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PutnamCountySchoolsCurriculumMap
Algebra12016-2017
Module:1Expressions,EquationsandInequalities
InstructionalWindow:August10-September28AssessmentWindow:September29-October12MAFSStandards
LessonA:UsingExpressionstoRepresentReal-WorldSituationsMAFS.912.A-APR.1.1-Understandthatpolynomialsformasystemanalogoustotheintegers,namely,theyareclosedundertheoperationsofaddition,subtraction,andmultiplication;add,subtract,andmultiplypolynomials.EngageNY,Module1,Lessons6-9/ShmoopUnit3&4.MathNationSection1,Topic1.MAFS.912.N-RN.1.1AssessedWithinN-RN.1.2-Explainhowthedefinitionofthemeaningofrationalexponentsfollowsfromextendingthepropertiesofintegerexponentstothosevalues,allowingforanotationforradicalsintermsofrationalexponents.Forexample,wedefine tobethecuberootof5becausewewant = tohold,so mustequal5.*NotcoveredinENY.*ShmoopUnit2.MathNationSection1,Topic6.MAFS.912.N-RN.1.2AlsoAssessesN-RN.2.3,N-RN.1.1-Rewriteexpressionsinvolvingradicalsandrationalexponentsusingthepropertiesofexponents.**NotcoveredinENY.ShmoopUnit2.MathNationSection1,Topics6-8.MAFS.912.N-RN.2.3AssessedWithinN-RN.1.2-Explainwhythesumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.ShmoopUnit1.MathNationSection1,Topic9.MAFS.912.A-SSE.1.1AssessedwithinA-SSE.2.3-Interpretexpressionsthatrepresentaquantityintermsofitscontext.
a. Interpretpartsofanexpression,suchasterms,factors,andcoefficients.b. Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.Forexample,interpret𝑃(1 + 𝑟)!
astheproductofPandafactornotdependingonP.EngageNY,Module1,Lessons25-28/ShmoopUnit4.MathNationSection1,Topics1&2.MAFS.912.A-SSE.1.2AssessedWithinA-SSE.2.3-Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4-y4as(x²)²–(y²)²,thusrecognizingitasadifferenceofsquaresthatcanbefactoredas(x²–y²)(x²+y²).EngageNY,Module4,Lessons1,2/ShmoopUnit2.MathNationSection1,Topics3-5.
LessonB:EquationsandInequalitiesMAFS.912.A-CED.1.1AlsoAssessesA-CED.1.4,REI.2.3-Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Includeequationsarisingfromlinearandquadraticfunctions,andsimplerational,absolute,andexponentialfunctions.EngageNY,Module1,Lessons25-28/ShmoopUnit4.MathNationSection2,Topics3,5,6,8.MAFS.912.A-CED.1.2AlsoAssessesA-REI.3.5,A-REI.3.6,A-REI.4.12-Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.EngageNY,Module1,Lessons25-28/ShmoopUnit4.MathNationSection2,Topic10.MAFS.912.A-CED.1.4AssessedwithinA-CED.1.1-Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.Forexample,rearrangeOhm’slawV=IRtohighlightresistanceR.EngageNY,Module1,Lessons12,19/ShmoopUnit4.MathNationSection2,Topic9.MAFS.912.A.REI.1.1Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbersassertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.Constructaviableargumenttojustifyasolutionmethod.EngageNY,Module1,Lessons12,13/ShmoopUnit4.MathNationSection2,Topics2&3.MAFS.912.A-REI.2.3AssessedwithinA-CED.1.1-Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.EngageNY,Module1,Lessons11-15/ShmoopUnit4.MathNationSection2,Topics1-8.MAFS.912.A-REI.4.10AssessedwithinA-REI.4.11-Understandthatthegraphofanequationintwovariablesisthesetofallitssolutionsplottedinthecoordinateplane,oftenformingacurve(whichcouldbealine).EngageNY,Module1,Lessons20/ShmoopUnit6.MathNationSection2,Topic10.MAFS.912.A-SSE.1.2AssessedWithinA-SSE.2.3-Usethestructureofanexpressiontoidentifywaystorewriteit.Forexample,seex4-y4as(x²)²–(y²)²,thusrecognizingitasadifferenceofsquaresthatcanbefactoredas(x²–y²)(x²+y²).EngageNY,Module4,Lessons1,2/ShmoopUnit2.MathNationSection2,Topic2.
ExpectationstobeLearned
UnpackingWhatdothesestandardsmeanachildwillknowandbeabletodo?
