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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Big Ideas in Euclidean and Non-Euclidean Geometries€¦ · Geometry allows us to think spatially, to see structure in art and form, and to create and visualize new “worlds” with

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BigIdeasinEuclideanandNon-EuclideanGeometries

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

(UpdatedSummer2017)

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John,Jason,Eric,Amy,Carol&Jen

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•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.

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CommonCoreStateStandardsforMathematicalPractice

Childrenshould…

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

4. Modelwithmathematics.

5. Useappropriatetoolsstrategically.

6. Attendtoprecision.

7. Lookforandmakeuseofstructure.

8. Lookforandexpressregularityinrepeatedreasoning

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CommonCoreStateStandardsforGeometry:

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

3. Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;use

coordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

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

3. Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

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Asmathematicianswewillalsoconveytoyouthebeautyofoursubject.Mathematiciansviewmathematicsasthestudyofpatternsandstructures.Wewanttoshowyouhowtoreasonlikeamathematician–andwewantyoutoshowthistoyourstudentstoo.Thiswayofreasoningisjust

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.

UnderstandandapplythePythagoreanTheorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsinacoordinatesystem.

Solvereal-worldandmathematicalproblemsinvolvingvolumeofcylinders,cones,andspheres.

9. Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.

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TableofContentsLetnoonedestituteofgeometryentermydoors.

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.

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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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1) axiom:arulethatthemathematicalcommunityhasdecidetoacceptastruewithoutproof.Anaxiomisanassumption.

3) inductivereasoning:comingtoaconclusionbasedonexamples.Imightnoticethatthesun

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

4) deductivereasoning:comingtoconclusionbasedontheaxiomsandlogic.Thistypeof

reasoningisthehallmarkofmathematicalargument.

5) counterexample:acounterexampleisaspecificexamplethatshowsthataconjectureisfalse.Giveanexampleofacounterexample.

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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B

A C

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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.

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.

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

a) Makeamodelforthisfinitegeometryusingdotsaspointsandsegmentsaslines.Can

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

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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.

d) Howmanypointsareoneachline?Howmanylinesoneachpoint?Howmanypointstotalareinaprojectiveplaneoforder2?Howmanylines?

e) Givenalineandapointnotontheline,howmanylinesaretherethroughthegivenpointthatareparalleltothegivenline?

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

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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:

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.

1) Everyclubhasatleastfourmembers.

2) Thereexistsaclubwithexactlyfourmembers.

3) Foreveryclubandforeachtownspersonwhoisnotamemberofthatclub,therewillexistoneandonlyoneclubthatpersonbelongstothathasnomembersincommonwiththefirstclub.

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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.

Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.

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

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5. Useappropriatetoolsstrategically.

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Homework:

Discoveryconsistsofseeingwhateverybodyhasseenandthinkingwhatnobodyhasthought.

section.

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

a) Statetheconverseofthistheorem.Isittrue?

b) Statethecontrapositiveofthistheorem.Isittrue?

8. Lookforandexpressregularityinrepeatedreasoning.

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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.

Affineplaneoforder3

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a) Checktoseethatourmodelforanaffineplaneoforder3satisfiesTheorems1and2above.

b) Checkyourprojectiveplaneoforder2fromtheclassactivityandseeifitsatisfiesboththeorems.

affineplaneisfourandtheminimumnumberoflinesissix.

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ClassActivity2B:PointsofPappus

Geometryenlightenstheintellectandsetsone’smindright. IbnKhaldun(MQS)

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

Undertherequirementthatthespecifiedlinesintersect,thisbecomesaEuclideanTheorem,meaningthatitistrueinthefamiliarflatinfiniteplaneofyourhighschooldays.

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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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.

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1) Auniquestraightlinesegmentcanbedrawnfromanypointtoanyotherpoint.

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

arelessthantworightangles,thenthelineswillmeetonthatside.

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

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1) Thingsequaltothesamethingarealsoequaltooneanother.

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• Twoanglesaresupplementsiftogethertheymaketworightangles.• Twoanglesarecomplementsiftogethertheymakearightangle.

