Algebra II Mod1 Module Overview and Assessments
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Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 1
2012
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Core,
Inc.
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reserved.
commoncore.org
NewYorkStateCommonCore
MathematicsCurriculum
ALGEBRAIIMODULE1
Tableof
Contents1
Polynomial,Rational,andRadical
Relationships
ModuleOverview.................................................................................................................................................. 3
TopicA: PolynomialsFromBaseTentoBaseX(ASSE.2,AAPR.4).................................................................... x
Lesson1: FromBaseTenArithmetictoPolynomialArithmetic................................................................ x
Lesson2: UsingtheAreaModeltoRepresentPolynomialMultiplication................................................ x
Lesson3: TheDivisionofPolynomials....................................................................................................... x
Lesson4: ComparingMethodsLongDivisionAgain?............................................................................. x
Lesson5: PuttingitAllTogether................................................................................................................ x
Lesson6: Dividingby andby ................................................................................................. x
Lesson7: MentalMath.............................................................................................................................. x
Lesson8: ThePowerofAlgebraFindingPrimes..................................................................................... x
Lesson9: ThePowerofAlgebraFindingPythagoreanTriples................................................................ x
Lesson10: FactoringandtheSpecialRoleofZero.................................................................................... x
TopicB: FactoringItsUseandItsObstacles(NQ.2,ASSE.2,AAPR.2,AAPR.3,AAPR.6,FIF.7c).................. x
Lesson11: OvercominganObstacleWhatifaFactorisNotGiventoYouFirst?.................................. x
Lesson12: MasteringFactoring................................................................................................................. x
Lesson13: GraphingFactoredPolynomials............................................................................................... x
Lesson14: StructureinGraphsofFactoredPolynomials.......................................................................... x
Lesson1516: ModelingwithPolynomialsAnIntroduction................................................................... x
Lesson17: OvercomingaSecondObstacleWhatifthereisaRemainder?........................................... x
Lesson18: TheRemainderTheorem......................................................................................................... x
Lesson19: ModelingRiverbedswithPolynomials.................................................................................... x
1EachlessonisONEdayandONEdayisconsidereda45minuteperiod.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 2
2012
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MidModuleAssessmentandRubric.................................................................................................................. 10
TopicsAthroughB(assessment1day,return1day,remediationorfurtherapplications3days)
TopicC: SolvingandApplyingEquationsPolynomial,Rational,andRadical(AAPR.6,AREI.1,
A
REI.2,
A
REI.4b,
A
REI.6,
A
REI.7,
G
GPE.2)
..........................................................................................
x
Lesson20: MultiplyingandDividingRationalExpressions........................................................................ x
Lesson21: AddingandSubtractingRationalExpressions......................................................................... x
Lesson22: SolvingRationalEquations...................................................................................................... x
Lesson23: SystemsofEquations............................................................................................................... x
Lesson24: GraphingSystemsofEquations............................................................................................... x
Lesson25: TheDefinitionofaParabola.................................................................................................... x
Lesson26: AreAllParabolasCongruent?.................................................................................................. x
Lesson27: AreAllParabolasSimilar?........................................................................................................ x
TopicD: ASurprisefromGeometryComplexNumbersOvercomeAllObstacles(NCN.1,NCN.2,
NCN.7,AAPR.6,AREI.2,AREI.4b)........................................................................................................ x
Lesson28: OvercomingaThirdObstacleWhatifthereareNoRealNumberSolutions?...................... x
Lesson29: ASurprisingBoostfromGeometry.......................................................................................... x
Lesson30: ComplexNumbersasSolutionstoEquations.......................................................................... x
Lesson31: FactoringExtendedtotheComplexRealm............................................................................. x
Lesson32: AFocusonSquareRoots......................................................................................................... x
Lesson33: SolvingRadicalEquations........................................................................................................ x
Lesson
34:
Obstacles
ResolvedA
Surprising
Result
................................................................................
x
EndofModuleAssessmentandRubric.............................................................................................................. 20
TopicsAthroughD(assessment1day,return1day,remediationorfurtherapplications4days)
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Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 3
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
AlgebraIIModule1
Polynomial,Rational,
and
Radical
Relationships
OVERVIEW
Inthismodule,studentsdrawontheirfoundationoftheanalogiesbetweenpolynomialarithmeticandbase
tencomputation,focusingonpropertiesofoperations,particularlythedistributiveproperty(AAPR.1,A
SSE.2). Studentsconnectmultiplicationofpolynomialswithmultiplicationofmultidigitintegers,anddivision
ofpolynomials
with
long
division
of
integers
(A
APR.1,
A
APR.6).
Students
identify
zeros
of
polynomials,
includingcomplexzerosofquadraticpolynomials,andmakeconnectionsbetweenzerosofpolynomialsand
solutionsofpolynomialequations(AAPR.3). Theroleoffactoring,asbothanaidtothealgebraandtothe
graphingofpolynomials,isexplored(ASSE.2,AAPR.2,AAPR.3,FIF.7c). Studentscontinuetobuildupon
thereasoningprocessofsolvingequationsastheysolvepolynomial,rational,andradicalequations,aswellas
linearandnonlinearsystemsofequations(AREI.1,AREI.2,AREI.6,AREI.7). Themoduleculminateswith
thefundamentaltheoremofalgebraastheultimateresultinfactoring. Connectionstoapplicationsinprime
numbersinencryptiontheory,Pythagoreantriples,andmodelingproblemsarepursued.
