supporting california standards high school standards with .... high school stds ca - int and...x,...

Supporting California Standards High School Standards with Courses – January 2013

1 Formatted by Educational Resource Services, Tulare County Office of Education, Visalia, California

(559) 651-3031 www.tcoe.org/ers

Number and Quantity CA Model Courses

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TheRealNumberSystem N-RN∘Extendthepropertiesofexponentstorationalexponents.N-RN.1.Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.Forexample,wedefine51/3tobethecuberootof5becausewewant(51/3)3=5(1/3)3tohold,so(51/3)3mustequal5.

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N-RN.2.Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.

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∘Usepropertiesofrationalandirrationalnumbers.N-RN.3.Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.

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Quantities★ N-Q∘Reasonquantitativelyanduseunitstosolveproblems.N-Q.1.Useunitsasawaytounderstandproblemsandtoguidethesolutionofmulti-stepproblems;chooseandinterpretunitsconsistentlyinformulas;chooseandinterpretthescaleandtheoriginingraphsanddatadisplays.

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N-Q.2.Defineappropriatequantitiesforthepurposeofdescriptivemodeling. X X N-Q.3.Choosealevelofaccuracyappropriatetolimitationsonmeasurementwhenreportingquantities.

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TheComplexNumberSystem N-CNPerformarithmeticoperationswithcomplexnumbers.N-CN.1.Knowthereisacomplexnumberisuchthati2=–1,andeverycomplexnumberhastheforma+biwithaandbreal.

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N-CN.2.Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,andmultiplycomplexnumbers.

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N-CN.3.(+)Findtheconjugateofacomplexnumber;useconjugatestofindmoduliandquotientsofcomplexnumbers.

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Representcomplexnumbersandtheiroperationsonthecomplexplane.N-CN.4.(+)Representcomplexnumbersonthecomplexplaneinrectangularandpolarform(includingrealandimaginarynumbers),andexplainwhytherectangularandpolarformsofagivencomplexnumberrepresentthesamenumber.

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N-CN.5.(+)Representaddition,subtraction,multiplication,andconjugationofcomplexnumbersgeometricallyonthecomplexplane;usepropertiesofthisrepresentationforcomputation.Forexample,(-1+Ö3i)3=8because(-1+Ö3i)hasmodulus2andargument120°.

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N-CN.6.(+)Calculatethedistancebetweennumbersinthecomplexplaneasthemodulusofthedifference,andthemidpointofasegmentastheaverageofthenumbersatitsendpoints.

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Usecomplexnumbersinpolynomialidentitiesandequations.N-CN.7.Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions. X X N-CN.8.(+)Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewritex2+4as(x+2i)(x–2i).

X X X N-CN.9.(+)KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadratic X X X

VectorandMatrixQuantities N-VMRepresentandmodelwithvectorquantities.N-VM.1.(+)Recognizevectorquantitiesashavingbothmagnitudeanddirection.Representvectorquantitiesbydirectedlinesegments,anduseappropriatesymbolsforvectorsandtheirmagnitudes(e.g.,v,|v|,||v||,v).

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CA Model Courses

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N-VM.2.(+)Findthecomponentsofavectorbysubtractingthecoordinatesofaninitialpointfromthecoordinatesofaterminalpoint.

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N-VM.3.(+)Solveproblemsinvolvingvelocityandotherquantitiesthatcanberepresentedbyvectors.

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Performoperationsonvectors.N-VM.4.(+)Addandsubtractvectors.

a. Addvectorsend-to-end,component-wise,andbytheparallelogramrule.Understandthatthemagnitudeofasumoftwovectorsistypicallynotthesumofthemagnitudes.

b. Giventwovectorsinmagnitudeanddirectionform,determinethemagnitudeanddirectionoftheirsum.

c. Understandvectorsubtractionv–wasv+(–w),where–wistheadditiveinverseofw,withthesamemagnitudeaswandpointingintheoppositedirection.Representvectorsubtractiongraphicallybyconnectingthetipsintheappropriateorder,andperformvectorsubtractioncomponent-wise.

