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GEOMETRY CCR MATH STANDARDS Mathematical Habits of Mind 1.Makesense ofproblemsandpersevere insolvingthem. 2.Reasonabstractlyandquantitatively. 3.Constructviableargumentsandcritiquethereasoning ofothers. 4.Modelwithmathematics. 5.Useappropriatetoolsstrategically. 6.Attendtoprecision 7.Lookforandmakeuseofstructure. 8.Lookforandexpress regularityinrepeatedreasoning. CONGRUENCE,PROOF,ANDCONSTRUCTIONS Cluster Experimentwithtransformationsintheplane. M.GHS.1 M.GHS.2 M.GHS.3 M.GHS.4 M.GHS.5 Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,basedonthe undefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc. Representtransformationsintheplaneusing,forexample,transparenciesandgeometrysoftware; describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsas outputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translation versushorizontalstretch).InstructionalNote:Buildonstudentexperiencewithrigidmotionsfromearlier grades.Pointoutthebasisofrigidmotionsingeometricconcepts,(e.g.,translationsmovepointsa specifieddistancealongalineparalleltoaspecifiedline;rotationsmoveobjectsalongacirculararcwitha specifiedcenterthroughaspecifiedangle). Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthat carryitontoitself.InstructionalNote:Buildonstudentexperiencewithrigidmotionsfromearliergrades. Pointoutthebasisofrigidmotionsingeometricconcepts,(e.g.,translationsmovepointsaspecified distancealongalineparalleltoaspecifiedline;rotationsmoveobjectsalongacirculararcwithaspecified centerthroughaspecifiedangle). Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicular lines,parallellines,andlinesegments.InstructionalNote:Buildonstudentexperiencewithrigidmotions fromearliergrades.Pointoutthebasisofrigidmotionsingeometricconcepts(e.g.,translationsmove pointsaspecifieddistancealongalineparalleltoaspecifiedline;rotationsmoveobjectsalongacircular arcwithaspecifiedcenterthroughaspecifiedangle). Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformedfigureusing,for example,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceoftransformationsthat willcarryagivenfigureontoanother.InstructionalNote:Buildonstudentexperiencewithrigidmotions fromearliergrades.Pointoutthebasisofrigidmotionsingeometricconcepts,(e.g.,translationsmove pointsaspecifieddistancealongalineparalleltoaspecifiedline;rotationsmoveobjectsalongacircular arcwithaspecifiedcenterthroughaspecifiedangle) Cluster Understandcongruenceintermsofrigidmotions. Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigid motiononagivenfigure;giventwofigures,usethedefinitionofcongruenceintermsofrigidmotionsto decideiftheyarecongruent.InstructionalNote:Rigidmotionsareatthefoundationofthedefinitionof M.GHS.6 congruence.Studentsreasonfromthebasicpropertiesofrigidmotions(thattheypreservedistanceand angle),whichareassumedwithoutproof.Rigidmotionsandtheirassumedpropertiescanbeusedto establishtheusualtrianglecongruencecriteria,whichcanthenbeusedtoproveothertheorems. Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifand onlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.InstructionalNote: Rigidmotionsareatthefoundationofthedefinitionofcongruence.Studentsreasonfromthebasic M.GHS.7 propertiesofrigidmotions(thattheypreservedistanceandangle),whichareassumedwithoutproof. Rigidmotionsandtheirassumedpropertiescanbeusedtoestablishtheusualtrianglecongruencecriteria, whichcanthenbeusedtoproveothertheorems. Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionof congruenceintermsofrigidmotions.InstructionalNote:Rigidmotionsareatthefoundationofthe M.GHS.8 definitionofcongruence.Studentsreasonfromthebasicpropertiesofrigidmotions(thattheypreserve distanceandangle),whichareassumedwithoutproof.Rigidmotionsandtheirassumedpropertiescanbe usedtoestablishtheusualtrianglecongruencecriteria,whichcanthenbeusedtoproveothertheorems. Cluster Provegeometrictheorems. Provetheoremsaboutlinesandangles.Theoremsinclude:verticalanglesarecongruent;whena transversalcrossesparallellines,alternateinterioranglesarecongruentandcorrespondinganglesare