Anadditionalthemeofthismoduleisthatthearithmeticofrationalexpressionsisgovernedbythesame
rulesasthearithmeticofrationalnumbers. Studentsuseappropriatetoolstoanalyzethekeyfeaturesofa
graphortableofapolynomialfunctionandrelatethosefeaturesbacktothetwoquantitiesintheproblem
thatthe
function
is
modeling
(F
IF.7c).
FocusStandards
Reasonquantitativelyanduseunitstosolveproblems.
NQ.22 Defineappropriatequantitiesforthepurposeofdescriptivemodeling.
Performarithmeticoperationswithcomplexnumbers.
NCN.1 Knowthereisacomplexnumberisuchthati2=1,andeverycomplexnumberhastheforma
+
biwith
aand
breal.
2ThisstandardwillbeassessedinAlgebraIIbyensuringthatsomemodelingtasks(involvingAlgebraIIcontentorsecurelyheldcontentfromprevious
gradesandcourses)requirethestudenttocreateaquantityofinterestinthesituationbeingdescribed(i.e.,thisisnotprovidedinthetask).For
example,inasituationinvolvingperiodicphenomena,thestudentmightautonomouslydecidethatamplitudeisakeyvariableinasituation,andthen
choosetoworkwithpeakamplitude.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 4
2012
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NCN.2 Usetherelationi2=1andthecommutative,associative,anddistributivepropertiestoadd,
subtract,andmultiplycomplexnumbers.
Usecomplexnumbersinpolynomialidentitiesandequations.
NCN.7 Solvequadraticequationswithrealcoefficientsthathavecomplexsolutions.
Interpretthestructureofexpressions.
ASSE.23 Usethestructureofanexpressiontoidentifywaystorewriteit. Forexample,seex4y4as
(x2)2(y2)2,thusrecognizingitasadifferenceofsquaresthatcanbefactoredas(x2y2)(x2+
y2).
Understandtherelationshipbetweenzerosandfactorsofpolynomials.
AAPR.2 KnowandapplytheRemainderTheorem: Forapolynomialp(x)andanumbera,the
remainder
on
division
by
x
a
is
p(a),
so
p(a)
=
0
if
and
only
if
(x
a)
is
a
factor
of
p(x).
AAPR.34 Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezerosto
constructaroughgraphofthefunctiondefinedbythepolynomial.
Usepolynomialidentitiestosolveproblems.
AAPR.4 Provepolynomialidentitiesandusethemtodescribenumericalrelationships. Forexample,
the
polynomial
identity
(x2
+
y2)2
=
(x2
y2)2
+
(2xy)2
can
be
used
to
generate
Pythagorean
triples.
Rewriterationalexpressions.
AAPR.6
Rewrite
simple
rational
expressions
in
different
forms;
write
a(x)/b(x)
in
the
form
q(x)
+r(x)/b(x),wherea(x),b(x),q(x),andr(x)arepolynomialswiththedegreeofr(x)lessthanthe
degreeofb(x),usinginspection,longdivision,or,forthemorecomplicatedexamples,a
computeralgebrasystem.
Understandsolvingequationsasaprocessofreasoningandexplainthereasoning.
AREI.15 Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbers
assertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasa
solution. Constructaviableargumenttojustifyasolutionmethod.
AREI.2 Solvesimplerationalandradicalequationsinonevariable,andgiveexamplesshowinghow
extraneoussolutionsmayarise.
3InAlgebraII,tasksarelimitedtopolynomial,rational,orexponentialexpressions.Examples:seex
4y
4as(x
2)2(y
2)2,thusrecognizingitasa
differenceofsquaresthatcanbefactoredas(x2y
2)(x
2+y
2). Intheequationx
2+2x+1+y
2=9,seeanopportunitytorewritethefirstthreetermsas
(x+1)2,thusrecognizingtheequationofacirclewithradius3andcenter(1,0). See(x
2+4)/(x
2+3)as((x
2+3)+1)/(x
2+3),thusrecognizingan
opportunitytowriteitas1+1/(x2+3).
4InAlgebraII,tasksincludequadratic,cubic,andquadraticpolynomialsandpolynomialsforwhichfactorsarenotprovided. Forexample,findthe
zerosof(x2 1)(x
2+1).
5InAlgebraII,tasksarelimitedtosimplerationalorradicalequations.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 5
2012
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Solveequationsandinequalitiesinonevariable.
AREI.46 Solvequadraticequationsinonevariable.
b. Solvequadraticequationsbyinspection(e.g.,forx2=49),takingsquareroots,completing
thesquare,thequadraticformulaandfactoring,asappropriatetotheinitialformoftheequation. Recognizewhenthequadraticformulagivescomplexsolutionsandwritethem
asabiforrealnumbersaandb.