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N-VM.5.(+)Multiplyavectorbyascalar.a. Representscalarmultiplicationgraphicallybyscalingvectorsandpossiblyreversingtheirdirection;performscalarmultiplicationcomponent-wise,e.g.,asc(vx,vy)=(cvx,cvy).

b. Computethemagnitudeofascalarmultiplecvusing||cv||=|c|v.Computethedirectionofcvknowingthatwhen|c|v≠ 0,thedirectionofcviseitheralongv(forc>0)oragainstv(forc<0).

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Performoperationsonmatricesandusematricesinapplications.N-VM.6.(+)Usematricestorepresentandmanipulatedata,e.g.,torepresentpayoffsorincidencerelationshipsinanetwork.

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N-VM.7.(+)Multiplymatricesbyscalarstoproducenewmatrices,e.g.,aswhenallofthepayoffsinagamearedoubled.

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N-VM.8.(+)Add,subtract,andmultiplymatricesofappropriatedimensions. X X N-VM.9.(+)Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquarematricesisnotacommutativeoperation,butstillsatisfiestheassociativeanddistributiveproperties.

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N-VM.10.(+)Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionandmultiplicationsimilartotheroleof0and1intherealnumbers.Thedeterminantofasquarematrixisnonzeroifandonlyifthematrixhasamultiplicativeinverse.

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N-VM.11.(+)Multiplyavector(regardedasamatrixwithonecolumn)byamatrixofsuitabledimensionstoproduceanothervector.Workwithmatricesastransformationsofvectors.

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N-VM.12.(+)Workwith2×2matricesasatransformationsoftheplane,andinterprettheabsolutevalueofthedeterminantintermsofarea.

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Algebra CA Model Courses

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SeeingStructureinExpressions A-SSE∙InterpretthestructureofexpressionsA-SSE.1.Interpretexpressionsthatrepresentaquantityintermsofitscontext.★

a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.Forexample,interpretP(1+r)nastheproductofPandafactornotdependingonP.

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A-SSE.2.Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4–y4as(x2)2–(y2)2,thusrecognizingitasadifferenceofsquaresthatcanbefactoredas(x2–y2)(x2+y2).

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∙WriteexpressionsinequivalentformstosolveproblemsA-SSE.3.Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesofthequantityrepresentedbytheexpression.★a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.b. Completethesquareinaquadraticexpressiontorevealthemaximumorminimumvalueofthefunctionitdefines.

c. Usethepropertiesofexponentstotransformexpressionsforexponentialfunctions.Forexampletheexpression1.15tcanberewrittenas(1.151/12)12t ≈1.01212ttorevealtheapproximateequivalentmonthlyinterestrateiftheannualrateis15%.

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A-SSE.4. Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusetheformulatosolveproblems.Forexample,calculatemortgagepayments.★

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ArithmeticwithPolynomialsandRationalExpressions A-APR∙PerformarithmeticoperationsonpolynomialsA-APR.1.Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials.

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UnderstandtherelationshipbetweenzerosandfactorsofpolynomialsA-APR.2.KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivisionbyx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x).

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A-APR.3.Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructaroughgraphofthefunctiondefinedbythepolynomial.

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UsepolynomialidentitiestosolveproblemsA-APR.4.Provepolynomialidentitiesandusethemtodescribenumericalrelationships.Forexample,thepolynomialidentity(x2+y2)2=(x2–y2)2+(2xy)2canbeusedtogeneratePythagoreantriples.

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A-APR.5.(+)KnowandapplytheBinomialTheoremfortheexpansionof(x+y)ninpowersofxandyforapositiveintegern,wherexandyareanynumbers,withcoefficientsdeterminedforexamplebyPascal’sTriangle.(note:ThebinomialTheoremcanbeprovedbymathematicalinductionorbyacombinatorialargument.)

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RewriterationalexpressionsA-APR.6.Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthedegreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem.

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A-APR.7.(+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunderaddition,subtraction,multiplication,anddivisionbyanonzerorationalexpression;add,subtract,multiply,anddividerationalexpressions.

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CreatingEquations★ A-CED∙Createequationsthatdescribenumbersorrelationships




CA Model Courses

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A.CED.1.Createequationsandinequalitiesinonevariableincludingoneswithabsolutevalueandusethemtosolveproblems.Includeequationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponentialfunctions.⋆

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A.CED.2.Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.⋆

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A.CED.3.Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornonviableoptionsinamodelingcontext.Forexample,representinequalitiesdescribingnutritionalandcostconstraintsoncombinationsofdifferentfoods.⋆

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A.CED.4.Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.Forexample,rearrangeOhm’slawV=IRtohighlightresistanceR.⋆

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ReasoningwithEquationsandInequalities A-REI∙UnderstandsolvingequationsasaprocessofreasoningandexplainthereasoningA-REI.1.Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod.