• Verticalanglesareanglesoppositeeachotherwhentwolinesintersectinapoint.

• Twolinesareparalleliftheylieinthesameplaneandsharenocommonpoint.

• Twolinesareperpendiculariftheyformrightverticalanglesatapointofintersection.

• 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?

[Noticethatwebeganbystatingwhatisassumedandwedrewapicturewithlabelstohelpothersfollowalong.Thisisagoodpracticethatyoushoulddoalso.]

Now,weknowsegmentACiscongruenttoBC. [Becausethetriangleisisosceles.]

WealsoknowthatsegmentCBiscongruenttoCA. [Strange.Weknow.Justbearwithus.]

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

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

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Homework: Youalwayspassfailureonthewaytosuccess. MickeyRooney

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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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).

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

43

NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematicsp.232ThegeometryyouwillteachinelementaryandmiddleschoolisthegeometryofEuclid.Thefocushoweverisnotanaxiomaticdevelopmentofthesubjectbutratherahands-onintuitiveapproach.

44

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

Onedayofpracticeislikeonedayofcleanliving.Itwon’tdoyouanygood.

AbeLemons

3) Writeaclearandcompletedescriptionofthestepsyouusedforeachoftheconstructions

intheClassActivity.

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

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

6) Isitpossibletobisectaline?Whyorwhynot?

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.)

47

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

48

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

49

50

Hereisaproblemthatmightfitthatstandard.Takethetimetodoitsothatyouseewhatwemeanhere.Youwillneedarulerandaprotractortomeasurelengthsandangles.

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

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

51

Homework:

Thedifferencebetweenasuccessfulpersonandothersisnotalackofstrength,notalackofknowledge,butratheralackofwill.

VinceLombardi

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

3) Learnalltheboldedandunderlinedtermsinthesection.

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

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SummaryofBigIdeasfromChapterOne Hey!What’sthebigidea? Sylvester

• Onedefinitionofgeometryisthatitisthestudyofidealshapesandtherelationshipsthatexistamongthem.

collectionsofpointscalled“lines,”andtherelationshipsbetweenpointsandlines.

• Axiomaticsystemsdefinetherulesgoverningtheparticulargeometry.

• Provenconsequencesofaparticularsetofaxiomsaretheorems.

• Afinitegeometryconsistsofafinitenumberofobjectsandtheirrelationships.

• Weconstructanobjectbycreatingitusingonlystraightlinesegmentsandcircles.

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CHAPTERTWO

LEARNINGANDTEACHINGEUCLIDEANGEOMETRY

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ClassActivity6:CircularReasoning

Natureisaninfinitesphereofwhichthecenteriseverywhereandthecircumferencenowhere.

1) Oneofthesepointsisspecialbecauseitisthecenterofmassofthetriangle(thebalancingpoint).Whichoneandwhy?

2) Oneofthesepointsisspecialbecauseitisthecenterofthecirclecontainingallthevertices

ofthetriangle.Whichoneandwhy?

3) Oneofthesepointsisspecialbecauseitisthecenterofthebiggestcirclethatcanbeplacedinsidethetriangle.(Thecirclethatistangenttoallthreesides.)Whichoneandwhy?

4) Whichthreeofthefourspecialpointsalwayslieonthesameline?

5) Whichofthepointscouldlieoutsideofthetriangle?Forwhattypeoftrianglesdoesthat

happen?Whydoesthismakesense?

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WenowhavesomebasictoolswithwhichtostudyEuclideanGeometry,andinthischapterwewilldevelopevenmore.

Amathematicalcircleisthesetofpointsthatareequidistantfromagivenpoint,calledthecenter(Ointhediagrambelow).Thediagramshowssomeoftheotherimportanttermsassociatedwithacircle.Becertainyouunderstandeachtermandcanexplainitsmathematicaldefinition.