Solvesystemsofequations.
AREI.67 Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingon
pairsoflinearequationsintwovariables.
AREI.7 Solveasimplesystemconsistingofalinearequationandaquadraticequationintwo
variablesalgebraicallyandgraphically. Forexample,findthepointsofintersectionbetween
theliney=3xandthecirclex2+y2=3.
Analyzefunctionsusingdifferentrepresentations.
FIF.7 Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimple
casesandusingtechnologyformorecomplicatedcases.
c. Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,
andshowingendbehavior.
Translatebetweenthegeometricdescriptionandtheequationforaconicsection.
GGPE.2 Derivetheequationofaparabolagivenafocusanddirectrix.
ExtensionStandards
The(+)standardsbelowareprovidedasanextensiontoModule1oftheAlgebraIIcoursetoprovide
coherencetothecurriculum. Theyareusedtointroducethemesandconceptsthatwillbefullycoveredin
thePrecalculuscourse. Note: Noneofthe(+)standardsbelowwillbeassessedontheRegentsExamor
PARRCAssessmentsuntilPrecalculus.
Usecomplexnumbersinpolynomialidentitiesandequations.
NCN.8 (+)Extendpolynomialidentitiestothecomplexnumbers. Forexample,rewritex2+4as(x+
2i)(x2i).
NCN.9 (+)KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.
6InAlgebraII,inthecaseofequationshavingrootswithnonzeroimaginaryparts,studentswritethesolutionsasabi,whereaandbarereal
numbers.7InAlgebraII,tasksarelimitedto3x3systems.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 6
2012
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Rewriterationalexpressions.
AAPR.7 (+)Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,
closedunderaddition,subtraction,multiplication,anddivisionbyanonzerorational
expression;add,
subtract,
multiply,
and
divide
rational
expressions.
FoundationalStandards
Usepropertiesofrationalandirrationalnumbers.
NRN.3 Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarational
numberandanirrationalnumberisirrational;andthattheproductofanonzerorational
numberandanirrationalnumberisirrational.
Reason
quantitatively
and
use
units
to
solve
problems.
NQ.1 Useunitsasawaytounderstandproblemsandtoguidethesolutionofmultistepproblems;
chooseandinterpretunitsconsistentlyinformulas;chooseandinterpretthescaleandthe
originingraphsanddatadisplays.
Interpretthestructureofexpressions.
ASSE.1 Interpretexpressionsthatrepresentaquantityintermsofitscontext.
a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.
b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.
For
example,
interpret
P(1+r)n
as
the
product
of
P
and
a
factor
not
depending
on
P.
Writeexpressionsinequivalentformstosolveproblems.
ASSE.3 Chooseandproduceanequivalentformofanexpressiontorevealandexplainpropertiesof
thequantityrepresentedbytheexpression.
a. Factoraquadraticexpressiontorevealthezerosofthefunctionitdefines.
Performarithmeticoperationsonpolynomials.
AAPR.1 Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyare
closedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,and
multiply
polynomials.
Createequationsthatdescribenumbersorrelationships.
ACED.1 Createequationsandinequalitiesinonevariableandusethemtosolveproblems. Include
equationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponential
functions.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
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ACED.2 Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;
graphequationsoncoordinateaxeswithlabelsandscales.
ACED.3 Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/or
inequalities,
and
interpret
solutions
as
viable
or
non
viable
options
in
a
modeling
context.
For
example,representinequalitiesdescribingnutritionalandcostconstraintsoncombinationsof
different
foods.
ACED.4 Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolving
equations. Forexample,rearrangeOhmslawV=IRtohighlightresistanceR.
Solveequationsandinequalitiesinonevariable.
AREI.3 Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficients
representedbyletters.
AREI.4 Solvequadraticequationsinonevariable.
a.
Usethe
method
of
completing
the
square
to
transform
any
quadratic
equation
in
x
into
anequationoftheform(xp)2=qthathasthesamesolutions. Derivethequadratic
formulafromthisform.
Solvesystemsofequations.
AREI.5 Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythe
sumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.
Representandsolveequationsandinequalitiesgraphically.
AREI.10 Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplotted
inthe
coordinate
plane,
often
forming
acurve
(which
could
be
aline).
AREI.11 Explainwhythexcoordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=
g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,
e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessive
approximations.Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolute
value,exponential,andlogarithmicfunctions.
Translatebetweenthegeometricdescriptionandtheequationforaconicsection.
GGPE.1 DerivetheequationofacircleofgivencenterandradiususingthePythagoreanTheorem;
completethesquaretofindthecenterandradiusofacirclegivenbyanequation.
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 8
2012
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FocusStandardsforMathematicalPractice
MP.1 Makesenseofproblemsandpersevereinsolvingthem. Studentsdiscoverthevalueof
equatingfactoredtermsofapolynomialtozeroasameansofsolvingequationsinvolving
polynomials.Students
solve
rational
equations
and
simple
radical
equations,
while
consideringthepossibilityofextraneoussolutionsandverifyingeachsolutionbeforedrawing
conclusionsabouttheproblem. Studentssolvesystemsoflinearequationsandlinearand
quadraticpairsintwovariables. Further,studentscometounderstandthatthecomplex
numbersystemprovidessolutionstotheequationx2+1=0andhigherdegreeequations.
