Big Ideas in Mathematics for Future Middle Grades Teachers and Elementary Math Specialists Big Ideas in Euclidean and Non-Euclidean Geometries John Beam, Jason Belnap, Eric Kuennen, Amy Parrott, Carol E. Seaman, and Jennifer Szydlik (Updated Summer 2017)
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BigIdeasinMathematicsforFutureMiddleGradesTeachersandElementaryMathSpecialists
BigIdeasinEuclideanandNon-EuclideanGeometries
JohnBeam,JasonBelnap,EricKuennen,AmyParrott,CarolE.Seaman,andJenniferSzydlik
(UpdatedSummer2017)
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ThisworkislicensedundertheCreativeCommonsAttribution-NonCommercial-NoDerivatives4.0InternationalLicense.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by-nc-nd/4.0/orsendalettertoCreativeCommons,POBox1866,MountainView,CA94042,USA.
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DearFutureTeacher,Wewrotethisbooktohelpyoutoseethestructurethatunderlieselementaryandmiddleschoolmathematics,togiveyouexperiencesreallydoingmathematics,andtoshowyouhowchildrenthinkandlearn.Wefullyintendthiscoursetotransformyourrelationshipwithmath.Asteachersoffuturemathteachers,wecreatedorgatheredtheactivitiesforthistext,andthenwetriedthemoutwithourownstudentsandmodifiedthembasedontheirsuggestionsandinsights.Weknowthatsomeoftheproblemsaretough–youwillgetstucksometimes.Pleasedon’tletthatdiscourageyou.There’smuchvalueinwrestlingwithanidea. Allourbest,
John,Jason,Eric,Amy,Carol&Jen
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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
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CHAPTERONE
ArguingfromAxioms
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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?
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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).
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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.
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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
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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.
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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.
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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.)
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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?
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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
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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.
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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
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• 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.
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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.
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ClassActivity4:ConstructionZone
Thehumanmindhasfirsttoconstructforms,independently,beforewecanfindtheminthings.
AlbertEinstein
1) ShowthatitispossibletoconstructaraywhichbisectsangleABC.Whatpropositionisthis?CheckAppendixAtosee.Thenprovethatyourconstructionworks.
2) Showthatitispossibletoconstructalinewhichbothbisectsandisperpendiculartolinesegment AB .(Wecallsuchalinetheperpendicularbisectorof AB .)Whichpropositionisthis?Checktosee.Thenprovethatyourconstructionworks.
B
A
C
A B
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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
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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
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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.
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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
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10) Provethateverypointontheanglebisectorofanangleisequidistantfromtheraysthatformthatangle.Inotherwords,provethatFDiscongruenttoED.
E
F
A
B
C
D
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ClassActivity5:IfYouBuildIt
Theshortestdistancebetweentwopointsisunderconstruction. NoelieAlitoInthisactivityyouwillperformandthenjustifytwomoreconstructionsofEuclideangeometry.Theseconstructions,alongwiththeothersinthissection,willgiveyousometoolswithwhichtoconstructotherobjectslateroninthecourse.Asusual,inyourjustification,youcanuseanypostulateorpropositionthatcomesbeforetheoneyouaretryingtoprove.
1) Showitispossibletocopyagivenanglesothattheraybelowisoneoftheraysoftheangle.Thenjustifythatyouhavedoneso.ThisisEuclid’sProposition23.
(Thisactivityiscontinuedonthenextpage.)
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2) Givenalineandapointontheline,showthatitispossibletoconstruct,throughthepoint,alinethatisperpendiculartothegivenline,andthenjustifythatyouhavedoneso.ThisisEuclid’sProposition11.
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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)
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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.
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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.
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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.
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SummaryofBigIdeasfromChapterOne Hey!What’sthebigidea? Sylvester
• Onedefinitionofgeometryisthatitisthestudyofidealshapesandtherelationshipsthatexistamongthem.
• Aseconddefinitionisthatgeometryisanaxiomaticsystemaboutobjectscalled“points,”
collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.
• Axiomaticsystemsdefinetherulesgoverningtheparticulargeometry.
• Provenconsequencesofaparticularsetofaxiomsaretheorems.
• Afinitegeometryconsistsofafinitenumberofobjectsandtheirrelationships.
• Mathematiciansareverycarefulaboutdistinctionswithinmathematicallanguage.
• Weconstructanobjectbycreatingitusingonlystraightlinesegmentsandcircles.
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CHAPTERTWO
LEARNINGANDTEACHINGEUCLIDEANGEOMETRY
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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?