Itisanamazingfactthatforanysizecircle,theratioofthecircumferencetothediameterisconstant.Wenowcallthisconstantpi(p).Overtheyearsmanymathematicianshavetriedtofind

Central Angle Ð AOC

Tangent

Secant

Chord DE Diameter AB

Arc DE

Sector O

C

A

B

D

E

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

).(Theaverageofthesetwovaluesisroughly3.1419,aprettydarngoodestimate.)It

decimalnamethatterminatesorrepeats.Studentscommonlyuseeither3.14or722 asan

C

A

B

D

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

O

C

A

B

D

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ThevanHieleLevelsofGeometricReasoning

59

Level4:Rigor:Studentsatthislevelwillunderstandthattherearemanygeometries,eachwithitsownaxiomaticsystemandmodels.Inthiscoursewewillgiveyouasenseofthis.

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

• Exploringthetruthofastatement,itsconverse,anditscontrapositive.Forexample,decide

whethereachofthefollowingistrueorfalse.Makeanargumentineachcase.

• Makingmodelsandpicturesofgeometricobjects.

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section(includingthosethatappearintheClassActivity).

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

5) SupposetheEarthisanidealsphereandyouhavewrappedaropetightlyaroundthe

6) Provethatthecircumcenterofatriangleisequidistantfromthethreeverticesofthe

triangle.Youwillhavetorelyonthewayyouconstructedthecircumcenter.YoumayuseanyofthepropositionsinBookIforthisargument.

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7) Provethattheincenterofatriangleisequidistantfromthethreesidesofthetriangle.

Againyouwillneedtorelyonhowyouconstructedtheincenter,andyoumayuseanyofthepropositionsinBookIforthisargument.

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ClassActivity7:FindingFormulas

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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Thedescriptionofrightlinesandcircles,uponwhichgeometryisfounded,belongstomechanics.Geometrydoesnotteachustodrawtheselines,butrequiresthemtobedrawn.

65

Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.

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

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Homework:

Learningwithoutthoughtislaborlost;thoughtwithoutlearningisperilous. Confucius

1) MakesurethatyoucanjustifyalloftheformulasfromtheClassActivity.

2) Findtheareaofthepentagoninatleast3differentways.Eachsquareisonecentimeterlong.

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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.

becausebothhavethesamebaseandthesameheight.Checkitout.

ImageusedwithpermissionfromWikapedia.com

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

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Thecowboyshaveawayoftrussingupasteerorapugnaciousbroncowhichfixesthebrutesothatitcanneithermovenorthink.Thisisthehog-tie,anditiswhatEucliddidtogeometry.

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

PythagoreanTheorem:Thesquareonthehypotenuseofarighttriangleisequaltothesumofthesquaresontheothertwosides.

Onthenextpageyouwillfindapuzzlethathelpstomakethepointsthattheareasofthesquaresonthelegsofarighttriangleexactlyfittofillupthesquareonthehypotenuse.

69

Tracethediagram,thencutoutthepartsofthesquaresonthelegsoftherighttriangleandseeifyoucanrearrangethepiecestofitthesquareonthehypotenuse(Hint:thetinysquaregoesinthemiddle).

PythagoreanPuzzle

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Noticehowstandard6expectsstudentstonotonlyunderstandaproofofthePythagoreanTheorem,butalsoproveitsconverse.WhatistheconverseofthePythagoreanTheorem?Stateitcarefully,thentrytoproveit!

Homework:

RobertJ.Shiller

1) DoalltheproblemsintheConnectionssectionincludingprovingtheconverseofthePythagoreanTheorem.

2) ThepuzzlefromtheConnectionssectiononlyworkswitharighttriangle.Ifthetriangleisacute,isthesumofthetwosmallersquaresbiggerorsmallerthanthesquareonthehypotenuse?Whatifthetriangleisobtuse?

UnderstandandapplythePythagoreanTheorem.

6. ExplainaproofofthePythagoreanTheoremanditsconverse.

7. ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.

8. ApplythePythagoreanTheoremtofindthedistancebetweentwopointsina

coordinatesystem.

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a) Theschoolis4milesdueeastofyourhouseandthemallis8milestothenorthofyourhouse.Howfarapart(asthecrowflies)aretheschoolandthemall?

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.

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Yougottoknowwhentohold‘em,knowwhentofold‘em… TheGamblerbyDonSchlitz

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

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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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Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.

• Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemtosolvereal-worldandmathematicalproblems.

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UsedwithpermissionfromWikipedia.com

YouwillsolvesomemoreproblemsinvolvingthevolumesoftheseobjectsaspartoftheHomeworksection.

Homework:

Doingisaquantumleapfromimagining. BarbaraSher

Herearesomeproblemsthatmightmeetthisstandard:

a) Arectangularroomis15feetlongby10feetwideandhasan8footceiling.Builda(scaleddown)modelfortheroomusinganet.

b) Youwanttopaintthewallsandceilingandsoneedtoestimatetheamountofpaintyouwillneed.Ifagallonofpaintcovers200squarefeet,howmanygallonsshouldyoupurchase?Explainyourwork.

3) Carefullymakeanetforarightcircularcylinder.Whatisaformulaforsurfaceareaofa

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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?

9) Goonlineandsearchfor“netsfortheregularpolyhedra.”Printoutandbuildeachofthefive.YouwillneedtheseforClassActivity13.

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

R

B S A C

(Thisactivityiscontinuedonthenextpage.)

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

NationalCouncilofTeachersofMathematics

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Homework:

Withregardtoexcellence,itisnotenoughtoknow,butwemusttrytohaveanduseit.

vectorSTtoA’B’C’D’.Ineachcaseconstructthetranslation.

S

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

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

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

a. Howdoestheplacementofthecenterpointaffecttheresultingshape?

b. Howdoestheshapechangeifthescalefactorisgreaterthanone?Between0and1?Equalto1?

c. Whathappensifq=0?q<0?

PaulValéry

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

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

• Side-Angle-SideSimilarityTheorem:Iftwotriangleshavetwopairsofcorrespondingsidesproportionalandtheincludedanglescongruent,thenthetrianglesaresimilar.

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

Thesetheoremsonlyworkfortriangles.Why?Whathappenswhenyoutrytoapplythemtootherpolygons?

Homework

Apupilfromwhomnothingiseverdemandedwhichhecannotdo,neverdoesallhecan.JohnStuartMill

2) Determineifthefollowingstatementsaretrueorfalse.Makesureyoucanexplainwhyineachcase.

3) Areallrectanglessimilar?Eitherprovethattheyareorprovideacounterexample

explainingwhytheyarenot.

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

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

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ClassActivity13:StrictlyPlatonic(Solids)

Themostgenerallawinnatureisequity–theprincipleofbalanceandsymmetrywhichguidesthegrowthofformsalongthelinesofthegreateststructuralefficiency.

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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

Theessenceofmathematicsisnottomakesimplethingscomplicated,buttomakecomplicatedthingssimple. S.Gudder

Thegeometricideaofsymmetryisdefinedintermsofrigidmotions.Hereistheofficialdefinition.Asymmetryofageometricobjectisarigidmotionoftheplaneinwhichtheimageoftheobjectcoincideswiththeoriginalobject.

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95

Now,seeifyoucansketchthepolyhedronthatcouldbeformedbyusingthemidpointsofthefacesoftheoctahedronasvertices.Howmanyverticeswouldthatnewpolyhedronhave?

Wecallobjectsthatarerelatedinthiswayduals.Thecubeandtheoctahedronaredualsofeachother,andthedodecahedronandtheicosahedrtonarealsodualsofeachother.Takeacloselookatyourmodelsofthedodecahedronandtheicosahedrontoseeifyoucantellthattheyareduals.Wemightevenimaginehowtheobjectswouldfitinsideoneanother.

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Dualshavethesamesymmetriesbecausetheywouldmovetogetherunderrotationsandreflections.Youmayhavenoticedthatwehaveleftoutthetetrahedron.Whatisitsdual?Seeifyoucansketchit.

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Homework:

Youteachbestwhatyoumostneedtolearn. RichardBach

2) Sketchanobjectintheplanethatmeetseachsetofcriteriaorexplainwhyitisimpossibletodoso:

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

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

noothersymmetries.e) Theobjecthasverticaltranslationsymmetry,360-degreerotationalsymmetryand

noothersymmetries.