MP.2 Reasonabstractlyandquantitatively. Studentsapplypolynomialidentitiestodetectprime
numbersanddiscoverPythagoreantriples. Studentsalsolearntomakesenseofremainders
inpolynomiallongdivisionproblems.
MP.4 Modelwithmathematics. Studentsuseprimestomodelencryption. Studentstransition
betweenverbal,numerical,algebraic,andgraphicalthinkinginanalyzingappliedpolynomial
problems. Studentsmodelacrosssectionofariverbedwithapolynomial,estimatefluidflow
withtheiralgebraicmodel,andfitpolynomialstodata. Studentsmodelthelocusofpointsatequaldistancebetweenapoint(focus)andaline(directrix)discoveringtheparabola.
MP.7 Lookforandmakeuseofstructure. Studentsconnectlongdivisionofpolynomialswiththe
longdivisionalgorithmofarithmeticandperformpolynomialdivisioninanabstractsettingto
derivethestandardpolynomialidentities. Studentsrecognizestructureinthegraphsof
polynomialsinfactoredformanddeveloprefinedtechniquesforgraphing. Studentsdiscern
thestructureofrationalexpressionsbycomparingtoanalogousarithmeticproblems.
Studentsperformgeometricoperationsonparabolastodiscovercongruenceandsimilarity.
MP.8 Lookforandexpressregularityinrepeatedreasoning. Studentsunderstandthat
polynomialsformasystemanalogoustotheintegers. Studentsapplypolynomialidentitiesto
detect
prime
numbers
and
discover
Pythagorean
triples.
Students
recognize
factors
of
expressionsanddevelopfactoringtechniques. Further,studentsunderstandthatall
quadraticscanbewrittenasaproductoflinearfactorsinthecomplexrealm.
Terminology
NeworRecentlyIntroducedTerms
Polynomial
StandardForm(ofapolynomial)
Degree
LeadingCoefficient
ConstantTerm
RationalExpression
Parabola
ComplexNumber
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M1ModuleOverviewNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Module1: Polynomial,Rational,andRadicalRelationships
Date: 5/10/13 9
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FamiliarTermsandSymbols8
Quadratic
Factor
SystemofEquations
SuggestedToolsandRepresentations
GraphingCalculator
WolframAlphaSoftware
GeometersSketchpadSoftware
Assessment
Summary
AssessmentType Administered Format StandardsAddressed
MidModule
AssessmentTaskAfterTopicB Constructedresponsewithrubric
NQ.2,ASSE.2,AAPR.2
AAPR.3,AAPR.4,
AREI.1,AREI.4b,
FIF.7c
EndofModule
AssessmentTask AfterTopicD Constructedresponsewithrubric
NQ.2,NCN.1,NCN.2,
NCN.7,NCN.8,NCN.9,
A.SSE.2,A.APR.2,
AAPR.3,AAPR.4,
AAPR.6,AAPR.7,AREI.1,AREI.2,
AREI.4b,AREI.6,
AREI.7,FIF.7c,GGPE.2
8Thesearetermsandsymbolsstudentshaveseenpreviously.
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Module1: Polynomial,Rational,andRadicalRelationships
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Name Date
1. Geographers,whilesittingatacaf,discusstheirfieldworksite,whichisahillandaneighboringriver
bed. Thehillisapproximately1050feethigh,800feetwide,withpeakabout300feeteastofthewestern
baseofthehill. Theriverisabout400feetwide. Theyknowtheriverisshallow,nomorethanabout
twentyfeetdeep.
Theymakethefollowingcrudesketchonanapkin,placingtheprofileofthehillandriverbedona
coordinatesystemwiththehorizontalaxisrepresentinggroundlevel.
Thegeographersdonothavewiththematthecafanycomputingtools,buttheynonethelessdecideto
computewithpenandpaperacubicpolynomialthatapproximatesthisprofileofthehillandriverbed.
a. Usingonlyapencilandpaper,writeacubicpolynomialfunction,thatcouldrepresentthecurve
shown(here,representsthedistance,infeet,alongthehorizontalaxisfromthewesternbaseof
thehill,andistheheightinfeetofthelandatthatdistancefromthewesternbase). Besure
thatyourformulasatisfies300 1050.
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Module1: Polynomial,Rational,andRadicalRelationships
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
b. Forthesakeofease,thegeographersmaketheassumptionthatthedeepestpointoftheriveris
halfwayacrosstheriver(recallthattheriverisknowntobeshallow,withadepthofnotmorethan
20feet). Underthisassumption,wouldacubicpolynomialprovideasuitablemodelforthishilland
riverbed? Explain.
2. Lukenoticedthatifyoutakeanythreeconsecutiveintegers,multiplythemtogether,andaddthemiddle
numbertotheresult,theansweralwaysseemstobethemiddlenumbercubed.
