∆DBPand,therefore,allcorrespondingpartsofthesetwotrianglesarecongruent.ThusÐADPiscongruenttoÐBDP,andsincethesetwoanglesaresupplementarytheyarebothrightangles(Why?).Wearedone!ConnectionstotheMiddleGrades:

Ingrades6-8allstudentsshouldpreciselydescribe,classify,andunderstandrelationshipsamongtypesoftwo-andthree-dimensionalobjectsusingtheirdefiningproperties.

NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematicsp.232ThegeometryyouwillteachinelementaryandmiddleschoolisthegeometryofEuclid.Thefocushoweverisnotanaxiomaticdevelopmentofthesubjectbutratherahands-onintuitiveapproach.

44

Youwillfocusonclassificationandpropertiesof2-and3-dimensionalobjects;transformationsandsymmetry,describingspatialrelationshipsusingmapsandcoordinategeometry,andgeometricproblemsolving.Homework:

Onedayofpracticeislikeonedayofcleanliving.Itwon’tdoyouanygood.

AbeLemons

1) DoalltheitalicizedthingsintheReadandStudysection.

2) CarefullyreadthroughEuclid’sPropositionsfromtheappendix.Whichonesareaboutconstructingobjects?Whatdoeseachmake?

3) Writeaclearandcompletedescriptionofthestepsyouusedforeachoftheconstructions

intheClassActivity.

4) Justifyyourconstruction#1inClassActivity,thatis,provethattherayyouconstructedcreatestwocongruentangles,eachhalfthemeasureofÐABC.

5) Justifyyourconstruction#2intheClassActivity,thatis,provethatthelineyouconstructedisperpendicularto AB atthemidpointof AB .

6) Isitpossibletobisectaline?Whyorwhynot?

7) YoumayhavenoticedthatEucliddidnottalkaboutmeasuringanglesin“degrees”aswe

oftendo.Wethinkofafullturnasbeingsplitinto360littleangles–eachcalledadegree.Wedon’tknowwhenthinkingindegreesbeganorwhichcivilizationbeganit–butwedohavesomeideasaboutwhy360waschosen.Whyis360suchagoodchoice?

8) ConstructalinesegmentBCsothatitiscongruenttoABandthemeasureofÐABCishalf

ofarightangle.(Youdon’tgettouseaprotractorhere.)

9) Provethateverypointontheperpendicularbisectorofalinesegmentisequidistantfromtheendpointsofthatsegment.

A B

45

10) Provethateverypointontheanglebisectorofanangleisequidistantfromtheraysthatformthatangle.Inotherwords,provethatFDiscongruenttoED.

E

F

A

B

C

D

46

ClassActivity5:IfYouBuildIt

Theshortestdistancebetweentwopointsisunderconstruction. NoelieAlitoInthisactivityyouwillperformandthenjustifytwomoreconstructionsofEuclideangeometry.Theseconstructions,alongwiththeothersinthissection,willgiveyousometoolswithwhichtoconstructotherobjectslateroninthecourse.Asusual,inyourjustification,youcanuseanypostulateorpropositionthatcomesbeforetheoneyouaretryingtoprove.

1) Showitispossibletocopyagivenanglesothattheraybelowisoneoftheraysoftheangle.Thenjustifythatyouhavedoneso.ThisisEuclid’sProposition23.


47

2) Givenalineandapointontheline,showthatitispossibletoconstruct,throughthepoint,alinethatisperpendiculartothegivenline,andthenjustifythatyouhavedoneso.ThisisEuclid’sProposition11.

48

ReadandStudy:

Thereisstilladifferencebetweensomethingandnothing,butitispurelygeometricalandthereisnothingbehindthegeometry. MartinGardner

Wehavebeentalkingabouttriangleswithoutofficiallydefiningthem,oranyotherpolygon,forthatmatter.Let’sfixthatnow.Apolygonisasimple,closedcurveintheplanemadeupentirelyoflinesegments.Thelinesegmentsarecalledsidesandthepointswheresegmentsmeetarethevertices.Let’shaveacloserlookatthepiecesofthisdefinition.Firstofallamathematicianusestheword“curve”totalkaboutprettymuchanypencillineyoucoulddrawwithoutliftingyourpencilfromapaper.Acurveneednotbecurvy;itcouldevenbeperfectlystraight.Acurveintheplaneissimpleifithasnoloopsandnobranches.Acurveisclosedifithasaboundarythatseparatesoutsidefrominside.Decidewhethereachofthefollowingisapolygonbasedonthedefinition(thisishowwealwaysmakesuchdecisionsinmathematics).

Doyouseethatonlyc)isapolygon?a)isnotclosed.b)isnotmadeonlyoflinesegments.d)isnotsimple.Noticetoothatasolidobjectlikethistrianglebelowisnotapolygon.Itistheboundaryoftheobjectthatisapolygon. Atriangleisapolygonwithexactlythreesides.Atriangleisequilateralifallitssidesarecongruent.Ifonlytwosidesarecongruent,thenwesayitisisosceles.Ifnosidesarecongruent,

(d)(c)(b)(a)

49

thenitisscalene.Atrianglewithananglebiggerthanarightangleiscalledobtuse.Ifallitsanglesarelessthanarightangle,wesaythetriangleisacute.YoulearnedlotsoftheoremsaboutEuclideantrianglesbackinhighschool.Somewerefactsabouteverytriangle(likethesumoftheinterioranglesofanytriangleisequaltotworightangles).Someofthemwerefortellingwhentwotriangleswerecongruent.Let’stakemomenttolookatthecongruencetheorems.Proposition4:Iftwotriangleshavetwosidesequaltotwosidesrespectively,andtheyhavetheanglescontainedbythestraightlinesequal,thenthetriangleequals(iscongruentto)thetriangle.(ThisistheSide-Angle-Sidecongruencetheorem.)Proposition8:Iftwotriangleshavetheirtwosidesequaltotwosidesrespectivelyandalsohavethebaseequaltothebase,thentheyalsohaveanglesequalwhicharecontainedbythestraightlines(andsoarecongruentbyProposition4).(ThisistheSide-Side-Sidecongruencetheorem.)Proposition26:Iftwotriangleshavetwoanglesequaltotwoanglesrespectively,andonesideequaltooneside,namely,eitherthesideadjoiningtheequalangles,orthatoppositeoneoftheequalangles,thenthetrianglesarecongruent.(ThisistheAngle-Angle-SideandalsotheAngle-Side-Anglecongruencetheorem.)Readthemagainandmakesketchestobesurethatyouunderstandwhateachoftheseissaying.YoumayhavenoticedthatthereisnoAngle-Side-Sidecongruencetheorem.Thisisbecausehavingacongruentangleandtwocongruentsides(unlesstheangleisbetweenthetwosides)isnotenoughtoguaranteecongruence.Here’stheproblem.Considertwotrianglesthateachhasasidethatmeasures4cm,anothersidethatmeasures1.5cm,andananglethatmeasures15degrees.Herearetwodifferenttrianglesthatmeetthoseconditions:Inotherwords,therearesometimestwochoicesforthethirdsidelength.NoticetoothatthereisnoAngle-Angle-Anglecongruencetheorem.Explainwhynot.

50

ConnectionstotheElementaryGrades: Whosoneglectslearninginhisyouth, losesthepastandisdeadforthefuture. EuripidesTheCommonCoreStateStandardsasksthatstudentsingradesevenexplorewhengiveninformationisenoughtospecifyatriangle.Inotherwords,theyadvocatethatthosestudentsshouldhaveanintuitiveintroductiontothetrianglecongruencetheoremswehavediscussedabove.

Hereisaproblemthatmightfitthatstandard.Takethetimetodoitsothatyouseewhatwemeanhere.Youwillneedarulerandaprotractortomeasurelengthsandangles.

SupposeyouaregivensomeinformationaboutatriangleABC.Inwhichofthefollowingcaseswilltheinformationbeenoughtoallowyoutodeterminetheexactsizeandshapeofthetriangle?Thatis,ifyouandapartnerindependentlymakethetriangleandthencutitout,willthetrianglescoincideifyoulaythemontopofeachother?Ifyouhaveenoughinformation,drawatriangleguaranteedtobecongruenttoDABC.Ifyoudonothaveenoughinformation,describetheproblemyouencounterinattemptingtodrawDABC.

a) AB =4cmandBC =5cmb) AB =8cmand AC =6cmandÐBAC=45°c) AB =8cmand AC =7cmandÐABC=45°d) ÐABC=75°,ÐBCA=80°,andÐCAB=25°e) BC =7cm, AC =8cm,and AB =9cmf) AB =9cm, BC =3cm,and AC =4cmg) AB =7cm,ÐABC=25°,andÐBAC=105°h) BC =11cm,ÐABC=75°,andÐBAC=40°

• Draw(freehand,withrulerandprotractor,andwithtechnology)geometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticingwhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.

51

Homework:

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackofwill.

VinceLombardi

1) DoallthethingsintheReadandStudysection.

2) DoalltheproblemsintheConnectionssection.WhichNCTMStandards(seep.4)dotheymeet?

3) Learnalltheboldedandunderlinedtermsinthesection.

4) AccordingtotheCommonCoreStateStandards,studentsingradeeightshouldbeabletodothefollowing.Readthisstandard–thendotheactivitydescribedintheirexample.

5) ProveProposition32,namely,thatthesumofthethreeinterioranglesinatriangleistworightangles.Recallthatyoucanuseanyofthepropositionsthatcomebefore32.Wesuggestthatyoufirstdrawanytriangleandthenconstructalineparalleltooneofthesidesofthetriangle,throughtheoppositevertex.

6) Apolygonisconvexifallofitsdiagonalslieintheinteriorofthepolygon.Adiagonalofa

polygonisalinesegmentthatjoinstwonon-adjacentvertices.Apolygonisconcaveifitisnotconvex.Usethesedefinitionstodecidewhethereachpolygonbelowisconvexorconcave.Ineachcase,explainyourthinking.

• Useinformalargumentstoestablishfactsabouttheanglesumandexteriorangleoftriangles,abouttheanglescreatedwhenparallellinesarecutbyatransversal,andtheangle-anglecriterionforsimilarityoftriangles.Forexample,arrangethreecopiesofthesametrianglesothatthesumofthethreeanglesappearstoformaline,andgiveanargumentintermsoftransversalswhythisisso.

52

SummaryofBigIdeasfromChapterOne Hey!What’sthebigidea? Sylvester

• Onedefinitionofgeometryisthatitisthestudyofidealshapesandtherelationshipsthatexistamongthem.

• Aseconddefinitionisthatgeometryisanaxiomaticsystemaboutobjectscalled“points,”

collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.

• Axiomaticsystemsdefinetherulesgoverningtheparticulargeometry.

• Provenconsequencesofaparticularsetofaxiomsaretheorems.

• Afinitegeometryconsistsofafinitenumberofobjectsandtheirrelationships.

• Mathematiciansareverycarefulaboutdistinctionswithinmathematicallanguage.

• Weconstructanobjectbycreatingitusingonlystraightlinesegmentsandcircles.

53

CHAPTERTWO

LEARNINGANDTEACHINGEUCLIDEANGEOMETRY

54

ClassActivity6:CircularReasoning

Natureisaninfinitesphereofwhichthecenteriseverywhereandthecircumferencenowhere.

BlaisePascal MathematicalQuotationsServerForthisactivity,eachpersoninyourgroupwillneedtodrawthreepointsofablanksheetofpaperasfollows:Onepersonshouldarrangethepointssothatthetriangleformedwiththepointsasitsverticesisanacutescalenetriangle.Anothershouldarrangeanobtusetriangle.Anothershouldarrangearighttriangle,andifthereisafourthperson,thatpersonshouldarrangehisorherpointstomakeanequilateraltriangle.Makeyourtrianglesfairlylarge,youaregoingtobedoinglotsofconstructing.First,afewdefinitions:Thecircumcenter(C)ofatriangleisintersectionpointoftheperpendicularbisectorsofthesides.Theincenter(I)ofatriangleistheintersectionpointoftheanglebisectors.Theorthocenter(O)istheintersectionofthealtitudes(heights)ofthetriangle.Thecentroid(M)isthepointofintersectionofthemedians(linesjoiningavertexwiththemidpointoftheoppositeside)ofthetriangle.Carefullyconstructeachofthesecentersforyourtriangle.(Youmaywanttousedifferentcoloredpencilsfordifferentconstructions.)Thenlabeleachspecialpoint.Whenyouaredone,compareyourresultsandanswerthefollowingquestions:

1) Oneofthesepointsisspecialbecauseitisthecenterofmassofthetriangle(thebalancingpoint).Whichoneandwhy?

2) Oneofthesepointsisspecialbecauseitisthecenterofthecirclecontainingallthevertices

ofthetriangle.Whichoneandwhy?

3) Oneofthesepointsisspecialbecauseitisthecenterofthebiggestcirclethatcanbeplacedinsidethetriangle.(Thecirclethatistangenttoallthreesides.)Whichoneandwhy?

4) Whichthreeofthefourspecialpointsalwayslieonthesameline?

5) Whichofthepointscouldlieoutsideofthetriangle?Forwhattypeoftrianglesdoesthat

happen?Whydoesthismakesense?

55

ReadandStudy: Itiseasiertosquarethecirclethantogetroundamathematician. AugustusDeMorgan MathematicalQuotationsServer

WenowhavesomebasictoolswithwhichtostudyEuclideanGeometry,andinthischapterwewilldevelopevenmore.

Amathematicalcircleisthesetofpointsthatareequidistantfromagivenpoint,calledthecenter(Ointhediagrambelow).Thediagramshowssomeoftheotherimportanttermsassociatedwithacircle.Becertainyouunderstandeachtermandcanexplainitsmathematicaldefinition.

Itisanamazingfactthatforanysizecircle,theratioofthecircumferencetothediameterisconstant.Wenowcallthisconstantpi(p).Overtheyearsmanymathematicianshavetriedtofind

Central Angle Ð AOC

Tangent

Secant

Chord DE Diameter AB

Arc DE

Radius CO

Sector O

C

A

B

D

E

56

approximationsforpi.Archimedes,ageniusoftheGreekmathematicians,foundapproximateboundsforitsvalueusingcircumscribedandinscribedpolygonswith96sides(heprovedthat''()*< 𝜋 < ''

).(Theaverageofthesetwovaluesisroughly3.1419,aprettydarngoodestimate.)It

isworthnotingthatevenwhenweusethepkeyonacalculatororaskacomputertocomputeit,weareusinganapproximatevaluebecausepisnotrationalandthereforedoesnothavea

decimalnamethatterminatesorrepeats.Studentscommonlyuseeither3.14or722 asan

approximatevalueforpwhencarryingoutcalculationsinvolvingcircles.Euclid’sworkcontainsmanytheoremsaboutcircles.WewilldiscusstwoofthemnowandaskyoutoexploresomemoreintheHomeworksection.Thefirsttheoremwe’lllookatsaysthattwochordsofacirclearecongruentifandonlyiftheircorrespondingarcshavethesamemeasure.

First,that“ifandonlyif”phrasemeansthatwearegettingtwotheoremsforthepriceofone.Boththestatementanditsconversearetrue.Thus,thistheoremgivesustwoif-thenstatements:iftwochordsofacirclearecongruentthentheircorrespondingarcshavethesamemeasure,andiftwoarcsofacirclehavethesamemeasure,thentheircorrespondingchordsarecongruent.Let’sillustratethesetheoremsinadiagram:ifchords AB andCD arecongruent,thenthearcsABandCD(shownindarkred)arealsocongruent,andviceversa.

Oursecondtheoremisoneaboutinscribedangleswhichstatesthatthemeasureofanangleinscribedinanarcisone-halfthemeasureofitsinterceptedarc.Tomakesureweunderstandwhatthistheoremissaying,weneeddistinguishbetweenaninscribedangle,aninterceptedarc,andacentralangle.Inthefollowingdiagram,ÐADBisinscribedinarcACBwhichiscalleditsinterceptedarc.ArcACBismeasuredbythecentralangleÐAOB.RestatethetheoremintermsofÐADBandÐAOB.

C

A

B

D

57

Notethatwehavenotprovedeithertheoreminthissection,buttakeafewminutesnowtodosomemeasurementssothatyoucanseethattheymightbetrue.Nowtakeoutyourcompassandstraightedgeandfollowalong.WritedowntwodistinctpointsandnamethemAandB.ConstructCircleABwithcenteratAandpointonthecircleB.Constructanychord(andnameitPQ)oncircleAB.

ConstructalinethroughAthatisperpendiculartochordPQandlabelthepointofintersectionofPQandthisperpendicularlineM.Thinkaboutwhatwouldhappenifyoucould“move”PandQaroundonthecircle?InotherwordswhathappensasyouchangethepositionsofPorQandkeeppointMastheintersectionpointofthatnewPQandthelinethatisperpendiculartoPQthroughA?WhatdoyounoticeaboutM?Whydoyouthinkthishappens?Canyoumakeageneralargumenttosupportyourconjecture?

FindtheintersectionpointsoftheperpendicularlineandcircleABandcallthemRandS.ConstructlinesegmentRSanderasetheperpendicularlineandpointM.Thiskindofasegmentwithendpointsonthecirclethatgoesthroughthecenterofthecircleiscalledadiameter.ForanychordPQthatisnotadiameter,whatcanwesayaboutthechord’slengthincomparisontothelengthofanydiameterforagivencircle?

ConstructtheperpendicularbisectorofchordPQ.ImaginemovingpointsPandQaroundonthecircle.Whathappenstotheresultingperpendicularbisector?Now,constructanewCircleAB(againwithcenterAandpointoncircleB).ConstructadiameterofthecircleABwithendpointsBandC.PickandlabelapointEonthecircle.MakesegmentsEBandEC.“Move”Earoundthecircle.WhatcanyouconjectureabouttriangleBEC?YouaregoingtoneedafinalnewCircleAB.PickapointonthecircleandlabelitP.ConstructradiusAP.ConstructalinethroughPperpendiculartosegmentAP.ImaginemovingParoundthecircle.Isthisperpendicularlineasecantoratangent?InbookfourofElements,Euclidprovedseveraltheoremsaboutcircles,oneofwhichisthatthreedistinct,non-collinearpointsdetermineauniquecircle(onethatpassesthroughallthreepoints).

O

C

A

B

D

58

Youconstructedthatcirclewhenyoufoundthecircumcenterofyourtriangle.IntheHomeworksection,youwilljustifythatconstruction.ConnectionstotheMiddleGrades: Geometryisanaturalplaceforthedevelopmentofstudents’reasoningand justificationskills. NCTM,PrinciplesandStandards,2000Perhapsthebestregardedmodelregardingchildren’sgeometricreasoningisthevanHieleLevels.PierrevanHieleandDinavanHiele-Geldofwerebrotherandsister,educators,andresearchersofchildren’sthinking.Theyassertedtherearefivedevelopmentallevelsofgeometricreasoning.Beforewediscussthelevels,we’llgiveyousomegeneralinformationaboutthemaccordingtotheresearch.First,thelevelsappeartobesequential;thatis,childrenmustpassthroughtheminorder.Second,theyarenotsomuchage-dependentasexperience-dependent.Itisgeometricactivityattheircurrentlevelthatprepareschildrenformoresophisticatedreasoning.Finally,itappearsthatinstructionandlanguageatlevelshigherthanthatofthechildwillactuallyinhibitlearning.That’salittleworrisomeforteachers–becauseitmeansthatyoucandoharmifyoudonottailorinstructiontothespecificlevelsofyourstudents.HereisthemodelasitisdescribedbyBattista(2007).

ThevanHieleLevelsofGeometricReasoning

Level0:Visual.Childrenrecognizegeometricobjectsbytheiroverallappearancebasedonafewprototypicalexamplesoftheobjects.Forexample,achildatthislevelmightrejectatrianglethatisorienteddifferentlythanthosesheisusedtoseeingoronethatisextremelylongandthin.Ifyouaskkindergartenerwhyashapeisatriangle,shewilllikelytellyou,“becauseitlookslikeone.”