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A-REI.2.Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneoussolutionsmayarise.

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∙SolveequationsandinequalitiesinonevariableA-REI.3.Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.

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A.REI.3.1 Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.

X X X A-REI.4.Solvequadraticequationsinonevariable.

a. Usethemethodofcompletingthesquaretotransformanyquadraticequationinxintoanequationoftheform(x–p)2=qthathasthesamesolutions.Derivethequadraticformulafromthisform.

b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completingthesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation.Recognizewhenthequadraticformulagivescomplexsolutionsandwritethemasa±biforrealnumbersaandb.

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SolvesystemsofequationsA-REI.5.Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.

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A-REI.6.Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.

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A-REI.7.Solveasimplesystemconsistingofalinearequationandaquadraticequationintwovariablesalgebraicallyandgraphically.Forexample,findthepointsofintersectionbetweentheliney=–3xandthecirclex2+y2=3.

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A-REI.8.(+)Representasystemoflinearequationsasasinglematrixequationinavectorvariable

X X A-REI.9.(+)Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(usingtechnologyformatricesofdimension3×3orgreater).

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∙RepresentandsolveequationsandinequalitiesgraphicallyA-REI.10.Understandthatthegraphofanequationintwovariablesisthesetofallitssolutions

plottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).X X

A-REI.11.Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfind

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CA Model Courses

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successiveapproximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.★

A-REI.12.Graphthesolutionstoalinearinequalityintwovariablesasahalfplane(excludingtheboundaryinthecaseofastrictinequality),andgraphthesolutionsettoasystemoflinearinequalitiesintwovariablesastheintersectionofthecorrespondinghalf-planes.

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Functions CA Model Courses

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InterpretingFunctions F-IF∙UnderstandtheconceptofafunctionandusefunctionnotationF-IF.1.Understandthatafunctionfromoneset(calledthedomain)toanotherset(calledtherange)assignstoeachelementofthedomainexactlyoneelementoftherange.Iffisafunctionandxisanelementofitsdomain,thenf(x)denotestheoutputoffcorrespondingtotheinputx.Thegraphoffisthegraphoftheequationy=f(x).

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F-IF.2.Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.

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F-IF.3.Recognizethatsequencesarefunctions,sometimesdefinedrecursively,whosedomainisasubsetoftheintegers.Forexample,theFibonaccisequenceisdefinedrecursivelybyf(0)=f(1)=1,f(n+1)=f(n)+f(n-1)forn³ 1.

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∙InterpretfunctionsthatariseinapplicationsintermsofthecontextF-IF.4.Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.★

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F-IF.5.Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofperson-hoursittakestoassemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainforthefunction.★

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F-IF.6.Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.★

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∙AnalyzefunctionsusingdifferentrepresentationsF-IF.7.Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.★ a. Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.b. Graphsquareroot,cuberoot,andpiecewise-definedfunctions,includingstepfunctionsandabsolutevaluefunctions.

c. Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,andshowingendbehavior.

d. (+)Graphrationalfunctions,identifyingzerosandasymptoteswhensuitablefactorizationsareavailable,andshowingendbehavior.

e. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.

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F-IF.8.Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferentpropertiesofthefunction.a. Usetheprocessoffactoringandcompletingthesquareinaquadraticfunctiontoshowzeros,extremevalues,andsymmetryofthegraph,andinterprettheseintermsofacontext.

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CA Model Courses

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b. Usethepropertiesofexponentstointerpretexpressionsforexponentialfunctions.Forexample,identifypercentrateofchangeinfunctionssuchasy=(1.02)t,y=(0.97)t,y=(1.01)12t,y=(1.2)t/10,andclassifythemasrepresentingexponentialgrowthordecay.

F-IF.9.Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).Forexample,givenagraphofonequadraticfunctionandanalgebraicexpressionforanother,saywhichhasthelargermaximum.

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F-IF.10. (+) Demonstrate an understanding of functions and equations defined parametrically and graph them.⋆ (CA Standard Math Analysis – 7.0)

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F-IF.11. (+) Graph polar coordinates and curves. Convert between polar and rectangular coordinate systems.