Forexample: 3 4 5 4 64 4
4 5 6 5 125 5
9 10 11 10 1000 10
a.
Inorderprovehisobservationtrue,Lukewritesdown 1 2 3 2.What
answerishehopingtoshowthisexpressionequals?
b. Lulu,uponhearingofLukesobservation,writesdownherownversionwithasthemiddlenumber.
Whatdoesherformulalooklike?
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Module1: Polynomial,Rational,andRadicalRelationships
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
c. UseLulusexpressiontoprovethataddingthemiddlenumbertotheproductofanythree
consecutivenumbersissuretoequalthatmiddlenumbercubed.
3. Acookiecompanypackagesitscookiesinrectangularprismboxesdesignedwithsquarebaseswhichhave
bothalength
and
width
of
4inches
less
than
the
height
of
the
box.
a. Writeapolynomialthatrepresentsthevolumeofaboxwithheightinches.
b. Findthedimensionsoftheboxifitsvolumeisequalto128cubicinches.
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Module1: Polynomial,Rational,andRadicalRelationships
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
c. Aftersolvingthisproblem,Juanwasverycleverandinventedthefollowingstrangequestion:
Abuilding,intheshapeofarectangularprismwithasquarebase,hasonitstoparadiotower. The
buildingis25timesastallasthetower,andthesidelengthofthebaseofthebuildingis100feetless
thanthe
height
of
the
building.
If
the
building
has
avolume
of
2million
cubic
feet,
how
tall
is
the
tower?
SolveJuansproblem.
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Module1: Polynomial,Rational,andRadicalRelationships
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
AProgressionTowardMastery
Assessment
TaskItem
STEP1
Missingorincorrect
answerand
little
evidenceof
reasoningor
applicationof
mathematicsto
solvetheproblem.
STEP2
Missingorincorrect
answerbut
evidenceofsome
reasoningor
applicationof
mathematicsto
solvetheproblem.
STEP3
Acorrectanswer
withsome
evidence
ofreasoningor
applicationof
mathematicsto
solvetheproblem,
oranincorrect
answerwith
substantial
evidenceofsolid
reasoningor
applicationof
mathematicsto
solvethe
problem.
STEP4
Acorrectanswer
supportedby
substantial
evidenceofsolid
reasoningor
applicationof
mathematicsto
solvetheproblem.
1 a
NQ.2
AAPR.2
AAPR.3
FIF.7c
Identifieszerosongraph. Useszerostowritea
factoredcubic
polynomialforH(x)
withoutaleading
coefficient.
Usesgivencondition
300 1050 tofind
avalue(leading
coefficient).
Writesacomplete
cubicmodelforH(x)in
factoredformwith
correctavalue(leading
coefficient).
b
NQ.2
AAPR.2
AAPR.3
FIF.7c
Findsthemidpointof
theriver.
EvaluatesH(x)usingthe
midpointexactanswer
isnotneeded,only
approximation.
Determinesifacubic
modelissuitableforthis
hillandriverbed.
Justifiesanswerusing
H(midpoint)in
explanation.
2 a
ASSE.2
AAPR.4
Answerdoesnot
indicateanyexpression
involvingnraisedtoan
exponentof3.
Answerinvolvesabase
involvingnbeingraised
toanexponentof3,but
doesnotchooseabase
of 2.
Answers, 2
withoutincluding
parenthesestoindicate
allof 2isbeing
cubedORhasanother
errorthatshowsgeneral
understanding,butis
technicallyincorrect.
Answerscorrectlyas
2.
bc
ASSE.2
AAPR.4
Bothpartsbandcare
missingOR
incorrect
OR
incomplete.
Answertopartbis
incorrectbut
student
usescorrectalgebraas
theyattempttoshow
equivalencetoORthe
answertopartbis
correct,butthestudent
mademajorerrorsor
Partbisanswered
correctlyas:
1 1
,butstudent
mademinorerrorsin
showingequivalenceto
.
Answeriscorrectly
writtenas:
1 1
ANDthe
studentcorrectly
multipliedtheleftside
andthencombinedlike
termstoshow
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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
wasunabletoshowits
equivalenceto.
equivalenceto.
3 ad
NQ.2
ASSE.2
AAPR.2
AAPR.3
AREI.1
AREI.4b
Determinesan
expressionforV(x).
SetsV(x)equaltogiven
volume.
Solvestheequation
understandingthatonly
realvaluesarepossible
solutionsforthe
dimensionsofabox.
Statesthe3dimensions
oftheboxwithproper
units.
c
NQ.2
ASSE.2
AAPR.2
AAPR.3
AREI.1
AREI.4b
Determinesan
expressionforV(h)and
setsitequaltothegiven
volume.
Simplifiestheequation
torevealthatisitexactly
thesameastheprevious
equation.
Solvestheequationor
statesthattheansweris
exactlythesameanswer
asintheprevious
exampleAND
understandsthat
only
realsolutionsare
possiblefortheheightof
atower.
Statestheheightofthe
towerwiththeproper
units.