Level1:DescriptiveorAnalytic.Childrenbegintoidentifypropertiesofgeometricobjectsanduseappropriatetermstodescribethoseproperties.Forexample,achildatthislevelcouldclassifytrianglesbasedonthepropertythattriangleshaveexactlythreesides.Level2:AbstractorRelational:Childrenrecognizerelationshipsbetweenandamongpropertiesofgeometricobjects,andwillmakeandfollowargumentsandclassifyshapesbasedontheseproperties.Level3:Deduction:Studentsconstructargumentsaboutgeometricobjectsusingdefinitions,axioms,anddeductivereasoning.Yourhighschoolgeometrycoursewasprobablytaughtatthislevel.

59

Level4:Rigor:Studentsatthislevelwillunderstandthattherearemanygeometries,eachwithitsownaxiomaticsystemandmodels.Inthiscoursewewillgiveyouasenseofthis.

UpperelementaryandmiddlegradesstudentstypicallytestatvanHieleLevel1orLevel2.Asateacherofthesegradesyourjobistogiveyourstudentslotsexperienceslikethefollowing:

• Classifyingobjectsbasedondefinitions.Forexample,youmightdefinearhombusasaparallelogramwithfourcongruentsides,andaskyourclasstodecidewhetherseveralshapesarerhombibasedonthatdefinition.

• Makingandtestingconjecturesaboutgeometricobjects.

• Usinginformaldeductivelanguage,wordslike“all,”“some,”“thereexists,”and“if-then”

statements.Forexample,youmightaskyourstudentstodecideifthefollowingstatementistrue:ifashapeisarectangle,thenitisarhombus.(Isittrue?)Oryoumightaskaquestionlike,doesthereexistarectanglethatisarhombus?(Doesthere?)

• Exploringthetruthofastatement,itsconverse,anditscontrapositive.Forexample,decide

whethereachofthefollowingistrueorfalse.Makeanargumentineachcase.

Ifaquadrilateralisasquare,thenithasfourcongruentsides.Ifaquadrilateralhasfourcongruentsides,thenitisasquare.Ifaquadrilateraldoesnothavefourcongruentsides,thenitisnotasquare.

• Problemsolvinginvolvinggeometricobjectsandrelationships.• Makinginformaldeductiveargumentsaboutobjectsandrelationshipsamongobjects.

• Makingmodelsandpicturesofgeometricobjects.

Hereisamiddlegradesactivityfocusedoncirclesthatallowsstudentstostudymodels,makeandtestconjectures,andmakeinformalarguments.Theideaistoestimatethevalueofπbymeasuringavarietyofcirclestofindthenumberoftimesthediameterofeachcirclefitsintoitscircumference.Takeamomenttodothatnowwiththetwocirclesbelowthenanswerthefollowingquestions:

60

1) Whyaren’tyouranswersexactlythesame?2) Doesthevalueofπdependontheunitsofmeasurementthatyouuse?

Explain.WhatvanHieleleveldochildrenneedtoreachinordertodovariouspartsoftheabovecircleactivity?Manymiddleschoolstudentshaveheardthatπisanirrationalnumber,buttheyarenotclearaboutwhatthatmeans.Itdoesnotmeanthatthenumberoftimesthediameterofacirclefitsintoitscircumferenceischanginginsomeway.Itdoesnotmeanthatthenumberoftimesthediameterofacirclefitsintoitscircumferenceisn’tanexactvalue.Itisanexactvalue,andwecallitπ.Itsimplymeansthatπhasadecimalnamethatneverendsnorrepeatsandsoanywaywewriteπwithadecimalorafractionnameismerelyanapproximationofthenumberoftimesthediameterofacirclefitsintoitscircumference.Homework: Theknowledgeofwhichgeometryaimsistheknowledgeoftheeternal. Plato,Republic,VII,52.

1) GobackanddoallthethingsinitalicsintheReadandStudysection.

2) DoalltheitalicizedthingsintheConnectionssection.3) Makeyourselfadefinitionsquizandlearnalltheboldedandunderlinedtermsinthe

section(includingthosethatappearintheClassActivity).

4) Hereisalistofactivities.ClassifyeachaccordingtothevanHieleLevelthatitbestfits:a) Sortingshapesbasedonthenumberofsides.b) Arguingthatallrectanglesareparallelograms.c) Identifyingcircleshapesintheclassroom.d) DoingtheactivityonestimatingπfromtheConnectionssection.e) DoingtheTwoFiniteGeometriesClassActivity.

5) SupposetheEarthisanidealsphereandyouhavewrappedaropetightlyaroundthe

equator.Nowsupposeyouaddedenoughslacktoraisetheropeuniformlyonefootoffthegroundallthewayaroundtheequator.Howmuchlongerropewouldyouneed?Explainwhythismakessense.

6) Provethatthecircumcenterofatriangleisequidistantfromthethreeverticesofthe

triangle.Youwillhavetorelyonthewayyouconstructedthecircumcenter.YoumayuseanyofthepropositionsinBookIforthisargument.

61

7) Provethattheincenterofatriangleisequidistantfromthethreesidesofthetriangle.

Againyouwillneedtorelyonhowyouconstructedtheincenter,andyoumayuseanyofthepropositionsinBookIforthisargument.

8) Calculatethenumberoftimesthediameterofthebelowcirclefitsintotheperimeteroftheinscribedsquareandthenintotheperimeteroftheinscribedhexagon.Whatisityouaredoinghere?Ifyoudidthesamethingusinga72-sidedpolygon,whatapproximatelywouldyouranswerbe?

62

ClassActivity7:FindingFormulas

Everythingshouldbemadeassimpleaspossible,butnotonebitsimpler. AlbertEinstein

1) Usingthedefinitionofareaasthenumberofsquareunitsittakestofillatwo-dimensional

space,explainitmakessensethatareaofarectangleis(base)×(height).2) Justifythatthefollowingformulasmakesense.Ifyourearrangeanyofthefigures,you

shouldarguethatthepiecesfittogetherasyouclaim.Forexample,ifyoucutarighttriangleoffoftheparallelogramandmoveittoformarectangle,youneedtoarguethatthenewfigureisactuallyarectangle.(YoumayassumetherectangleareaformulaandanyofthepostulatesandpropositionsinBookIofElements.)

a)Areaofatriangle=½(base)×(height)


63

b)Areaofaparallelogram=(base)×(height)

c) Areaofatrapezoid=½(baseI+baseII)×(height)

64

ReadandStudy:

Thedescriptionofrightlinesandcircles,uponwhichgeometryisfounded,belongstomechanics.Geometrydoesnotteachustodrawtheselines,butrequiresthemtobedrawn.

IssacNewton,PrincipiaMathematicaInElementsEucliddidnotexplicitlydefinelength,areaorvolume–butitseemsasthoughhethoughtoftheseconstructsmuchaswedotoday–forexample,helikelythoughtof“area”astheamountoftwo-dimensionalspaceoccupiedbyanobject.Allthedefinitionswewilluseinthissectiondependoncomparinganobjecttoaunitofmeasurement.Infact,theyareallabouthowmanyunits“fit”insideanobject.Thelengthofanobjectisthenumberof1-dimensionalunits(likealinesegment)thatfitina1-dimensionalobject.Alengthunitmightlooklikethis: ____Theareaofanobjectisameasureofthenumberof2-dimensionalunits(likesolid(filled-in)squares)thatfitina2-dimensionalobject.Anareaunitmightlooklikethis:Thevolumeofanobjectisameasureofthenumberof3-dimensionalunits(solidcubesperhaps)fitina3-dimensionalspace.Avolumeunitmightlooklikethissolidblock:Thismaysoundsimple,butwecan’tbegintotellyouhowoftenstudentsareconfusedaboutthis.Askpeoplewhatareais,forexample,andmostwillrespondthatareais“basetimesheight.”Butthisisn’ttheideaofarea,itissimplyaformulaforfindinganareaofsomeveryspecificobjects(namelyparallelograms:theformuladoesn’tevenworkforotherthings).Whenyouareaskedtocomputeanarea,pleasedon’tresorttoacoupleofmemorizedformulaswithoutthinkingaboutwhatareameansandwhetherthoseformulasapply,andpleasehelpyourstudentstounderstandtheideaofmeasurement.

65


Ingrades6-8allstudentsshoulddevelopanduseformulastodeterminethecircumferenceofcircles,andtheareasoftriangles,parallelograms,trapezoids,andcircles,anddevelopstrategiestofindtheareaofmorecomplexshapes. NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematics,p.240

Tohelpyourstudentstothinkofareaasthenumberofsolidsquaresthatfillorcovera2-dimensionalobject,youmightstartbyhavingthemtracetheobjectonsquare-grid-paperandthenaskthemtoestimateandthencountthenumberofsquaresittakestofill(cover)theobject.Similarly,theycanlearntothinkofvolumeasthenumberofsolidcubesthatittakestofillathree-dimensionalobject.HerearetwooftherelevantCommonCoreStateStandardsforchildreningradesix.Readthesecarefully.

Infact,agreatwaytohelpchildrenunderstandareaformulasistohavethemseetheformulas(bycuttingandpasting)basedonformulastheyalreadyknowlikeyoudidintheClassActivity.

Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.

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

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

66

Homework:

Learningwithoutthoughtislaborlost;thoughtwithoutlearningisperilous. Confucius

1) MakesurethatyoucanjustifyalloftheformulasfromtheClassActivity.

2) Findtheareaofthepentagoninatleast3differentways.Eachsquareisonecentimeterlong.

3) Middleschoolstudentsshouldhaveavarietyofopportunitiestoseewhyitmakessensethattheareaofacircleshouldbeπ×r2(whereristheradius).Belowisapicturethatgivestheideaofanargumentforthatfact.Whatistheideahere?Whyisthisjustanideaoftheargument?

67

ClassActivity8:PlayingPythagoras

Everythingyoucanimagineisreal. PabloPicasso(TQP)

1) StatethePythagoreanTheorem.(It’snotjusta2+b2=c2.Whataretheconditionsona,bandc?Youneedanif-thenstatement.)Now,stateitsconverse.

2) YouwillconsiderwhatislikelyEuclid’sownproofofthistheoremnow.Wearegoingtoexplainthebigideasandyourgroupshouldtofollowalongandsupplythedetails.

First,Euclidclaimedthat∆FBChadthesameareaastriangle∆FBA(halfthepinksquare)becausebothtriangleshavethesamebaseandthesameheight.Makesureeveryoneinyourgroupseesandunderstandsthat.NowEuclidarguedthat∆FBCwasequalto(congruentto)∆DBA.Makethatargument.

Next,Euclidarguedthat∆DBAhadthesameareaas∆BDK(halfofthepinkrectangle)

becausebothhavethesamebaseandthesameheight.Checkitout.

ImageusedwithpermissionfromWikapedia.com

Sothatmeansthatthepinksquarehasthesameareaasthepinkrectangle.Asimilarargumentshowsthatthebluesquarehasthesameareaasthebluerectangle.Gothroughthedetailsofthatnowtobesureeveryoneunderstandsit.Sotheareasofthesquaresontherighttriangle’ssidessumtotheareaofthesquareonthehypotenuse.Now,whereintheproofdidyouneedthefactthatthetrianglewasarighttriangle?Explain.

68

ReadandStudy:

Thecowboyshaveawayoftrussingupasteerorapugnaciousbroncowhichfixesthebrutesothatitcanneithermovenorthink.Thisisthehog-tie,anditiswhatEucliddidtogeometry.

EricTempleBell InR.Crayshaw-Williams,TheSearchforTruth ThePythagoreanswereagroupofmysticsandscholarswholivedinGreeceabout400BC.Whilethereisnowrittenrecordoftheirbeliefsorwork,theyarethoughttohaveascribedtoabeliefinthemathematicalorderoftheuniverse.Theyarealsothoughttohaveprovedthetheoremthatbearstheirname–althoughtherelationshipamongthesidesofarighttrianglewasknownearlierinBabyloniaandperhapsinChina.ThePythagoreanTheoremisakeymilestoneinEuclid’sElements.EuclidarrivesatthistheoremanditsconverseasthefinalpropositionsofBook1.(Thereare13booksthatmakeuptheElements).Sowethinkthathemusthaveconsidereditofgreatsignificance,ifnotthewholepurposefordevelopingthepropositionsthatprecedeit.It’sabigdealbecauseitisthekeytodefiningEuclideandistance.We’lltalkmoreaboutthatlaterinthistext.ConnectionstotheMiddleGrades:

Iconstantlymeetpeoplewhoaredoubtful,generallywithoutduereason,abouttheirpotentialcapacity[asmathematicians].Thefirsttestiswhetheryougotanythingoutofgeometry.Tohavedislikedorfailedtogetonwithother[mathematical]subjectsneedmeannothing.

J.E.Littlewood,AMathematician’sMiscellanyThePythagoreanTheoremisoneofthoseusefultoolsforsolvingproblems;unfortunately,studentsusuallyrememberonlythea2+b2=c2part,asthoughit’sjustaformulaandnotarelationshipamongtheareasofthesquaresonthesidesofarighttriangle.Yourjobistohelpyourstudentstoseethistheorem.Theinitialstatementofthistheoremwasalwaysgivenintermsofareas.Itwentsomethinglikethis:

PythagoreanTheorem:Thesquareonthehypotenuseofarighttriangleisequaltothesumofthesquaresontheothertwosides.

Onthenextpageyouwillfindapuzzlethathelpstomakethepointsthattheareasofthesquaresonthelegsofarighttriangleexactlyfittofillupthesquareonthehypotenuse.

69

Tracethediagram,thencutoutthepartsofthesquaresonthelegsoftherighttriangleandseeifyoucanrearrangethepiecestofitthesquareonthehypotenuse(Hint:thetinysquaregoesinthemiddle).

PythagoreanPuzzle

70

HerearethreeoftheCommonCoreStateStandardsforchildreningradeeight.Readthesecarefully.

Noticehowstandard6expectsstudentstonotonlyunderstandaproofofthePythagoreanTheorem,butalsoproveitsconverse.WhatistheconverseofthePythagoreanTheorem?Stateitcarefully,thentrytoproveit!

Homework:

Theabilitytofocusattentiononimportantthingsisadefiningcharacteristicofintelligence.

RobertJ.Shiller

1) DoalltheproblemsintheConnectionssectionincludingprovingtheconverseofthePythagoreanTheorem.

2) ThepuzzlefromtheConnectionssectiononlyworkswitharighttriangle.Ifthetriangleisacute,isthesumofthetwosmallersquaresbiggerorsmallerthanthesquareonthehypotenuse?Whatifthetriangleisobtuse?

UnderstandandapplythePythagoreanTheorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsina

coordinatesystem.

71

3) TheCommonCoreStandardsforGeometryadvocatethatstudentsineighthgradelearntodothefollowingregardingthePythagoreanTheorem.Wehaveaddressedthefirststandardinthissectionandwewilldothethirdinalatersectionwhenwestudyanalyticgeometry.Herearesomeproblemstogiveyoumorepracticewiththesecond:solvingreal-worldandmathematicalproblems.

a) Theschoolis4milesdueeastofyourhouseandthemallis8milestothenorthofyourhouse.Howfarapart(asthecrowflies)aretheschoolandthemall?

b) Asquarehasadiagonaloflength10inches.Whatisitsarea?c) Forarectangularshoeboxwithsidesoflengtha,bandc,explainwhythe

diagonaldsatisfiesthe“three-dimensionalPythagoreantheorem”givenbytheequation: 2222 dcba =++ .

4) StudythediagrambelowandthenuseittoprovideanotherproofofthePythagoreanTheorem.Youmayassumethatallfourtrianglesarecongruentrighttriangles.

72

ClassActivity9:NothingbutNet

I’vefailedoverandoveragaininmylifeandthatiswhyIsucceed. MichaelJordanIfyouhaveaprismwithasquarebasewithsidelengthbandaheighth,thenitssurfaceareaandvolumearegivenbytheformulasbelow:

Volume=b2h

SurfaceArea=2b2+4bh

1) Buildarightprismwithasquarebaseoutofpaperandverifytheaboveformulas.

2) Anon-rightprismiscalledanobliqueprism.Hereisapictureofone:

Supposethatyouhaveanobliqueprismwithheighthandasquarebasewithsidelengthb.Doestheaboveformulaforvolumestillhold?Buildsomeobliqueprismsandexplainwhatyousee.

Doestheformulaforsurfaceareastillhold?Explain.

3) Seeifyoucanmakeanetforanoblique(non-rightcylinder)liketheoneshownbelow.Whatareyourconjecturesaboutthevolumeandsurfaceareaofanobliquecylindercomparedtoarightcylinderwiththesameheightandradius?

73

ReadandStudy:

Yougottoknowwhentohold‘em,knowwhentofold‘em… TheGamblerbyDonSchlitz

Anetforathree-dimensionalobjectisatwo-dimensionalpatternthatcanbefoldedtomaketheobject.So,forexample,hereisapictureofanetthatcanbefoldedtomakeacube.Mentallyfolditup.

Thereareseveralnetsthatfoldtomakeacube.IntheHomework,yougettofindthemall.Netsareusefulforstudyingobjectslikepolyhedra.Apolyhedronisasurfaceofathreedimensionalobject.Inordertobeapolyhedron,thatsurfacemustbeclosed,simple(haveonlyonechamber),andcomposedentirelyofpolygons.Theprismsandpyramidsthatyouworkedwithintheclassactivitywerebothexamplesofpolyhedra.Thepolygons(andtheirinteriors)thatcomposethesurfaceofthepolyhedraarecalledfaces.Thefacesmeetpairwisealongedgesandtheedgesmeetotheredgesatvertices.Howmanyofeach:faces,edgesandvertices,doesthecubehave?Aregularpolyhedronisapolyhedronmadeupofentirelyofcongruentregularpolygonfacesinsuchawaythatallthevertexarrangementsarethesame.Sothecubeaboveisanexampleofaregularpolyhedronbecauseitiscomposedentirelyofcongruentsquarefaceswithexactlythreefacesmeetingateachvertex.Itturnsoutthereareonlyfiveregularpolyhedra.Inordertounderstandthisargumentyouwillneedtocutoutallofthetriangle,square,pentagonandhexagonfacesinAppendixCandfindsometape.Takeafewminutestodothosethingsnow.Startwiththeequilateraltriangles.Noticethatyouneedtoarrangeatleastthreeatavertexinordertofoldathree-dimensionalobject.Maketheregularpolyhedronthathasexactlythreetrianglefacesmeetingateachvertex.Itiscalledatetrahedron.

74

Nowseeifyoucanfitfourtrianglesateachvertex.Buildthatregularpolyhedron.Itiscalledanoctahedron.Finallynoticethatyouhaveroomtofitfivetrianglesatavertexandstillbeabletofolditup–butwithsixtrianglesatavertexthethingliesflatontheplaneandcannotbefolded.Sothatmeansthatthereareonlythreeregularpolyhedrathatcanbebuiltofequilateraltriangles.Hereiswiremodeloftheregularpolyhedronwithfivetrianglesatavertex.Itiscalledanicosahedron.

Okay.Let’smoveontosquares.Weknowwecanfitthreeatavertexandthatgetsusthecube.Canyoubuildsomethingwithfouratavertex?Morethanfour?Ineachcase,eitherdoit,orexplainwhynot.Thereisonemoreregularguythatiscomposedentirelyofpentagons.

Wecannotbuildaregularpolyhedronwithonlyhexagonsbecausethreeatavertexlieflatandcannotbefolded.(Tryit.)Polygonswithevenmoresidesthanahexagondonotworkeitherbecausetheycannotbefoldedintothreedimensions.Sothatmeanstherecanbeonlyfiveregularpolyhedra.Makesurethatyouunderstandthisargument.

75


Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.

CommonCoreStateStandardsforMathematics,p.6.TheCommonCoreStandardsforgradeeightrequirethatstudentssolvereal-worldproblemsinvolvingvolumeofcylinders,conesandspheres.

Acylinderissimilartoaprisminform.Ithasacircularbaseofradiusrandaheighth.YouprobablyarguedaspartoftheClassActivitythatthevolumeofacylinderisπ×r2×h.Nowimagineacompatiblecone(onewiththesameradiusandheight)livinginsidethecylinder.Itsvolumeisonethirdofthecylinder’svolumeor1/3×π×r2×h.Thisformulaisdifficulttoderive–butyoucanhelpyourstudentstoseetherelationshipbetweenthevolumesoftheseobjectsbypurchasingcompatibleplasticmodelsandhavingthestudentsseethatittakesthewaterfromthreeconestofillthecylinder.Asphereisthesurfaceofaball.Thevolumeofasolid(filled)sphereisgivenbytheformula4/3×π×r3whereristheradiusofthesphere.Imagineaspherelivinginsidetherightcylinder.Youcanshowyourstudentsthatinordertofillthecylinder,youneedthewaterfromonecompatibleconeandonecompatiblesphere.Sincethevolumeoftheconeis1/3×π×r2×h,thevolumeofthespheremustbetherest.Dothecalculationtoshowthatthe(volumeoftheshowncylinder)–(volumeofaconeofthesameheight)doesyougiveyouthevolumeoftheshownsphere.

• Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.

76

UsedwithpermissionfromWikipedia.com

YouwillsolvesomemoreproblemsinvolvingthevolumesoftheseobjectsaspartoftheHomeworksection.

Homework:

Doingisaquantumleapfromimagining. BarbaraSher

1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.

2) TheCommonCoreStateStandardsforstudentsingradesixincludethefollowing:

Herearesomeproblemsthatmightmeetthisstandard:

a) Arectangularroomis15feetlongby10feetwideandhasan8footceiling.Builda(scaleddown)modelfortheroomusinganet.

b) Youwanttopaintthewallsandceilingandsoneedtoestimatetheamountofpaintyouwillneed.Ifagallonofpaintcovers200squarefeet,howmanygallonsshouldyoupurchase?Explainyourwork.

3) Carefullymakeanetforarightcircularcylinder.Whatisaformulaforsurfaceareaofa

rightcircularcylinder?Whatistheformulaforitsvolume?Explainyouranswerineachcaseasyouwouldtomiddlegradesstudents.

4) Ifyoudoubledeachlineardimensionofyourcylinder(radiusandheight),whatwouldhappentothesurfacearea?Thevolume?Explain.

• Representthree-dimensionalfiguresusingnetsmadeupofrectanglesandtriangles,andusethenetstofindthesurfaceareaofthesefigures.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

77

5) Howmanydifferentnetsarepossibleforacubethatmeasures1inchonaside?Sketch

themandarguethatyouhavethemall.

6) Ifyoudoubledeachlineardimensionofyourcube(i.e.,gofrom1×1×1to2×2×2)whatwouldhappentothesurfacearea?Thevolume?Explain.

7) Arecones,spheres,orcylindersexamplesofpolyhedra?Whyorwhynot?

8) Anicecreamsnackiscomposedofaconewithhalfasphereontop.Whatisthevolumeofthesnackiftheconehasradius3cmandaheightof8cm?

9) Goonlineandsearchfor“netsfortheregularpolyhedra.”Printoutandbuildeachofthefive.YouwillneedtheseforClassActivity13.

78

ClassActivity10:Slides,TurnsandFlips ThelawsofnaturearebutthemathematicalthoughtsofGod. EuclidTherearethreebasicrigidmotionsoftheplane–waystomovetheplanewithoutdistortingit.Youmayhavelearnedabouttheminformallyinmiddleorhighschoolorperhapsinanearliercourse.Hereyouwillstudytheprecisedefinitionsforthoserigidmotionsandyouwillusethosedefinitionstofigureouthowtoconstructeachrigidmotion.AtranslationbyavectorRSisamotionoftheplanesothatifAisanypointintheplaneandwecallA’theimageofA,thenvectorAA’andvectorRShavethesamelengthanddirection.WewilldenotethistranslationTRS.

1) Construct∆A’B’C’(theimageof∆ABCunderthetranslationTRS)andthenprove,usingthedefinitionofatranslation,thatyouhavedoneso.

R

B S A C


79

Arotation(aboutcenterpointPofangleφ)isamotionoftheplaneinwhichtheimageofPisitself,andiftheimageofAisA’thenPA’iscongruenttoPAandthemeasureofangleAPA’=φ.WewilldenotethisrotationR(P,φ).

2) Construct∆A’B’C’(theimageof∆ABCundertheclockwiserotationR(P,φ))andthenprove,usingthedefinitionofarotation,thatyouhavedoneso.

B φ A C


P

80

Areflection(inlinem)isamotionoftheplaneinwhichtheimageofapointonmisitself,andifAisnotonmandA’istheimageofA,thenmistheperpendicularbisectorofAA’.WewilldenotethisreflectionMm.(Mformirror.)

3) Construct∆A’B’C’(theimageof∆ABCunderthereflectionMm)andthenprove,usingthedefinitionofareflection,thatyouhavedoneso.

m

B C A

81

ReadandStudy:

Saywhatyouknow,dowhatyoumust,comewhatmay.SonjaKovalevsky(Mottoonherpaper"OntheProblemoftheRotationofaSolidBodyaboutaFixedPoint.")

Informally,arigidmotionoftheplaneisonethatdoesnotcausedistortion.Youcanthinkofrigidmotionslikethis:supposethatyousetaninfinitepieceofpaperonthetableinfrontofyouandpicturethatpaperasrepresentingthesetofpointsontheplane.Now,whatcanyoudotomovethispapersothatintheenditisbackflatonthetable?Well,youcouldspinitaround(i.e.,performarotation);youcouldflipitover(i.e.,performareflection);youcouldslideitinsomedirection(i.e.,performatranslation);oryoucoulddosomecombinationofthesemoves.Ifyoustretchthepaper,crumpleit,ortearit,thatyouhavenotperformedarigidmotion.Hereisamoretechnicaldefinition.ArigidmotiononasetSisatypeoffunction(transformation)fromSbacktoSthatpreservesthedistancebetweenpoints.So,iftwopointsPandQwere3unitsapartbeforetherigidmotion,thentheirimagesP’andQ’are3unitsapartafterwards.Thisensuresthe‘nodistortion’rule.Itisimportanttonotethatwhenweperformarigidmotion,theentireplanemoves–notjusttheobjectsontheplane.Forexample,whenyouconstructedtheimageofthetriangleunderthereflection,thenewtrianglethatresultedfromtherigidmotion(oftencalledtheimageoftherigidmotion)justshowedwhereintheplanetheoriginaltrianglemoved.Itdidnotresultinasecondtrianglebeingplacedontheplane.Thisisaveryimportantideamakesureyouunderstandit.ApointPisafixedpointoftherigidmotioniftheimageofPisPitself.Makessenseright?Fixedpointsarethosethatdonot“move”undertherigidmotion,or,saidanotherway,fixedpointsarethepointsthatgetpairedwiththemselvesunderthefunction.Adilationisanexampleofamotionoftheplanethatisnotrigid–informally,adilationisastretchingorashrinkingoftheplane(andalloftheobjectsontheplane)inauniformmanner.Youcanperformrigidmotionsoneaftertheother.Inthatcasewesayyouhaveperformedacompositionofrigidmotions.Forexample,youcoulddoarotationfollowedbyareflection.Oratranslationfollowedbyanothertranslation.Youcanalsocomposerigidmotionswithdilations.Thisbringsustotwoimportantdefinitions:Twogeometricobjectsarecongruentifoneistheimageoftheotherunderarigidmotion(orcompositionofrigidmotions)oftheplane.Twogeometricobjectsaresimilarifoneistheimageoftheotherunderacompositionofrigidmotionsanddilations.(Inotherwords,objectsaresimilarifonecanmovedandshrunk(ormade

82

larger(dilated))sotocoincidewiththeother.)Similarobjectsaretheshapebutnotnecessarilythesamesize.Forexample,thesesnowflakesaresimilarbutnotcongruent.Wewilldiscussthisideafurtherinanupcomingsection.


Ingrades6-8allstudentsshoulddescribesizes,positions,andorientationsofshapesunderinformaltransformationssuchasflips,turns,slides,andscaling.

NationalCouncilofTeachersofMathematics

PrinciplesandStandardsforSchoolMathematics,p.232TheCommonCoreStateStandardsrecommendthatstudentsinGrade8learntodoandunderstandthefollowing.Readthesecarefully.

83

Middlegradesstudentswilllikelynotconstructrigidmotions;rathertheywillusegraphpaperortracingpapertostudytheminformally.Toseewhatwemean,getafewpiecesoftracingpaperandre-dotheclassactivitybytracingthefiguresandmovingyourpaper.

Homework:

Withregardtoexcellence,itisnotenoughtoknow,butwemusttrytohaveanduseit.

Aristotle1) DoalltheitalicizedthingsintheReadandStudyandConnectionssections.2) First,translatequadrilateralABCDbythetranslationvectorRS,thenapplythetranslation

vectorSTtoA’B’C’D’.Ineachcaseconstructthetranslation.

S

GeometryGradeEight:Understandcongruenceandsimilarityusingphysicalmodels,transparencies,orgeometrysoftware.

1. Verifyexperimentallythepropertiesofrotations,reflections,andtranslations:a. Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.b. Anglesaretakentoanglesofthesamemeasure.c. Parallellinesaretakentoparallellines.

2. Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.

3. Describetheeffectofdilations,translations,rotations,andreflectionsontwo-dimensionalfiguresusingcoordinates.

4. Understandthatatwo-dimensionalfigureissimilartoanotherifthesecondcanbe

obtainedfromthefirstbyasequenceofrotations,reflections,translations,anddilations;giventwosimilartwo-dimensionalfigures,describeasequencethatexhibitsthesimilaritybetweenthem.

84

T R IdentifythesingletransformationthattakesABCDtoA’B’C’D’

3) First,rotateABCDclockwisearoundPby90°,thenrotateA’B’C’D’clockwisearoundPby60°.Ineachcase,constructtherotation.

IdentifythesingletransformationthattakesABCDtoA’B’C’D’

PC

A

B

D

85

4) First,reflectABCDoverlinem,thenreflectA’B’C’D’overlinek.Assumethatlinemisparalleltolinek.Ineachcase,constructyourreflection.

IdentifythesingletransformationthattakesABCDtoA’B’C’D’.

5) First,translateABCDbythetranslationvectorRS,thenreflectA’B’C’D’overlineRS.Youdonotneedtoconstructtheserigidmotions.Usethegridtoperformthem.

Wecalltheresultaglidereflection,alsoknownasa“slideflip.”Inaglidereflectionthetranslationvectorisalwaysparalleltothelineofreflection.Acommonexampleofaglidereflectionisasetoffootprintsinsand.

km

C

AD

B

R

C

A

B

D

S

86

6) Howmanyfixedpointsdoeseachofthefollowingrigidmotionhave?Ineachcase,explain.

a) Translationb) Rotationc) Reflectiond) GlideReflection

ClassActivity11:TransformativeThinking

Themathematicalsciencesparticularlyexhibitorder,symmetry,andlimitation;andthesearethegreatestformsofthebeautiful.

AristotleInthisactivityyouwillcontinuetoinvestigatetheresultofperformingtworigidmotionsoftheplane,onefollowingtheother.Thefirstrigidmotionwillbeappliedtotheoriginalobject.Thesecondwillbeappliedtoimageofthefirst.Recallthatthisprocessofapplyingtwomotionsconsecutivelyiscalledcomposition.Yourjobistoclassifyallpossiblecompositionsoftherigidmotionsoftheplane.

°

TranslationTRS

RotationR(P,φ)

ReflectionMl

GlideReflectionG(RS,l)

Tran

slatio

nT P

Q

PQandRSparallel

PQandRSnotparallel

87

Itturnsoutthateveryrigidmotionoftheplaneendsupbeingarotation,areflection,atranslationoraglidereflection.Discusswhatwemightmeanbythis,andexplainitinyourownwords.

ClassActivity12:ExpandingandContracting

GivemeextensionandmotionandIwillconstructtheuniverse. ReneDescartesAnothermotionoftheplaneisadilation.Youhaveexperienceddilationswheneveryoushrinkorenlargeaphotographwithoutdistortingtheimage.Formally,adilation(aboutpointPwithscalefactorq)oftheplaneisamotionoftheplaneinwhichtheimageofPisitselfandiftheimageofAisA’thenPA’=q(PA)andP,A,andA’arecollinear.Inadilation,Pisthecenterofthedilationandqisthescalefactor.

1) Unliketherigidmotions,dilationsarenotalwaysconstructible.(i.e.Youcannotalwaysmakethemwithacompassandstraightedgealone.)Whyisthisthecase?

RotationR (

Q,θ

)

P=Q

P≠Q

ReflectionM

n

nandlparallel nandlintersect nandlparallel nandlintersect

GlideRe

flectionG (

PQ,n)

nandlparallel nandlintersect

88

2) Onaseparatesheetofpaper,drawseveraldilationsofrABC.Experimentwiththe

locationofthecentralpointandthevalueofthescalefactor.Then,answerthequestionsbelow.

a. Howdoestheplacementofthecenterpointaffecttheresultingshape?

b. Howdoestheshapechangeifthescalefactorisgreaterthanone?Between0and1?Equalto1?

c. Whathappensifq=0?q<0?

ReadandStudyInthephysicalworld,onecannotincreasethesizeorquantityofanythingwithoutchangingitsquality.Similarfiguresexistonlyinpuregeometry.

PaulValéry

Dilationsareanexampleofamovementoftheplanethatisnotarigidmotion.Whenyoucreatedyourdilationsintheclassactivity,youwerecreatingshapesthatweresimilartotheoriginalfigure.Recallthattwofiguresaresimilarifoneistheimageoftheotherunderacompositionofrigidmotionsanddilations.Forexample,thefollowingfiguresaresimilar.Seeifyoucandetermineasequenceofrigidmotionsanddilationswhichmaponeofthefiguresbelowontotheother.

89

Asaconsequenceofthedefinitionofsimilar,weknowthattwopolygonsaresimilariftheircorrespondingvertexanglesarecongruentandcorrespondingsidesareproportional.Takeamomentandthinkaboutwhythisisthecase.Trianglesarereallyspecialpolygonsinthefactthatwedonothavetocheckallthesidesandalltheanglestodetermineiftwotrianglesaresimilar.Weonlyhavetocheckoneofthefollowing:

• Angle-Angle-AngleSimilarityTheorem:Iftwotriangleshavecorrespondinganglescongruent,thenthetrianglesaresimilar.(ThistheoremissometimescalledtheAAtheorembecausecheckingtwoanglesissufficientforprovingthattwotrianglesaresimilar.Whyisthisthecase?)

• Side-Angle-SideSimilarityTheorem:Iftwotriangleshavetwopairsofcorrespondingsidesproportionalandtheincludedanglescongruent,thenthetrianglesaresimilar.

• Side-Side-SideSimilarityTheorem:Iftwotriangleshaveallthreepairsofcorrespondingsidesproportional(withthesameconstantofproportionality),thenthetrianglesaresimilar.

Thesetheoremsonlyworkfortriangles.Why?Whathappenswhenyoutrytoapplythemtootherpolygons?

Homework

Apupilfromwhomnothingiseverdemandedwhichhecannotdo,neverdoesallhecan.JohnStuartMill


2) Determineifthefollowingstatementsaretrueorfalse.Makesureyoucanexplainwhyineachcase.

a. Therearenofixedpointsinadilation.b. Anglesarepreservedunderadilation.c. Iftwolinesegmentsareparallelbeforeadilation,theywillbeparallelafterthe

dilation.d. Linesegmentlengthsarepreservedunderadilation.

3) Areallrectanglessimilar?Eitherprovethattheyareorprovideacounterexample

explainingwhytheyarenot.

4) Determineifeachpairoftrianglesbelowaresimilar.Iftheyaresimilar,findthemissingparts,ifnot,explainwhynot.

90

5) Ifthescalefactorinadilationisk,whatistheratiooftheareaoftheresultingshapeascomparedtotheoriginalshape?

6) InthefollowingfigureassumethatÐACBisarightangleandlinesegmentCDisperpendiculartolinesegmentAB.Whyare∆ABC,∆ACD,and∆CBDallsimilar?Showthatcy=a2andcx=b2andthenusethesefactstodevelopacarefulproofofthePythagoreanTheorem.

91

ClassActivity13:StrictlyPlatonic(Solids)

Themostgenerallawinnatureisequity–theprincipleofbalanceandsymmetrywhichguidesthegrowthofformsalongthelinesofthegreateststructuralefficiency.

HerbertRead Youwillneedtobuildmodelsoftheregularpolyhedrainordertocompletethisactivity–netsareavailableonline.

92

Three-dimensionalobjects,includingtheregularpolyhedra,canhaverotationalandreflectionalsymmetries.Forrotationalsymmetry,thecenterofrotationisreallyalineofrotation(calledtheaxisofsymmetry).Therecanbemorethanoneaxisofsymmetryforathree-dimensionalobject.Forexample,thecubehasthreeaxesofsymmetryoforder4connectingthecentersofoppositefaces,fouraxesofsymmetryoforder3connectingdiagonallyoppositevertices,andsixaxesoforder2connectingmidpointsofoppositeedges.Theorderofalineofsymmetryisthenumberofturnsthatputtheobjectbackonitself.Herearethethreeorder-4axesofsymmetryforacube.Takeaminuteinyourgroupstobesurethateveryoneseeswhyeachofthesehasorder4.Thensketchtherestoftheaxesofsymmetryforacube.

Thecubealsohasreflectionalsymmetry.Thelineofreflectionbecomesaplaneofreflectionthatdividesthecubeintotwomirrorimages.Therearenineplanesofreflectionalsymmetry,twovertical,onehorizontalandtwothroughthediagonalsofeachpairofoppositefaces.Findeachplaneofsymmetryonyourmodelofthecube.Imagineslicingyourcubealongeachplane.Youshouldbeabletovisualizethetwocongruent“half-cubes”thatwouldresult.YourjobforthisClassActivityistofindanddescribealltheplanesofreflectionalsymmetryandalltheaxesofrotationalsymmetryfortheotherfourregularpolyhedra.Completethetableandthendescribeanypatternsyousee.

Polyhedron #anddescriptionofplanesofreflectionsymmetry

#anddescriptionoflinesofrotationsymmetry

RegularTetrahedron

93

Cube

RegularOctahedron

RegularDodecahedron

RegularIcosahedron

ReadandStudy:

Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple. S.Gudder

Thegeometricideaofsymmetryisdefinedintermsofrigidmotions.Hereistheofficialdefinition.Asymmetryofageometricobjectisarigidmotionoftheplaneinwhichtheimageoftheobjectcoincideswiththeoriginalobject.

94

Stopandthinkaboutthisdefinitiontobesureitmakessensetoyou.Let’scharacterizeanobjectintheplanebasedonitssymmetries.Havealookatthetwo-sidedarrowbelow.Thisobjecthastworeflectionsymmetriesbecausereflectionsovereitherlineshownbelowwillresultintheimagecoincidingexactlywiththeoriginalobject.Thetwo-sidedarrowalsohas180-degreerotationalsymmetryaroundthecenter(wherethetwolinesaboveintersect).Italsohas360-degreerotationalsymmetry(wecallthatthetrivialsymmetrybecauseeveryobjecthasit).Drawanobjectthathas90,180,270and360rotationalsymmetriesandnoothersymmetries.Whattypesofobjectswillhavetranslationalsymmetries?Havealookbackatthetableofsymmetriesyoumadefortheregularpolyhedra.Whatdoyounotice?Onethingthatwenoticedwasthatthecubeandtheoctahedronhaveexactlythesamesetofsymmetries,andthatthedodecahedronandtheicosahedronalsosharethesamesymmetries.Sowhatisitaboutthesepairsofobjectsthatwouldhavethatbethecase?Let’sstartwiththecubeandtheoctahedron.Imaginetakingthemidpointofeachfaceofthecubeandthinkingofthoseastheverticesofanewpolyhedron.Thenthatnewpolyhedronwouldhave6vertices.Seeifyoucansketchthatnewpolyhedroninsideofthecube.

95

Now,seeifyoucansketchthepolyhedronthatcouldbeformedbyusingthemidpointsofthefacesoftheoctahedronasvertices.Howmanyverticeswouldthatnewpolyhedronhave?

Wecallobjectsthatarerelatedinthiswayduals.Thecubeandtheoctahedronaredualsofeachother,andthedodecahedronandtheicosahedrtonarealsodualsofeachother.Takeacloselookatyourmodelsofthedodecahedronandtheicosahedrontoseeifyoucantellthattheyareduals.Wemightevenimaginehowtheobjectswouldfitinsideoneanother.

96

Dualshavethesamesymmetriesbecausetheywouldmovetogetherunderrotationsandreflections.Youmayhavenoticedthatwehaveleftoutthetetrahedron.Whatisitsdual?Seeifyoucansketchit.

