algebra ii mod1 module overview and assessments

8/10/2019 Algebra II Mod1 Module Overview and Assessments

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Module1: Polynomial,Rational,andRadicalRelationships

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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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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M1MidModuleAssessmentTaskNYSCOMMONCOREMATHEMATICS CURRICULUM

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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ALGEBRAII

c. Findtheequationoftheparabolawiththisfocusanddirectrix.

d.

Whatis

the

interceptofthisparabola?

e. Demonstratethatyouranswerfrom(d)iscorrectbyshowingthatthe

interceptyouhave

identifiedisindeedequidistantfromthefocusandthedirectrix.


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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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ALGEBRAII

2. Thegraphofthepolynomialfunction 4 6 4isshownbelow.

a. Basedontheappearanceofthegraph,whatseemstobetherealsolutiontotheequation:4 6 4? Jijudoesnottrusttheaccuracyofthegraph. Provetoheralgebraicallythatyouranswerisinfactazeroof .

b. Writeasaproductofalinearfactorandaquadraticfactor,eachwithrealnumbercoefficients.


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ALGEBRAII

c. Whatisthevalueof10? Explainhowknowingthelinearfactorofestablishesthat10isamultipleof12.

d.

Findthe

two

complex

number

zeros

of

.

e. Writeasaproductofthreelinearfactors.


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ALGEBRAII

3. Alinepassesthroughthepoints1,0and 0,

and forsomerealnumberandintersectsthecircle 1atapointdifferentfrom1,0.

a.

If , sothatthepointhascoordinates0,,findthecoordinatesofthepoint.


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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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ALGEBRAII

e. Supposeweset

and , forarealnumber. Showthat, isthenapointonthe

circle 1.

f. Set intheformulas and

.Whichpointonthecircle

1doesthisgive?WhatistheassociatedPythagoreantriple?


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ALGEBRAII

4.

a. Writeasystemoftwoequationsintwovariableswhereoneequationisquadraticandtheotheris

linearsuchthatthesystemhasnosolution. Explain,usinggraphs,algebraand/orwords,whythe

systemhasnosolution.

b. Provethat 5 6hasnosolution.


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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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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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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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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.


andequation

of

line

correct,butfailsto

identifythecorrectvalue

oft.

(e)Substitutes

coordinatesintox2+y

2=

1andsimplifiestoshow

theysatisfythe

equation.

(f)IdentifiespointQand

thetriplecorrectly.

NOTE:Oneormoreparts

maycontainminor

calculationerrors.


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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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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algebra ii mod1 module overview and assessments

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