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ALGEBRAII
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ALGEBRAII
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ALGEBRAII
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ALGEBRAII
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
Name Date
1. Aparabolaisdefinedasthesetofpointsintheplanethatareequidistantfromafixedpoint(calledthe
focusoftheparabola)andafixedline(calledthedirectrixoftheparabola).
Considertheparabolawithfocuspoint1,1anddirectrixthehorizontalline 3.
a. Whatwillbethecoordinatesofthevertexoftheparabola?
b. Plotthefocusanddrawthedirectrixonthegraphbelow. Thendrawaroughsketchoftheparabola.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
c. Findtheequationoftheparabolawiththisfocusanddirectrix.
d.
Whatis
the
interceptofthisparabola?
e. Demonstratethatyouranswerfrom(d)iscorrectbyshowingthatthe
interceptyouhave
identifiedisindeedequidistantfromthefocusandthedirectrix.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
f. Istheparabolainthisquestion(withfocuspoint1,1anddirectrix 3)congruenttoaparabolawithfocus2,3anddirectrix 1? Explain.
g. Istheparabolainthisquestion(withfocuspoint(1,1)anddirectrix 3)congruenttotheparabola
with
equation
given
by
? Explain.
h. Arethetwoparabolasfrompartgsimilar?Whyorwhynot?
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
2. Thegraphofthepolynomialfunction 4 6 4isshownbelow.
a. Basedontheappearanceofthegraph,whatseemstobetherealsolutiontotheequation:4 6 4? Jijudoesnottrusttheaccuracyofthegraph. Provetoheralgebraicallythatyouranswerisinfactazeroof .
b. Writeasaproductofalinearfactorandaquadraticfactor,eachwithrealnumbercoefficients.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
c. Whatisthevalueof10? Explainhowknowingthelinearfactorofestablishesthat10isamultipleof12.
d.
Findthe
two
complex
number
zeros
of
.
e. Writeasaproductofthreelinearfactors.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
3. Alinepassesthroughthepoints1,0and 0,
and forsomerealnumberandintersectsthecircle 1atapointdifferentfrom1,0.
a.
If , sothatthepointhascoordinates0,,findthecoordinatesofthepoint.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
APythagoreantripleisasetofthreepositiveintegers,,andsatisfying . Forexample,setting 3, 4,and 5givesaPythagoreantriple.
b. Supposethat,
isapointwithrationalnumbercoordinateslyingonthecircle 1.
Explainwhythen,,andformaPythagoreantriple.
c.
WhichPythagoreantripleisassociatedwiththepoint ,onthecircle?
d. If ,,whatisthevalueofsothatthepointhascoordinates0, ?
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
e. Supposeweset
and , forarealnumber. Showthat, isthenapointonthe
circle 1.
f. Set intheformulas and
.Whichpointonthecircle
1doesthisgive?WhatistheassociatedPythagoreantriple?
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
4.
a. Writeasystemoftwoequationsintwovariableswhereoneequationisquadraticandtheotheris
linearsuchthatthesystemhasnosolution. Explain,usinggraphs,algebraand/orwords,whythe
systemhasnosolution.
b. Provethat 5 6hasnosolution.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
c. Doesthefollowingsystemofequationshaveasolution? Ifso,findone. Ifnot,explainwhynot.
2 4
3 2
2
x y z
x y z
x y z
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
AProgressionTowardMastery
Assessment
TaskItem
STEP1
Missingor
incorrectanswer
andlittleevidence
ofreasoningor
applicationof
mathematicsto
solvetheproblem.
STEP2
Missingorincorrect
answerbut
evidenceofsome
reasoningor
applicationof
mathematicsto
solvetheproblem.
STEP3
Acorrectanswer
withsome
evidence
ofreasoningor
applicationof
mathematicsto
solvetheproblem,
oranincorrect
answerwith
substantial
evidenceofsolid
reasoningor
applicationof
mathematicsto
solvethe
problem.
STEP4
Acorrectanswer
supportedby
substantial
evidenceofsolid
reasoningor
applicationof
mathematicsto
solvetheproblem.
1 ac
NQ.2
FIF.7c
GGPE.2
(a)Bothvertex
coordinatesincorrect.
(b)Paraboladoesnot
openuporis
horizontal.
(c)Equationisnotin
theformofavertical
parabola.
(a)Eitherx ory
coordinateisincorrect
(b)Minimalsketchofa
verticalparabolawith
littleornoscaleor
labels.
(c)Incorrectequation
usingthevertexfrom
part(a). avalueis
incorrectdueto
conceptualerrors.
(a)Correctvertex.
(b)Parabolasketch
opensupwithcorrect
vertex. Sketchmaybe
incompleteorlack
sufficientlabelsorscale.
(c)Parabolaequation
withcorrectvertex.
Workshowingavalue
calculationmaycontain
minorerrors.
(a)Correctvertex.
(b)Welllabeledand
accuratesketchincludes
focus,directrix,vertex,
andparabolaopening
up.