X X BuildingFunctions F-BF∙BuildafunctionthatmodelsarelationshipbetweentwoquantitiesF-BF.1.Writeafunctionthatdescribesarelationshipbetweentwoquantities.★

a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.b. Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthatmodelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.

c. (+)Composefunctions.Forexample,ifT(y)isthetemperatureintheatmosphereasafunctionofheight,andh(t)istheheightofaweatherballoonasafunctionoftime,thenT(h(t))isthetemperatureatthelocationoftheweatherballoonasafunctionoftime.

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F-BF.2.Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituations,andtranslatebetweenthetwoforms.★

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BuildnewfunctionsfromexistingfunctionsF-BF.3.Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenandoddfunctionsfromtheirgraphsandalgebraicexpressionsforthem.

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F-BF.4.Findinversefunctions.a. Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverseandwriteanexpressionfortheinverse.Forexample,f(x)=2x3orf(x)=(x+1)/(x–1)forx≠1.

b. (+)Verifybycompositionthatonefunctionistheinverseofanother.c. (+)Readvaluesofaninversefunctionfromagraphoratable,giventhatthefunctionhasaninverse.

d. (+)Produceaninvertiblefunctionfromanon-invertiblefunctionbyrestrictingthedomain.

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F-BF.5. (+)Understandtheinverserelationshipbetweenexponentsandlogarithmsandusethisrelationshiptosolveproblemsinvolvinglogarithmsandexponents.

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Linear,Quadratic,andExponentialModels★ F-LEConstructandcomparelinearandexponentialmodelsandsolveproblemsF-LE.1.Distinguishbetweensituationsthatcanbemodeledwithlinearfunctionsandwithexponentialfunctions.

a. Provethatlinearfunctionsgrowbyequaldifferencesoverequalintervals,andthatexponentialfunctionsgrowbyequalfactorsoverequalintervals.

b. Recognizesituationsinwhichonequantitychangesataconstantrateperunitintervalrelativetoanother.

c. Recognizesituationsinwhichaquantitygrowsordecaysbyaconstantpercentrateperunitintervalrelativetoanother.

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F-LE.2.Constructlinearandexponentialfunctions,includingarithmeticandgeometricsequences,givenagraph,adescriptionofarelationship,ortwoinput-outputpairs(includereadingthesefromatable).

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CA Model Courses

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F-LE.3.Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.

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F-LE.4.Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersandthebasebis2,10,ore;evaluatethelogarithmusingtechnology.

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F-LE.4.1.Provesimplelawsoflogarithms.⋆(CAStandardAlgebraII-11.0) X X F-LE.4.2.Usethedefinitionoflogarithmstotranslatebetweenlogarithmsinanybase.⋆(CAStandardAlgebraII-13.0)

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F-LE.4.3.Understandandusethepropertiesoflogarithmstosimplifylogarithmicnumericexpressionsandtoidentifytheirapproximatevalues.⋆(CAStandardAlgebraII-14.0)

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InterpretexpressionsforfunctionsintermsofthesituationtheymodelF-LE.5.Interprettheparametersinalinearorexponentialfunctionintermsofacontext. X X F-LE.6. Apply quadratic functions to physical problems, such as the motion of an object

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TrigonometricFunctions F-TFExtendthedomainoftrigonometricfunctionsusingtheunitcircleF-TF.1.Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle.

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F-TF.2.Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoallrealnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.

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F-TF.2.1.Graphall6basictrigonometricfunctions. X X F-TF.3.(+)Usespecialtrianglestodeterminegeometricallythevaluesofsine,cosine,tangentforp/3,p/4andp/6,andusetheunitcircletoexpressthevaluesofsine,cosines,andtangentforp-x,p+x,and2p–xintermsoftheirvaluesforx,wherexisanyrealnumber.

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F-TF.4.(+)Usetheunitcircletoexplainsymmetry(oddandeven)andperiodicityoftrigonometricfunctions.

X X ModelperiodicphenomenawithtrigonometricfunctionsF-TF.5.Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,andmidline.★

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F-TF.6.(+)Understandthatrestrictingatrigonometricfunctiontoadomainonwhichitisalwaysincreasingoralwaysdecreasingallowsitsinversetobeconstructed.

