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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,
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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
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ontheline.InstructionalNote:Buildonpriorstudentexperiencewithsimpleconstructions.Emphasize
theabilitytoformalizeandexplainhowtheseconstructionsresultinthedesiredobjects.Someofthese
constructionsarecloselyrelatedtopreviousstandardsandcanbeintroducedinconjunctionwiththem.
Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.InstructionalNote:
M.GHS.1 Buildonpriorstudentexperiencewithsimpleconstructions.Emphasizetheabilitytoformalizeandexplain
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howtheseconstructionsresultinthedesiredobjects.Someoftheseconstructionsarecloselyrelatedto
previousstandardsandcanbeintroducedinconjunctionwiththem.
SIMILARITY,PROOF,ANDTRIGONOMETRY
Cluster Understandsimilarityintermsofsimilaritytransformations.
Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor.
M.GHS.1 a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassing
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throughthecenterunchanged.
b.
Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.
Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformationstodecideiftheyare
M.GHS.1
similar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofall
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correspondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.
M.GHS.1
UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.
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Cluster Provetheoremsinvolvingsimilarity.
M.GHS.1 Provetheoremsabouttriangles.Theoremsinclude:alineparalleltoonesideofatriangledividestheother
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twoproportionally,andconversely;thePythagoreanTheoremprovedusingtrianglesimilarity.
M.GHS.1 Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsin
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geometricfigures.
Cluster Definetrigonometricratiosandsolveproblemsinvolvingrighttriangles.
M.GHS.1
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M.GHS.2
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M.GHS.2
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Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,
leadingtodefinitionsoftrigonometricratiosforacuteangles.
Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles.
UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.
Cluster Applytrigonometrytogeneraltriangles.
M.GHS.2 DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertex
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perpendiculartotheoppositeside.
ProvetheLawsofSinesandCosinesandusethemtosolveproblems.InstructionalNote:Withrespectto
M.GHS.2
thegeneralcaseoftheLawsofSinesandCosines,thedefinitionsofsineandcosinemustbeextendedto
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obtuseangles.
UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightand
M.GHS.2
non-righttriangles.InstructionalNote:WithrespecttothegeneralcaseoftheLawsofSinesandCosines,
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thedefinitionsofsineandcosinemustbeextendedtoobtuseangles.
EXTENDINGTOTHREEDIMENSIONS
Cluster Explainvolumeformulasandusethemtosolveproblems.
Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofa
cylinder,pyramid,andcone.Usedissectionarguments,Cavalieri’sprinciple,andinformallimitarguments.
M.GHS.2 InstructionalNote:Informalargumentsforareaandvolumeformulascanmakeuseofthewayinwhich
areaandvolumescaleundersimilaritytransformations:whenonefigureintheplaneresultsfromanother
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byapplyingasimilaritytransformationwithscalefactork,itsareaisk2timestheareaofthefirst.Similarly,
volumesofsolidfiguresscalebyk3underasimilaritytransformationwithscalefactork.
Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.InstructionalNote:
Informalargumentsforareaandvolumeformulascanmakeuseofthewayinwhichareaandvolume
M.GHS.2 scaleundersimilaritytransformations:whenonefigureintheplaneresultsfromanotherbyapplyinga
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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
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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)
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liesonthecirclecenteredattheoriginandcontainingthepoint(0,2).
Provetheslopecriteriaforparallelandperpendicularlinesandusesthemtosolvegeometricproblems.
M.GHS.3 (e.g.,Findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint.)
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InstructionalNote:RelateworkonparallellinestoworkinHighSchoolAlgebraIinvolvingsystemsof
equationshavingnosolutionorinfinitelymanysolutions.
M.GHS. Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagiven
ratio.
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Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthe
M.GHS.3
distanceformula.Thisstandardprovidespracticewiththedistanceformulaanditsconnectionwiththe
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Pythagoreantheorem.
Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection.
M.GHS.3 Derivetheequationofaparabolagivenafocusanddirectrix.InstructionalNote:Thedirectrixshouldbe
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paralleltoacoordinateaxis.
CIRCLESWITHANDWITHOUTCOORDINATES
Cluster Understandandapplytheoremsaboutcircles.
M.GHS.3
Provethatallcirclesaresimilar.