DOKLevel
MAFS.912.A-APR.1.1:ITEMSPECIFICATIONS:ItemTypesEditingTaskChoice–Mayrequirecompletinganinformalargumentonclosure.EquationEditor–Mayrequirecreatingavalueoranexpression.GRID–Mayrequiredragginganddroppingexpressions/statementstocompleteaninformalargument.HotText–Mayrequiredragginganddroppingvalues/expressionstocompleteapolynomial.MatchingItem–Mayrequirematchingequivalentpolynomials.MultipleChoice–Mayrequireselectingavalueoranexpressionfromalist.Multiselect–Mayrequireselectingallequivalentexpressions.OpenResponse–Mayrequirecreatingawrittenexplanation.ClarificationsStudentswillrelatetheaddition,subtraction,andmultiplicationofintegerstotheaddition,subtraction,andmultiplicationofpolynomialswithintegralcoefficientsthroughapplicationofthedistributiveproperty.Studentswillapplytheirunderstandingofclosuretoadding,subtracting,andmultiplyingpolynomialswithintegralcoefficients.Studentswilladd,subtract,andmultiplypolynomialswithintegralcoefficients.AssessmentLimitsItemssetinareal-worldcontextshouldnotresultinanonrealanswerifthepolynomialisusedtosolvefortheunknown.Initemsthatrequireadditionandsubtraction,polynomialsarelimitedtomonomials,binomials,andtrinomials.Thesimplifiedpolynomialshouldcontainnomorethansixterms.Itemsrequiringmultiplicationofpolynomialsarelimitedtoaproductof:twomonomials,amonomialandabinomial,amonomialandatrinomial,twobinomials,andabinomialandatrinomial.StimulusAttributesItemsmaybesetinamathematicalorreal-worldcontext.Itemsmayusefunctionnotation.ResponseAttributesItemsmayrequirethestudenttowritetheanswerinstandardform.Itemsmayrequirethestudenttorecognizeequivalentexpressions.Itemsmayrequirethestudenttorewriteexpressionswithnegativeexponents,butitemsmustnotrequirethestudenttorewriterationalexpressionasseeninthestandardMAFS.912.A-APR.4.7.
MAFS.912.A-APR.1.1ContentComplexity:Level1:Recall
CalculatorNo
MAFS.912.A-CED.1.1:AlsoAssess:MAFS.912.A-REI.2.3:Solvelinearequationsandinequalitiesinonevariable,includingequationswithcoefficientsrepresentedbyletters.MAFS.912.A-CED.1.4:Rearrangeformulastohighlightaquantityofinterest,usingthesamereasoningasinsolvingequations.Forexample,rearrangeOhm’slaw,V=IR,tohighlightresistance,R.ITEMSSPECIFICATIONS:ItemTypes:EditingTaskChoice–Mayrequirechoosingacorrectequationorthecorrectdefinitionofavariable.EquationEditor–Mayrequirecreatinganequation,aninequality,oravalue.GRID–Mayrequiredragginganddroppingexpressions/statementstocompleteamodel.
MAFS.912.A-CED.1.1Complexity:Level2:BasicApplicationofSkills&Concepts
HotText–Mayrequiredragginganddroppingvaluesand/orexpressionstocreatelinearequationsandinequalitiesorrearrangingequations.MultipleChoice–Mayrequireidentifyinganequation,aninequality,oravaluefromalistoffourchoices.Multiselect–Mayrequireselectinganequationandidentifyingavariable.OpenResponse–Mayrequirecreatingawrittenexplanation.Clarifications:Studentswillwriteanequationinonevariablethatrepresentsareal-worldcontext.Studentswillwriteaninequalityinonevariablethatrepresentsareal-worldcontext.Studentswillsolvealinearequation.Studentswillsolvealinearinequality.Studentswillsolvemulti-variableformulasorliteralequationsforaspecificvariable.Studentswillsolveformulasandequationswithcoefficientsrepresentedbyletters.AssessmentLimitsInitemsthatrequirethestudenttowriteanequation,equationsarelimitedtoexponentialfunctionswithonetranslation,linearfunctions,orquadraticfunctions.Itemsmayincludeequationsorinequalitiesthatcontainvariablesonbothsides.Initemsthatrequirethestudenttowriteanexponentialfunctiongivenorderedpairs,atleastonepairofconsecutivevaluesmustbegiven.Initemsthatrequirethestudenttowriteorsolveaninequality,variablesarerestrictedtoanexponentofone.ItemsthatinvolveformulasshouldnotincludeoverusedcontextssuchasFahrenheit/Celsiusorthree-dimensionalgeometryformulas.Initemsthatrequirethestudenttosolveliteralequationsandformulas,alineartermshouldbethetermofinterest.Itemsshouldnotrequiremorethanfourproceduralstepstoisolatethevariableofinterest.ItemsmayrequirethestudenttorecognizeequivalentexpressionsbutmaynotrequireastudenttoperformanalgebraicoperationoutsidethecontextofAlgebra1.StimulusAttributesItemsassessingA-CED.1.1andA-CED.1.4mustbeplacedinreal-worldcontext.ItemsassessingREI.2.3donothavetobeinareal-worldcontext.ResponseAttributesItemsassessingREI.2.3shouldnotrequirethestudenttowritetheequation.Itemsmayrequirethestudenttochooseanappropriatelevelofaccuracy.Itemsmayrequirethestudenttochooseandinterpretunits.ForA-CED.1.1andA-CED.1.4,itemsmayrequirethestudenttoapplythebasicmodelingcycle.