97

Homework:

Youteachbestwhatyoumostneedtolearn. RichardBach


2) Sketchanobjectintheplanethatmeetseachsetofcriteriaorexplainwhyitisimpossibletodoso:

a) Theobjecthasonly360-rotationalsymmetry.b) Theobjecthas120,240and360-degreerotationalsymmetriesandnoother

symmetries.c) Theobjecthas120and360-degreerotationalsymmetryandnoothersymmetries.d) Theobjecthasverticalreflectionsymmetry,360-degreerotationalsymmetryand

noothersymmetries.e) Theobjecthasverticaltranslationsymmetry,360-degreerotationalsymmetryand

noothersymmetries.

3) Anobjectintheplanehastwolinesofsymmetry.Iftheselinesareparallel,whatothersymmetriesmustthisobjecthave?Why?

4) Findallofthesymmetriesofthethree-dimensionalsquare-basedpyramidshownbelow.

5) Provethatifanobjectintheplanehastwointersectinglinesofsymmetry,thenitmustalsohaverotationalsymmetry.

6) Describeconditionswhichwouldguaranteethatarightprismhasexactlyoneplaneofreflectionalsymmetry.Whereistheplanelocated?

7) Describeconditionswhichwouldguaranteethatanobliqueprismhasexactlyoneplaneofreflectionalsymmetry.Whereistheplanelocated?

98

8) Buildthefollowingmodels.Thenfindtheirsurfaceareas,volumes,anddescribealltheirsymmetries.

a) Arightcircularcylinderwithradius2cmandheight7cm.b) Asquare-basedpyramidwitha4cmby4cmbaseandaheightof5cm.c) Arightprismwiththebelowregularhexagonasthebase,andaheightof8cm.

99

ClassActivity14:BuriedTreasure

I'mverywellacquaintedtoowithmattersmathematical,Iunderstandequations,boththesimpleandquadratical,AboutbinomialtheoremI'mteemingwithalotofnews--Withmanycheerfulfactsaboutthesquareofthehypotenuse.

Gilbert&Sullivan,"ThePiratesofPenzance"

Thesneakypirateandthefirstmateburiedtreasureonanislandwithtwolargerocksandpalmtreeneartheshore.You’vefoundthetop-secretmapthatexplainsthelocationofthebountyasfollows:Me captain started at the palm tree and paced off the distance to the first rock, turned 90º in a counterclockwise direction and paced off an equal distance. Argh. I, the matey, started at the palm tree and paced off the distance to the second rock, then turned 90º in a clockwise direction and paced off an equal distance. We then buried the treasure halfway between us two. Youarestandingontheislandandtherocksarestillthere,but,sadly,thepalmtreehaslongsincediedandyouhavenoideawhereitwas.Findthetreasure.

100

ReadandStudy:

Equationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsofgeometry. StephenHawking

Inthe1700’sRenéDescartes(pronouncedDay-cart)hadtheideathatwecouldsolvesomegeometricproblemsmoreeasilybytranslatingthemintoalgebraicproblems.Hisideawastoplaceastructure(agrid)ontopoftheEuclideanplaneandtogivenames(like(-3,-1))tothepoints.Oneversionofthestorygoeslikethis:Descarteswasnotanearlyriser,butheenjoyedlyingaroundinbedandthinkingdeeply.(Descartesiscreditedwiththequote,“Ithink,thereforeIam.”)Onemorning,whileponderingtheceilingofhisbedroom,henoticedaflywalkingacross.Ashementallytracedthepathofthefly’swalkheconsideredhowhecoulddescribethepathmathematically.Hereasonedthathecouldlabelanyonepointonthepathbyhowfartheflywasfromthesouthwallandhowfaritwasfromthewestwallofhisroom.Thuswasborntheideaofthecoordinateplaneuponwhichwecan“see”the“path”ofafunction’sgraph.Thecoordinateplane(alsocalledtheCartesianplaneinDescartes’honor)isafamiliarfeatureofmiddleschoolandhighschoolalgebracoursesasstudentslearntographlinear,quadratic,exponential,andotherfunctions.Youmayrecallthatitfeaturestwoperpendicularaxes,thehorizontalx-axisandtheverticaly-axis,whichintersectatapointcalledtheorigin.Wethenlabeleachpointontheplanewithanorderedpairofcoordinates(x,y),wherethex-coordinatetellsushowfarthepointisfromtheorigin(0,0)inthehorizontaldirectionandthey-coordinategivesthedistancefromtheoriginintheverticaldirection.Forexample,thepoint(-3,-1)islocated3unitstotheleftand1unitdownfromtheorigin.

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

D: (2, - 3)

C: (4, 0)

B (-2, 5)

A (- 3, -1)

origin

y-axis

x-axis

101

UsingthePythagoreanTheoremwecanfindthedistancebetweenanytwopointsontheCartesianplane.Forexample,let’sfindthedistancebetweenpointsAandDinthepictureabove.ThelinesegmentADisthehypotenuseofarighttrianglewithahorizontallegoflength5(2–(-3))andaverticallegoflength2((-1)–(-3)).SothesquareofthedistancebetweenAandDis52+22=25+4=29andthedistancebetweenAandDis 29 .Findthedistancebetween(2,-3)and(4,0).

ThereareseveralfactsaboutlinesontheCartesianplanethatareusefultorecall.Oneisthateverylinehasaslope,whichisameasureofitsinclinationwiththex-axis.Theideaofslopeisthatitistheamountyouneedtomoveinthey-directiontostayonthelineforaoneunitchangeinthex-direction.Sothinkaboutthis.Whatdoesaslopeof7mean?Sketchalinewiththatslope.Whatdoesaslopeof-¼mean?Sketchalinewiththatslope.Wecancalculatetheslope(m)ofalinebyusingthecoordinatesoftwopointsthatlieonthelinewiththeformula

𝑚 = /01/230132

where ),( 11 yx and ),( 22 yx arethecoordinatesofthetwopoints.Justtojogyourmemory,computetheslopeofthelinecontainingthepoints(4,0)and(-2,5).Iftwolinesareparallel,thentheywillmakethesameanglewiththex-axis(atransversal)andsowillhavethesameslope–andviceversa,iftwolineshavethesameslope,thentheyareparallel.Thinkabouthowyoucouldmakeanargumentforthisfact.Thisturnsouttobeaveryusefulobservation.Ifweneedtoshowthattwolinesareparallel,wecansimplycalculatetheirslopesandshowthattheyareequal.(Rememberthiswhenyougettothehomeworkproblems.)Whatiftwolinesareperpendicular?Howaretheirslopesrelated?Itturnsoutthattheslopesofperpendicularlinesalsohaveanumericalrelationship.Theproductoftheslopesofperpendicularlinesisalways-1.Thinkabouthowyoucouldmakeanargumentforthisfact.Whatwouldbetheslopeoftheperpendiculartothelinecontainingthepoints(4,0)and(-2,5)?Andhereisthelastuseful“fact”aboutusingcoordinatesontheCartesianplanethatweneedforourwork.Thecoordinatesofthemidpointofthelinesegmentconnecting ),( 11 yx and ),( 22 yx are

++2

,2

2121 yyxx .

Makeanargumentforthisfact.Whatarethecoordinatesofthemidpointofthelinesegmentconnectingthepoints(4,0)and(-2,5)?

102

Sowhatdoesallofthishavetodowithusingalgebratosolvegeometricproblems?ThatwasthegeniusofDescartes’invention.We’llshowyouanexample.Considerthefollowinggeometricproblem:Showthatthesegmentsjoiningthemidpointsoftheoppositesidesofaquadrilateralbisecteachother.Sowehaveanyquadrilateral,nothingspecialaboutit,butifweconnectthemidpointsofitsoppositesides,thosesegmentswillcuteachotherintotwoequallengthpieces.Drawasketchtoseethatthisseemstrue.Now,let’sseeifwecanprovethisusingthestructureoftheCartesianplanetohelpusout.Ourfirststepistochoosefourrandompointsandletthembetheverticesofourquadrilateral–remembernothingspecialallowed–noparallelsides,nocongruentsides,etc.Butwecanchoosesomeeasy-to-usepoints(suchastheorigin)fortwoofourpoints.(Theproblem-solvercanlaydownthestructurewhereverwelike.)Wewilllabelourpointswithcoordinates(0,0),(a,0),(b,c),and(d,e).Eventhoughwehavetoplacethesepointsinparticularspotsonourdiagram,wearemakingnoassumptionabouttheactualvaluesofa,b,c,d,ande.Nextwe’llconnectourfourpointstomakethequadrilateralandthencalculatethecoordinatesofthemidpointsofeachofthesidesusingthemidpointcoordinateformulawetalkedaboutearlier.Carryoutthesecalculationsforyourself.Doyougetthesameresults?Ourdiagramnowlookslikethis:

M3: ((b+d)/2, (c+e)/2)

M4: (d/2, e/2)M1: (a/2,0)

M2: ((a+b)/2, c/2)

O: (0, 0) A: (a, 0)

B (b, c)

A (d, e)

y-axis

x-axis

103

Lastly,weareinterestedinthetwosegmentswhichjoinmidpointsofoppositesides,thatissegmentM1M3andsegmentM2M4.Weneedtoshowthattheybisecteachotherattheirpointofintersection.Thinkcarefullyforafewminutes–howcanweshowthis?Thereareseveralapproachesthatwilldothejob,butsomeareeasierthanothers.Decideonamethodthatmakessensetoyouandfinishtheproofbeforereadinganyfurther.(Hey,reallydoit.)OnewaytoshowthatM1M3andM2M4bisecteachotheristofindtheequationofeachlineandsolvethissystemoftwoequationsforthecoordinatesoftheircommonpoint(let’scallitM).ThenwewouldfindthedistancefromthatpointtoeachofthepointsM1,M2,M3,andM4.IfthedistancefromMtoM1andthedistancefromMtoM3wereequalandifthedistancefromMtoM2

andthedistancefromMtoM4wereequal,wearedone.ExplainwhyshowingthatthesepairsofdistancesareequaldoshowthatM1M3andM2M4bisecteachother.Anothereasierwaymightbetoarguelikethis:IfM1M3andM2M4bisecteachother,thenthemidpointofeachsegmentmusthavethesamecoordinates.Explainthelogicofthisstatementbeforecontinuing.Sowecansimplyfindthecoordinatesofthemidpointofeachsegmentanddemonstratethatthesetwomidpointsareindeedthesamepoint.Welikethisapproachbecauseitissimplertocarryout.Sohereareourcalculations.Makecertainyoucangetthesameresults.

CoordinatesofthemidpointofM1M3= +++=

++

++

4,

422

0,

222 ecdba

ecdba.

CoordinatesofthemidpointofM2M4= +++=

+++

4,

4222,

222 ecdba

ecdba.

Sohereisthepoint:wejusttookapurelygeometricproblem,translatedittoanalgebraicproblem(or,saidanotherway,weimposedanalgebraicstructure)andthenweusedalgebratosolveit.Wecallthisapproachanalyticgeometry.

104


Ingrades6-8allstudentsshouldusecoordinategeometrytorepresentandexaminethepropertiesofgeometricshapes.

NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematics,p.232

Representationalsystemsarestructuresthathelpustomakesenseofgeometricobjects.Examplesincludegrids,thecoordinateplane,linesoflongitudeandlatitudeonaglobe,mapsandcontourmaps.Ofcourse,themostimportantoftheseinmiddlegradesmathematicsisthecoordinateplanebecauseitlaysthegroundworkforgraphingalgebraicrelationships.HereistherelevantCommonCoreStateStandardforGeometryforstudentsingradesix:

Makeupanexampleofareal-worldproblemthatwouldhaveyourstudentsapplythetechniquesdescribedabove.Youwillteachsomeanalyticgeometry.Youwillteachyourmiddlegradesstudentsthatthegeometricobjectcalledalinecanbedescribedbyanequation.Ithastheformy=mx+bwheremtellshowsteepthelineisandbgiveitspositiononthecoordinateaxes(biscalledthey-intercept).Youwillalsoteachyourstudentsthataparabola(alsoageometricobject-justwaituntilthenextClassActivity)hasanequationoftheformy=a(x–k)2+h,where(k,h)givesthevertexoftheparabolaandatellshow“fat”theparabolais.Fillinthevaluesofa,kandh(justmakethemup)andthengraphtheequationonyourcalculator.Nowchangeavalueanddoitagain.

• Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

105

Homework:

EachproblemthatIsolvedbecamearulewhichservedafterwardstosolveotherproblems.

ReneDescartes


2) DotheConnectionsproblems.

3) Ifyouhaven’talreadydoneso,provethattwolinesareparallelifandonlyiftheyhavethesameslope.

4) ApplythePythagoreanTheoremtothepoints ),( 11 yx and ),( 22 yx toderivetheformulaforfindingthedistancebetweentwopointsonacoordinategrid:

𝒅 = 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐

5) Usethemethodsofanalyticgeometrytoshowthatthefourmidpointsofanyquadrilateral

alwaysformaparallelogram.

6) Useanalyticgeometrytodeterminethecurvethatthemidpointofaladdermakesasthetopoftheladderslipsdownawallandthebottomoftheladdermovesawayfromthewall.(Hint:Drawadiagram.Wouldthisbethesameasfindingthesetofallmidpointsofsegmentsofladderlengthwhoseendpointsareonthex-andy-axes?Youcanhavetheladderbeoflength1tosimplifyyourcalculations.)

7) Usethemethodsofanalyticgeometrytoshowthatthediagonalsofarectangleare

congruent.

8) Usethemethodsofanalyticgeometrytoshowthatthediagonalsofarhombusareperpendicular.

9) Ifyouhaven’talreadydoneso,usethemethodsofanalyticgeometrytofindthesolution

totheBuriedTreasureproblemfromtheClassActivity.

106

ClassActivity15:PlagueofLocus

Therearenosectsingeometry. Voltaire

1) Imaginetwoinfinite(hollow)coneswiththeirtipstouchingatonepoint.Nowthinkofallthe

waysyoucouldslicethroughthoseconeswithaplane.Whatarethepossiblecurves(orotherobjects)thatcouldresult(don’tlookonthebackofthissheetuntilyou’vedonethis).Sketchapictureofeach.

2) Eachoftheseobjectshasageometricdefinition(thatwecallthelocusdefinition),andifyou

applyanalyticgeometrytothatdefinitionyougetthefamiliaralgebraicformulafortheobject.Here’sanexample:Youprobablydecidedthatacirclewouldresultifyouslicedthroughjustoneoftheconeswithyourplaneparalleltothe“base”.Thelocusdefinitionofacircleisthis:Acircleisthesetofpointsintheplanethatareequidistantfromagivenpointintheplane(calledthelocus).Now,ifweputdownaCartesiancoordinatesystemonthatplaneandcallthecenterofourcircle(h,k)andtheradiusr,wecanfindanequationthatmustbesatisfiedbyallthepoints(x,y)thatlieonthatcircle.Drawasketchandthenderivethatequation.


107

3) Thelocusdefinitionforaparabolaisthis:Aparabolaisthesetofallpointsintheplanethatareequidistantfromagivenpoint(thefocus)andagivenline(calledthedirectrix).

Usethedefinitionabovetosketchaparabolawithfocus(3,6)anddirectrix,y=2onthegraphpaperbelow.Nowfindtheequationforthatparabola.

4) Thelocusdefinitionforanellipseisthesetofallpointsintheplanesuchthatthesumofthe

distancesfromtwogivenpoints(thefoci–that’spluralforfocus)isconstant.

Usethedefinitiontosketchapictureofanellipsewithfoci(-2,0)and(2,0)andaconstantsumof7onyourgraphpaper.Youdonotneedtofinditsequation.


108

5) Thelocusdefinitionofahyperbolaisthesetofallpointsintheplanesuchthatthedifferenceofthedistancesfromapointonthehyperbolaandtwogivenfociisconstant.

Usethedefinitiontosketchapictureofanhyperbolawithfoci(0,0)and(6,0)andaconstantdifferenceof4onyourgraphpaper.Youdonotneedtofinditsequation.

6) Youmayhavedecidedthatapointandalinecouldalsobeformedbyslicingyourinfinite

conesinproblem#1.Wecanthinkofthoseobjectsas“degenerateforms”ofthesefourobjectswe’vealreadylisted.Forexample,apointisadegeneratecircle(thecirclewithzeroradius).Whatisaline?Explain.

109

ReadandStudy:

Inspirationisneededingeometry,justasmuchasinpoetry. AleksandrSergeyevichPushkinTheconicsectionswerenamedandstudiedaslongagoas200BC,whenApolloniusofPergaundertookasystematicstudyoftheirproperties.Theyarethefourcurves(thecircle,ellipse,hyperbola,andparabola)thatareformedwhenaplaneintersectsadoublecone.Byvaryingtheangleatwhichtheplaneintersectstheconewecanproduceeachofthem,asshownbelow.

Circle

Ellipse

Parabola

Hyperbola

(Illustrationstakenfromhttp://math2.org/math/algebra/conics.htm.)

IntheClassActivityyoufoundthatofthesecurvescanbedefinedusingalocusdefinition(adefinitionthatdescribesthecurveasasetofpointsintheplane).Forexample,youwereaskedtousetheEuclideandistanceformulaandthisdefinitionofacircletodeterminethegeneralequationofthecircleofradiusrandcenter(h,k).Nowwewillfurtherdiscuss(withillustrations)theellipse,thehyperbola,andtheparabola:GiventwopointsF1andF2,anellipseisthesetofpointsPintheplanesuchthatthesumofthedistancesfromPtoF1andF2isconstant.ThismeansthatifwetakeanypointPontheellipseandmeasurethedistancebetweenPandF1andthedistancebetweenPandF2,thenwhenweaddthesetwodistancestogetherwewillalwaysgetthesamesum.WhatwouldhappentotheshapeofthisellipseifwemovedF1andF2closertogetherbutkeptthegivendistanceconstant?

P

F1 F2

110

AhyperbolaisthesetofpointsPintheplanesuchthatthedifferenceofthedistancesfromPtoF1andF2isconstant.WhatwouldhappentotheshapeofthehyperbolaifwemovedF1andF2closertogetherbutkeptthegivendifferenceconstant?

AparabolaisthesetofpointsPintheplanesuchthatthedistancefromPtoagivenpointFisequaltothedistancefromPtoagivenlinem.(RecallthatPointFiscalledthefocusoftheparabolaandlinemisthedirectrix.)Whatwouldhappentotheshapeoftheparabolaifwemovedthedirectrixfurtherfromthefocus?Whatwouldhappentotheparabolaifwechangedthedirectrixtoaverticalline?

F1 F2

P

P

F

Directrix

111

Homework:

Alltruthsareeasytounderstandoncetheyarediscovered;thepointistodiscoverthem. GalileoGalilei

1) DoalloftheitalicizedthingsintheReadandStudysection.

2) TheCommonCoreStateStandardsliststhefollowingstandardforstudentsingradeseven.Howdoesthisstandardfitwiththeideasdescribedinthissection?

3) Explainhowtoformalinebyintersectingaplanewithapairofinfinitecones.

4) Inanalyticgeometryalineisthesetofallpoints(x,y)thatsatisfytheequationax+by+c=0,wherebothaandbarenotzero.Findtheslopeandy-interceptofthelineintermsoftherealnumberparametersa,bandc.Whathappenswhena=0?Whenb=0?Whenc=0?

5) Findtheequationofacirclewithcenter(2,-4)andradius5.6) Supposetheequationofthedirectrixofaparabolaisy=–3andthepointF=(0,5)isits

focus.Finditsequation.

7) Anellipsecanbemodeledusingtwostickpins(oneateachfocus)andalengthofstring(equaltothesumofdistancesfromtheellipsetothefoci).Experimentwiththismethodtocreatevariousellipses.Whathappenswhenthelengthofstringstaysthesamebutyouvarythepositionofthefoci?Whathappenswhenyoukeepthefocifixedbutvarythelengthofthestring?Isthereaminimumlengthofstringnecessary?

• Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

112

ClassActivity16:ComparingStandards

TheseStandardsdefinewhatstudentsshouldunderstandandbeabletodointheirstudyofmathematics. CommonCoreStateStandards

YouwillfindtheStandardsforGeometryinGrades6–8fromtheNationalCouncilofTeachersofMathematics(2000)onpage4ofthistextandtheCommonCoreStateStandardsinGeometryforthatsamegradebandonpages6-7.Rereadallofthose.