(c)Correctparabolain
vertexorstandardform
withorwithoutwork
showinghowtheygot
a=1/8.
d
e
NQ.2
FIF.7c
GGPE.2
(d)Incorrect
yintercept. Nowork
shownorconceptual
error.
(e)Bothdistancesare
incorrectornot
attempted.
(d)yintercept
is
incorrect. Nowork
shownorconceptual
error(e.g.,triestomake
y=0notx).
(e)Onedistanceis
correct,butnotboth
usingstudentsy
interceptandthegiven
focusanddirectrix.
OR
Correctyinterceptin
part(d),butstudentis
unabletocomputeone
orboth
distances
betweentheyintercept
andthegivenfocusand
directrix.
(d)Substitutes
x
=0to
determineyintercept,
butmaycontainminor
calculationerror.
(e)Correctdistanceto
directrixusingtheiry
intercept. Correct
distancebetweenfocus
andyinterceptusing
theiryintercept.
NOTE: Ifthesearenot
equal,studentsolution
shouldindicatethatthey
shouldbebasedonthe
definitionof
aparabola.
(d)Correct
yintercept
(e)Correctdistanceto
directrix. Applies
distanceformulato
calculatedistancefrom
focusandyintercept.
Bothareequalto17/8.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
fh
NQ.2
F
IF.7c
GGPE.2
Twoormoreparts
incorrectwithno
justificationORthree
partsincorrectwith
faulty
or
no
justification.
Twoormoreparts
incorrect.Minimal
justificationthatincludes
areferencetothea
value.
Correctanswertoall
threepartswithno
justificationORtwoout
ofthreepartscorrect
with
correct
justification.
Correctanswertoall
threeparts. Justification
statesthatparabolas
withequalavaluesare
congruent
but
all
parabolasaresimilar.
2 ab
NCN.1
NCN.2
NCN.7
NCN.8
NCN.9
ASSE.2
AAPR.1
AAPR.2
AAPR.3
AREI.1
AREI.4b
FIF.7c
(a)Concludesthatx=
2isNOTazerodueto
conceptualerroror
majorcalculationerrors
(e.g.incorrect
applicationofdivision
algorithm)orshowsno
workatall.
(b)Factoredformis
incorrectormissing.
(a)Concludesthatx= 2
isNOTazerodueto
minorcalculationerrors
intheapproach. Limited
justificationfortheir
solution.
(b)Factoredformis
incorrectormissing.
(a)Concludesthatx=2
isazero,butmaynot
supportmathematically
orverbally.
(b)Evidenceshowing
bothcorrectfactors,but
polynomialmaynotbe
writteninfactoredform
((x+2)(x2+2x+2)).
(a)Concludesthatx=2
isazerobyshowingf(2)
=0orusingdivisionand
gettingaremainder
equalto0.Workor
writtenexplanation
supportsconclusion.
(b)fiswrittenincorrect
factoredformwithwork
showntosupportthe
solution.
NOTE:
Work
may
be
doneinpart(a).
c
NCN.1
NCN.2
NCN.7
NCN.8
NCN.9
ASSE.2
AAPR.2
AAPR.3
AREI.1
AREI.4b
FIF.7c
f(10)incorrectanda
conclusionregarding12
beingafactorismissing
orunsupportedbyany
mathematicalworkor
explanation.
f(10)incorrectand
factoredformoff
incorrect. Solutiondoes
notattempttofindthe
numericalfactorsof
f(10)ordividef(10)by
12toseeifthe
remainder
is
0.
Completesolution,but
maycontainminor
calculationerrorsonthe
valueoff(10).
OR
Concludesthat12isNOT
afactoroff(10)because
the
solution
used
an
incorrectlyfactoredform
offinthefirstplaceor
anincorrectvaluefor
f(10).
f(10)=1464
Explanationclearly
communicatesthat12is
afactoroff(10)because
whenx=10,(x+2)is12.
de
NCN.1
NCN.2
NCN.7
NCN.8
NCN.9
ASSE.2
AAPR.1
AAPR.2
(d)Doesnotuse
quadraticformulaor
usesincorrectformula.
(e)Incorrectcomplex
rootsandsolutionnota
cubicequivalent
to
(x+2)(xr1)(xr2),
wherer1andr2are
complexconjugates.
(d)Minorerrorsinthe
quadraticformula.
(e)Incorrectrootsfrom
(d),butsolutionisa
cubicequivalentto(x+
2)(x
r1)(x
r2),
where
r1andr2arethestudent
solutionstod.
(d)Correctcomplex
rootsusingthequadratic
formula(doesnothave
tobeinsimplestform).
(e)Cubicpolynomial
using2,
and
complex
rootsfrom(d).May
containminorerrors
(e.g.leavingout
parentheseson(x(1+
i))oramultiplication
errorwhenwritingthe
(d)Correctcomplex
rootsexpressedas(1
i).
(e)Cubicpolynomial
equivalentto
(x+2)(x
(1+
i))(x
(1
i)). OKtoleavein
factoredform.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
AAPR.3
AREI.1
AREI.4b
F
IF.7c
polynomialinstandard
form.