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F-TF.7.(+)Useinversefunctionstosolvetrigonometricequationsthatariseinmodelingcontexts;evaluatethesolutionsusingtechnology,andinterpretthemintermsofthecontext.★

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ProveandapplytrigonometricidentitiesF-TF.8.ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ), cos(θ),ortan(θ)givensin(θ),cos(θ),ortan(θ)andthequadrantoftheangle.

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F-TF.9.(+)Provetheadditionandsubtractionformulasforsine,cosine,andtangentandusethemtosolveproblems.

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F-TF.10. (+) Prove the half angle and double angle identities for sine and cosine and use them to solve problems.

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Geometry CA Model Courses

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Congruence G-COExperimentwithtransformationsintheplaneG-CO.1.Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc.

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G-CO.2.Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).

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G-CO.3.Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.

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G-CO.4.Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.

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G-CO.5.Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformedfigureusing,e.g.,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceoftransformationsthatwillcarryagivenfigureontoanother.

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UnderstandcongruenceintermsofrigidmotionsG-CO.6.Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure;giventwofigures,usethedefinitionofcongruenceintermsofrigidmotionstodecideiftheyarecongruent.

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G-CO.7.Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.

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G-CO.8.Explainhowthecriteriafortrianglecongruence(ASA,SAS,andSSS)followfromthedefinitionofcongruenceintermsofrigidmotions.

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ProvegeometrictheoremsG-CO.9.Provetheoremsaboutlinesandangles.Theoremsinclude:verticalanglesarecongruent;whenatransversalcrossesparallellines,alternateinterioranglesarecongruentandcorrespondinganglesarecongruent;pointsonaperpendicularbisectorofalinesegmentareexactlythoseequidistantfromthesegment’sendpoints.

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G-CO.10.Provetheoremsabouttriangles.Theoremsinclude:measuresofinterioranglesofatrianglesumto180°;baseanglesofisoscelestrianglesarecongruent;thesegmentjoiningmidpointsoftwosidesofatriangleisparalleltothethirdsideandhalfthelength;themediansofatrianglemeetatapoint.

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G-CO.11.Provetheoremsaboutparallelograms.Theoremsinclude:oppositesidesarecongruent,oppositeanglesarecongruent,thediagonalsofaparallelogrambisecteachother,andconversely,rectanglesareparallelogramswithcongruentdiagonals.

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MakegeometricconstructionsG-CO.12.Makeformalgeometricconstructionswithavarietyoftoolsandmethods(compassandstraightedge,string,reflectivedevices,paperfolding,dynamicgeometricsoftware,etc.).Copyingasegment;copyinganangle;bisectingasegment;bisectinganangle;constructingperpendicularlines,includingtheperpendicularbisectorofalinesegment;andconstructingalineparalleltoagivenlinethroughapointnotontheline.

X X

G-CO.13.Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle. X X Similarity,RightTriangles,andTrigonometry G-SRTUnderstandsimilarityintermsofsimilaritytransformationsG-SRT.1.Verifyexperimentallythepropertiesofdilationsgivenbyacenterandascalefactor:

a. Adilationtakesalinenotpassingthroughthecenterofthedilationtoaparallelline,andleavesalinepassingthroughthecenterunchanged.

b. Thedilationofalinesegmentislongerorshorterintheratiogivenbythescalefactor.

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G-SRT.2.Giventwofigures,usethedefinitionofsimilarityintermsofsimilaritytransformationstodecideiftheyaresimilar;explainusingsimilaritytransformationsthemeaningofsimilarityfortrianglesastheequalityofallcorrespondingpairsofanglesandtheproportionalityofallcorrespondingpairsofsides.

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G-SRT.3.UsethepropertiesofsimilaritytransformationstoestablishtheAAcriterionfortwotrianglestobesimilar.

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ProvetheoremsinvolvingsimilarityG-SRT.4.Provetheoremsabouttriangles.Theoremsinclude:alineparalleltoonesideofatriangledividestheothertwoproportionally,andconversely;thePythagoreanTheoremprovedusingtrianglesimilarity.

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G-SRT.5.Usecongruenceandsimilaritycriteriafortrianglestosolveproblemsandtoproverelationshipsingeometricfigures.