1) InwhatwaysdotheNCTMStandardsandtheCommonCoreStateStandardsoverlap?Whatthingsmentionedbyonegrouparemissingfromtheother?

2) Asteachers,whichwouldyoufindbemoreeasytoimplement?Explain.

3) Considerthefollowinglistofgeometrictasks.Wheredoeseachfit(ifatall)withineachframework?

a) CollectingdataonseveralcirclestoseethattheratioofCircumferencetoDiameter

isalwaysaconstant.b) Understandingthedefinitionofacircle.c) Cuttingandrearrangingaparallelogramtofindaformulaforitsarea.d) Classifyingquadrilaterals.e) Usinggridpapertoperformatranslation.f) Findingtheequationofaline.g) Findingthecostofpaintingaroom.h) Makingascalemodelofaship.

113

SummaryofBigIdeasfromChapterTwo Ifanidea’sworthhavingonce,it’sworthhavingtwice. TomStoppard

• ThevanHielelevelsdescribeaprogressionofgeometricunderstanding.

• Itisimportantforyourstudentstomakesenseoftheformulasforareaandvolume.

• Therearethreerigidmotionsoftheplane:rotation,reflection,andtranslation.

• Ifglidereflectionisconsidereditsownmotion,thenthecompositionofanytworigidmotionsisanotherrigidmotion.

• Twofiguresaresimilarifthereisasequenceofrigidmotionsandadilationoftheplane

whichmapsonefigureontotheother.

• Analyticgeometryinvolvestakingageometricproblemandtranslatingitintoanalgebraicproblem.Itisaveryusefulprooftechnique.

• Wecanuseanalyticgeometrytohelpusdescribefigureslikeparabolasandhyperbolas.

114

CHAPTER3

EXPLORINGSTRANGENEWWORLDS:NON-EUCLIDEANGEOMETRIES

115

ClassActivity17:LifeonaOne-SidedWorldOnlythosewhoattempttheabsurdwillachievetheimpossible.Ithinkit'sinmybasement...letmegoupstairsandcheck.

M.C.Escher

Cutseveral1-inchwidestrips(thelongway)fromablanksheetof8½by11inchpaper.Withonestrip,tapetheone-inchendstogethertoformacylinder.Withanother,makeahalf-twistandthentapetheone-inchendstogethertoformatwo-dimensionalsurfacecalledaMöbiusstrip.Savetheremainingstripsforadditionalexamples,asneeded.

1) Howmany“sides”doesthecylinderhave?TheMöbiusstrip?Whatmakesthedifference?Howmanysidesdoesastripmadewith2half-twistshave?Onewith3half-twists?Howmanysidesdoesastripwith46halftwistshave?Onewith511?

2) Howmanyedgesdoesthecylinderhave?TheMöbiusstrip?Howmanyedgesdothestrips

with2,3,46,or511halftwistshave?Explainthedifference.

3) Isthereaconnectionbetweenthenumberofsidesandthenumberofedges?Whataboutbetweenthenumberofhalf-twistsandthenumberofsides?Betweenthenumberofhalf-twistsandthenumberofedges?Explainallofthis.

4) Whathappenswhenyoucutacylinderdownthemiddle?Whathappenswhenyoucuta

Möbiusstripdownthemiddle?Thinkaboutitbeforeyoudoit!Thenexplainprecisely.(e.g.,Whatpiecesresult?Howaretheylinked?Howarethesizesrelated?)Whatifyoucutthesepiecesdownthemiddle(i.e.,cuttheoriginalstripintofourths)?

116

ReadandStudy:

…bynaturalselectionourmindhasadapteditselftotheconditionsoftheexternalworld.Ithasadoptedthegeometrymostadvantageoustothespeciesor,inotherwords,themostconvenient.Geometryisnottrue,itisadvantageous.

HenriJulesPoincareTheMöbiusstripisnamedafterAugustFerdinandMöbius,anineteenthcenturyGermanmathematicianandastronomer,whowasapioneerinthefieldoftopology.(Bytheway,topologyisthestudyofspaceswherethequestionsofinterestarethingslike:Doesthespacehaveholesinit?Isitconnected?Sometimestopologyiscalledrubber-sheet-geometrybecauseintopologyonespaceisconsideredthesameasanotherspaceifitcanbebentorstretchedintotheotherspace.Forexample,intopology,acubeandaspherearethesamespacebutadonutandaspherearenot.Whynot?)Möbius,alongwithhiscontemporariesBolyai,Lobachevsky,andRiemann,turnedtheworldofEuclideangeometryupsidedown,insideout,andeverywhichwaybutflat.TheMöbiusstripisasimplesurfacewithsurprisingproperties.AtrueMöbiusstripisatwo-dimensionalsurface(asifourstripofpaperhadnothicknesswhatsoever)withonlyonesideandonlyoneboundaryedge.IfwerestrictourselvestoasmallsectionoftheMöbiusstrip,thegeometrythereisthesameasitisontheflat(Euclidean)stripofpaperfromwhichitwasformed.However,whenweconsidertheentireMöbiusstrip,thegeometryisquitedifferent.Notonlyisitasurfacewithonlyonesideandoneedge,butitisalsowhatwecallnon-orientable.IfanamoebalivingonthesurfacemadeatriparoundtheentireMöbiusstrip,itwouldreturntoitsstartingpointasamirrorimageofitself!Thinkaboutthis.Thiskindofthingdoesn’thappenonaEuclideansurfacesuchasthecylinder.ThesesurprisingpropertiesmaketheMöbiusstripquiteusefulinthe“realworld.”GiantMöbiusStripshavebeenusedasconveyorbelts(tomakethemlastlonger,since"eachside"getsthesameamountofwear)andascontinuous-looprecordingtapes(todoubletheplayingtime).Inthe1960'sSandiaLaboratoriesusedMöbiusStripsinthedesignofversatileelectronicresistors.Free-styleskiershavechristenedoneoftheiracrobaticstuntstheMöbiusFlip.TheinternationalsymbolforrecyclingisaMöbiusstrip.

117

WegetevenmoresurprisingresultsifwegluethetwoedgesofacylinderoraMöbiusstriptogether.Trytoimaginebringingthetwoopenendsofthecylindertowardseachother(ithelpsifyouareimaginingalongskinnycylinder–likeapapertoweltube).Whatshapewouldresult?Mathematicianscallthisshapeatorus.Bagelsandinnertubesaretwoexamples.NowimaginegluingtwoMöbiusstripstogetheredgetoedge.Youhavetojustimagineit–itisphysicallyimpossibletoaccomplishthegluinginthree-dimensionalspacewithouttearingtheMöbiusstrips.WhatresultsiscalledtheKleinBottle–asurfacewhoseinsideisitsoutside!Theapparentself-intersectionyouseeinthefollowingpictureismisleading–theKleinbottleexistsin4-dimensionalspacewithnoselfintersections.Itwasfirstdescribedin1882bytheGermanmathematicianFelixKlein.

Illustrationfromhttp://www.geom.uiuc.edu/zoo/toptype/klein/standard/gifs/trans.gif.Cylinders,Möbiusstrips,tori(pluraloftorus),andKleinbottlescanallberepresentedbya“flat”rectanglewithappropriategluinginstructionsfortheoppositeedges.Theoppositeedgeswitharrowsaretobegluedtogetherwitharrowsmatching.Wecalltheseidentificationspacesfortheobjects.Thisisthesameideausedinvideogameswherethespacecraftorrobotorwhateverleavesthescreenontheleftsideandreturnsontherightorleavesonthetopandreturnsfromthebottomandviceversa.Studythegluingdirectionsforeachobjectandexplainhowtheymatchthephysicalmodelsyouhavemade(orimagined,inthecaseoftheKleinbottle).

Klein bottleMobius stripTorusCylinder

118


Ifyouwouldthoroughlyknowanything,teachittoothers. TyronEdwards

Youmiddlegradesstudentswillliketoexplorethegeometryofthecylinder,torus(donut),MöbiusstripandKleinbottle(amongothers)throughactivities,puzzlesandgames.Forexample,hereisawordsearchonatorus.Seeifyoucanfindallofthesewords:possum,panda,jaguar,camelandllama.

Homework:

Wheneveryouareaskedifyoucandoajob,tell‘em,‘CertainlyIcan!’Thengetbusyandfindouthowtodoit.

TheodoreRoosevelt


2) DothewordsearchfromtheConnectionssection.Thenseeifyoucanmakeupawordsearch(thesamesizeastheoneabovewithatleastfivewordstofind)onaKleinBottle.

3) UseanidentificationspacetopredictwhatwouldhappenifyoucutaMöbiusstripintothirds.Then,checkitout.Ifyourpredictionswerewrong,trytofigureoutwhereyoumadeanerrorinthinking.Whydotheactualresultsmakesense?

4) Predictwhatwouldhappenifyoucutastripwiththreehalf-twistsinhalfdownthemiddle.

Checkitout.Ifyourpredictionwaswrong,trytofigureoutwhereyoumadeanerrorinthinking.Whydotheactualresultsmakesense?Theresultingobjectisknownasatrefoilknot.(Knottheoryisanotherfunareaofmathematicsrelatedtogeometricideas.)

h l m e a i

n a l n b r

j l d a e a

c a t w m t

x e g i p a

o s s u m p

119

5) Usetheflatmodelsofthecylinder,theMöbiusstrip,thetorus,andtheKleinbottletocreatetic-tac-toegameboards.Playseveralgamesoneachsurface.Don’tforgettoincludethegluinginstructionsinyourstrategy.Howdoesthegamechangeoneachsurface?Whatstrategiescanyouusetowinineachcase?Isthereasurfaceonwhichyoucanguaranteeawinbygoingfirst?Bygoingsecond?Isthereasurfaceonwhichthegamealwaysresultsinatie?(Assumetwocompetentplayersandthatneithermakesamistake.)

6) Threeamoebas,Apox,Brillo,andCheesy,lineupforaraceonavirtualMöbiusstrip

swimmingpool.Allthreeswimupthemiddleoftheirlanesatexactlythesamespeed.Whichamoebawillreturntohisorherownstartingpointfirst?Why?

c

B

A

120

ClassActivity18:LifeinaTaxicabWorld

TofullyappreciateEuclideangeometry,oneneedstohavesomecontactwithanon-Euclideangeometry.

EugeneF.Krause,TaxicabGeometryTerranceandSashaliveinPerfectionCitywhereallstreetsintersectatrightanglesandareevenlyspaced.AmodelofPerfectionCityistheCartesianplanewithstreetsrepresentedbyverticallinesatallintegervaluesofthex-axisandavenuesrepresentedbyhorizontallinesatallintegervaluesofthey-axis.UnliketheCartesianplane,PerfectionCityisnotinfiniteinsize;wewillfocusontheheartofthecitycontainedwithinthegrid-10£x£10and-10£y£10.Terranceworksatthepubliclibrarylocatedatthecornerof3rdStreetEastand1stAvenueSouthandSashateachesmathatPerfectionHighSchoollocatedatthecornerof5thStreetWestand9thAvenueNorth.(Noticethatonlytheevennumberedstreetsandavenuesareshownonthisgrid.)LocatethelibraryandtheHighSchoolonthegrid.

1) Howfarapartarethelibraryandthehighschool(asthecrowflies)?Stayingonthe

streets,howfarmusteitherofthemwalktomeettheotherattheirworkplace?Sashalikestowalkadifferentrouteeachday,butshealsowantstowalktheshortestdistancepossible.Forhowmanydayscanshemakethewalkwithoutrepeatingaroute?

2) TerranceandSashadecidetomeethalfwayforlunch.Whereisthishalfwaypoint?Is

theremorethanonehalfwaypoint?MarkallofthehalfwaypointsonthegridwiththeletterM.Nowsupposethatnoneoftheseintersectionscontainaneatingplacesatisfactorytobothofthem,whereelsecouldtheymeetforlunchsothateachofthemhasthesamelengthwalk?MarkallofthesepointsonthegridwiththeletterP.Whatisthemathematicaldescriptionofthe“line”whichjoinsallofthepointslabeledMorP?

121

ReadandStudy:

Geometry,whichistheonlysciencethatithathpleasedGodhithertotobestowonmankind.

ThomasHobbesSupposewetaketheEuclideanplaneandchangenothingexceptourdefinitionofdistance.Pointsarestillpoints;linesarestilllines;andanglesarestillmeasuredinthefamiliarway.Butwewillnolongerusethe“asthecrowflies”definitionofdistancebasedonthePythagoreanTheorem.Insteadwewillmeasurethedistancebetweentwopointsbyfindingthesumoftheverticaldistanceandthehorizontaldistancebetweenthetwopoints.Inotherwords,wewillmeasuredistanceinthesamewaythatwemeasuredthelengthofTerranceandSasha’swalksintheclassactivity.WecanusethefollowingformulatodeterminethisnewdistancedTbetweenthetwopoints(x,y)and(u,v):

𝑑< = 𝑥 − 𝑢 + 𝑦 − 𝑣 Ageometrywiththisnewwayofmeasuringdistanceisoftencalledtaxicabgeometrybecausethisformulagivesthedistanceataxigoesifittravelsonlyalongnorth-southandeast-weststreets,asinPerfectionCity.Whywouldwesuddenlywanttochangethedefinitionofdistance?Afterall,Euclideangeometryhasserveduswellforthelast2000years.Thereareafewpossibleanswerstothisquestion.Themostobviousoneissuggestedbythenameoftaxicabgeometry.Euclideangeometrymeasuresdistance"asthecrowflies,"butthisdoesn’talwaysprovideagoodmodelforareal-lifesituation,particularlyincities,whereoneisonlyconcernedwiththedistancetheircarwillneedtotravel.Anotherreasonforstudyingtaxicabgeometryisthatitisasimplenon-Euclideangeometry.Taxicabgeometryisfairlyintuitiveandrequireslessmathematicalbackgroundthanothergeometries;inshort,itisagoodexampleofanon-Euclideangeometryformiddleschoolstudents.Let’sexaminethisnewdefinitionofdistancemoreclosely.Reallydothesethingsinitalicsbelow.First,calculatethenormalEuclideandistancebetweenpoints(2,5)and(4,1)andthenfindthetaxicabdistancebetweenthesetwopoints.Whataboutthepoints(2,5)and(2,1)?Thepoints(2,5)and(4,5)?Okay,whatdidyoufind?Aretherepairsofpointsforwhichthe“normal”distanceandthetaxicabdistancebetweenthemareequal?Ifso,generalizetherelationshipbetweenpairsofpointsforwhichthisistrue.WhenthetaxicabdistanceandtheEuclideandistancearenotequal,whichoneisgreater?Willthisalwaysbethecase?Why?

122

AllofEuclid’spostulatesholdintaxicabgeometry,butdefinitionsbasedondistancecanlookdifferent.Forexample,let’sconsidercircles.Whatwouldacircleofradius5centeredattheoriginlooklikeintaxicabgeometry?Thinkaboutthedefinitionofacircleandthetaxicabdefinitionofdistanceandsketchthetaxicabcircleofradius5onthefollowingpairofaxes.

Whatisthecircumferenceofthistaxicabcircle?(Remembertomeasureittoousingtaxicabdistance.)Now,recallthatpisdefinedtobetheratioofthecircumferenceofacircletoitsdiameter.InEuclideangeometry,pisanirrationalnumber(approximatelyequalto3.1416).Whatisareasonablevaluefortheratio“p”intaxicabcircles?Why?Willitbeaconstantvalueforalltaxicabcircles?Explain.Manyotherfamiliarobjectsalso“look”differentintaxicabgeometry.Inthehomeworkyouwillbeaskedtoexploretheshapeoftaxicabsquares,equilateraltriangles,andtheconicsections.SomeofourfamiliarEuclideanresultsarenolongervalidinTaxicabgeometry.Forexample,considerthetrianglecongruencetheorem,Side-Angle-Side(SAS).UsethefollowingtwotrianglesandtheformulafordTtocreateacounterexampleshowingthatSASisnottrueintaxicabgeometry.

123

WhatdoesthisexamplesayaboutthePythagoreanTheoremintaxicabgeometry?Doesithold?Whatabouttheothertrianglecongruencetheorems?Willanyofthembevalidorcanyoufindcounterexamplesforthemaswell?Checkitoutforsomeexamples. InEuclideangeometry,thesetofallpointsequidistantfromtwogivenpointsistheperpendicularbisectorofthelinesegmentjoiningthetwopoints.Whatwillthe“perpendicularbisector”ofalinesegmentlooklikeintaxicabgeometry?Inthesecondpartoftheclassactivity,allofthepointslabeledMandPwereequidistantfromthe(L)ibraryandthehighschool(HS).Sothelinesegmentsjoiningthesepointsformthe“perpendicularbisector”ofthelinesegmentjoiningLandHS.Nowconsiderthetaxicabperpendicularbisectorofthesegmentjoining(2,2)and(-1,-1).Howdoesthis“perpendicularbisector”differfromtheEuclideanone?Inwhatwaysisitsimilar?Whenwillataxicab“perpendicularbisector”looklikeaEuclideanperpendicularbisector?Whenwillitbedifferent?


Whoeverceasestobeastudenthasneverbeenastudent. GeorgIlesTypicallystudyofnon-Euclideangeometriesisnotexplicitlypartoftheupperelementaryormiddlegradescurricula.However,therearepiecesofthesegeometriesthatwillhelpstudentstounderstandmapsandmap-making.TherearemanygoodproblemideasinTaxicabgeometryforthemiddle-gradesatthewebsitehttp://emat6000taxicab.weebly.com/teacher-resources.html.

6

4

2

-2

-4

-6

-5 5

A

B C

A'

C'

B'

124

Homework:

Saynot,‘Ihavefoundthetruth,’butrather,‘Ihavefoundatruth.’ KahlilGibran

1) Ifyouhaven’talreadydoneso,gobackanddoalltheitalicizedthingsintheReadandStudyandtheConnectionssectionsabove.

2) PerfectionCityactuallyhasthreehighschools:PerfectionHighSchoollocatedat(-5,9),

IdealHighSchoollocatedat(8,-1)andIdyllicHighSchoollocatedat(0,-7).Drawtheschoolboundariessothateachstudentattendstheschoolclosesttohisorherhome,asthetaxi(orschoolbus)drives.

3) ModelBurger,thefast-foodchain,wantstoopenanewrestaurantthatiscentrallylocated

sothatitisthesametaxicabdistancefromeachofthethreehighschools.Whereshoulditbelocated?

4) TerranceandSashaneedtofindanapartmentsothatthatthesumofthedistancesthat

thetwoofthemwillwalktoworkshouldbenomorethantwenty-fourblocks.Drawtheboundaryoftheirsearcharea.Whichoftheconicsectionsaretheyusingtodefinethesearcharea?

5) WhenTerranceandSashawereunabletofindanapartment,theynextagreedthatneither

ofthemshouldhavetowalkmorethanfourblocksfartherthantheotherinordertogettowork.Nowwherecantheylook?Whichoftheconicsectionsaretheyusingtodefinethesearchareathistime?

6) Inthereading,youdiscoveredthattaxicabcircleslooklikeEuclideansquares.Whatdo

taxicabsquareslooklike?UsethedefinitionofasquareandthetaxicabdefinitionofdistanceanddrawthreetaxicabsquareswithasidelengthoffoursuchthatthefiguresarenotcongruentasEuclideanfigures.WhatEuclideanshapedothetaxicabsquareshave?Whydoesthishappen?

7) Nowexperimentwithtaxicabtriangles.Canyoudrawaregulartriangleintaxicab

geometry?Whyorwhynot?Howaboutarighttrianglewithsidesofequallength?Howaboutanisoscelestrianglewhosebaseanglesarenotcongruent?

125

ClassActivity19:LifeonaSphericalWorld Youcan’tcombthehaironaball! MaryEllenRudinInthisactivityyouwillexploregeometryonthesurfaceofaEuclideanspherebyworkingwithaphysicalmodelofasphere(aball)andaphysicalmodelofaline(apieceofstring).Youmayneedmarkerstodrawlinesonthesphere(orrubberbandstomodellines)andaregularprotractortomeasureangles.Assumethattheradiusofyoursphereisoneunit.

