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MATH IV 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. BUILDINGRELATIONSHIPSAMONGCOMPLEXNUMBERS,VECTORS,ANDMATRICIES Cluster Performarithmeticoperationswithcomplexnumbers M.4HSTP.1 Findtheconjugateofacomplexnumber;useconjugatestofindmoduli(magnitude)andquotientsof complexnumbers.InstructionalNote:InMathIIstudentsextendedthenumbersystemtoinclude complexnumbersandperformedtheoperationsofaddition,subtraction,andmultiplication. Cluster Representcomplexnumbersandtheiroperationsonthecomplexplane. M.4HSTP.2 M.4HSTP.3 M.4HSTP.4 Representcomplexnumbersonthecomplexplaneinrectangularandpolarform(includingrealand imaginarynumbers),andexplainwhytherectangularandpolarformsofagivencomplexnumber representthesamenumber. Representaddition,subtraction,multiplicationandconjugationofcomplexnumbersgeometricallyonthe complexplane;usepropertiesofthisrepresentationforcomputation.(e.g.,(–1+√3i)3 =8because(–1+ √3i)hasmodulus2andargument120°. Calculatethedistancebetweennumbersinthecomplexplaneasthemodulusofthedifferenceandthe midpointofasegmentastheaverageofthenumbersatitsendpoints. Cluster Representandmodelwithvectorquantities. M.4HSTP.5 M.4HSTP.6 M.4HSTP.7 Recognizevectorquantitiesashavingbothmagnitudeanddirection.Representvectorquantitiesby directedlinesegmentsanduseappropriatesymbolsforvectorsandtheirmagnitudes(e.g.,v,|v|,||v||, v).InstructionalNote:Thisisthestudent’sfirstexperiencewithvectors.Thevectorsmustbe representedbothgeometricallyandincomponentformwithemphasisonvocabularyandsymbols. Findthecomponentsofavectorbysubtractingthecoordinatesofaninitialpointfromthecoordinatesof aterminalpoint. Solveproblemsinvolvingvelocityandotherquantitiesthatcanberepresentedbyvectors. Cluster Performoperationsonvectors. M.4HSTP.8 M.4HSTP.9 Addandsubtractvectors. a.Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthe magnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes. b.Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheir sum. c.Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesame magnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallyby connectingthetipsintheappropriateorderandperformvectorsubtractioncomponent-wise. Multiplyavectorbyascalar. a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection; performscalarmultiplicationcomponent-wise,e.g.,asc(vx,vy)=(cvx,cvy). b.Computethemagnitudeofascalarmultiplecvusing Computethedirectionofcvknowingthatwhen|c|v≠0,thedirectionofcviseitheralongv(forc>0)or againstv(forc<0). Cluster Performoperationsonmatricesandusematricesinapplications. M.4HSTP.10 M.4HSTP.11 M.4HSTP.12 Usematricestorepresentandmanipulatedata(e.g.,torepresentpayoffsorincidencerelationshipsina network). Multiplymatricesbyscalarstoproducenewmatrices(e.g.,aswhenallofthepayoffsinagameare doubled. Add,subtractandmultiplymatricesofappropriatedimensions. M.4HSTP.13 M.4HSTP.14 M.4HSTP.15 M.4HSTP.16 Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquarematricesisnota commutativeoperation,butstillsatisfiestheassociativeanddistributiveproperties.InstructionalNote: Thisisanopportunitytoviewthealgebraicfieldpropertiesinamoregenericcontext,particularlynoting thatmatrixmultiplicationisnotcommutative. Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionandmultiplicationsimilarto theroleof0and1intherealnumbers.Thedeterminantofasquarematrixisnonzeroifandonlyifthe matrixhasamultiplicativeinverse. Multiplyavector(regardedasamatrixwithonecolumn)byamatrixofsuitabledimensionstoproduce anothervector.Workwithmatricesastransformationsofvectors. Workwith2×2matricesastransformationsoftheplaneandinterprettheabsolutevalueofthe determinantintermsofarea.InstructionalNote:Matrixmultiplicationofa2x2matrixbyavectorcan beinterpretedastransformingpointsorregionsintheplanetodifferentpointsorregions.Inparticulara matrixwhosedeterminantis1or-1doesnotchangetheareaofaregion. Cluster Solvesystemsofequations M.4HSTP.17 Representasystemoflinearequationsasasinglematrixequationinavectorvariable. M.4HSTP.18 Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(usingtechnologyfor matricesofdimension3×3orgreater).InstructionalNote:Studentshaveearliersolvedtwolinear equationsintwovariablesbyalgebraicmethods. ANALYSISANDSYNTHESISOFFUNCTIONS Cluster Analyzefunctionsusingdifferentrepresentations. M.4HSTP.19 Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesand usingtechnologyformorecomplicatedcases.Graphrationalfunctions,identifyingzerosandasymptotes whensuitablefactorizationsareavailable,andshowingendbehavior.InstructionalNote:Thisisan extensionofgraphicalanalysisfromMathIIIorAlgebraIIthatdevelopsthekeyfeaturesofgraphswiththe exceptionofasymptotes.Studentsexaminevertical,horizontal,andobliqueasymptotesbyconsidering limits.Studentsshouldnotethecasewhenthenumeratoranddenominatorofarationalfunctionsharea commonfactor.Utilizeaninformalnotionoflimittoanalyzeasymptotesandcontinuityinrational functions.Althoughthenotionoflimitisdevelopedinformally,propernotationshouldbefollowed. Cluster Buildafunctionthatmodelsarelationshipbetweentwoquantities. M.4HSTP.20 Writeafunctionthatdescribesarelationshipbetweentwoquantities,includingcompositionoffunctions. Forexample,ifT(y)isthetemperatureintheatmosphereasafunctionofheight,andh(t)istheheightofa weatherballoonasafunctionoftime,thenT(h(t))isthetemperatureatthelocationoftheweather balloonasafunctionoftime. Cluster Buildnewfunctionsfromexistingfunctions. M.4HSTP.21 M.4HSTP.22 Findinversefunctions.InstructionalNote:ThisisanextensionofconceptsfromMathIIIwheretheidea ofinversefunctionswasintroduced. a.Verifybycompositionthatonefunctionistheinverseofanother. b. Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse. InstructionalNote:Studentsmustrealizethatinversescreatedthroughfunctioncompositionproducethe samegraphasreflectionabouttheliney=x.) c. Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.Instructional Note:Systematicproceduresmustbedevelopedforrestrictingdomainsofnon-invertiblefunctionsso thattheirinversesexist.) Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethisrelationshiptosolve problemsinvolvinglogarithmsandexponents. TRIGONOMETRICANDINVERSETRIGONOMETRICFUNCTIONSOFREALNUMBERS Cluster Extendthedomainoftrigonometricfunctionsusingtheunitcircle. M.4HSTP.23 Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforπ/3,π/4andπ/6, andusetheunitcircletoexpressthevaluesofsine,cosine,andtangentforπ–x,π+x,and2π–xintermsof theirvaluesforx,wherexisanyrealnumber.InstructionalNote:Studentsusetheextensionofthe domainofthetrigonometricfunctionsdevelopedinMathIIItoobtainadditionalspecialanglesandmore generalpropertiesofthetrigonometricfunctions. M.4HSTP.24 Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunctions. Cluster Modelperiodicphenomenawithtrigonometricfunctions. M.4HSTP.25 M.4HSTP.26 M.4HSTP.27 Understandthatrestrictingatrigonometricfunctiontoadomainonwhichitisalwaysincreasingoralways decreasingallowsitsinversetobeconstructed. Useinversefunctionstosolvetrigonometricequationsthatariseinmodelingcontexts;evaluatethe solutionsusingtechnology,andinterpretthemintermsofthecontext.InstructionalNote:Students shoulddrawanalogiestotheworkwithinversesinthepreviousunit. Solvemoregeneraltrigonometricequations.