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∘DefinetrigonometricratiosandsolveproblemsinvolvingrighttrianglesG-SRT.6.Understandthatbysimilarity,sideratiosinrighttrianglesarepropertiesoftheanglesinthetriangle,leadingtodefinitionsoftrigonometricratiosforacuteangles.

X X

G-SRT.7.Explainandusetherelationshipbetweenthesineandcosineofcomplementaryangles. X X G-SRT.8.UsetrigonometricratiosandthePythagoreanTheoremtosolverighttrianglesinappliedproblems.«

X X

G-SRT 8.1 Derive and use the trigonometric ratios for special right triangles (30°, 60°, 90° and 45°, 45°, 90°)

X X ApplytrigonometrytogeneraltrianglesG-SRT.9.(+)DerivetheformulaA=1/2absin(C)fortheareaofatrianglebydrawinganauxiliarylinefromavertexperpendiculartotheoppositeside.

X X X X

G-SRT.10.(+)ProvetheLawsofSinesandCosinesandusethemtosolveproblems. X X X X G-SRT.11.(+)UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknownmeasurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).

X X X X

Circles G-CUnderstandandapplytheoremsaboutcirclesG-C.1.Provethatallcirclesaresimilar. X X G-C.2.Identifyanddescriberelationshipsamonginscribedangles,radii,andchords.Includetherelationshipbetweencentral,inscribed,andcircumscribedangles;inscribedanglesonadiameterarerightangles;theradiusofacircleisperpendiculartothetangentwheretheradiusintersectsthecircle.

X X

G-C.3.Constructtheinscribedandcircumscribedcirclesofatriangle,andprovepropertiesofanglesforaquadrilateralinscribedinacircle.

X X

G-C.4.(+)Constructatangentlinefromapointoutsideagivencircletothecircle. X X FindarclengthsandareasofsectorsofcirclesG-C.5.Deriveusingsimilaritythefactthatthelengthofthearcinterceptedbyanangleisproportionaltotheradius,anddefinetheradianmeasureoftheangleastheconstantofproportionality;derivetheformulafortheareaofasector.Convertbetweendegreesandradians.

X X

ExpressingGeometricPropertieswithEquations G-GPETranslatebetweenthegeometricdescriptionandtheequationforaconicsectionG-GPE.1.DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;completethesquaretofindthecenterandradiusofacirclegivenbyanequation.

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G-GPE.2.Derivetheequationofaparabolagivenafocusanddirectrix. X X G-GPE.3.(+)Derivetheequationsofellipsesandhyperbolasgiventhefoci,usingthefactthatthesumordifferenceofdistancesfromthefociisconstant.

X X G-GPE.3.1. Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, use the X X X X




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method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, or a parabola, and graph the equation (ellipse or hyperbola +).

UsecoordinatestoprovesimplegeometrictheoremsalgebraicallyG-GPE.4.Usecoordinatestoprovesimplegeometrictheoremsalgebraically.Forexample,proveordisprovethatafiguredefinedbyfourgivenpointsinthecoordinateplaneisarectangle;proveordisprovethatthepoint(1,Ö3)liesonthecirclecenteredattheoriginandcontainingthepoint(0,2).

X X X

G-GPE.5.Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolvegeometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint).

X X

G-GPE.6.Findthepointonadirectedlinesegmentbetweentwogivenpointsthatpartitionsthesegmentinagivenratio.

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G-GPE.7.Usecoordinatestocomputeperimetersofpolygonsandareasoftrianglesandrectangles,e.g.,usingthedistanceformula.★

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GeometricMeasurementandDimension G-GMDExplainvolumeformulasandusethemtosolveproblems G-GMD.1.Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofacircle,volumeofacylinder,pyramid,andcone.Usedissectionarguments,Cavalieri’sprinciple,andinformallimitarguments.

X X

G-GMD.2.(+)GiveaninformalargumentusingCavalieri’sprinciplefortheformulasforthevolumeofasphereandothersolidfigures.

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G-GMD.3.Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems.★ X X Visualizerelationshipsbetweentwo-dimensionalandthree-dimensionalobjects G-GMD.4.Identifytheshapesoftwo-dimensionalcross-sectionsofthree-dimensionalobjects,andidentifythree-dimensionalobjectsgeneratedbyrotationsoftwo-dimensionalobjects.

