Download CCR High School Math IV

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.