congruent;pointsonaperpendicularbisectorofalinesegmentareexactlythoseequidistantfromthe M.GHS.9 segment’sendpoints.InstructionalNote:Encouragemultiplewaysofwritingproofs,suchasinnarrative paragraphs,usingflowdiagrams,intwo-columnformat,andusingdiagramswithoutwords.Students shouldbeencouragedtofocusonthevalidityoftheunderlyingreasoningwhileexploringavarietyof formatsforexpressingthatreasoning. Provetheoremsabouttriangles.Theoremsinclude:measuresofinterioranglesofatrianglesumto180°; baseanglesofisoscelestrianglesarecongruent;thesegmentjoiningmidpointsoftwosidesofatriangleis paralleltothethirdsideandhalfthelength;themediansofatrianglemeetatapoint.InstructionalNote: M.GHS.1 Encouragemultiplewaysofwritingproofs,suchasinnarrativeparagraphs,usingflowdiagrams,intwo0 columnformat,andusingdiagramswithoutwords.Studentsshouldbeencouragedtofocusonthevalidity oftheunderlyingreasoningwhileexploringavarietyofformatsforexpressingthatreasoning. Implementationofthisstandardmaybeextendedtoincludeconcurrenceofperpendicularbisectorsand anglebisectorsaspreparationforM.GHS.36. Provetheoremsaboutparallelograms.Theoremsinclude:oppositesidesarecongruent,oppositeangles arecongruent,thediagonalsofaparallelogrambisecteachother,andconversely,rectanglesare M.GHS.1 parallelogramswithcongruentdiagonals.InstructionalNote:Encouragemultiplewaysofwritingproofs, 1 suchasinnarrativeparagraphs,usingflowdiagrams,intwo-columnformat,andusingdiagramswithout words.Studentsshouldbeencouragedtofocusonthevalidityoftheunderlyingreasoningwhileexploring avarietyofformatsforexpressingthatreasoning. Cluster Makegeometricconstructions. Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassandstraightedge, string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).Copyingasegment;copyingan angle;bisectingasegment;bisectinganangle;constructingperpendicularlines,includingthe M.GHS.1 perpendicularbisectorofalinesegment;andconstructingalineparalleltoagivenlinethroughapointnot 2 ontheline.InstructionalNote:Buildonpriorstudentexperiencewithsimpleconstructions.Emphasize theabilitytoformalizeandexplainhowtheseconstructionsresultinthedesiredobjects.Someofthese constructionsarecloselyrelatedtopreviousstandardsandcanbeintroducedinconjunctionwiththem. Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.InstructionalNote: M.GHS.1 Buildonpriorstudentexperiencewithsimpleconstructions.Emphasizetheabilitytoformalizeandexplain 3 howtheseconstructionsresultinthedesiredobjects.Someoftheseconstructionsarecloselyrelatedto previousstandardsandcanbeintroducedinconjunctionwiththem. SIMILARITY,PROOF,ANDTRIGONOMETRY Cluster Understandsimilarityintermsofsimilaritytransformations. Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor. M.GHS.1 a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassing 4 throughthecenterunchanged. b. Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor. Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformationstodecideiftheyare M.GHS.1 similar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofall 5 correspondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides. M.GHS.1 UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar. 6 Cluster Provetheoremsinvolvingsimilarity. M.GHS.1 Provetheoremsabouttriangles.Theoremsinclude:alineparalleltoonesideofatriangledividestheother 7 twoproportionally,andconversely;thePythagoreanTheoremprovedusingtrianglesimilarity. M.GHS.1 Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsin 8 geometricfigures. Cluster Definetrigonometricratiosandsolveproblemsinvolvingrighttriangles. M.GHS.1 9 M.GHS.2 0 M.GHS.2 1 Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle, leadingtodefinitionsoftrigonometricratiosforacuteangles. Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles. UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems. Cluster Applytrigonometrytogeneraltriangles. M.GHS.2 DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertex 2 perpendiculartotheoppositeside. ProvetheLawsofSinesandCosinesandusethemtosolveproblems.InstructionalNote:Withrespectto M.GHS.2 thegeneralcaseoftheLawsofSinesandCosines,thedefinitionsofsineandcosinemustbeextendedto 3 obtuseangles. UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightand M.GHS.2 non-righttriangles.InstructionalNote:WithrespecttothegeneralcaseoftheLawsofSinesandCosines, 4 thedefinitionsofsineandcosinemustbeextendedtoobtuseangles. EXTENDINGTOTHREEDIMENSIONS Cluster Explainvolumeformulasandusethemtosolveproblems. Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofa cylinder,pyramid,andcone.Usedissectionarguments,Cavalieri’sprinciple,andinformallimitarguments. M.GHS.2 InstructionalNote:Informalargumentsforareaandvolumeformulascanmakeuseofthewayinwhich areaandvolumescaleundersimilaritytransformations:whenonefigureintheplaneresultsfromanother 5 byapplyingasimilaritytransformationwithscalefactork,itsareaisk2timestheareaofthefirst.Similarly, volumesofsolidfiguresscalebyk3underasimilaritytransformationwithscalefactork. Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.InstructionalNote: Informalargumentsforareaandvolumeformulascanmakeuseofthewayinwhichareaandvolume M.GHS.2 scaleundersimilaritytransformations:whenonefigureintheplaneresultsfromanotherbyapplyinga 6 similaritytransformationwithscalefactork,itsareaisk2timestheareaofthefirst.Similarly,volumesof solidfiguresscalebyk3underasimilaritytransformationwithscalefactork. Cluster Visualizetherelationbetweentwodimensionalandthree-dimensionalobjects. M.GHS.2 Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,andidentifythree7 dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects. Cluster Applygeometricconceptsinmodelingsituations. Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunk M.GHS.2 orahumantorsoasacylinder).InstructionalNote:Focusonsituationsthatrequirerelatingtwo-and 8 three-dimensionalobjects,determiningandusingvolume,andthetrigonometryofgeneraltriangles. CONNECTINGALGEBRAANDGEOMETRYTHROUGHCOORDINATES (Thisunithasacloseconnectionwiththeunit,CirclesWithandWithoutCoordinates.Reasoningwithtrianglesinthisunitislimitedto righttriangles;e.g.,derivetheequationforalinethroughtwopointsusingsimilarrighttriangles.Relateworkonparallellinestoworkin HighSchoolAlgebraIinvolvingsystemsofequationshavingnosolutionorinfinitelymanysolutions.M.GHS.32providespracticewiththe distanceformulaanditsconnectionwiththePythagoreanTheorem.) Cluster Usecoordinatestoprovesimplegeometrictheoremsalgebraically. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.(e.g.,Proveordisprovethatafigure M.GHS.2 definedbyfourgivenpointsinthecoordinateplaneisarectangle;proveordisprovethatthepoint(1,√3) 9 liesonthecirclecenteredattheoriginandcontainingthepoint(0,2). Provetheslopecriteriaforparallelandperpendicularlinesandusesthemtosolvegeometricproblems. M.GHS.3 (e.g.,Findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint.) 0 InstructionalNote:RelateworkonparallellinestoworkinHighSchoolAlgebraIinvolvingsystemsof equationshavingnosolutionorinfinitelymanysolutions. M.GHS. Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagiven ratio. 31 Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthe M.GHS.3 distanceformula.Thisstandardprovidespracticewiththedistanceformulaanditsconnectionwiththe 2 Pythagoreantheorem. Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection. M.GHS.3 Derivetheequationofaparabolagivenafocusanddirectrix.InstructionalNote:Thedirectrixshouldbe 3 paralleltoacoordinateaxis. CIRCLESWITHANDWITHOUTCOORDINATES Cluster Understandandapplytheoremsaboutcircles. M.GHS.3 Provethatallcirclesaresimilar. 4 Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Includetherelationship M.GHS.3 betweencentral,inscribed,andcircumscribedangles;inscribedanglesonadiameterarerightangles;the 5 radiusofacircleisperpendiculartothetangentwheretheradiusintersectsthecircle. M.GHS.3 Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesfora 6 quadrilateralinscribedinacircle. M.GHS.3 Constructatangentlinefromapointoutsideagivencircletothecircle. 7 Cluster Findarclengthsandareasofsectorsofcircles. Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltothe radius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformula M.GHS.3 fortheareaofasector.InstructionalNote:Emphasizethesimilarityofallcircles.Reasonthatbysimilarity 8 ofsectorswiththesamecentralangle,arclengthsareproportionaltotheradius.Usethisasabasisfor introducingradianasaunitofmeasure.Itisnotintendedthatitbeappliedtothedevelopmentofcircular trigonometryinthiscourse. Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection. M.GHS.3 DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethe 9 squaretofindthecenterandradiusofacirclegivenbyanequation. Cluster Usecoordinatestoprovesimplegeometrictheoremsalgebraically. Usecoordinatestoprovesimplegeometrictheoremsalgebraically.(e.g.,Proveordisprovethatafigure M.GHS.4 definedbyfourgivenpointsinthecoordinateplaneisarectangle;proveordisprovethatthepoint(1,√3) 0 liesonthecirclecenteredattheoriginandcontainingthepoint(0,2).)InstructionalNote:Includesimple proofsinvolvingcircles. Cluster Applygeometricconceptsinmodelingsituations. Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunk M.GHS.4 orahumantorsoasacylinder).InstructionalNote:Focusonsituationsinwhichtheanalysisofcirclesis 1 required. APPLICATIONSOFPROBABILITY Cluster Understandindependenceandconditionalprobabilityandusethemtointerpretdata. M.GHS.4 Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)of 2 theoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”). M.GHS.4 UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristhe 3 productoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent. RecognizetheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpretindependenceofAand M.GHS.4 BassayingthattheconditionalprobabilityofAgivenBisthesameastheprobabilityofA,andthe 4 conditionalprobabilityofBgivenAisthesameastheprobabilityofB.InstructionalNote:Buildonwork withtwo-waytablesfromAlgebraItodevelopunderstandingofconditionalprobabilityandindependence. Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheach objectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentand toapproximateconditionalprobabilities.Forexample,collectdatafromarandomsampleofstudentsin M.GHS.4 yourschoolontheirfavoritesubjectamongmath,science,andEnglish.Estimatetheprobabilitythata 5 randomlyselectedstudentfromyourschoolwillfavorsciencegiventhatthestudentisintenthgrade.Do thesameforothersubjectsandcomparetheresults.InstructionalNote:Buildonworkwithtwo-way tablesfromAlgebraItodevelopunderstandingofconditionalprobabilityandindependence. Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageand M.GHS.4 everydaysituations.Forexample,comparethechanceofhavinglungcancerifyouareasmokerwiththe 6 chanceofbeingasmokerifyouhavelungcancer. Cluster Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobabilitymodel. M.GHS.4 FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,and 7 interprettheanswerintermsofthemodel. M.GHS.4 ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthe 8 model. M.GHS.4 ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B), 9 andinterprettheanswerintermsofthemodel. M.GHS.5 Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems. 0 Useprobabilitytoevaluateoutcomesofdecisions. InstructionalNote:ThisunitsetsthestageforworkinAlgebraII,wheretheideasofstatisticalinferenceare Cluster introduced.Evaluatingtherisksassociatedwithconclusionsdrawnfromsampledata(i.e.incompleteinformation) requiresanunderstandingofprobabilityconcepts. M.GHS.5 Useprobabilitiestomakefairdecisions(e.g.,drawingbylotsand/orusingarandomnumbergenerator). 1 M.GHS.5 Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,and/or 2 pullingahockeygoalieattheendofagame). MODELINGWITHGEOMETRY Cluster Visualizerelationshipsbetweentwodimensionalandthree-dimensionalobjectsandapplygeometricconceptsin modelingsituations. M.GHS.5 Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunk 3 orahumantorsoasacylinder). M.GHS.5 Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile, 4 BTUspercubicfoot). M.GHS.5 Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfy 5 physicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).