3) Anobjectintheplanehastwolinesofsymmetry.Iftheselinesareparallel,whatothersymmetriesmustthisobjecthave?Why?

4) Findallofthesymmetriesofthethree-dimensionalsquare-basedpyramidshownbelow.

5) Provethatifanobjectintheplanehastwointersectinglinesofsymmetry,thenitmustalsohaverotationalsymmetry.

6) Describeconditionswhichwouldguaranteethatarightprismhasexactlyoneplaneofreflectionalsymmetry.Whereistheplanelocated?

7) Describeconditionswhichwouldguaranteethatanobliqueprismhasexactlyoneplaneofreflectionalsymmetry.Whereistheplanelocated?

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8) Buildthefollowingmodels.Thenfindtheirsurfaceareas,volumes,anddescribealltheirsymmetries.

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ClassActivity14:BuriedTreasure

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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Equationsarejusttheboringpartofmathematics.Iattempttoseethingsintermsofgeometry. StephenHawking

8

6

4

2

-2

-4

-6

-8

-10 -5 5 10

D: (2, - 3)

C: (4, 0)

B (-2, 5)

A (- 3, -1)

origin

y-axis

x-axis

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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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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

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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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NationalCouncilofTeachersofMathematics PrinciplesandStandardsforSchoolMathematics,p.232

• Drawpolygonsinthecoordinateplanegivencoordinatesforthevertices;usecoordinatestofindthelengthofasidejoiningpointswiththesamefirstcoordinateorthesamesecondcoordinate.Applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.

105

Homework:

EachproblemthatIsolvedbecamearulewhichservedafterwardstosolveotherproblems.

ReneDescartes

2) DotheConnectionsproblems.

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

𝒅 = 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐

alwaysformaparallelogram.

7) Usethemethodsofanalyticgeometrytoshowthatthediagonalsofarectangleare

congruent.

8) Usethemethodsofanalyticgeometrytoshowthatthediagonalsofarhombusareperpendicular.

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

(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.

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5) Thelocusdefinitionofahyperbolaisthesetofallpointsintheplanesuchthatthedifferenceofthedistancesfromapointonthehyperbolaandtwogivenfociisconstant.

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

6) Youmayhavedecidedthatapointandalinecouldalsobeformedbyslicingyourinfinite

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Circle

Ellipse

Parabola

Hyperbola

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

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

3) Explainhowtoformalinebyintersectingaplanewithapairofinfinitecones.

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

focus.Finditsequation.

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

• Describethetwo-dimensionalfiguresthatresultfromslicingthree-dimensionalfigures,asinplanesectionsofrightrectangularprismsandrightrectangularpyramids.

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ClassActivity16:ComparingStandards

TheseStandardsdefinewhatstudentsshouldunderstandandbeabletodointheirstudyofmathematics. CommonCoreStateStandards

1) InwhatwaysdotheNCTMStandardsandtheCommonCoreStateStandardsoverlap?Whatthingsmentionedbyonegrouparemissingfromtheother?

2) Asteachers,whichwouldyoufindbemoreeasytoimplement?Explain.

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.

whichmapsonefigureontotheother.

• Analyticgeometryinvolvestakingageometricproblemandtranslatingitintoanalgebraicproblem.Itisaveryusefulprooftechnique.

• Wecanuseanalyticgeometrytohelpusdescribefigureslikeparabolasandhyperbolas.

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CHAPTER3

EXPLORINGSTRANGENEWWORLDS:NON-EUCLIDEANGEOMETRIES

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

M.C.Escher

2) Howmanyedgesdoesthecylinderhave?TheMöbiusstrip?Howmanyedgesdothestrips

with2,3,46,or511halftwistshave?Explainthedifference.

4) Whathappenswhenyoucutacylinderdownthemiddle?Whathappenswhenyoucuta

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117

Klein bottleMobius stripTorusCylinder

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Ifyouwouldthoroughlyknowanything,teachittoothers. TyronEdwards

Homework:

TheodoreRoosevelt

2) DothewordsearchfromtheConnectionssection.Thenseeifyoucanmakeupawordsearch(thesamesizeastheoneabovewithatleastfivewordstofind)onaKleinBottle.