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G-GMD.5. Know that the effect of a scale factor K greater than zero on length, area, and volume is to multiply each by k, k2, and k3, respectively; determine length, area and volume measures using scale factors.

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G-GMD.6. Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.

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ModelingwithGeometry G-MGApplygeometricconceptsinmodelingsituations G-MG.1.Usegeometricshapes,theirmeasures,andtheirpropertiestodescribeobjects(e.g.,modelingatreetrunkorahumantorsoasacylinder).★

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G-MG.2.Applyconceptsofdensitybasedonareaandvolumeinmodelingsituations(e.g.,personspersquaremile,BTUspercubicfoot).★

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G-MG.3.Applygeometricmethodstosolvedesignproblems(e.g.,designinganobjectorstructuretosatisfyphysicalconstraintsorminimizecost;workingwithtypographicgridsystemsbasedonratios).★

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Statistics and Probability CA Model Courses

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InterpretingCategoricalandQuantitativeData S-ID∘Summarize,represent,andinterpretdataonasinglecountormeasurementvariableS-ID.1.Representdatawithplotsontherealnumberline(dotplots,histograms,andboxplots). X X S-ID.2.Usestatisticsappropriatetotheshapeofthedatadistributiontocomparecenter(median,mean)andspread(interquartilerange,standarddeviation)oftwoormoredifferentdatasets.

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S-ID.3.Interpretdifferencesinshape,center,andspreadinthecontextofthedatasets,accountingforpossibleeffectsofextremedatapoints(outliers).

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S-ID.4.Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandtoestimatepopulationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureisnotappropriate.Usecalculators,spreadsheets,andtablestoestimateareasunderthenormalcurve.

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Summarize,represent,andinterpretdataontwocategoricalandquantitativevariablesS-ID.5.Summarizecategoricaldatafortwocategoriesintwo-wayfrequencytables.Interpretrelativefrequenciesinthecontextofthedata(includingjoint,marginal,andconditionalrelativefrequencies).Recognizepossibleassociationsandtrendsinthedata.

X X

S-ID.6.Representdataontwoquantitativevariablesonascatterplot,anddescribehowthevariablesarerelated.a. Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsinthecontextofthedata.Usegivenfunctionsorchooseafunctionsuggestedbythecontext.Emphasizelinearandexponentialmodels.

b. Informallyassessthefitofafunctionbyplottingandanalyzingresiduals.c. Fitalinearfunctionforascatterplotthatsuggestsalinearassociation.

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InterpretlinearmodelsS-ID.7.Interprettheslope(rateofchange)andtheintercept(constantterm)ofalinearmodelinthecontextofthedata.

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S-ID.8.Compute(usingtechnology)andinterpretthecorrelationcoefficientofalinearfit. X X S-ID.9.Distinguishbetweencorrelationandcausation. X X

MakingInferencesandJustifyingConclusions S-ICUnderstandandevaluaterandomprocessesunderlyingstatisticalexperimentsS-IC.1.Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandomsamplefromthatpopulation.

X X

S-IC.2.Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,usingsimulation.Forexample,amodelsaysaspinningcoinfallsheadsupwithprobability0.5.Wouldaresultof5tailsinarowcauseyoutoquestionthemodel?

X X

Makeinferencesandjustifyconclusionsfromsamplesurveys,experiments,andobservationalstudiesS-IC.3.Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies;explainhowrandomizationrelatestoeach.

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S-IC.4.Usedatafromasamplesurveytoestimateapopulationmeanorproportion;developamarginoferrorthroughtheuseofsimulationmodelsforrandomsampling.

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S-IC.5.Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideifdifferencesbetweenparametersaresignificant.

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S-IC.6.Evaluatereportsbasedondata. X X ConditionalProbabilityandtheRulesofProbability S-CPUnderstandindependenceandconditionalprobabilityandusethemtointerpretdataS-CP.1.Describeeventsassubsetsofasamplespace(thesetofoutcomes)usingcharacteristics(orcategories)oftheoutcomes,orasunions,intersections,orcomplementsofotherevents(“or,”“and,”“not”).

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S-CP.2.UnderstandthattwoeventsAandBareindependentiftheprobabilityofAandBoccurringtogetheristheproductoftheirprobabilities,andusethischaracterizationtodetermineiftheyareindependent

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S-CP.3.UnderstandtheconditionalprobabilityofAgivenBasP(AandB)/P(B),andinterpretindependenceofAandBassayingthattheconditionalprobabilityofAgivenBisthesameastheprobabilityofA,andtheconditionalprobabilityofBgivenAisthesameastheprobabilityofB.

