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ALGEBRA II 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. POLYNOMIAL,RATIONAL,ANDRADICALRELATIONSHIPS Cluster Performarithmeticoperationswithcomplexnumbers. M.A2HS.1 Knowthereisacomplexnumberisuchthati2=−1,andeverycomplexnumberhastheforma+biwitha andbreal. M.A2HS.2 Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,and multiplycomplexnumbers. Cluster Usecomplexnumbersinpolynomialidentitiesandequations. M.A2HS.3 M.A2HS.4 M.A2HS.5 Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.InstructionalNote:Limitto polynomialswithrealcoefficients. Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewritex2 +4as(x+2i)(x–2i). InstructionalNote:Limittopolynomialswithrealcoefficients. KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.Instructional Note:Limittopolynomialswithrealcoefficients. Cluster Interpretthestructureofexpressions. Interpretexpressionsthatrepresentaquantityintermsofitscontext. a. M.A2HS.6 Interpretpartsofanexpression,suchasterms,factors,andcoefficients. b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.Forexample,interpret P(1+r)nastheproductofPandafactornotdependingonP. InstructionalNote:Extendtopolynomialandrationalexpressions. Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4–y4as(x2)2–(y2)2,thus M.A2HS.7 recognizingitasadifferenceofsquaresthatcanbefactoredas(x2–y2)(x2+y2).InstructionalNote:Extend topolynomialandrationalexpressions. Cluster Writeexpressionsinequivalentformstosolveproblems. M.A2HS.8 Derivetheformulaforthesumofafinitegeometricseries(whenthecommonratioisnot1),andusethe formulatosolveproblems.Forexample,calculatemortgagepayments.InstructionalNote:Consider extendingthisstandardtoinfinitegeometricseriesincurricularimplementationsofthiscourse description. Cluster Performarithmeticoperationsonpolynomials. M.A2HS.9 Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedunderthe operationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials. InstructionalNote:ExtendbeyondthequadraticpolynomialsfoundinAlgebraI. Cluster Understandtherelationshipbetweenzerosandfactorsofpolynomials. M.A2HS.10 M.A2HS.11 KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivision byx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x). Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructa roughgraphofthefunctiondefinedbythepolynomial. Cluster Usepolynomialidentitiestosolveproblems. Provepolynomialidentitiesandusethemtodescribenumericalrelationships.Forexample,thepolynomial M.A2HS.12 identity(x2+y2)2=(x2–y2)2+(2xy)2 canbeusedtogeneratePythagoreantriples.InstructionalNote:This clusterhasmanypossibilitiesforoptionalenrichment,suchasrelatingtheexampleinM.A2HS.10tothe solutionofthesystemu2 +v2=1,v=t(u+1),relatingthePascaltrianglepropertyofbinomialcoefficientsto (x+y)n+1=(x+y)(x+y)n,derivingexplicitformulasforthecoefficients,orprovingthebinomialtheoremby induction. M.A2HS.13 KnowandapplytheBinomialTheoremfortheexpansionof(x+y)n inpowersofxandyforapositive integern,wherexandyareanynumbers,withcoefficientsdeterminedforexamplebyPascal’sTriangle. InstructionalNote:Thisclusterhasmanypossibilitiesforoptionalenrichment,suchasrelatingthe exampleinM.A2HS.10tothesolutionofthesystemu2 +v2=1,v=t(u+1),relatingthePascaltriangle propertyofbinomialcoefficientsto(x+y)n+1=(x+y)(x+y)n,derivingexplicitformulasforthecoefficients, orprovingthebinomialtheorembyinduction. Cluster Rewriterationalexpressions. M.A2HS.14 M.A2HS.15 Rewritesimplerationalexpressionsindifferentforms;writea(x)/b(x)intheformq(x)+r(x)/b(x),where a(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthedegreeofb(x),using inspection,longdivision,or,forthemorecomplicatedexamples,acomputeralgebrasystem.Instructional Note:Thelimitationsonrationalfunctionsapplytotherationalexpressions. Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,closedunder addition,subtraction,multiplication,anddivisionbyanonzerorationalexpression;add,subtract,multiply, anddividerationalexpressions.InstructionalNote:Thisstandardrequiresthegeneraldivisionalgorithm forpolynomials. Cluster Understandsolvingequationsasaprocessofreasoningandexplainthereasoning. M.A2HS.16 Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneous solutionsmayarise.InstructionalNote:Extendtosimplerationalandradicalequations. Cluster Representandsolveequationsandinequalitiesgraphically. M.A2HS.17 Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x) intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,using technologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.Instructional Note:Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential, andlogarithmicfunctions.InstructionalNote:Includecombinationsoflinear,polynomial,rational, radical,absolutevalue,andexponentialfunctions. Cluster Analyzefunctionsusingdifferentrepresentations. M.A2HS.18 Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesand usingtechnologyformorecomplicatedcases.Graphpolynomialfunctions,identifyingzeroswhensuitable factorizationsareavailable,andshowingendbehavior.InstructionalNote:Relatethisstandardtothe relationshipbetweenzerosofquadraticfunctionsandtheirfactoredforms. TRIGONOMETRICFUNCTIONS Cluster Extendthedomainoftrigonometricfunctionsusingtheunitcircle. M.A2HS.19 Understandradianmeasureofanangleasthelengthofthearcontheunitcirclesubtendedbytheangle. M.A2HS.20 Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoall realnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle. Cluster Modelperiodicphenomenawithtrigonometricfunctions. M.A2HS.21 Choosetrigonometricfunctionstomodelperiodicphenomenawithspecifiedamplitude,frequency,and midline. Cluster Proveandapplytrigonometricidentities. M.A2HS.22 ProvethePythagoreanidentitysin2(θ)+cos2(θ)=1anduseittofindsin(θ),cos(θ),ortan(θ),givensin (θ),cos(θ),ortan(θ),andthequadrantoftheangle.InstructionalNote:AnAlgebraIIcoursewithan additionalfocusontrigonometrycouldincludethestandard“Provetheadditionandsubtractionformulas forsine,cosine,andtangentandusethemtosolveproblems.”Thiscouldbelimitedtoacuteanglesin AlgebraII. MODELINGWITHFUNCTIONS Cluster Createequationsthatdescribenumbersorrelationships. M.A2HS.23 M.A2HS.24 M.A2HS.25 M.A2HS.26 Createequationsandinequalitiesinonevariableandusethemtosolveproblems.InstructionalNote: Includeequationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponential functions. Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequations oncoordinateaxeswithlabelsandscales. InstructionalNote:Whilefunctionswilloftenbelinear,exponential,orquadraticthetypesofproblems shoulddrawfrommorecomplexsituationsthanthoseaddressedinAlgebraI.(e.g.,Findingtheequation ofalinethroughagivenpointperpendiculartoanotherlineallowsonetofindthedistancefromapointto aline). Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,and interpretsolutionsasviableornon-viableoptionsinamodelingcontext.(e.g.,Representinequalities describingnutritionalandcostconstraintsoncombinationsofdifferentfoods.)InstructionalNote:While functionswilloftenbelinear,exponential,orquadraticthetypesofproblemsshoulddrawfrommore complexsituationsthanthoseaddressedinAlgebraI.Forexample,findingtheequationofalinethrougha givenpointperpendiculartoanotherlineallowsonetofindthedistancefromapointtoaline. Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations. (e.g.,RearrangeOhm’slawV=IRtohighlightresistanceR.)Whilefunctionswilloftenbelinear, exponential,orquadraticthetypesofproblemsshoulddrawfrommorecomplexsituationsthanthose addressedinAlgebraI.Forexample,findingtheequationofalinethroughagivenpointperpendicularto anotherlineallowsonetofindthedistancefromapointtoaline.Thisexampleappliestoearlier instancesofthisstandard,nottothecurrentcourse. Cluster Interpretfunctionsthatariseinapplicationsintermsofacontext. M.A2HS.27 M.A2HS.28 M.A2HS.29 Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsand tablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionofthe relationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing, positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity. InstructionalNote:Emphasizetheselectionofamodelfunctionbasedonbehaviorofdataandcontext. Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipit describes.(e.g.,Ifthefunctionh(n)givesthenumberofperson-hoursittakestoassemblenenginesina factory,thenthepositiveintegerswouldbeanappropriatedomainforthefunction.)Note:Emphasize theselectionofamodelfunctionbasedonbehaviorofdataandcontext. Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)over aspecifiedinterval.Estimatetherateofchangefromagraph.Note:Emphasizetheselectionofamodel functionbasedonbehaviorofdataandcontext. Cluster Analyzefunctionsusingdifferentrepresentations. Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesand usingtechnologyformorecomplicatedcases. a. Graphsquareroot,cuberoot,andpiecewise-definedfunctions,includingstepfunctionsandabsolutevalue functions. M.A2HS.30 b. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions, showingperiod,midline,andamplitude. InstructionalNote:Focusonapplicationsandhowkeyfeaturesrelatetocharacteristicsofasituation, makingselectionofaparticulartypeoffunctionmodelappropriate. M.A2HS.31 M.A2HS.32 Writeafunctiondefinedbyanexpressionindifferentbutequivalentformstorevealandexplaindifferent propertiesofthefunction.InstructionalNote:Focusonapplicationsandhowkeyfeaturesrelateto characteristicsofasituation,makingselectionofaparticulartypeoffunctionmodelappropriate. Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically, numericallyintables,orbyverbaldescriptions).(e.g.,Givenagraphofonequadraticfunctionandan algebraicexpressionforanother,saywhichhasthelargermaximum.)InstructionalNote:Focuson applicationsandhowkeyfeaturesrelatetocharacteristicsofasituation,makingselectionofaparticular typeoffunctionmodelappropriate. Cluster Buildafunctionthatmodelsarelationshipbetweentwoquantities. M.A2HS.33 Writeafunctionthatdescribesarelationshipbetweentwoquantities.Combinestandardfunctiontypes usingarithmeticoperations.(e.g.,Buildafunctionthatmodelsthetemperatureofacoolingbodyby addingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.) InstructionalNote:Developmodelsformorecomplexorsophisticatedsituationsthaninpreviouscourses. Cluster Buildnewfunctionsfromexistingfunctions. M.A2HS.34 M.A2HS.35 Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk (bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustratean explanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenandoddfunctionsfrom theirgraphsandalgebraicexpressionsforthem.InstructionalNote:Usetransformationsoffunctionsto findmodelsasstudentsconsiderincreasinglymorecomplexsituations.Observetheeffectofmultiple transformationsonasinglegraphandthecommoneffectofeachtransformationacrossfunctiontypes. Findinversefunctions.Solveanequationoftheformf(x)=cforasimplefunctionfthathasaninverse andwriteanexpressionfortheinverse.(e.g.,f(x)=2x3 orf(x)=(x+1)/(x-1)forx≠1.)InstructionalNote: Usetransformationsoffunctionstofindmodelsasstudentsconsiderincreasinglymorecomplex situations.Extendthisstandardtosimplerational,simpleradical,andsimpleexponentialfunctions; connectthisstandardtoM.A2HS.34. Cluster Constructandcomparelinear,quadratic,andexponentialmodelsandsolveproblems. M.A2HS.36 Forexponentialmodels,expressasalogarithmthesolutiontoabct=dwherea,c,anddarenumbersand thebasebis2,10,ore;evaluatethelogarithmusingtechnology. InstructionalNote:Considerextendingthisunittoincludetherelationshipbetweenpropertiesof logarithmsandpropertiesofexponents,suchastheconnectionbetweenthepropertiesofexponentsand thebasiclogarithmpropertythatlogxy=logx+logy. INFERENCESANDCONCLUSIONSFROMDATA Cluster Summarize,represent,andinterpretdataonasinglecountormeasurementvariable. M.A2HS.37 Usethemeanandstandarddeviationofadatasettofitittoanormaldistributionandtoestimate populationpercentages.Recognizethattherearedatasetsforwhichsuchaprocedureisnotappropriate. Usecalculators,spreadsheets,andtablestoestimateareasunderthenormalcurve.InstructionalNote: Whilestudentsmayhaveheardofthenormaldistribution,itisunlikelythattheywillhaveprior experienceusingittomakespecificestimates.Buildonstudents’understandingofdatadistributionsto helpthemseehowthenormaldistributionusesareatomakeestimatesoffrequencies(whichcanbe expressedasprobabilities).Emphasizethatonlysomedataarewelldescribedbyanormaldistribution. Cluster Understandandevaluaterandomprocessesunderlyingstatisticalexperiments. M.A2HS.38 M.A2HS.39 Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandom samplefromthatpopulation.InstructionalNote:Includecomparingtheoreticalandempiricalresultsto evaluatetheeffectivenessofatreatment. Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,using simulation.(e.g.,Amodelsaysaspinningcoinfallsheadsupwithprobability0.5.Wouldaresultof5tails inarowcauseyoutoquestionthemodel?) Cluster Makeinferencesandjustifyconclusionsfromsamplesurveys,experiments,andobservationalstudies. M.A2HS.40 M.A2HS.41 M.A2HS.42 M.A2HS.43 Recognizethepurposesofanddifferencesamongsamplesurveys,experiments,andobservationalstudies; explainhowrandomizationrelatestoeach.InstructionalNote:Inearliergrades,studentsareintroduced todifferentwaysofcollectingdataandusegraphicaldisplaysandsummarystatisticstomake comparisons.Theseideasarerevisitedwithafocusonhowthewayinwhichdataiscollecteddetermines thescopeandnatureoftheconclusionsthatcanbedrawnfromthatdata.Theconceptofstatistical significanceisdevelopedinformallythroughsimulationasmeaningaresultthatisunlikelytohave occurredsolelyasaresultofrandomselectioninsamplingorrandomassignmentinanexperiment. Usedatafromasamplesurveytoestimateapopulationmeanorproportion;developamarginoferror throughtheuseofsimulationmodelsforrandomsampling.InstructionalNote:Inearliergrades, studentsareintroducedtodifferentwaysofcollectingdataandusegraphicaldisplaysandsummary statisticstomakecomparisons.Theseideasarerevisitedwithafocusonhowthewayinwhichdatais collecteddeterminesthescopeandnatureoftheconclusionsthatcanbedrawnfromthatdata.The conceptofstatisticalsignificanceisdevelopedinformallythroughsimulationasmeaningaresultthatis unlikelytohaveoccurredsolelyasaresultofrandomselectioninsamplingorrandomassignmentinan experiment.Focusonthevariabilityofresultsfromexperiments—thatis,focusonstatisticsasawayof dealingwith,noteliminating,inherentrandomness. Usedatafromarandomizedexperimenttocomparetwotreatments;usesimulationstodecideif differencesbetweenparametersaresignificant.InstructionalNote:Inearliergrades,studentsare introducedtodifferentwaysofcollectingdataandusegraphicaldisplaysandsummarystatisticstomake comparisons.Theseideasarerevisitedwithafocusonhowthewayinwhichdataiscollecteddetermines thescopeandnatureoftheconclusionsthatcanbedrawnfromthatdata.Theconceptofstatistical significanceisdevelopedinformallythroughsimulationasmeaningaresultthatisunlikelytohave occurredsolelyasaresultofrandomselectioninsamplingorrandomassignmentinanexperiment.Focus onthevariabilityofresultsfromexperiments—thatis,focusonstatisticsasawayofdealingwith,not eliminating,inherentrandomness. Evaluatereportsbasedondata.InstructionalNote:Inearliergrades,studentsareintroducedtodifferent waysofcollectingdataandusegraphicaldisplaysandsummarystatisticstomakecomparisons.These ideasarerevisitedwithafocusonhowthewayinwhichdataiscollecteddeterminesthescopeandnature oftheconclusionsthatcanbedrawnfromthatdata.Theconceptofstatisticalsignificanceisdeveloped informallythroughsimulationasmeaningaresultthatisunlikelytohaveoccurredsolelyasaresultof randomselectioninsamplingorrandomassignmentinanexperiment. Cluster Useprobabilitytoevaluateoutcomesofdecisions. M.A2HS.44 M.A2HS.45 Useprobabilitiestomakefairdecisions(e.g.,drawingbylotsorusingarandomnumbergenerator). InstructionalNote:Extendtomorecomplexprobabilitymodels.Includesituationssuchasthoseinvolving qualitycontrol,ordiagnosticteststhatyieldbothfalsepositiveandfalsenegativeresults. Analyzedecisionsandstrategiesusingprobabilityconcepts(e.g.,producttesting,medicaltesting,and/or pullingahockeygoalieattheendofagame).InstructionalNote:Extendtomorecomplexprobability models.Includesituationssuchasthoseinvolvingqualitycontrol,ordiagnosticteststhatyieldbothfalse positiveandfalsenegativeresults.