1) Talkwithyourgroupanddecidehowyoucanuseapieceofstringtomakeastraightline–firstonaflatsheetofpaper(Euclideanmodel)andthenonthesphere.Takethisseriously–itisimportanttohaveavalidmodelofastraightlinebeforeproceeding.Relatewhatyouhavedecidedaboutstraightlinesonthespheretothe“lines”oflongitudeandlatitudemarkingsonaglobe.

2) Drawastraightlineonyoursphere(notasegmentbutaline).Howlongisit?Nowfindadifferentstraightlinethatisparalleltoit.(Recallthatlinesareparalleliftheyhavenopointsincommon.)Howmanylinesparalleltoyouroriginallinecanyoufind?IsthegeometryofthesurfaceofasphereEuclidean?Whyorwhynot?

3) Marktwopointsanywhereonthesphere.Drawthelinesegment(usingthestringmethod)betweenthesetwopoints.Whatdoyounotice?Howmanylinesegmentscanyoufind?Doesitmatterwherethetwopointsareinrelationtoeachother?Experimentwithvariouspairsofpointsandformaconjectureaboutlinesegmentsonasphere.


126

4) Drawasmalltriangleandalargetriangle(onethatcoversatleast1/8ofthesurfaceareaonyoursphere).Makecertainthatthesidesofyourtrianglesareactuallystraightlinesegmentsbyusingthestringmethodtoconstructthetriangle.Determineamethodtomeasuretheanglesofthetrianglesusingyourprotractorandthenmeasureeachoftheanglesinbothofthetriangles.Whatistheanglesumofthesmalltriangle?Thelargetriangle?Nowdrawamedium-sizedtriangleandareallybigtriangleandmeasuretheiranglesums.Makeaconjectureabouttheanglesumofasphericaltriangle.

5) Drawarighttriangleonyoursphere.Howcanyoumakecertainthatyouhavearightangle?Howmanyrightanglescanyouhaveinonetriangle?Canyouadrawatrianglethathastworightangles?Canyoudrawatrianglethathasthreerightangles?DoyouthinkthePythagoreanTheoremholdsonasphere?Whyorwhynot?

6) Drawalineonthesphereandchooseapointthatisnotonthatline.Howmanyperpendicularlinestoyouroriginallinecanyoudrawthroughthatpoint?Aretherepointsyoucanchoosewheretherewouldbemanyperpendicularlinesthroughthatpoint?Ifso,describethesepointsandexplainwhyyouhavemorethanoneperpendiculartothelinethroughthosepoints.

127

ReadandStudy:

Mathematics,rightlyviewed,possessesnotonlytruth,butsupremebeauty–abeautycoldandaustere,likethatofasculpture

BertrandRussellSofaryouhavestudiedtwoinfinitenon-Euclideangeometries,eachcreatedbyonesimplechangetothefamiliarEuclideangeometryoftheflatplane.IntheMöbiusstripweintroducedahalf-twistbeforegluingtogetheronepairofoppositesidesofaflatrectangle.Intaxicabgeometrywechangedthedefinitionofdistanceontheflatplane.Nowwe’llconsiderwhathappenswhenweintroduceaconstantpositivecurvaturetotheflatplane.Thefactthatthecurvatureispositivecausestheplanetocloseupintoaball–thefactthatthecurvatureisconstantmeansthatourballisperfectlyround(likeabasketballandnotafootball).Infact,theCartesianplanewithconstantpositivecurvaturebecomesthesurfaceofasphere–andthissinglechangeagainaffectsthegeometryindrasticways.Forstarters,wenolongerhaveaninfiniteplane.Thesurfaceareaofasphereisfiniteanddependsontheradius.Remembertheformula,𝐴 = 4𝜋𝑟',forsurfaceareaofaspherewithradiusr?Thisareacanbequitesmall,asonabeachball,oritcanbequitelarge,asontheplanetJupiter,butitisalwaysfinite.Thiseffectivelymeansthatthesizeofeverygeometricobjectdrawnonthesurfaceofaspherehasalimitingsize–thereisalargestcircle,thereisalongestlinesegment,andthereisabiggesttriangle.Thenthereisthestoryaboutlines.Inordertomaintaintheconceptofstraightnessonthesphere,wehavetousethefactthatonaflatplaneastraightlineistheshortestdistancebetweentwopoints(representedbypullingastringtightbetweenthosetwopoints).Whenyoupulledthestringtightagainstthesurfaceofthesphereandwentallthewayaroundthespherebacktoyourstartingpoint,youcreatedamodelofastraightlineonthesphere.This“straightline”isagreatcircle,acircleformedonthesurfaceofthespherebytheintersectionofaplanethatgoesthroughthecenterofthesphere.Youwillknowthatacircleonthesphereisagreatcircleifitcutsthesphereintotwohalvesofequalarea(twohemispheres).Theequatoronaglobeisanexampleofagreatcircle.Soarethelinesoflongitude-butnotthelatitudemarkings.Onasphere,greatcirclesarelines;allothercirclesarejustcircles.Solinesarealsofiniteinlength–infact,alllineshavethesamelength.Whatistheformulaforthelengthofalineonaspherewithradiusr?Anothersurprisingfindingaboutlinesonasphereisthattherearenoparallellines.Alllinesintersect,andinfacttheyallintersectinexactlytwoantipodalpoints.(Antipodalpointsarepointsthatareatoppositeendsofadiameterofthesphere,likethenorthandsouthpoles.)Thussphericalgeometryisnon-Euclideaninamostbasicway–itdoesnotsatisfyEuclid’s5thPostulateaboutparallellines.Sincetherearenoparallellines,therecanbenoparallelograms,rhombi,

128

rectangles,orsquareseither.Alittlebitlaterwewillexploreanotherargumentforthefactthatrectanglesandsquaresdonotexistinsphericalgeometry.Intheactivityyoufoundthattherearealwaystwolinesegmentsbetweenanytwopointsonthesphere–andwhenthosepointsareantipodal,thereareaninfinitenumberoflinesegmentsbetweenthem.Again,thisisnotatalllikewhathappensintheflatplane.Ifthepointsarenotantipodal,thenoneofthesegmentsisshorterthantheotherandtogetherthetwosegmentscomposetheentirelinebetweenthetwopoints.(Theshorteroneiscalledtheminorsegment.Thelongeroneisthemajorsegment.)Ifthepointsareantipodal,theneverysegmentbetweenthemisequalinlengthtohalfthecircumferenceofthesphere.Sincewecanformlinesegmentsbetweenpoints,wedohavetrianglesonthesphere.Butifwestartwiththreepoints,thereismorethanonetrianglewecanformwiththosethreepointsasvertices.Sotwotrianglesmaysharethesamevertices,buthavedifferentlengthsides,differentanglemeasures,anddifferentareas.Andinfact,evenifwespecifythatthesidesaretobetheminorsegmentsbetweenthepoints,westillhavetwotrianglesofdifferentareaformedbythosesegments.Didyouseethiswhenyouwereformingyourtrianglesintheclassactivity?Stopnowanduseaballtovisualizeexactlywhatwearesaying.Thisisagoodtimetopointoutagaintheimportanceofcarefullywordeddefinitions.Ontheflatplaneitissufficienttosaythatatriangleisthreenon-collinearpointsandthelinesegmentsjoiningthosepoints.Onthesphericalplanewemustrefineourdefinitiontosaythatatriangleisthreenon-collinearpointsandtheminorlinesegmentsjoiningthosepoints,takingtheinteriorofthetriangletobethesmallerofthetwoareasenclosedbythosesegments.Isitnecessarytoincludetherequirementthatthepointsbenon-collinearinthesphericaldefinition?Canweplacethreepointsonthesamelineandchooselinesegmentsbetweenthemtoformatriangle?Whatwouldbetheareaofsuchatriangle?Sinceanylineonthesphereisagreatcircle,wecandefinetheanglebetweentwolinesastheangleformedbytheintersectionofthetwoplanesthatcreatethegreatcirclesthatarethoselines.Sincethosetwoplanescanintersectinanyanglebetween0°and180°,wehavethesameanglemeasuresonthesphere.Inparticular,wehaveanglesof90°betweenlinesonthesphereandsowehaveperpendicularlinesandrighttriangles.Intheclassactivity,youinvestigatedrighttriangles,andinparticular,whetherornotitwaspossibletohavetwooreventhreerightangleswithinonetriangle.Whatconclusionsdidyoumake?Canyoudescribeatriangleonthespherethathastworightangles?Thathasthreerightangles?Atrianglewiththreerightangleswouldhaveananglesumof270°sothefactthattrianglesinEuclideangeometryhaveanglessumsof180°mustcomefromthe5thpostulate.Changeyouraxioms,andyouchangeyourtheorems.Whatdidyoufindtobetheanglesumsofthetrianglesyouformedintheclassactivity?Whatwasthesmallestanglesumyoufound?Thelargest?Whatwouldbethelargestanglesumpossible?Why?Ofcourse,yourmeasurementswithaprotractorwereapproximate,asareallmeasurements,butyoushouldhavefoundthatyouranglesumswerealllargerthan180°andthatastheareaofthetrianglebecamelarger,sodidtheanglesum.

129

Infact,itisanamazingfeatureofsphericalgeometrythattheanglesumofanytriangleisgreaterthan180°andthattheareaofatriangleisequaltoitsanglesum(inradians)minusp.Totryandunderstandthis,consideratypeofpolygonthatdoesnotexistinEuclideangeometry,atwo-sidedpolygoncalledabiangleoralune.Sinceeverypairoflinesonthesphereintersectsintwopoints,wedohaveapolygonwithtwosidesandtwovertices(whichwillbeantipodal).Whyisitcalledalune?ThenamecomesfromtheLatinwordluna,whichmeansmoon.Thinkaboutthepartofthemoonthatisseenatanytime.Thatportionhastobebothinthehemispherewhichisilluminatedbythesunandinthehemispherethatisvisiblefromtheearth.Theintersectionoftwohemispheresispreciselyalune.Everypairoflineswillformtwopairsofcongruentlunes(similartothetwopairsofcongruentverticalanglesformedbyintersectinglinesontheflatplane).Studythediagrambelowtomakecertainyouunderstandthisdefinition.Oneofthefourlunesformedisshadedwithverticalhatching.Doyouseethelunecongruenttoit?Seetheotherpairofcongruentlunes?Whatwillbetheareaoftheshadedluneiftheanglebetweenthetwosidesis30°(p/6radians)andtheradiusofthesphereisoneunit?

[Hereyoumightneedaquickreminderaboutradiananglemeasure.Aswementionedearlier,assigning360degreestoonefullrotationisjustarbitrary.Thereisanotherstandardwaytomeasureanglesandthatisbythelengthofthearcthattheanglesweepsoutwithan“arm”ofradiusone. 1

130

Havealookattheangleabove.Itmeasuresabout80degrees.Inradiansthemeasureoftheangleisthelengthofthearcshown.Nowinafullrotationthelengthofthearcis2πradians.(Whyisthat?)Sothisangleisalittlelessthan½πradians.Whatistheradianmeasureofananglethatmeasures45degrees?180degrees?]Nowwewillgetbacktofindingtheareaofasphericaltriangle.Itwillhelpalotifyouhaveapingpongballortennisballorsomeotherballthatyoucanwriteontofollowalong(andtothinkalong)withus.StudythefigurebelowuntilyouarecomfortableexplaininghowtriangleABCisformedbytheintersectionofluneAA’,luneBB’,andluneCC’.NoticethatthereisamirrorimagetriangleA’B’C’formedonthebacksideofthesphere.WewillassumethattheradiusofthesphereisoneunitandthatÐCAB=aradians,ÐABC=bradians,andÐBCA=gradians.

Theareaoftheentiresphereis4p.Theareaofeachluneisequaltotwiceitsanglemeasure.(Forexample,areaofluneAA’is(a/2p)timesthetotalareaofthesphere,or(a/2p)*4p=2a.)Ifweaddtheareaofeachpairoflunestogetherwewillcounttheareaof∆ABCthreetimesandtheareaof∆A’B’C’threetimes.(Explainwhy.)Ofcourse,∆ABCand∆A’B’C’arecongruent.Whenweuseallofthisinformationwecansaythatthesumoftheareasofthelunesisequaltotheareaofthesphereplusfourtimestheareaof∆ABC(Besureyoucanexplainwhyweaddfourtimestheareaof∆ABC.),givingtheequation:

131

2 2𝛼 + 2 2𝛽 + 2 2𝛾 = 4𝜋 + 4(𝑎𝑟𝑒𝑎𝑜𝑓Δ𝐴𝐵𝐶)whichsimplifiesto:

𝛼 + 𝛽 + 𝛾 − 𝜋 = 𝑎𝑟𝑒𝑎𝑜𝑓Δ𝐴𝐵𝐶YoucanfindaninteractiveversionofthisproofatawebsitewrittenbyanauthorandDr.StephenSzydlik-http://www.uwosh.edu/faculty_staff/szydliks/elliptic/elliptic.htm.ConnectionstotheMiddleGrades:

Ihaveneverletmyschoolinginterferewithmyeducation. MarkTwain

Whyshouldstudentsstudynon-Euclideangeometries?Wethinktherearemanyreasons,thefirstofwhichisthatonewaywelearnaboutwhatsomethingisisbyseeingwhatitisnot;non-Euclideangeometrygivesusausefulcontrasttoourstandardhighschoolgeometry.Italsobringsintosharpfocustheimportanceofaxioms(onechangeinoneaxiomandyougetawholenewgeometrywithdifferenttheorems)andmathematicaldefinitions(forexamplethinkaboutwhathappenedwhenwechangedourdefinitionofdistanceinthecaseofTaxi-cabgeometry).Finally,scientistsaregainingmoreandmoreevidencethatouruniverseisnotEuclideanspace.NonEuclideangeometryisthusbecomingincreasinglyimportanttoanunderstandingofastronomy.

Thesegeometriesprovidestudentsopportunitiestomodelandexploredifferenttypesofspaces.IntheirpaperinthejournalMathematicsTeachingintheMiddleSchool,SharpandHeimer(2002)describetheirexperiencehavingasixth-gradeclassexploregeometryonasphereusingbeachballsinmuchthewayyoudidintheclassactivity.Studentsdefinedwhatwasmeantbyalineonasphere,andexploredlunes,trianglesandotherpolygons.Finally,studentsappliedwhatthey’dlearnedtomeasurementonaglobe.Forexample,sixthgraderslearnedtousegreatcircles(ratherthanlinesoflatitude)tofindtheshortestroutebetweenvariouscitiesontheplanet.Whatdoestypicallyhappenwhentheglobeismadeintoaflatmap?Whatisdistortedandinwhatway?SharpandHeimerclaimedthatanexperiencewithanon-Euclideangeometryhelpedtheirstudentstobroadentheirunderstandingsofgeometry.Forexample,childrenobservedthatparallellinesareimpossibleonasphereandtheauthorsarguedthatthissortofobservation“…laysthefoundationfortheformationofinformaldeductions,avitalskillingeometricthinking,whetherontheplaneorthesphere”(p.185).Whatdotheymeanbythis?

132

Homework:

DonotworryaboutyourdifficultiesinMathematics.Icanassureyouminearestillgreater.

AlbertEinstein1) DoalloftheitalicizedthingsinReadandStudysection.

2) DoalltheitalicizedthingsintheConnectionssection.3) Insphericalgeometryhowmanyperpendicularlinescanbedrawntoagivenlinethrougha

pointnotonthatline?Doestheanswertothisquestiondependuponthelocationofthepointinrelationtotheline?Ifso,describethedifferentcasesandexplainwhyyouhavemorethanoneperpendiculartothelineinsomecases.

4) DoesthePythagoreanTheoremholdinsphericalgeometry?Ifyes,supportyouranswerwith

aproof.Ifno,supportyouranswerwithacounterexample.5) Giventhattheanglesumofanysphericaltriangleisgreaterthan180°,makeanargument

(differentfromtheonegiveninthereading)thatrectanglesdonotexistinsphericalgeometry.

6) DeterminewhichoftheEuclideantrianglecongruencetheoremsaretrueinspherical

geometry.Supportyouranswerswithanargumentoracounterexample.7) Usethewebsiteathttp://www.uwosh.edu/faculty_staff/szydliks/elliptic/elliptic.htmto

exploresimilartrianglesinsphericalgeometry.Cansphericaltrianglesbesimilarbutnotcongruent?Makeanargumenttosupportyouranswer.WhatdoesthissayabouttheAAATheorem?

8) DoestheIsoscelesTriangleTheoremholdforsphericaltriangles?Supportyouranswer.

9) OnthespheredrawalineyoucanconsidertheequatorandletNbethepointthatwouldbe

thenorthpole.Marktwopoints,AandB,ontheequatorsuchthatthemeasureofÐANBis90degrees.LetC,D,andEbethemidpointsofAB,AN,andBN(theminorsegments),respectively.

a)ExplainandillustratewhyCN,DB,andAEintersectinacommonpoint,F. b)FindtheanglesumofthesphericaltriangleACF.

133

ClassActivity20:LifeonaHyperbolicWorld

Geometryisaskilloftheeyesandthehandsaswellasofthemind.JeanPedersen

Inthisactivityyouwillexploresomeofthepropertiesofthegeometrythatresultswhen“flatness”isreplacedby“constantnegativecurvature.”Todosoweneedaphysicalmodeltoplaywith–andfirstyouwillneedtomakethismodel.TakethetwosheetsofregularheptagonsfromAppendixD,carefullycutouteachheptagonandthentapetheheptagonstogetherattheedges,threetoavertex.Don’tbesurprisedthattheydonotlieflat-recallthatthevertexanglemeasureinaregularheptagonis»128.57°andsothreeheptagonssumtomorethan360°.Workwithapartnerandtogethermakeonesheetofhyperbolicpapertouseintheseexplorations.(Youwillalsoneedalengthofstring,aprotractor,andcoloredmarkersorpencils.)Yourfinalresultshouldlooklikethis:

1) Whatwillastraightlinelooklikeonthehyperbolicplane?Usethesameconceptofstraightnessthatweusedonthesphere(thestringmethod)anddrawseveralstraightlinesonyourhyperbolicpaper.Doyouthinktheselinesarefiniteinlengthlikethoseonthesphere–oraretheyinfinitelikelinesontheflatEuclideanplane?(Rememberthereisnothingexcepttimetokeepyoufromaddingmoreheptagonstoalltheedgesofyourhyperbolicpaper.Youareworkingwithapieceofthehyperbolicplane,justlikearegular8½x11sheetofpaperisapieceoftheCartesianplane.)


134

2) Canyoudrawparallellinesonyourhyperbolicpaper?(Makecertainyouareusingthestringmethodtodrawlines.)Canyoumakeanargumentthatthelinesyoudrewdonotintersectsomewhereonanextensionofyourpaper?Whatdoyounoticeaboutthedistancebetweenhyperboliclinesthatdonotintersect?HowdoesthisdifferfromEuclideanparallellines?

3) Nowchooseapointononeofyourparallellines.Canyoudrawanotherlinethroughthatpointthatisalsoparalleltothefirstline?Howmanyhyperboliclinescanbedrawnparalleltothefirstlinethroughthissamepoint?(Ifyouroriginalpairofparallellinesarequiteclosetogether,itwillbeeasiertoanswerthisquestionifyouchooseapointfartherawayfromoneofthelinesandseehowmanyparallellinesyoucandrawthroughthatpoint.)

4) ThereisaEuclideantheoremstatingthattwolinesthatarebothparalleltothesameline

arealsoparalleltoeachother.Doyouthinkthistheoremholdsinhyperbolicgeometry?Whyorwhynot?

5) Drawasmalltriangleandalargetriangle(onethatcoversatleast1/4ofthepaper)onyourhyperbolicpaper.Makecertainthatthesidesofyourtrianglesareactuallystraightlinesegmentsbyusingthestringmethodtomakethetriangle.Determineamethodtomeasuretheanglesofthetrianglesusingyourprotractorandthenmeasureeachoftheanglesinbothofthetriangles.Whatistheanglesumofthesmalltriangle?Ofthelargetriangle?Makeaconjectureabouttheanglesumofahyperbolictriangle.