CalculatorNeutral
STANDARDDECONSTRUCTION:Equationscanrepresentreal-worldandmathematicalproblems.Includeequationsandinequalitiesthatarisewhencomparingthevaluesoftwodifferentfunctions,suchasonedescribinglineargrowthandonedescribingexponentialgrowth.Studentsshouldbeabletointerpretwordproblemsandformequationsandinequalitiesinordertosolvetheproblem.Thatmeanstranslatingawordproblemtoanalgebraicequation.Let’sbereal,here.Mathisanotherlanguage,justlikeSpanish,Japanese,orIcelandic.Whenyoustartlearningalanguage,youdon’tstartbytranslatingwordslike“absquatulate”or“loquacious”or“pneumonoultramicroscopicsilicovolcanoconiosis”(andyes,thatisarealword).It’sbettertostarteasier,withwordslike“cat”and“girl”andslowlyworkyourwayup.Justthesame,ifyouusesimplelinearequationsthatarefamiliartostudents,theycanfocusonthetranslationprocessandit’llallgoalotsmoother.Translationisausefulanalogyinandofitselfbecauseitemphasizesthatthealgebraicequationisthesameasthewordproblem,justpresentedinadifferentway.Inadditiontohelpingstudentstounderstandtheprocess,the
translationanalogycanalsohelpreassurestrugglinglearnersandencouragepractice.Afterthey’vegottenahangofthebasics,studentscanstartlearningquadratic,rational,andexponentialfunctionstoaddressallaspectsofthisstandard.Oncestudentsarefamiliarwiththeseoperationsindividually,theyshouldbeaskedtodistinguishthemfromeachanother.Asstudentsgainexperience,thereareadditionalstrategiesthatshouldbeintroduced.Oneexperiencedproblemsolverstrategyistoreadthequestiontwicebeforebeginning.It’sausefulpieceofadviceingeneral,actually.Writingalistofwhatisknownandalistofwhatneedstobecalculatedisalsoanexcellentstrategy.Suchlistsareespeciallyusefulwhensortingoutunnecessaryinformation,identifyinganappropriateformulatoutilize,orconstructingaproof.Thesestrategiesshouldbesuggestedandshowntostudentsaftertheyareproficientwiththebasictranslationprocess.Tostartoff,thechartbelowmaybepresentedasadictionarytosupportwordtosymboltranslation.Studentscanalsoaddtothechartastheyfindotherkeywordsorphrases.
AlgebraSymbols KeyWords=equals 1.all
2.equals3.gives4.is,are,was,were,willbe5.results6.same7.yields
<islessthan 1.below2.lessthan
≤islessthanorequalto
1.maximumof2.notmorethan
>isgreaterthan 1.greaterthan2.morethan3.over
≥isgreaterthanorequalto
1.atleast2.minimumof3.notlessthan
+addition 1.add2.and3.combine4.increase5.more6.plus7.raise8.sum9.together10.total
–subtraction 1.decrease2.difference3.fewer4.less5.lose6.minus7.reduce
xmultiplication 1.directlyproportional2.double(x2),triple(x3),etc.3.groupof4.linear5.multiplied6.product7.times
/division 1.average2.cut3.dividedby/into4.each5.inverselyproportional6.outof7.per8.pieces9.quotient10.ratio12.share13.split
𝑥!power 1.power2.square(n=2),cube(n=3),etc.
𝑛!exponential 1.decays2.doubles(n=2),triples(n=3),quadruples(n=4),etc.3.grows4.rateofnperx
MAFS.912.A-CED.1.2AlsoassessesMAFS.912.A-REI.3.5Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.MAFS.912.A-REI.3.6Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.MAFS.912.A-REI.4.12Graphthesolutionstoalinearinequalityintwovariablesasahalf-plane(excludingtheboundaryinthecaseofastrictinequality),andgraphthesolutionsettoasystemoflinearinequalitiesintwovariablesastheintersectionofthecorrespondinghalf-planes.ITEMSSPECIFICATIONS:ItemTypesEditingTaskChoice–Mayrequirechoosingthecorrectdefinitionofavariableorcompletinganexplanationoraproof.EquationEditor–Mayrequirecreatingasetofequations,creatingasetofinequalities,orgivinganorderedpair.GRID–Mayrequiregraphingarepresentationofasetofequations,asetofinequalities,oranorderedpair;selectingasolutionregion;ordragginganddroppingtexttocompleteaproof.HotText–Mayrequireselectingasolutionordragginganddroppingtexttocompleteaproof.MultipleChoice–Mayrequireidentifyingasetofequations,asetofinequalities,avalue,anorderedpair,oragraph.Multiselect–Mayrequireidentifyingequationsorinequalities.OpenResponse–Mayrequirewritinganexplanation.ClarificationsStudentswillidentifythequantitiesinareal-worldsituationthatshouldberepresentedbydistinctvariables.Studentswillwriteasystemofequationsgivenareal-worldsituation.
