Download ALGEBRA II CCR Math

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.
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Knowthereisacomplexnumberisuchthati2=−1,andeverycomplexnumberhastheforma+biwitha
andbreal.
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Usetherelationi2=–1andthecommutative,associative,anddistributivepropertiestoadd,subtract,and
multiplycomplexnumbers.
Cluster Usecomplexnumbersinpolynomialidentitiesandequations.
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Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.InstructionalNote:Limitto
polynomialswithrealcoefficients.
Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewritex2 +4as(x+2i)(x–2i).
InstructionalNote:Limittopolynomialswithrealcoefficients.
KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.Instructional
Note:Limittopolynomialswithrealcoefficients.
Cluster Interpretthestructureofexpressions.
Interpretexpressionsthatrepresentaquantityintermsofitscontext.
a.
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Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.Forexample,interpret
P(1+r)nastheproductofPandafactornotdependingonP.
InstructionalNote:Extendtopolynomialandrationalexpressions.
Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4–y4as(x2)2–(y2)2,thus
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recognizingitasadifferenceofsquaresthatcanbefactoredas(x2–y2)(x2+y2).InstructionalNote:Extend
topolynomialandrationalexpressions.
Cluster Writeexpressionsinequivalentformstosolveproblems.
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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.
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KnowandapplytheRemainderTheorem:Forapolynomialp(x)andanumbera,theremainderondivision
byx–aisp(a),sop(a)=0ifandonlyif(x–a)isafactorofp(x).
Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerostoconstructa
roughgraphofthefunctiondefinedbythepolynomial.
Cluster Usepolynomialidentitiestosolveproblems.
Provepolynomialidentitiesandusethemtodescribenumericalrelationships.Forexample,thepolynomial
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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.
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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.
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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.
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Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghowextraneous
solutionsmayarise.InstructionalNote:Extendtosimplerationalandradicalequations.
Cluster Representandsolveequationsandinequalitiesgraphically.
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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.
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Explainhowtheunitcircleinthecoordinateplaneenablestheextensionoftrigonometricfunctionstoall
realnumbers,interpretedasradianmeasuresofanglestraversedcounterclockwisearoundtheunitcircle.
Cluster Modelperiodicphenomenawithtrigonometricfunctions.
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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.
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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.
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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.
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b. Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,
showingperiod,midline,andamplitude.
InstructionalNote:Focusonapplicationsandhowkeyfeaturesrelatetocharacteristicsofasituation,
makingselectionofaparticulartypeoffunctionmodelappropriate.
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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.
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Understandstatisticsasaprocessformakinginferencesaboutpopulationparametersbasedonarandom
samplefromthatpopulation.InstructionalNote:Includecomparingtheoreticalandempiricalresultsto
evaluatetheeffectivenessofatreatment.
Decideifaspecifiedmodelisconsistentwithresultsfromagivendata-generatingprocess,e.g.,using
simulation.(e.g.,Amodelsaysaspinningcoinfallsheadsupwithprobability0.5.Wouldaresultof5tails
inarowcauseyoutoquestionthemodel?)
Cluster Makeinferencesandjustifyconclusionsfromsamplesurveys,experiments,andobservationalstudies.
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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.