135

ReadandStudy:

OutofnothingIhavecreatedastrangenewuniverse. JanosBolyaiThestoryofthedevelopmentofhyperbolicgeometryreallybeginswithEuclid.Recallthathechosefivepostulatesforhisaxiomaticsystem–thefirstfourweregenerallyaccepted,butthefifthpostulate(theParallelPostulate)causedproblemsfromtheverybeginning.First,itwaslotsmorecomplicatedthantheothers.Second,itdidnotseemas‘self-evident.’ManymathematicianshavetriedtoprovetheParallelPostulatefromtheotherfour,thinkingittoocomplexastatementtoacceptwithoutproof.Inthe1700’s,theItalianmathematicianSaccherimountedaconcertedefforttoshowthatiftheParallelPostulatewasreplacedbyonethatallowedmorethanoneparallel,theresultingtheoremswouldcontradictthemselves.Whilehefoundmanyinterestingresults,hedidnotfindthecontradictionhesought.However,hewassosurethattheParallelPostulateofEuclidwastheonlytruecase,heconcludedhisworkbysaying(withoutproof)thatanyotherreplacementpostulateisabsolutelyfalsebecauseitis“repugnanttothenatureofthestraightline.”AcenturylaterthefamousGermanmathematicianGausscametotheconclusionthatthe5thpostulateistrulyindependentoftheothers.Inotherwordsitcannotbeprovedusingtheotherpostulates(axioms)andnordoesitcontradictthem.Furthermore,itcanbereplacedbyalternativepostulateswhichwillyieldinterestingandconsistentgeometriesdifferentfromEuclideangeometry.Readthisparagraphagain.Itisimportant.However,Gausswasnotwillingtoriskhissignificantmathematicalreputationbypublishinghisresults.Andsoitwaslefttotwounknowns,HungarianJánosBolyaiandRussianNikolaiLobachevsky,toindependentlypublishtheirfindingsofthisstrangenewgeometrywenowcallhyperbolicgeometry.Asanaxiomsystem,hyperbolicgeometryretainsalltheaxiomsofEuclideangeometryexcepttheParallelPostulate,replacingitwiththeHyperbolicParallelPostulate:Givenalineandapointnotonthatline,thereareatleasttwolinesthroughthatpointparalleltothegivenline.(InSphericalGeometry,PostulatesIandIIIofEuclidareviolatedaswellastheParallelPostulate.Explainhow.)Ofcourse,aswehavealreadyseeninourlookattaxicabgeometry,makingjustonechangecanresultinaverydifferentgeometry.Hyperbolicgeometryisnoexception.Physically,wecanunderstandthedifferencebetweenEuclidean,hyperbolic,andsphericalgeometrybyconsideringthecurvatureofthesurfaceofaplaneineach.TheEuclideanplaneisflat;thesphericalplaneiscurvedpositivelysothatitclosesuponitself;andthehyperbolicplaneiscurvednegativelysothatstandingatanyonepointthesurfacecurvesupalongonedirectionandcurvesdownalongtheperpendiculardirection,likestandinginthemiddle

136

ofasaddleoraPringlepotatochip.Herearesomeotherpicturesofobjectswithnegativecurvature.Thesecomefromhttp://xahlee.org/surface/gallery_o.html.

Sohowdoesthischangeincurvature(orequivalently,thischangeintheparallelpostulate)changethegeometry?LikeEuclideangeometry,thehyperbolicplaneisinfiniteandunboundedandsoarehyperboliclines.Ifweweretowalkalongahyperboliclineinonedirection,wewouldneverreturntoourstartingpoint,aswedoinsphericalgeometry.Wehaveanabundanceofparallellines,but,unlikeEuclideangeometry,notwoparallellinesareequidistant.Thereareactuallytwotypesofparallellines.Inonecase,twoparallellineswillbeclosesttoeachotherattheirsinglecommonperpendicularandthendivergefromeachotherasyoumoveawayfromthatcommonperpendicularineitherdirection.Intheothercase,twoparallellinesareasymptoticinonedirectionanddivergentintheother.Thinkaboutthis.Ifwehaveapairof“lines”thatareequidistant,oneofthe“lines”isnotaline,butacurve.Thisissimilartothesituationonthespherewheretheequatorandthe10°latitudemarkingareequidistant,butonlytheequatorisaline.Wehavetrianglesandotherpolygonsinhyperbolicgeometry,but,onceagain,theybehavedifferently.Thereisonlyonehyperboliclinesegmentbetweentwopointssohyperbolictrianglesarewell-definedusingtheEuclideandefinition.Buttheanglesumofahyperbolictriangleisnotconstantandisalwayslessthan180°.Furthermore,theareaofahyperbolictrianglegetslargerastheanglesumgetssmaller.Andwecanmaketheanglesumsmallerbymakingthesidelengthslonger.(CheckoutyourresultsfromtheClassActivity.Dotheysupporttheseclaims?)Asinsphericalgeometry,thereisaformulaforfindingtheareaofahyperbolictrianglethatdependsonlyonthemeasuresofitsangles:𝐴 = 𝜋 − (𝛼 + 𝛽 + 𝛾).(Noticetherelationshipwiththeareaformulaforasphericaltriangle.)Thisformulashowsusthatthelargestareathatahyperbolictrianglecanhaveisp.Astheanglesumapproacheszero,theareaapproachesp,andthesidelengthsapproachinfinitelength.Soas

137

oursidesgetlongerandlonger,theanglesgetsmallerandsmallerandourareanevergetslargerthanp.Thismeansthattheanglemeasuredeterminesnotonlytheshapeofthetrianglebutalsoitssize.

Homework:

ForGod’ssake,pleasegive[hyperbolicgeometry]up.Fearitnolessthanthesensualpassion,becauseittoo,maytakeupallyourtimeanddepriveyouofyourhealth,peaceofmindandhappinessinlife.

WolfgangBolyai(Janos’Father)


2) Giventhattheanglesumofanyhyperbolictriangleislessthan180°,arguethatrectanglesdonotexistinhyperbolicgeometry.

3) Dosimilarbutnotcongruenttrianglesexistinhyperbolicgeometry?WhatabouttheAAA

Theorem?Supportyouranswer.

4) DoestheIsoscelesTriangleTheoremholdforhyperbolictriangles?Supportyouranswer.

5) ThereisaEuclideantheoremstatingthattwolinesthatarebothparalleltothesamelinearealsoparalleltoeachother.Doesthistheoremholdinhyperbolicgeometry?Supportyouranswer.

6) Canwebuildasetofrailroadtracksonahyperbolicplane?Supportyouranswer?

7) Canaright-angledregularpentagonexistonthehyperbolicplane?Supportyouranswer.

138

ClassActivity21:LifeinaFractalWorld

Themostexcitingphrasetohearinscience,theonethatheraldsnewdiscoveriesisnot‘Eureka!’but‘That’sfunny…’

IsaacAsimovInthisactivityyouwillcreateafamousfractal,theKochSnowflake,andtheninvestigateseveralofitsproperties.Tocreateanyfractalwemustapplyaprocesstoaninitialgeometricobjectandthenapplythesameprocesstotheresultingobjectandthenapplythesameprocesstotheresultingobjectandthenapplythesameprocesstotheresultingobjectandthen…yougettheidea.Wecallsuchaprocedureaniterativeprocessandtheobjectineachstepiscalledaniteration.Whentheiterativeprocessproducesobjectsthatareincreasinglycomplex,butsimilartothefirstiterationonasmallerandsmallerscale,the‘final’iterationisafractal.Intheory,theprocessisrepeatedindefinitely,sotherereallyisnofinaliterationbutratherlimitingobjectthatistheactualfractal.Don’tworry;we’llonlyproducethreeiterationsoftheKochSnowflake.TocreatetheKochSnowflake,takeanequilateraltriangle(theinitialgeometricobject)andapplythefollowingiterativeprocesstoeachsideofthetriangle.

Step1:Divideeachlinesegmentintothirdsandremove(erase)themiddlethird.Step2:Replacethemiddlethirdwithtwosidesofanequilateraltrianglewhosesidelengthisthesameasthelengthofthemiddlethirdyouremoved.

Thefollowingpictureshowstheprocessappliedoncetoonesideoftheoriginaltriangle.

(Thisactivityiscontinuedonthenextpage)

139

1) Constructanequilateraltrianglewithsidesapproximately2incheslonganduseittocreate

thefirstthreeiterationsoftheKochSnowflake.Whenyouhavefinishedyoushouldhaveaseparatedrawingforeachiteration.Assumethesidelengthoftheoriginaltriangleisoneunitinansweringthefollowingquestions.

2) Whatistheperimeteroftheoriginaltriangle?Thefirstiteration?Theseconditeration?Thethirditeration?Lookforapatternandmakeaconjecturefortheperimeterofthenthiteration.WhatabouttheperimeteroftheKochSnowflake(the“infinite”iteration)?

3) Whatistheareaoftheoriginaltriangle?Thefirstiteration?Theseconditeration?Thethird

iteration?(Itwillbesimplertoseeapatternifyouuseanon-standardunitforarea–wesuggestusingtheareaofthesmallesttriangleinthethirditerationastheunit.)Lookforapatternandmakeaconjecturefortheareaofthenthiteration.WhatabouttheareaoftheKochSnowflake(the“infinite”iteration)?

140

ReadandStudy:

Biggasketsaremadeoflittlegaskets,Thebitsintowhichweslice‘em.AndlittlegasketsaremadeoflessergasketsAndsoadinfinitum.

Fromhttp://classes.yale.edu/fractals/ Takeacloselookattheclouds,mountainridges,lakeshoresandicebergsinthetwopicturesbelow(takenbyanauthorinAlaska).Whatgeometricshapecanbeusedtoadequatelydescribetheintricaciesoftheirboundaries?Asphericaltriangle?AEuclideancircle?Ahyperbolicpolygon?No.Nothingwehavestudiedthusfarcomesclosetoapproximatingthecomplexityofthesenaturalshapes,particularlywhentheyareexaminedonasmallscale.

BenoitMandelbrotisthemathematiciancreditedwithfindingthegeometricstructureunderlyingthesecomplicatednaturalshapes.In1975hecoinedthewordfractal(fromtheLatinwordfractusmeaningbrokenorfractured)todescribetheconvolutedcurvesandsurfacesthatcanbeusedto

141

modelnaturalshapes.Thekeytohisunderstandingwashisobservationthatmanyrealphenomena,suchascoastlines,mountainsandlungs,havearoughlyself-similarshape:Thesmallerfeaturesoftheseobjectshaveapproximatelythesameshapeandcomplexityasthelargerfeaturesdo.Thatis,asmallportionofamountainridgewilllookapproximatelylikeanentiremountainridgewhenmagnified.Thinkaboutthis.Belowisacomputer-generatedfractalpicture(notarealpicture)ofridgescutbyastream.Itlooksrealdoesn’tit?

Mandelbrotusedtheconceptsofself-similarityandcomplexityundermagnificationtodescribecertainmathematicalsetsthatarefractal.Afamousexample,calledtheMandelbrotset,hasaboundarythatisamathematicalfractal.

TheMandelbrotSetfromhttp://en.wikipedia.org/wiki/Fractal

142

Approximatefractalsareeasilyfoundinnature.Theseobjectsdisplayself-similarstructureovermanymagnifications.Examplesincludeclouds,snowflakes,mountains,rivernetworks,andbroccoli.Treesandfernsarealsofractalinnatureandcanbemodeledonacomputerbyusingarecursive(iterative)algorithm.Thisrecursivenatureisobviousintheseexamples—abranchfromatreeorafrondfromafernisaminiaturereplicaofthewhole:notidentical,butsimilarinnature.Fractalsprovideagoodmodelformanyorgansofthebody,suchasthelungs.Thetracheasplitsintothebronchialtubes,whichinturnsplitintoshorterandnarrowertubes.Eventheembryonicdevelopmentofthelungisaniterativeprocess.Theconvolutedsurfaceofthelunggreatlyincreasesitsareawhilekeepingitsoverallvolumesmall.Thelargesurfaceareaisbiologicallyessentialbecausetheamountofcarbondioxideandoxygenthatthelungscanexchangeisroughlyproportionaltotheirsurfacearea.Usingalightmicroscope,biologistsfoundapproximately80m2ofsurfaceareainalung(roughlythefloorspaceofasmallhouse).Thehighermagnificationofanelectronmicroscopeyieldedapproximately140m2.Scientistshaveestimatedthefractaldimensionofalungtobe2.17(ThomasQ.Sibley.TheGeometricViewpoint.p.220–221).We’lltellyouwhatwemeanbythatinaminute.Alloftheseexamplespointoutthreenecessarycharacteristicsofafractal:

1) itisself-similar(atleastapproximately);2) itcanbedefinedbyaniterativeprocess;and3) ithasanon-integerdimensionthatitlargerthanitsgeometricdimension.

(Notethatnotallself-similarobjectsarefractal.Forexamplealineisself-similar,butitsdimensionisone,soitisnotafractal.)

Let’stalksomemoreaboutdimensionforaself-similarobject.Wewilldetermine“dimension”bydoublingitslengthandseeinghowmanycopiesoftheoriginalobjectweget.Thedimensionistheexponenttowhichyoumustraisethescalingfactor(2fordoubling)inordertogetthenumberofcopiesproducedbythatscaling.Alinesegmenthasdimensiononebecausewhenyoudoublethelengthofthesegmentyougettwocopiesofthesegmentand21=2.Asquarehasdimension2becausewhenyoudoublethelengthofthesideyougetfourcopiesofthesquareand22=4.Acubehasdimension3becausewhenyoudoublethelengthoftheedgeyougeteightcopiesofthecubeand23=8.Wecanwritethisrelationshipasaformulaasfollows:

sndornsd

loglog

==

Heresiscalledthescalingfactor,disthedimension,andnisthenumberofcopiesproduced.(Trytoexplainthesecondversionoftheformula.Howdowesolvethefirstequationford?)

143

IfweusethisformulaontheKochSnowflake,wehaves=3,n=4,andd= 26.13log4log= .

TheKochSnowflakehasadimensionof1.26.(Weird,huh?Makesureyoucanexplainwhys=3andn=4.)Insomeway,thedimensionisameasureofthecomplexityofthefractal.TheKochSnowflakeismorecomplexthanastraightline,butnotascomplexasasquare(includingtheinterior).IntheclassactivityyouexploredtheperimeterandareaoftheKochSnowflake(namedfortheSwedishmathematicianwhofirstcreateditin1904).Didyoudiscovertheamazingfactthatthisfractalhasaninfiniteperimeterbutafinitearea?Inotherwords,youcandrawacirclearoundtheentirefractalenclosingitwithinafinitearea,buttheboundaryoftheenclosedfractalisinfiniteinlength.Inthehomeworkproblemsyouwilldeterminetheperimeter,area,anddimensionofseveralotherfractals.Beonthelookouttoseeifinfiniteperimeterandfiniteareajustmightbeapropertyofallfractals.ConnectionstotheMiddleGrades:

Learningisnotcompulsory…neitherissurvival. W.EdwardsDemingYourfuturestudentswilllikedoinggeometryonaMöbiusstrip,abeachballoratorus–buttheywilllovefractals.Notonlydofractalslookcool,buttheinfiniteaspectisfascinatingtoMiddle-Schoolers.Wewillpresentjustonefractalactivityhere;youcancertainlyfindmany,manymoreonline.Forasample,govisitthewebsiteathttp://math.rice.edu/~lanius/frac/foranonlinelessononfractalsdesignedforstudentsinGrades4to8.BecertaintoinvestigatetheKochSnowflakeandthesectiononfractalproperties.Answerthequestionsfoundatthewebsite.Visitthewebsiteathttp://classes.yale.edu/fractals/foracompleteandaccessiblediscussionoffractalswithlotsoffascinatingpictureswrittenbyMandelbrothimself.Thereareevenlessonplansformiddleschoolclassrooms.Besuretocheckoutthefractallandscapesfoundunder“MoreExamplesofSelf-Similarity.”TheSierpinskiTriangleisanotherfractalthatyourstudentscaninvestigate.

144

SowhatistheSierpinskiTriangle?Here’stheidea.Beginwithanequilateraltriangle:

Locatethemidpointofeachsideandcreateanewtrianglebyconnectingthosemidpoints.Thenremovethatmiddletriangle.

Nowdothesamethingtoeachofthethreeresulting‘outside’triangles.

Keepongoingforever.(Recallthatmathematicalobjectsareidealobjects–sotheideaofimagingwhatwouldhappenifaprocessisrepeatedforeverdoesnotbothermathematicians.)TheresultingfractalistheSierpinskiTriangle.Whyisitafractal?Usethedefinitiontoexplainthis.Supposethattheoriginaltrianglehadanareaof1u2.Findaformulafortheareaatthenthstep.

Step 0

Step 1

Step 2

145

Homework:…sincegeometryistherightfoundationofallpainting,Ihavedecidedtoteachitsrudimentsandprinciplestoallyoungsterseagerforart.

AlbrechtDurer,CourseintheArtofMeasurement


2) DotheproblemsintheConnectionssection.

3) Carefullysketchthreeiterationsofeachfractalidea.a) Astylizedtree,whereeachbranchsplitsintothreeothershalfaslong.Beginwith

onetrunkandthreebranches.b) AmodifiedKochcurve,withasquareonthemiddlethirdofalinesegment,rather

thanatriangle.Applythisiterativeprocesstoeachsideofasquare.4) FindtheperimeterofeachfractalinProblem3.

5) FindthelimitingareaofthefractalinProblem3b.

6) FindthedimensionofeachfractalinProblem3.

7) Inthereadingwediscussedtheconceptofself-similarity.Anotherwaytodescribethis

propertyistosaythataself-similarobjectcanbecomposedofsmallersimilarcopiesofitself.Whichofthefollowinggeometricobjectsareself-similar:alinesegment,atriangle,asquare,atrapezoid,ahexagon,acircle?Whichoftheself-similarobjectsarealsofractals?Why?

8) Picturedbelowarethefirstfouriterationsoftheboxfractal.Writetheinstructionsforthe

iterativeprocessthatcreatesit.Whatistheperimeterandareaofthelastiterationshownifthesideoftheoriginalsquareisoflengthone?Whatisthedimensionoftheboxfractal?

146

SummaryofBigIdeasfromChapterThree Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes

• Wecanchangeourgeometrybychangingthespace,liketheKleinbottle,sphere,ortorus,thewaywemeasuredistance,likeintaxi-cabgeometry,orbyadjustinganaxiom,likeinhyperbolicgeometry.

• Taxi-cabgeometryisanon-Euclideangeometrythatmiddlegradesstudentscanexplore.

• SeveraltheoremsfromEuclideangeometryfailwhenappliedtoSphericalandHyperbolicgeometries.

• AFractalisageometricfigurethatisself-similar,thatcanbedefinedbyaniterative

process,andhasanon-integerdimension.

147

APPENDICES

148

References:

• Adams,T.L.&Aslan-Tutak,F.(2005)ServingUpSierpinkski!MathematicsTeachingintheMiddleSchool,11(5),p.248-253.

• Battista,M.(2007).Thedevelopmentofgeometricthinking.IntheSecondHandbookofResearchonMathematicsTeachingandLearning,F.Lester(Ed.).NCTM:InformationAgePublishing.

• CommonCoreStateStandardsasfoundinJanuary2012athttp://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

• MathematicalQuotationsServer(MQS)atmath.furman.edu.

• NationalCouncilofTeachersofMathematics.(2006).CurriculumFocalPointsfor

PrekindergartenthroughGrade8Mathematics:AQuestforCoherence.Reston,VA:NCTM.

• NationalCouncilofTeachersofMathematics.(2000).PrinciplesandStandardsforSchoolMathematics.Reston,VA:NCTM.

• Poole,J.T.(2002).Elements.FoundonJanuary10,2012athttp://math.furman.edu/~jpoole/euclidselements/euclid.htmDepartmentofMathematics,FurmanUniversity,Greenville,SC.

• Sharp,J.&Heimer,C.(2002).Whathappenstogeometryonasphere?MathematicsTeachingintheMiddleSchool,8(4),p.182.

• Shulman,L.S.(1985).Onteachingproblemsolvingandsolvingtheproblemsofteaching.In

E.A.Silver(Ed.),TeachingandLearningMathematicalProblemSolving:multipleresearchperspectives(pp.439-450).Hillsdale,NJ:Erlbaum.

• Sibley,T.Q.(1997)TheGeometricViewpoint:ASurveyofGeometries.Addison-Wesley.

149

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.

150

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.

151

Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.

Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.

Proposition14.Ifwithanystraightline,andatapointonit,twostraightlinesnotlyingonthesamesidemaketheadjacentanglesequaltotworightangles,thetwostraightlineswillbeinastraightlinewithoneanother.