(e.g.,2sin2 x+sinx-1=0canbesolvedusingfactoring. Cluster Proveandapplytrigonometricidentities. M.4HSTP.28 Provetheadditionandsubtractionformulasforsine,cosine,andtangentandusethemtosolveproblems. Cluster Applytransformationsoffunctiontotrigonometricfunctions. M.4HSTP.29 Graphtrigonometricfunctionsshowingkeyfeatures,includingphaseshift.InstructionalNote:InMathIII, studentsgraphedtrigonometricfunctionsshowingperiod,amplitudeandverticalshifts.) DERIVATIONSINANALYTICGEOMETRY Cluster Translatebetweenthegeometricdescriptionandtheequationforaconicsection. M.4HSTP.30 Derivetheequationsofellipsesandhyperbolasgiventhefoci,usingthefactthatthesumordifferenceof distancesfromthefociisconstant.InstructionalNote:InMathIIstudentsderivedtheequationsofcircles andparabolas.Thesederivationsprovidemeaningtotheotherwisearbitraryconstantsintheformulas.) Cluster Explainvolumeformulasandusethemtosolveproblems. M.4HSTP.31 GiveaninformalargumentusingCavalieri’sprinciplefortheformulasforthevolumeofasphereand othersolidfigures.InstructionalNote:StudentswereintroducedtoCavalieri’sprincipleinMathII. MODELINGWITHPROBABILITY Cluster Calculateexpectedvaluesandusethemtosolveproblems. M.4HSTP.32 Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeacheventina samplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphicaldisplaysasfor datadistributions.InstructionalNote:Althoughstudentsarebuildingontheirpreviousexperiencewith probabilityinmiddlegradesandinMathIIandIII,thisistheirfirstexperiencewithexpectedvalueand probabilitydistributions. M.4HSTP.33 Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanoftheprobabilitydistribution. M.4HSTP.34 M.4HSTP.35 Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoretical probabilitiescanbecalculated;findtheexpectedvalue.(e.g.,Findthetheoreticalprobabilitydistribution forthenumberofcorrectanswersobtainedbyguessingonallfivequestionsofamultiple-choicetest whereeachquestionhasfourchoices,andfindtheexpectedgradeundervariousgradingschemes.) Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilities areassignedempirically;findtheexpectedvalue.Forexample,findacurrentdatadistributiononthe numberofTVsetsperhouseholdintheUnitedStates,andcalculatetheexpectednumberofsetsper household.HowmanyTVsetswouldyouexpecttofindin100randomlyselectedhouseholds? InstructionalNote:Itisimportantthatstudentscaninterprettheprobabilityofanoutcomeasthearea underaregionofaprobabilitydistributiongraph. Cluster Useprobabilitytoevaluateoutcomesofdecisions. M.4HSTP.36 Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpected values. a. Findtheexpectedpayoffforagameofchance.(e.g.,Findtheexpectedwinningsfromastatelottery ticketoragameatafastfoodrestaurant.) b.Evaluateandcomparestrategiesonthebasisofexpectedvalues.(e.g.,Compareahigh-deductible versusalow-deductibleautomobileinsurancepolicyusingvarious,butreasonable,chancesofhavinga minororamajoraccident.) Cluster Usesigmanotationstoevaluatefinitesums. Developsigmanotationanduseittowriteseriesinequivalentform.Forexample,write M.4HSTP.37 Applythemethodofmathematicalinductiontoprovesummationformulas.Forexample,verifythat M.4HSTP.38 InstructionalNote:SomestudentsmayhaveencounteredinductioninMathIIIinprovingtheBinomial ExpansionTheorem,butformanythisistheirfirstexperience. Cluster Extendgeometricseriestoinfinitegeometricseries. M.4HSTP.39 M.4HSTP.40 Developintuitivelythatthesumofaninfiniteseriesofpositivenumberscanconvergeandderivethe formulaforthesumofaninfinitegeometricseries.InstructionalNote:InMathI,studentsdescribed geometricsequenceswithexplicitformulas.FinitegeometricseriesweredevelopedinMathIII. Applyinfinitegeometricseriesmodels.Forexample,findtheareaboundedbyaKochcurve.Instructional Note:Relyontheintuitiveconceptoflimitdevelopedinunit2tojustifythatageometricseriesconverges ifandonlyiftheratioisbetween-1and1.