4) Predictwhatwouldhappenifyoucutastripwiththreehalf-twistsinhalfdownthemiddle.

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

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

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122

123

6

4

2

-2

-4

-6

-5 5

A

B C

A'

C'

B'

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Homework:

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

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?

taxicabsquareslooklike?UsethedefinitionofasquareandthetaxicabdefinitionofdistanceanddrawthreetaxicabsquareswithasidelengthoffoursuchthatthefiguresarenotcongruentasEuclideanfigures.WhatEuclideanshapedothetaxicabsquareshave?Whydoesthishappen?

7) Nowexperimentwithtaxicabtriangles.Canyoudrawaregulartriangleintaxicab

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(Thisactivityiscontinuedonthenextpage.)

126

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

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Mathematics,rightlyviewed,possessesnotonlytruth,butsupremebeauty–abeautycoldandaustere,likethatofasculpture

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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.

129

130

131

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

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

Ihaveneverletmyschoolinginterferewithmyeducation. MarkTwain

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Homework:

2) DoalltheitalicizedthingsintheConnectionssection.3) Insphericalgeometryhowmanyperpendicularlinescanbedrawntoagivenlinethrougha

6) DeterminewhichoftheEuclideantrianglecongruencetheoremsaretrueinspherical

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

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

(Thisactivityiscontinuedonthenextpage.)

134

4) ThereisaEuclideantheoremstatingthattwolinesthatarebothparalleltothesameline

arealsoparalleltoeachother.Doyouthinkthistheoremholdsinhyperbolicgeometry?Whyorwhynot?

135

136

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

137

oursidesgetlongerandlonger,theanglesgetsmallerandsmallerandourareanevergetslargerthanp.Thismeansthattheanglemeasuredeterminesnotonlytheshapeofthetrianglebutalsoitssize.

Homework:

WolfgangBolyai(Janos’Father)

2) Giventhattheanglesumofanyhyperbolictriangleislessthan180°,arguethatrectanglesdonotexistinhyperbolicgeometry.

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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)

139

1) Constructanequilateraltrianglewithsidesapproximately2incheslonganduseittocreate

3) Whatistheareaoftheoriginaltriangle?Thefirstiteration?Theseconditeration?Thethird

140

BenoitMandelbrotisthemathematiciancreditedwithfindingthegeometricstructureunderlyingthesecomplicatednaturalshapes.In1975hecoinedthewordfractal(fromtheLatinwordfractusmeaningbrokenorfractured)todescribetheconvolutedcurvesandsurfacesthatcanbeusedto

141

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.)

sndornsd

loglog

==

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

143

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

144

SowhatistheSierpinskiTriangle?Here’stheidea.Beginwithanequilateraltriangle:

Locatethemidpointofeachsideandcreateanewtrianglebyconnectingthosemidpoints.Thenremovethatmiddletriangle.

Nowdothesamethingtoeachofthethreeresulting‘outside’triangles.

Step 0

Step 1

Step 2

145

Homework:…sincegeometryistherightfoundationofallpainting,Ihavedecidedtoteachitsrudimentsandprinciplestoallyoungsterseagerforart.

AlbrechtDurer,CourseintheArtofMeasurement

2) DotheproblemsintheConnectionssection.

3) Carefullysketchthreeiterationsofeachfractalidea.a) Astylizedtree,whereeachbranchsplitsintothreeothershalfaslong.Beginwith

onetrunkandthreebranches.b) AmodifiedKochcurve,withasquareonthemiddlethirdofalinesegment,rather

thanatriangle.Applythisiterativeprocesstoeachsideofasquare.4) FindtheperimeterofeachfractalinProblem3.

5) FindthelimitingareaofthefractalinProblem3b.

6) FindthedimensionofeachfractalinProblem3.

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

8) Picturedbelowarethefirstfouriterationsoftheboxfractal.Writetheinstructionsforthe

iterativeprocessthatcreatesit.Whatistheperimeterandareaofthelastiterationshownifthesideoftheoriginalsquareisoflengthone?Whatisthedimensionoftheboxfractal?