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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
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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
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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
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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.
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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:
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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.
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7) Provethattheincenterofatriangleisequidistantfromthethreesidesofthetriangle.
Againyouwillneedtorelyonhowyouconstructedtheincenter,andyoumayuseanyofthepropositionsinBookIforthisargument.
8) Calculatethenumberoftimesthediameterofthebelowcirclefitsintotheperimeteroftheinscribedsquareandthenintotheperimeteroftheinscribedhexagon.Whatisityouaredoinghere?Ifyoudidthesamethingusinga72-sidedpolygon,whatapproximatelywouldyouranswerbe?
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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)
(Thisactivityiscontinuedonthenextpage.)
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b)Areaofaparallelogram=(base)×(height)
c) Areaofatrapezoid=½(baseI+baseII)×(height)
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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.
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ConnectionstotheMiddleGrades:
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.
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Homework:
Learningwithoutthoughtislaborlost;thoughtwithoutlearningisperilous. Confucius
1) MakesurethatyoucanjustifyalloftheformulasfromtheClassActivity.
2) Findtheareaofthepentagoninatleast3differentways.Eachsquareisonecentimeterlong.
3) Middleschoolstudentsshouldhaveavarietyofopportunitiestoseewhyitmakessensethattheareaofacircleshouldbeπ×r2(whereristheradius).Belowisapicturethatgivestheideaofanargumentforthatfact.Whatistheideahere?Whyisthisjustanideaoftheargument?
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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.
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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.
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Tracethediagram,thencutoutthepartsofthesquaresonthelegsoftherighttriangleandseeifyoucanrearrangethepiecestofitthesquareonthehypotenuse(Hint:thetinysquaregoesinthemiddle).
PythagoreanPuzzle
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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.
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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.
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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?
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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.
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Nowseeifyoucanfitfourtrianglesateachvertex.Buildthatregularpolyhedron.Itiscalledanoctahedron.Finallynoticethatyouhaveroomtofitfivetrianglesatavertexandstillbeabletofolditup–butwithsixtrianglesatavertexthethingliesflatontheplaneandcannotbefolded.Sothatmeansthatthereareonlythreeregularpolyhedrathatcanbebuiltofequilateraltriangles.Hereiswiremodeloftheregularpolyhedronwithfivetrianglesatavertex.Itiscalledanicosahedron.
Okay.Let’smoveontosquares.Weknowwecanfitthreeatavertexandthatgetsusthecube.Canyoubuildsomethingwithfouratavertex?Morethanfour?Ineachcase,eitherdoit,orexplainwhynot.Thereisonemoreregularguythatiscomposedentirelyofpentagons.
Wecannotbuildaregularpolyhedronwithonlyhexagonsbecausethreeatavertexlieflatandcannotbefolded.(Tryit.)Polygonswithevenmoresidesthanahexagondonotworkeitherbecausetheycannotbefoldedintothreedimensions.Sothatmeanstherecanbeonlyfiveregularpolyhedra.Makesurethatyouunderstandthisargument.
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ConnectionstotheMiddleGrades:
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.
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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.
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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.
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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
(Thisactivityiscontinuedonthenextpage.)
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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
(Thisactivityiscontinuedonthenextpage.)
P
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Areflection(inlinem)isamotionoftheplaneinwhichtheimageofapointonmisitself,andifAisnotonmandA’istheimageofA,thenmistheperpendicularbisectorofAA’.WewilldenotethisreflectionMm.(Mformirror.)
3) Construct∆A’B’C’(theimageof∆ABCunderthereflectionMm)andthenprove,usingthedefinitionofareflection,thatyouhavedoneso.
m
B C A
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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
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larger(dilated))sotocoincidewiththeother.)Similarobjectsaretheshapebutnotnecessarilythesamesize.Forexample,thesesnowflakesaresimilarbutnotcongruent.Wewilldiscussthisideafurtherinanupcomingsection.
ConnectionstotheMiddleGrades:
Ingrades6-8allstudentsshoulddescribesizes,positions,andorientationsofshapesunderinformaltransformationssuchasflips,turns,slides,andscaling.
NationalCouncilofTeachersofMathematics
PrinciplesandStandardsforSchoolMathematics,p.232TheCommonCoreStateStandardsrecommendthatstudentsinGrade8learntodoandunderstandthefollowing.Readthesecarefully.
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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.
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T R IdentifythesingletransformationthattakesABCDtoA’B’C’D’
3) First,rotateABCDclockwisearoundPby90°,thenrotateA’B’C’D’clockwisearoundPby60°.Ineachcase,constructtherotation.