Proposition15.Iftwostraightlinescutoneanother,theymaketheverticalanglesequaltooneanother.

Proposition16.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisgreaterthaneitheroftheinteriorandoppositeangles.

Proposition17.Inatriangletwoanglestakentogetherinanymannerarelessthantworightangles.

Proposition18.Inanytrianglethegreatersidesubtendsthegreaterangle.

Proposition19.Inanytrianglethegreaterangleissubtendedbythegreaterside.

Proposition20.Inanytriangletwosidestakentogetherinanymanneraregreaterthantheremainingone.

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

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

Proposition23.Onagivenstraightlineandatapointonittoconstructarectilinealangleequaltoagivenrectilinealangle.

152

Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.

Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.

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

Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.

Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.

Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.

Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.

Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.

Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.

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

153

Proposition34.Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.

Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.

Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.

Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.

Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.

Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.

Proposition46.Onagivenstraightlinetodescribeasquare.

154

Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.

Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.

155

Glossary:Acuteangle–ananglewithmeasurelessthanthemeasureofarightangle

Acutetriangle–atrianglewiththreeacuteangles

Adjacentangles–twonon-overlappinganglesthatshareavertexandacommonray

Affineplane–ageometrywithparallellinesbasedontheaffinesetofaxioms

Algorithm–asetofstepsusedtocarryoutaprocedure

Alternateexteriorangles–twoangles(formedbyatransversalofapairoflines)thatlieoutside

thelinesandonoppositesidesofthetransversal

Alternateinteriorangles–twoangles(formedbyatransversalofapairoflines)thatliebetween

thelinesandonoppositesidesofthetransversal

Altitude(ofatriangle)–thelinethroughavertexthatisperpendiculartotheoppositeside

Altitude(ofapyramid)–thelinesegmentfromtheapexperpendiculartothebaseofthe

pyramid;alsocalledtheheight

Altitude(ofaprism)–alinesegmentperpendiculartothebasesoftheprism;alsoinformally

calledthe“height”

Analyticgeometry–theuseofacoordinatesystemtotranslategeometricproblemsintoalgebraic

problems

Angle–thefigureformedbytworayswithacommonendpoint

Anglebisector–thelinethroughthevertexofananglethatdividestheangleintotwocongruent

angles

Antipodalpoints–pointsthataretheendpointsofadiameterofasphere

Apex(ofapyramid)–thecommonpointofthenon-basefacesofapyramid

Apex(ofacone)–thecommonpointofthelinesegmentsthatcreateacone

Arc–thesetofpointsonacirclebetweentwogivenpointsofthecircle(Thereareactuallytwo

arcsbetweenanytwogivenpoints;theshorteroneiscalledtheminorarcandthelonger

oneiscalledthemajorarc.)

Area–thequantityoftwo-dimensionalspaceenclosedbyaplanefigure

Attribute–apropertyofageometricobjectthatcanbemeasured(suchaslength)orcategorized

(suchascolor)

156

Axiom–astatementthatistruebyassumption

Axiomaticsystem–asetofundefinedterms,definitions,axioms,andtheoremsthatcreatea

mathematicalstructure

Axis(ofacone)–thelinejoiningtheapextothecenterofthe(circle)base

Axisofsymmetry–alineinspacearoundwhichathree-dimensionalobjectisrotated

Baseangles(ofanisoscelestriangle)–theanglesthatareoppositethecongruentsidesofan

isoscelestriangle

Bilateralsymmetry–anobjecthasbilateralsymmetrywhenithasexactlyonelineofreflectional

symmetry

Bisect–todivideageometricobject(suchasalinesegmentoranangle)intotwocongruent

pieces

Boundary–thesetofpointsthatseparatetheinsideofaclosedplanarobjectfromtheoutside

Center(ofacircle)–thepointthatisequidistantfromallpointsonthecircle

Centralangle–ananglewhosevertexisacenterofageometricobject

Centroid–thepointofintersectionofthethreemediansofatriangle;alsoknowntobethecenter

ofmassofthetriangle

Chord–alinesegmentwhoseendpointsaredistinctpointsonagivencircle

Circle–thesetofpointsthatarethesamedistancefromagivenpoint,calledthecenter

Circumcenter–thepointofintersectionofthethreeperpendicularbisectorsofatriangle;alsothe

centerofthecirclethatcircumscribesthetriangle

Circumscribedcircle–thecirclethatcontainsalltheverticesofapolygon

Closedcurve–acurvethatstartsandstopsatthesamepoint

Closure(ofasetunderanoperation)–thepropertythattheresultoftheoperationonanytwo

elementsofthesetisalsoanelementoftheset

Collinearpoints–pointsthatlieonthesameline

Complementaryangles–twoangleswhosemeasuressumtothemeasureofonerightangle

Compositionofrigidmotions–thecombinedactionsoftworigidmotionswiththesecondmotion

appliedtotheimageofthefirstmotion

Concavepolygon–apolygonforwhichatleastonediagonalliesoutsidethepolygon

157

Concurrentlines–threeormorelinesthatintersectinthesamepoint

Cone(circular)-athree-dimensionalgeometricobjectconsistingofalllinesegmentsjoininga

singlepoint(calledtheapex)toeverypointofacircle(calledthebase)

Congruentobjects–twogeometricobjectsarecongruentifoneobjectistheimageoftheother

underarigidmotionoftheplane.

Conicsections–thefourcurves(circleellipse,hyperbola,andparabola)formedwhenaplane

intersectsadoublecone.

Conjecture–aguessorahypothesis

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

Convexpolygon–apolygonallofwhosediagonalslieinsidethepolygon

Consistent(setofaxioms)–oneinwhichitisimpossibletodeducefromtheseaxiomsatheorem

thatcontradictsanyaxiomorpreviouslyprovedtheorem

Construction–creatingageometricobjectusingonlystraightlinesegmentsandcircles(Euclid’s

first,second,andthirdaxioms)

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

Coordinate(Cartesian)plane–amodelofEuclideangeometryinwhicheachpointisidentifiedby

twocoordinates,thefirstofwhichrepresentsthehorizontaldistanceofthepointfromthe

y-axisandthesecondofwhichrepresentsverticaldistancefromthex-axis.(Thex-andy-

axesareperpendicularandlieinthesameplane.)

Coplanarlines–linesthatlieinthesameplane

Correspondingangles-twoangles(formedbyatransversalofapairoflines)thatlieonthesame

sideofthetransversalandalsolieonthesamesideofthepairoflines

Correspondingpoints–apairofpoints,oneofwhichistheoriginalpointandtheotherofwhichis

theimageofthatpointunderarigidmotion

Counterexample–anexamplethatshowsaconjectureisfalse

Curve–asetofpointsdrawnwithasinglecontinuousmotion

Cylinder(circular)–athree-dimensionalgeometricobjectconsistingoftwoparallelandcongruent

circles(andtheirinteriors)andtheparallellinesegmentsthatjoincorrespondingpointson

thecircles

158

Deductivereasoning–theprocessofcomingtoaconclusionbasedonlogic

Definition–astatementofthemeaningofaterm,word,orphrase

Degree–aunitofanglemeasureforwhichafullturnaboutapointequals360degrees

Diagonal–thelinesegmentjoiningtwonon-adjacentverticesofapolygon

Diameter–alinesegmentthroughthecenterofacirclewhoseendpointslieonthecircle

Dimension(ofarealspace)–thenumberofmutuallyperpendiculardirectionsneededtodescribe

thelocationofthesetofpointsinthatspace

Edge–thelinesegment(side)thatissharedbytwofacesofapolyhedron

Ellipse–thesetofpointsPintheplanesuchthatthesumofthedistancesfromPtotwogiven

pointsF1andF2isconstant.ThepointsF1andF2arecalledthefocioftheellipse.

Equiangular(polygon)–apolygonallofwhosevertexanglesarecongruent

Equilateral(polygon)–apolygonallofwhosesidesarecongruent

Euclideanmodel–amodelofthegeometryoftheinfiniteflatplanebasedontheaxiomsystem

firstestablishedbyEuclid

Euler’sline–thelinecontainingthecircumcenter,thecentroid,andtheorthocenterofatriangle

Exteriorangle–theangleformedbyasideofapolygonandtheextensionofanadjacentside

Face–apolygon(withinterior)thatformsaportionofthetwo-dimensionalsurfaceofa

polyhedron

Finitegeometry–ageometrythatconsistsofafinitenumberofpointsandtheirrelationships

Fixedpoint–apointPwhoseimageunderarigidmotionisP

Fractal–anobjectthatresultsfromapplyinganiterativeprocessinwhicheachiterationis

increasinglycomplex,butself-similar

Function–arulethatassignstoeachelementofasetSanelementofsetTinsuchawaythat

everyelementinSispairedwithanelementofTandnoelementofSisassignedtomore

thanoneelementofT

Glidereflection–arigidmotionthatisthecompositionofatranslationandareflectioninwhich

thelineofreflectionandthetranslationvectorareparallel

Greatcircle–theintersectionofasphereandaplanethatcontainsthecenterofthesphere

Height(ofatriangle)–lengthofthelinesegmentfromavertexperpendiculartotheoppositeside

159

Hyperbola–thesetofpointsPintheplanesuchthatthedifferenceofthedistancesfromPtotwo

givenpointsF1andF2isconstant

Hypotenuse–thesideofarighttriangleoppositetherightangle

Identificationspace–atwodimensionalmodelofanobjectthatlivesinhigherdimensions.The

modelshowshowsidesareidentified(“gluedtogether”)

Image(ofarigidmotion)–thesetofpointsthatresultfromthemotionofanobjectbyarigid

motionoftheplane

Incenter–thepointofintersectionofthethreeanglebisectorsofatriangle;alsothecenterofthe

inscribedcircle

Incircle(inscribedcircle)–thecirclethatistangenttoallsidesofapolygon

Inductivereasoning–theinformalprocessofcomingtoaconclusionbasedonexamples

Inscribedcircle–thecirclethatistangenttoeachsideofapolygon

Intersection(oftwolines)–thepoint(s)thelineshaveincommon

Intersection(oftwosets)–thesetofelementsthatarecommontobothsets

Isosceles–havingatleastonepairofcongruentsides

Iterativeprocess–analgorithmappliedtoanobjectandthentotheresultandthentotheresult

andsoforth.Theobjectineachstepoftheprocessiscalledaniteration

Justification–anargumentbasedonaxioms,definitions,andpreviouslyprovenresultstoshow

thataconjectureistrue

Leg–asideofarighttriangleoppositeanacuteangle

Length–themeasureofa1-dimensionalobject

Line–anundefinedone-dimensionalsetofpointsunderstoodtofollowtheshortestpath

(betweeneverypairofpointsontheline)andtoextendinoppositedirectionsindefinitely

Lineofreflection–thelineaboutwhichanobjectisreflectedtoformitsmirrorimage

Linesegment–thesetofpointsonalinebetweentwogivenpoints,calledtheendpoints

Locus(definition)–adefinitionthatdescribesacurveasasetofpointsintheplane

Logicallyequivalent(statements)–statementsthathavethesametruthvalueineverycase

Lune–aconcaveplaneregionboundedbytwoarcsofdifferentradii

160

Majorsegment(ofagreatcircle)–thelargerofthetwoarcsdeterminedbytwodistinctpointson

agreatcircle

Measure–todeterminethequantityofanattribute(orofafundamentalconceptsuchastime)

usingagivenunit

Median–thelinesegmentjoiningavertexofatriangletothemidpointoftheoppositeside

Midpoint–thepointonalinesegmentthatdividesitintotwocongruentlinesegments

Minorsegment(ofagreatcircle)–thesmallerofthetwoarcsdeterminedbytwodistinctpoints

onagreatcircle

Model–arepresentationofanaxiomsysteminwhicheachundefinedtermisgivenaconcrete

interpretationwhichallowtheaxiomstomakesense

Net–atwo-dimensionalmodelthatcanbefoldedintoathree-dimensionalobject

Obtuseangle–ananglewithmeasuregreaterthanthemeasureofarightangle

Obtusetriangle–atrianglewithoneobtuseangle

One-to-one(function)–afunctionfromasetStoasetTinwhichnoelementofTisassignedto

morethanoneelementofS

Onto(function)–afunctionfromasetStoasetTinwhicheveryelementofTisassignedtosome

elementfromS

Order(ofanaffineplane)–thenumberofpointsoneachlineoftheplane

Order(ofaprojectiveplane)–thenumberofpointsoneachlineoftheplanelessone

Order(ofarotationalsymmetry)–thenumberofdifferentrotationsthatareasymmetryofan

object

Orientation–thedirection,clockwiseorcounterclockwise,ofthereadingoftheverticesofa

polygoninalphabeticalorder

Orthocenter–thepointofintersectionofthethreealtitudesofatriangle

Orthogonal(circles)–intersectingcircleswhoserespectiveradii(orrespectivetangents)are

perpendicularatthepointsofintersection

Parabola–thesetofpointsPintheplanesuchthatthedistancefromPtoagivenpointFisequal

tothedistancefromPtoagivenlinem.PointFiscalledthefocusoftheparabolaandline

misthedirectrix

161

Parallellines–coplanarlineswithnopointsincommon

Parallelogram–aquadrilateralinwhichbothpairsofoppositesidesareparallel

Partition–adivisionofageometricobjectintoasetofnon-overlappingobjectswhoseunionis

theoriginalobject

Perimeter(ofaplaneobject)–thelengthoftheboundaryoftheobject

Perpendicularbisector–thelinethroughthemidpointofalinesegmentthatisalsoperpendicular

tothelinesegment

Perpendicularlines–twolinesthatintersecttoformfourrightangles

Pi–theratioofthecircumferenceofacircletoitsdiameter;thisratioisanirrationalnumberthat

isconstantforallsizecirclesandisapproximatelyequalto3.1415926

Planarcurve–acurvethatliesentirelywithinaplane

Plane–anundefinedtwo-dimensionalsetofpointsunderstoodtoextendinalldirections

indefinitely

Planeofsymmetry–aplaneinspaceaboutwhichathree-dimensionalobjectisreflected

Point–anundefinedzero-dimensionalobjectunderstoodtobealocationwithnosize

Polygon–asetoflinesegmentsthatformasimpleclosedplanarcurve

Polyhedron(plural:polyhedra)–afinitesetofpolygonsjoinedpair-wisealongthesidesofthe

polygonstoencloseafiniteregionofspacewithinonechamber

Postulate–anaxiom

Prism–apolyhedroninwhichtwoofthefacesareparallelandcongruent(calledthebases)and

theremainingfacesareparallelograms

Projectiveplane–ageometryinwhichtherearenoparallellinesbasedontheprojectivesetof

axioms

Proof–ajustificationwritteninformalmathematicallanguage

Pyramid–apolyhedroninwhichallbutoneofthefacesistrianglesthatshareacommonvertex

(calledtheapex);theremainingfacemaybeanypolygonandiscalledthebase

Quadrilateral–apolygonwithexactlyfoursides

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

thesettowhichthestatementapplies

162

Radius(plural:radii)–thelinesegmentjoiningapointonacircletothecenterofthecircle

Ray–thesetofpointsonalinebeginningatagivenpoint(calledtheendpoint)andextendingin

onedirectiononthelinefromthatpoint

Rectangle–aquadrilateralwithfourrightangles

Rectilinearangle–anangleformedbystraightlines(asopposedtocurves),nowadayssimply

referredtobyangle

Redundant(setofaxioms)–asetofaxiomsinwhichitispossibletoproveatleastoneofthe

axiomsfromtheotheraxioms

Reflection(inalinel)–arigidmotionoftheplaneinwhichtheimageofapointPonlisP,andif

A¹PandiftheimageofAis 'A ,thenlistheperpendicularbisectorof 'AA .

Reflectionalsymmetry(2-dimensional)–areflectioninwhichanobjectisdividedbythelineof

reflectionintotwopartsthataremirrorimagesofeachother

Reflectionalsymmetry(3-dimensional)–areflectioninwhichanobjectisdividedbytheplaneof

reflectionintotwopartsthataremirrorimagesofeachother

Regularpolygon–apolygonwithallsidescongruentandallvertexanglescongruent

Regularpolyhedron–apolyhedronwhosefacesareallthesameregularpolygonwiththesame

numberoffacesmeetingateachvertex

Rhombus(plural:rhombi)–aquadrilateralwithfourcongruentsides

Rightangle–ananglethatformsexactlyonefourthofacompleteturnaboutapoint

Righttriangle–atrianglewithonerightangle

RigidMotionoftheplane–amotionoftheplanethatpreservesthedistancesbetweenpoints

Rotation(aboutapointPthroughanangleq)–arigidmotionoftheplaneinwhichtheimageof

PisPand,iftheimageofAis 'A ,then PA@ 'PA and 'm APA =q.PointPiscalledthe

centeroftherotation

Rotationalsymmetry(2-dimensional)–arotationaboutapointinwhichtheimagecoincideswith

theoriginalobject

Rotationalsymmetry(3-dimensional)–arotationaboutanaxisofsymmetryinwhichtheimage

coincideswiththeoriginalobject

Scalenetriangle–atrianglenoneofwhosesidesarecongruent

163

Scaling–atransformationoftheplanethatcauseseitheramagnificationorashrinkingofan

objectinwhichtheimageremainssimilartotheoriginalobject

Scalingfactor–thefactorbywhichanobjectismagnifiedorcontractedinascaling

Secant–alinethatintersectsacircleintwodistinctpoints

Sector–theportionofacircleanditsinteriorbetweentworadii

Shearing–atransformationoftheplanethatchangestheshapeofanobject

Side–oneofthelinesegmentsthatmakeupapolygon

Similar(polygons)–polygonswhosecorrespondingvertexanglesarecongruentandwhose

correspondingsidesareproportional

Simplecurve–acurvethatdoesnotintersectitself

Slope(ofalineontheCartesianplane)–thetangentoftheangleofinclinationthelinemakes

withthepositivex-axis

Space–anundefinedtermthatdenotesthesetofpointsthatextendsindefinitelyinthree

dimensions

Sphere–thesetofpointsin(three-dimensional)spacethatareequidistantfromagivenpoint,

calledthecenter

Square–aquadrilateralwithfourrightanglesandfourcongruentsides

Supplementaryangles–twoangleswhosemeasuressumtothemeasureoftworightangles

Surface–thesetofpointsthatformtheboundaryofasolidthree-dimensionalobject

Surfacearea–thesumoftheareasofthefacesofaclosed3-dimensionalobject

Symmetry(ofanobject)–arigidmotionoftheobjectinwhichtheimagecoincideswiththe

original

Tangent(toacircle)–alinethatintersectsacircleinexactlyonepoint

Taxicabgeometry–ageometryoftheinfiniteflatplaneinwhichdistancebetweenpointsis

measuredasthesumoftheverticalandhorizontaldistancesbetweenthetwopoints

Theorem–amathematicalstatementthatisproventrue

Translation(byavectorRS)–amotionoftheplanesothatifAisanypointintheplaneandwe

callA’theimageofA,thenvectorAA’andvectorRShavethesamelengthanddirection

Transversal–alinewhichintersectstwoormorelines(eachatadifferentpoint)

164

Trapezoid–aquadrilateralwithexactlyonepairofparallelsides

Triangle–apolygonwithexactlythreesides

Trivialrotation–therotationof360°;itisarotationalsymmetryofeveryobject

Undefinedterm–atermwhichhasanintuitivemeaning,butnoformaldefinition

Union(ofsets)–thesetcontainingeveryelementofeachset

Vertex(plural:vertices)–thecommonendpointoftwoadjacentsidesofapolygon

Vertexangle–theangleformedbyadjacentsidesofapolygon

Vertex(ofapolyhedron)–theintersectionoftwoormoreedgesofapolyhedron

Verticalangles–anonadjacentpairofanglesformedbytwointersectinglines

Volume–ameasureofthecapacityofa3-dimensionalobjector,alternatively,thequantityof

spaceenclosedbya3-dimensionalobject

165

166

PolygonsforReadandStudy10:

167

168

169

170

171

172

HyperbolicPapertemplate:

173

174

175

Date post:	08-Jul-2020
Category:	Documents
Upload:	others
View:	16 times
Download:	0 times

Big Ideas in Euclidean and Non-Euclidean Geometries€¦ · Geometry allows us to think spatially,...

Documents