146

SummaryofBigIdeasfromChapterThree Man’smind,oncestretchedbyanewidea,neverregainsitsoriginaldimensions. OliverWendellHolmes

• SeveraltheoremsfromEuclideangeometryfailwhenappliedtoSphericalandHyperbolicgeometries.

• AFractalisageometricfigurethatisself-similar,thatcanbedefinedbyaniterative

process,andhasanon-integerdimension.

147

APPENDICES

148

References:

• 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

• 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.

149

Euclid’sPostulatesandPropositions:

Euclid'sElementsThispresentationofElementsistheworkofJ.T.Poole,

BookI

POSTULATES

Letthefollowingbepostulated:1.Todrawastraightlinefromanypointtoanypoint.2.Toproduceafinitestraightlinecontinuouslyinastraightline.3.Todescribeacirclewithanycenteranddistance.4.Thatallrightanglesareequaltooneanother.5.That,ifastraightlinefallingontwostraightlinesmaketheinterioranglesonthesamesidelessthantworightangles,thetwostraightlines,ifproducedindefinitely,meetonthatsideonwhicharetheangleslessthanthetworightangles.

150

BOOKIPROPOSITIONSProposition1.

Onagivenfinitestraightlinetoconstructanequilateraltriangle.

Proposition2.Toplaceatagivenpoint(asanextremity)astraightlineequaltoagivenstraightline.

Proposition3.Giventwounequalstraightlines,tocutofffromthegreaterastraightlineequaltotheless.

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

Proposition5.Inisoscelestrianglestheanglesatthebaseareequaltooneanother,and,iftheequalstraightlinesbeproducedfurther,theanglesunderthebasewillbeequaltooneanother.

Proposition6.Ifinatriangletwoanglesbeequaltooneanother,thesideswhichsubtendtheequalangleswillalsobeequaltooneanother.

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

Proposition8.Iftwotriangleshavethetwosidesequaltotwosidesrespectively,andhavealsothebaseequaltothebase,theywillalsohavetheanglesequalwhicharecontainedbytheequalstraightlines.

Proposition9.Tobisectagivenrectilinealangle.

Proposition10.Tobisectagivenfinitestraightline.

Proposition11.Todrawastraightlineatrightanglestoagivenstraightlinefromagivenpointonit.

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Proposition12.Toagiveninfinitestraightline,fromagivenpointwhichisnotonit,todrawaperpendicularstraightline.

Proposition13.Ifastraightlinesetuponastraightlinemakeangles,itwillmakeeithertworightanglesoranglesequaltotworightangles.

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.

Proposition27.Ifastraightlinefallingontwostraightlinesmakethealternateanglesequaltooneanother,thestraightlineswillbeparalleltooneanother.

Proposition28.Ifastraightlinefallingontwostraightlinesmaketheexteriorangleequaltotheinteriorandoppositeangleonthesameside,ortheinterioranglesonthesamesideequaltotworightangles,thestraightlineswillbeparalleltooneanother.

Proposition29.Astraightlinefallingonparallelstraightlinesmakesthealternateanglesequaltooneanother,theexteriorangleequaltotheinteriorandoppositeangle,andtheinterioranglesonthesamesideequaltotworightangles.

Proposition30.Straightlinesparalleltothesamestraightlinearealsoparalleltooneanother.

Proposition31.Throughagivenpointtodrawastraightlineparalleltoagivenstraightline.

Proposition32.Inanytriangle,ifoneofthesidesbeproduced,theexteriorangleisequaltothetwointeriorandoppositeangles,andthethreeinterioranglesofthetriangleareequaltotworightangles.

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

153

Proposition34.Inparallelogrammicareastheoppositesidesandanglesareequaltooneanother,andthediameterbisectstheareas.

Proposition35.Parallelogramswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition36.Parallelogramswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition37.Triangleswhichareonthesamebaseandinthesameparallelsareequaltooneanother.

Proposition38.Triangleswhichareonequalbasesandinthesameparallelsareequaltooneanother.