IdentifythesingletransformationthattakesABCDtoA’B’C’D’
PC
A
B
D
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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
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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
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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
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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
1) DoalltheitalicizedthingsintheReadandStudysection.
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.
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ClassActivity13:StrictlyPlatonic(Solids)
Themostgenerallawinnatureisequity–theprincipleofbalanceandsymmetrywhichguidesthegrowthofformsalongthelinesofthegreateststructuralefficiency.
HerbertRead Youwillneedtobuildmodelsoftheregularpolyhedrainordertocompletethisactivity–netsareavailableonline.
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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
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Cube
RegularOctahedron
RegularDodecahedron
RegularIcosahedron
ReadandStudy:
Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple. S.Gudder
Thegeometricideaofsymmetryisdefinedintermsofrigidmotions.Hereistheofficialdefinition.Asymmetryofageometricobjectisarigidmotionoftheplaneinwhichtheimageoftheobjectcoincideswiththeoriginalobject.
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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.
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Now,seeifyoucansketchthepolyhedronthatcouldbeformedbyusingthemidpointsofthefacesoftheoctahedronasvertices.Howmanyverticeswouldthatnewpolyhedronhave?
Wecallobjectsthatarerelatedinthiswayduals.Thecubeandtheoctahedronaredualsofeachother,andthedodecahedronandtheicosahedrtonarealsodualsofeachother.Takeacloselookatyourmodelsofthedodecahedronandtheicosahedrontoseeifyoucantellthattheyareduals.Wemightevenimaginehowtheobjectswouldfitinsideoneanother.
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Dualshavethesamesymmetriesbecausetheywouldmovetogetherunderrotationsandreflections.Youmayhavenoticedthatwehaveleftoutthetetrahedron.Whatisitsdual?Seeifyoucansketchit.
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Homework:
Youteachbestwhatyoumostneedtolearn. RichardBach
1) DoalltheitalicizedthingsintheReadandStudysection.
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?
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8) Buildthefollowingmodels.Thenfindtheirsurfaceareas,volumes,anddescribealltheirsymmetries.
a) Arightcircularcylinderwithradius2cmandheight7cm.b) Asquare-basedpyramidwitha4cmby4cmbaseandaheightof5cm.c) Arightprismwiththebelowregularhexagonasthebase,andaheightof8cm.
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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.
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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)?
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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.
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ConnectionstotheMiddleGrades:
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.
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Homework:
EachproblemthatIsolvedbecamearulewhichservedafterwardstosolveotherproblems.
ReneDescartes
1) DoalltheitalicizedthingsintheReadandStudysection.
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.
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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.
(Thisactivityiscontinuedonthenextpage.)
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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.
(Thisactivityiscontinuedonthenextpage.)
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5) Thelocusdefinitionofahyperbolaisthesetofallpointsintheplanesuchthatthedifferenceofthedistancesfromapointonthehyperbolaandtwogivenfociisconstant.
Usethedefinitiontosketchapictureofanhyperbolawithfoci(0,0)and(6,0)andaconstantdifferenceof4onyourgraphpaper.Youdonotneedtofinditsequation.
6) Youmayhavedecidedthatapointandalinecouldalsobeformedbyslicingyourinfinite
conesinproblem#1.Wecanthinkofthoseobjectsas“degenerateforms”ofthesefourobjectswe’vealreadylisted.Forexample,apointisadegeneratecircle(thecirclewithzeroradius).Whatisaline?Explain.
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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
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AhyperbolaisthesetofpointsPintheplanesuchthatthedifferenceofthedistancesfromPtoF1andF2isconstant.WhatwouldhappentotheshapeofthehyperbolaifwemovedF1andF2closertogetherbutkeptthegivendifferenceconstant?
AparabolaisthesetofpointsPintheplanesuchthatthedistancefromPtoagivenpointFisequaltothedistancefromPtoagivenlinem.(RecallthatPointFiscalledthefocusoftheparabolaandlinemisthedirectrix.)Whatwouldhappentotheshapeoftheparabolaifwemovedthedirectrixfurtherfromthefocus?Whatwouldhappentotheparabolaifwechangedthedirectrixtoaverticalline?
F1 F2
P
P
F
Directrix
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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.
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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.
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SummaryofBigIdeasfromChapterTwo Ifanidea’sworthhavingonce,it’sworthhavingtwice. TomStoppard
• ThevanHielelevelsdescribeaprogressionofgeometricunderstanding.
• Itisimportantforyourstudentstomakesenseoftheformulasforareaandvolume.