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Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Includetherelationship
M.GHS.3
betweencentral,inscribed,andcircumscribedangles;inscribedanglesonadiameterarerightangles;the
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radiusofacircleisperpendiculartothetangentwheretheradiusintersectsthecircle.
M.GHS.3 Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesfora
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quadrilateralinscribedinacircle.
M.GHS.3
Constructatangentlinefromapointoutsideagivencircletothecircle.
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Cluster Findarclengthsandareasofsectorsofcircles.
Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltothe
radius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformula
M.GHS.3 fortheareaofasector.InstructionalNote:Emphasizethesimilarityofallcircles.Reasonthatbysimilarity
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ofsectorswiththesamecentralangle,arclengthsareproportionaltotheradius.Usethisasabasisfor
introducingradianasaunitofmeasure.Itisnotintendedthatitbeappliedtothedevelopmentofcircular
trigonometryinthiscourse.
Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection.
M.GHS.3 DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethe
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squaretofindthecenterandradiusofacirclegivenbyanequation.
Cluster Usecoordinatestoprovesimplegeometrictheoremsalgebraically.
Usecoordinatestoprovesimplegeometrictheoremsalgebraically.(e.g.,Proveordisprovethatafigure
M.GHS.4 definedbyfourgivenpointsinthecoordinateplaneisarectangle;proveordisprovethatthepoint(1,√3)
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liesonthecirclecenteredattheoriginandcontainingthepoint(0,2).)InstructionalNote:Includesimple
proofsinvolvingcircles.
Cluster Applygeometricconceptsinmodelingsituations.
Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunk
M.GHS.4
orahumantorsoasacylinder).InstructionalNote:Focusonsituationsinwhichtheanalysisofcirclesis
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required.
APPLICATIONSOFPROBABILITY
Cluster Understandindependenceandconditionalprobabilityandusethemtointerpretdata.
M.GHS.4 Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)of
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theoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”).
M.GHS.4 UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristhe
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productoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent.
RecognizetheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpretindependenceofAand
M.GHS.4 BassayingthattheconditionalprobabilityofAgivenBisthesameastheprobabilityofA,andthe
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conditionalprobabilityofBgivenAisthesameastheprobabilityofB.InstructionalNote:Buildonwork
withtwo-waytablesfromAlgebraItodevelopunderstandingofconditionalprobabilityandindependence.
Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheach
objectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentand
toapproximateconditionalprobabilities.Forexample,collectdatafromarandomsampleofstudentsin
M.GHS.4
yourschoolontheirfavoritesubjectamongmath,science,andEnglish.Estimatetheprobabilitythata
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randomlyselectedstudentfromyourschoolwillfavorsciencegiventhatthestudentisintenthgrade.Do
thesameforothersubjectsandcomparetheresults.InstructionalNote:Buildonworkwithtwo-way
tablesfromAlgebraItodevelopunderstandingofconditionalprobabilityandindependence.
Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageand
M.GHS.4
everydaysituations.Forexample,comparethechanceofhavinglungcancerifyouareasmokerwiththe
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chanceofbeingasmokerifyouhavelungcancer.
Cluster Usetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobabilitymodel.
M.GHS.4 FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,and
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interprettheanswerintermsofthemodel.
M.GHS.4 ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthe
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model.
M.GHS.4 ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),
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andinterprettheanswerintermsofthemodel.
M.GHS.5
Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.
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Useprobabilitytoevaluateoutcomesofdecisions.
InstructionalNote:ThisunitsetsthestageforworkinAlgebraII,wheretheideasofstatisticalinferenceare
Cluster
introduced.Evaluatingtherisksassociatedwithconclusionsdrawnfromsampledata(i.e.incompleteinformation)
requiresanunderstandingofprobabilityconcepts.
M.GHS.5
Useprobabilitiestomakefairdecisions(e.g.,drawingbylotsand/orusingarandomnumbergenerator).
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M.GHS.5 Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,and/or
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pullingahockeygoalieattheendofagame).
MODELINGWITHGEOMETRY
Cluster
Visualizerelationshipsbetweentwodimensionalandthree-dimensionalobjectsandapplygeometricconceptsin
modelingsituations.
M.GHS.5 Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunk
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orahumantorsoasacylinder).
M.GHS.5 Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile,
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BTUspercubicfoot).
M.GHS.5 Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfy
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physicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).