Proposition39.Equaltriangleswhichareonthesamebaseandonthesamesidearealsointhesameparallels.

Proposition40.Equaltriangleswhichareonequalbasesandonthesamesidearealsointhesameparallels.

Proposition41.Ifaparallelogramhavethesamebasewithatriangleandbeinthesameparallels,theparallelogramisdoubleofthetriangle.

Proposition42.Toconstruct,inagivenrectilinealangle,aparallelogramequaltoagiventriangle.

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

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

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

Conjecture–aguessorahypothesis

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

Convexpolygon–apolygonallofwhosediagonalslieinsidethepolygon

Consistent(setofaxioms)–oneinwhichitisimpossibletodeducefromtheseaxiomsatheorem

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

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

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

Linesegment–thesetofpointsonalinebetweentwogivenpoints,calledtheendpoints

Logicallyequivalent(statements)–statementsthathavethesametruthvalueineverycase

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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

polygoninalphabeticalorder

Orthocenter–thepointofintersectionofthethreealtitudesofatriangle

perpendicularatthepointsofintersection

Parabola–thesetofpointsPintheplanesuchthatthedistancefromPtoagivenpointFisequal

tothedistancefromPtoagivenlinem.PointFiscalledthefocusoftheparabolaandline

misthedirectrix

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Parallellines–coplanarlineswithnopointsincommon

theoriginalobject

Perimeter(ofaplaneobject)–thelengthoftheboundaryoftheobject

Perpendicularbisector–thelinethroughthemidpointofalinesegmentthatisalsoperpendicular

tothelinesegment

Perpendicularlines–twolinesthatintersecttoformfourrightangles

Pi–theratioofthecircumferenceofacircletoitsdiameter;thisratioisanirrationalnumberthat

isconstantforallsizecirclesandisapproximatelyequalto3.1415926

Planarcurve–acurvethatliesentirelywithinaplane

Plane–anundefinedtwo-dimensionalsetofpointsunderstoodtoextendinalldirections

indefinitely

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

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

thesettowhichthestatementapplies

162

Ray–thesetofpointsonalinebeginningatagivenpoint(calledtheendpoint)andextendingin

onedirectiononthelinefromthatpoint

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

Righttriangle–atrianglewithonerightangle

RigidMotionoftheplane–amotionoftheplanethatpreservesthedistancesbetweenpoints

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

centeroftherotation

theoriginalobject

coincideswiththeoriginalobject

Scalenetriangle–atrianglenoneofwhosesidesarecongruent

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Scaling–atransformationoftheplanethatcauseseitheramagnificationorashrinkingofan

objectinwhichtheimageremainssimilartotheoriginalobject

Scalingfactor–thefactorbywhichanobjectismagnifiedorcontractedinascaling

Secant–alinethatintersectsacircleintwodistinctpoints

Shearing–atransformationoftheplanethatchangestheshapeofanobject

Side–oneofthelinesegmentsthatmakeupapolygon

Similar(polygons)–polygonswhosecorrespondingvertexanglesarecongruentandwhose

correspondingsidesareproportional

Simplecurve–acurvethatdoesnotintersectitself

Slope(ofalineontheCartesianplane)–thetangentoftheangleofinclinationthelinemakes

withthepositivex-axis

Space–anundefinedtermthatdenotesthesetofpointsthatextendsindefinitelyinthree

dimensions

Sphere–thesetofpointsin(three-dimensional)spacethatareequidistantfromagivenpoint,

calledthecenter

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

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Triangle–apolygonwithexactlythreesides

Trivialrotation–therotationof360°;itisarotationalsymmetryofeveryobject

Undefinedterm–atermwhichhasanintuitivemeaning,butnoformaldefinition

Union(ofsets)–thesetcontainingeveryelementofeachset

Vertex(ofapolyhedron)–theintersectionoftwoormoreedgesofapolyhedron

Volume–ameasureofthecapacityofa3-dimensionalobjector,alternatively,thequantityof

spaceenclosedbya3-dimensionalobject

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167

168

169

170

171

172

HyperbolicPapertemplate:

173

174

175