• Therearethreerigidmotionsoftheplane:rotation,reflection,andtranslation.
• Ifglidereflectionisconsidereditsownmotion,thenthecompositionofanytworigidmotionsisanotherrigidmotion.
• Twofiguresaresimilarifthereisasequenceofrigidmotionsandadilationoftheplane
whichmapsonefigureontotheother.
• Analyticgeometryinvolvestakingageometricproblemandtranslatingitintoanalgebraicproblem.Itisaveryusefulprooftechnique.
• Wecanuseanalyticgeometrytohelpusdescribefigureslikeparabolasandhyperbolas.
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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)?
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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
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ConnectionstotheMiddleGrades:
Ifyouwouldthoroughlyknowanything,teachittoothers. TyronEdwards
Youmiddlegradesstudentswillliketoexplorethegeometryofthecylinder,torus(donut),MöbiusstripandKleinbottle(amongothers)throughactivities,puzzlesandgames.Forexample,hereisawordsearchonatorus.Seeifyoucanfindallofthesewords:possum,panda,jaguar,camelandllama.
Homework:
Wheneveryouareaskedifyoucandoajob,tell‘em,‘CertainlyIcan!’Thengetbusyandfindouthowtodoit.
TheodoreRoosevelt
1) DoalltheitalicizedthingsintheReadandStudysection.
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
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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
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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?
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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?
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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.
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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?
ConnectionstotheMiddleGrades:
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'
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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?
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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.
(Thisactivityiscontinuedonthenextpage.)
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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.
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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,
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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.
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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
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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:
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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?
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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.
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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.)
(Thisactivityiscontinuedonthenextpage.)
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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.
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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
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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
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oursidesgetlongerandlonger,theanglesgetsmallerandsmallerandourareanevergetslargerthanp.Thismeansthattheanglemeasuredeterminesnotonlytheshapeofthetrianglebutalsoitssize.
Homework:
ForGod’ssake,pleasegive[hyperbolicgeometry]up.Fearitnolessthanthesensualpassion,becauseittoo,maytakeupallyourtimeanddepriveyouofyourhealth,peaceofmindandhappinessinlife.
WolfgangBolyai(Janos’Father)
1) DoalltheitalicizedthingsintheReadandStudysection.
2) Giventhattheanglesumofanyhyperbolictriangleislessthan180°,arguethatrectanglesdonotexistinhyperbolicgeometry.
3) Dosimilarbutnotcongruenttrianglesexistinhyperbolicgeometry?WhatabouttheAAA
Theorem?Supportyouranswer.
4) DoestheIsoscelesTriangleTheoremholdforhyperbolictriangles?Supportyouranswer.
5) ThereisaEuclideantheoremstatingthattwolinesthatarebothparalleltothesamelinearealsoparalleltoeachother.Doesthistheoremholdinhyperbolicgeometry?Supportyouranswer.
6) Canwebuildasetofrailroadtracksonahyperbolicplane?Supportyouranswer?
7) Canaright-angledregularpentagonexistonthehyperbolicplane?Supportyouranswer.
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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)
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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)?
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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
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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
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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.
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SowhatistheSierpinskiTriangle?Here’stheidea.Beginwithanequilateraltriangle:
Locatethemidpointofeachsideandcreateanewtrianglebyconnectingthosemidpoints.Thenremovethatmiddletriangle.
Nowdothesamethingtoeachofthethreeresulting‘outside’triangles.
Keepongoingforever.(Recallthatmathematicalobjectsareidealobjects–sotheideaofimagingwhatwouldhappenifaprocessisrepeatedforeverdoesnotbothermathematicians.)TheresultingfractalistheSierpinskiTriangle.Whyisitafractal?Usethedefinitiontoexplainthis.Supposethattheoriginaltrianglehadanareaof1u2.Findaformulafortheareaatthenthstep.
Step 0
Step 1
Step 2
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Homework:…sincegeometryistherightfoundationofallpainting,Ihavedecidedtoteachitsrudimentsandprinciplestoallyoungsterseagerforart.
AlbrechtDurer,CourseintheArtofMeasurement
1) DoalltheitalicizedthingsintheReadandStudysection.
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?
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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.
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Euclid’sPostulatesandPropositions:
Euclid'sElementsThispresentationofElementsistheworkofJ.T.Poole,
DepartmentofMathematics,FurmanUniversity,Greenville,SC.©2002J.T.Poole.Allrightsreserved.