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S-CP.4.Constructandinterprettwo-wayfrequencytablesofdatawhentwocategoriesareassociatedwitheachobjectbeingclassified.Usethetwo-waytableasasamplespacetodecideifeventsareindependentandtoapproximateconditionalprobabilities.Forexample,collectdatafromarandomsampleofstudentsinyourschoolontheirfavoritesubjectamongmath,science,andEnglish.Estimatetheprobabilitythatarandomlyselectedstudentfromyourschoolwillfavorsciencegiventhatthestudentisintenthgrade.Dothesameforothersubjectsandcomparetheresults.

X X

S-CP.5.Recognizeandexplaintheconceptsofconditionalprobabilityandindependenceineverydaylanguageandeverydaysituations.Forexample,comparethechanceofhavinglungcancerifyouareasmokerwiththechanceofbeingasmokerifyouhavelungcancer.

X X

UsetherulesofprobabilitytocomputeprobabilitiesofcompoundeventsinauniformprobabilitymodelS-CP.6.FindtheconditionalprobabilityofAgivenBasthefractionofB’soutcomesthatalsobelongtoA,andinterprettheanswerintermsofthemodel.

X X

S-CP.7.ApplytheAdditionRule,P(AorB)=P(A)+P(B)–P(AandB),andinterprettheanswerintermsofthemodel.

X X

S-CP.8.(+)ApplythegeneralMultiplicationRuleinauniformprobabilitymodel,P(AandB)=P(A)P(B|A)=P(B)P(A|B),andinterprettheanswerintermsofthemodel.

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S-CP.9.(+)Usepermutationsandcombinationstocomputeprobabilitiesofcompoundeventsandsolveproblems.

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UsingProbabilitytoMakeDecisions S-MDCalculateexpectedvaluesandusethemtosolveproblemsS-MD.1.(+)Definearandomvariableforaquantityofinterestbyassigninganumericalvaluetoeacheventinasamplespace;graphthecorrespondingprobabilitydistributionusingthesamegraphicaldisplaysasfordatadistributions.

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S-MD.2.(+)Calculatetheexpectedvalueofarandomvariable;interpretitasthemeanoftheprobabilitydistribution.

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S-MD.3.(+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichtheoreticalprobabilitiescanbecalculated;findtheexpectedvalue.Forexample,findthetheoreticalprobabilitydistributionforthenumberofcorrectanswersobtainedbyguessingonallfivequestionsofamultiple-choicetestwhereeachquestionhasfourchoices,andfindtheexpectedgradeundervariousgradingschemes.

X X

S-MD.4.(+)Developaprobabilitydistributionforarandomvariabledefinedforasamplespaceinwhichprobabilitiesareassignedempirically;findtheexpectedvalue.Forexample,findacurrentdatadistributiononthenumberofTVsetsperhouseholdintheUnitedStates,andcalculatetheexpectednumberofsetsperhousehold.HowmanyTVsetswouldyouexpecttofindin100randomlyselectedhouseholds?

X X

UseprobabilitytoevaluateoutcomesofdecisionsS-MD.5.(+)Weighthepossibleoutcomesofadecisionbyassigningprobabilitiestopayoffvaluesandfindingexpectedvalues.a. Findtheexpectedpayoffforagameofchance.Forexample,findtheexpectedwinningsfromastatelotteryticketoragameatafastfoodrestaurant.

b. Evaluateandcomparestrategiesonthebasisofexpectedvalues.Forexample,compareahigh-deductibleversusalow-deductibleautomobileinsurancepolicyusingvarious,butreasonable,chancesofhavingaminororamajoraccident.

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S-MD.6.(+)Useprobabilitiestomakefairdecisions(e.g.,drawingbylots,usingarandomnumbergenerator).

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S-MD.7.(+)Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,pullingahockeygoalieattheendofagame).

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∙ Priority Cluster Identified for SBAC Grade 11 Summative Assessment ∘Supporting Cluster Identified for SBAC Grade 11 Summative Assessment

supporting california standards high school standards with .... high school stds ca - int and...x,...

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