3 a
NCN.8
AAPR.4
AAPR.6
AAPR.7
AREI.2
AREI.6
AREI.7
Equationofthelineis
incorrectandsolution
showsmajor
mathematicalerrorsin
attemptingtosolvea
systemofalinearand
nonlinearequation.
Equationofthelinemay
beincorrect,butsolution
showssubstitutionof
studentlinearequation
intothecircleequation.
Solutiontothesystem
mayalsocontainminor
calculationerrors.
OREquationofthelineis
correctbutstudentis
unabletosolvethe
systemduetomajor
mathematicalerrors.
Equationofthelineis
correct. Solutiontothe
systemmaycontain
minorcalculationerrors.
Correctsolutionthatis
notexpressedasan
orderedpair.
ORSolutiononly
includesacorrectxory
valueforpointQ.
Equationoflineis
correct. Correctsolution
tothesystemof
equations. Solution
expressedasanordered
pairQ(3/5,4/5).
bc
NCN.8
AAPR.4
AAPR.6
AAPR.7
AREI.2
AREI.6
AREI.7
(b)Missingorincorrect
answershowinglimited
understandingofthe
task.
(c)Incorrectormissing
triple.
(b)Substitutes(a/c,b/c)
buttherestofthe
solutionislimited.
(c)Incorrecttriple.
(b)Incompletesolution
mayincludeminor
algebramistakes.
(c)Identifies5,12,13as
thetriple.
(b)Completesolution
showingsubstitutionof
(a/c,b/c)intoequation
ofcircle.Workclearly
establishesthisequation
equivalencetoa2+b
2=
c2.
(c)Identifies5,12,13as
thetriple.
df
NCN.8
AAPR.4
AAPR.6
AAPR.7
AREI.2
AREI.6
AREI.7
Missingorincomplete
solutionto
(d),
(e),
and
(f)withmajor
mathematicalerrors.
(d)Slopeoflinecorrect
butfails
to
identify
correctvalueoft.
(e)Substitutes
coordinatesintox2+y
2=
1,butmajorerrorsin
attempttoshowthey
satisfytheequation.
(f)Substitutesfort,
butsolutionisincorrect.
(d)Slopeoflinecorrect
andequation
of
line
correct,butfailsto
identifythecorrectvalue
oft.
(e)Substitutes
coordinatesintox2+y
2=
1andsimplifiestoshow
theysatisfythe
equation.
(f)IdentifiespointQand
thetriplecorrectly.
NOTE:Oneormoreparts
maycontainminor
calculationerrors.
(d)Slopeoflinecorrect
andequation
of
line
correct. Correct
identificationoftvalue.
(e)Substitutes
coordinatesintox2+y
2=
1andsimplifiestoshow
theysatisfythe
equation.
(f)IdentifiespointQand
thetriplecorrectly.
NOTE:Solutionsuse
propermathematical
notationandclearly
communicatesstudent
thinking.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
ALGEBRAII
g
NCN.8
A
APR.4
AAPR.6
AAPR.7
AREI.2
AREI.6
AREI.7
Missingorincomplete
solutionwithmajor
mathematicalerrors.
Limitedworktoa
completesolutionmay
includeanaccurate
sketchandtheequation
of
the
line
y
=
tx
+
t,
but
littleadditionalwork.
Attemptstosolvethe
systembysubstitutingy
=tx+tintothecircle
equationandrecognizes
the
need
to
apply
the
quadraticequationto
solveforx.Maycontain
algebraicerrors.
Completeandcorrect
solutionshowing
sufficientworkand
calculationofboththei
and
y
coordinate
of
the
point.
4 a
NCN.7
AREI.2
AREI.4b
AREI.6
AREI.7
Missingorincomplete
work. Systemdoesnot
includealinearanda
quadraticequation.
Givensystemhasa
solution,butstudent
workindicates
understandingthatthe
graphsoftheequations
shouldnotintersector
thatalgebraicallythe
systemhasnoreal
numbersolutions.
Givensystemhasno
solution,butthe
justificationmayreveal
minorerrorsinstudents
thoughtprocesses. Ifa
graphicaljustificationin
theonlyoneprovided
thegraphmustbescaled
sufficientlytoprovidea
convincingargument
thatthetwoequations
donotintersect.
Givensystemhasno
solution. Justification
includesagraphical,
verbalexplanation,or
algebraicexplanation
thatclearly
demonstratesstudent
thinking.
bc
NCN.7
AREI.2
AREI.4b
AREI.6
AREI.7
Incorrectsolutionsand
littleornotsupporting
workshown.
Incorrectsolutionsto(b)
and(c). Solutionsare
limitedandrevealmajor
mathematicalerrorsin
thesolutionprocess.
Incorrectsolutionsto(b)
or(c). Solutionsshow
considerable
understandingofthe
processes,butmay
containminorerrors.
Correctsolutionswith
sufficientworkshown.
MathematicalworkOR
verbalexplanationshow
why2and3areNOT
solutionstopartb.
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M1EndofModuleAssessmentTaskNYSCOMMONCOREMATHEMATICSCURRICULUM
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