BookI
POSTULATES
Letthefollowingbepostulated:1.Todrawastraightlinefromanypointtoanypoint.2.Toproduceafinitestraightlinecontinuouslyinastraightline.3.Todescribeacirclewithanycenteranddistance.4.Thatallrightanglesareequaltooneanother.5.That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.
COMMONNOTIONS1.Thingswhichareequaltothesamethingarealsoequaltooneanother.2.Ifequalsbeaddedtoequals,thewholesareequal.3.Ifequalsbesubtractedfromequals,theremaindersareequal.4.Thingswhichcoincidewithoneanotherareequaltooneanother.5.Thewholeisgreaterthanthepart.
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BOOKIPROPOSITIONSProposition1.
Onagivenfinitestraightlinetoconstructanequilateraltriangle.
Proposition2.Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.
Proposition3.Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.
Proposition4.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhaveanglescontainedbytheequalstraightlinesequal,theywillalsohavethebaseequaltothebase,thetrianglewillbeequaltothetriangle,andtheremainingangleswillbeequaltotheremaininganglesrespectively,namelythosewhichtheequalsidessubtend.
Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.
Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.
Proposition7.Giventwostraightlinesconstructedonastraightline(fromitsextremities)andmeetinginapoint,therecannotbeconstructedonthesamestraightline(fromitsextremities),andonthesamesideofit,twootherstraightlinesmeetinginanotherpointandequaltotheformertworespectively,namelyeachtothatwhichhasthesameextremitywithit.
Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.
Proposition9.Tobisectagivenrectilinealangle.
Proposition10.Tobisectagivenfinitestraightline.
Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.
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Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.
Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.
Proposition14.Ifwithanystraightline,andatapointonit,twostraightlinesnotlyingonthesamesidemaketheadjacentanglesequaltotworightangles,thetwostraightlineswillbeinastraightlinewithoneanother.
Proposition15.Iftwostraightlinescutoneanother,theymaketheverticalanglesequaltooneanother.
Proposition16.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisgreaterthaneitheroftheinteriorandoppositeangles.
Proposition17.Inatriangletwoanglestakentogetherinanymannerarelessthantworightangles.
Proposition18.Inanytrianglethegreatersidesubtendsthegreaterangle.
Proposition19.Inanytrianglethegreaterangleissubtendedbythegreaterside.
Proposition20.Inanytriangletwosidestakentogetherinanymanneraregreaterthantheremainingone.
Proposition21.Ifononeofthesidesofatriangle,fromitsextremities,therebeconstructedtwostraightlinesmeetingwithinthetriangle,thestraightlinessoconstructedwillbelessthantheremainingtwosidesofthetriangle,butwillcontainagreaterangle.
Proposition22.Outofthreestraightlines,whichareequaltothreegivenstraightlines,toconstructatriangle:thusitisnecessarythattwoofthestraightlinestakentogetherinanymannershouldbegreaterthantheremainingone.[I.20]
Proposition23.Onagivenstraightlineandatapointonittoconstructarectilinealangleequaltoagivenrectilinealangle.
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Proposition24.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthantheother,theywillalsohavethebasegreaterthanthebase.
Proposition25.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,buthavethebasegreaterthanthebase,theywillalsohavetheoneoftheanglescontainedbytheequalstraightlinesgreaterthattheother.
Proposition26.Iftwotriangleshavethetwoanglesequaltotwoanglesrespectively,andonesideequaltooneside,namely,eitherthesideadjoiningtheequalangles,ofthatsubtendingoneoftheequalangles,theywillalsohavetheremainingsidesequaltotheremainingsidesandtheremainingangletotheremainingangle.
Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.
Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.
Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.
Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.
Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.
Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.
Proposition33.Thestraightlinesjoiningequalandparallelstraightlines(attheextremitieswhichare)inthesamedirections(respectively)arethemselvesalsoequalandparallel.
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Proposition34.Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.
Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.
Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.
Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.
Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.
Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.
Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.
Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.
Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.
Proposition43.Inanyparallelogramthecomplementsoftheparallelogramsaboutthediameterareequaltooneanother.
Proposition44.Toagivenstraightlinetoapply,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.
Proposition45.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagivenrectilinealfigure.
Proposition46.Onagivenstraightlinetodescribeasquare.
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Proposition47.Inright-angledtrianglesthesquareonthesidesubtendingtherightangleisequaltothesquaresonthesidescontainingtherightangle.
Proposition48.Ifinatrianglethesquareononeofthesidesbeequaltothesquaresontheremainingtwosidesofthetriangle,theanglecontainedbytheremainingtwosidesofthetriangleisright.
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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)
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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PolygonsforReadandStudy10:
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168
169
170
171
172
HyperbolicPapertemplate:
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