MAFS.912.A-CED.1.2ContentComplexity:Level2:BasicApplicationofSkills&Concepts
Studentswillgraphasystemofequationsthatrepresentsareal-worldcontextusingappropriateaxislabelsandscale.Studentswillsolvesystemsoflinearequations.Studentswillprovidestepsinanalgebraicproofthatshowsoneequationbeingreplacedwithanothertofindasolutionforasystemofequations.Studentswillidentifysystemswhosesolutionswouldbethesamethroughexaminationofthecoefficients.Studentswillidentifythegraphthatrepresentsalinearinequality.Studentswillgraphalinearinequality.Studentswillidentifythesolutionsettoasystemofinequalities.Studentswillidentifyorderedpairsthatareinthesolutionsetofasystemofinequalities.Studentswillgraphthesolutionsettoasystemofinequalities.AssessmentLimitsItemsthatrequirethestudenttowriteasystemofequationsusingareal-worldcontextarelimitedtoasystemof2x2linearequationswithintegralcoefficientsiftheequationsarewrittenintheformAx+By=C.Itemsthatrequirethestudenttosolveasystemofequationsarelimitedtoasystemof2x2linearequationswithintegralcoefficientsiftheequationsarewrittenintheformAx+By=C.Itemsthatrequirethestudenttographasystemofequationsorinequalitiestofindthesolutionarelimitedtoa2x2system.Itemsthatrequirethestudenttowriteasystemofinequalitiesusingareal-worldcontextarelimitedtointegercoefficients.StimulusAttributesItemsassessingA-CED.1.2mustbeplacedinareal-worldcontext.ItemsassessingA-REI.3.5mustbeplacedinamathematicalcontext.ItemsassessingA-REI.3.6andA-REI.4.12maybesetinareal-worldormathematicalcontext.Itemsmayresultininfinitelymanysolutionsornosolution.ResponseAttributesItemsmayrequirethestudenttochooseanappropriatelevelofaccuracy.Itemsmayrequirethestudenttochooseandinterpretthescaleinagraph.Itemsmayrequirethestudenttochooseandinterpretunits.ForA-CED.1.2itemsmayrequirethestudenttoapplythebasicmodelingcycle.CalculatorNeutral
STANDARDDECONSTRUCTION:Thisstandardhastwosignificantcomponents.Thefirstistranslatingwordproblemsintoequationswithtwoormorevariables.Themorethemerrier.Well,maybenotinthiscase.Translatingwordproblemstocreatesimpleequationswithtwoormorevariablesisnotthatdifferentconceptuallyfromcreatingequationswithonevariable.Themaindifferenceisthatmorecomplicatedmathematicalrelationshipssuchassystemsofequations,functions,andproportionsmaydevelop(alongwithnausea,headaches,andspontaneousyodeling).Inanycase,thisaspectofthisstandardshouldbetaughtwiththepreviousone.Thesecondcomponentiscreatinggraphsofequationsoncoordinateaxes,whichincorporatesmultipleskillssuchasvisualperception,interpretingdata,andsynthesizinginformation.Suchgraphsrelatetoequationswithmultipleequationsbyrelatingonevariabletoanother.Takelines,forexample.Intheformy=mx+b,wecanlookateitherxoryandanydefinedvalueforxwillgiveusadefinedvaluefory,andviceversa.Graphscanhelpvisualizetheserelationshipsbetweenvariablesandfacilitatetheconnectionofequationstothegraphsthatrepresentthem.Yearnin’formoregraphin’?Don’tworry.There’llbemoredowntheline.
MAFS.912.A-CED.1.4AssessedwithinA-CED.1.1STANDARDDECONSTRUCTION:Studentscanrearrangeformulasuntiltheirpencilswhittledowntotoothpicks,butitwon’thelpthemonebitiftheformulastheyusewon’tenablethemactuallysolvetheproblem.Thatmeansstudentsshouldbeabletomatchcommonlyencounteredformulastocontextinwordproblemsaswellasrearrangingthemtosolveforwhatevervaluetheywant.MatchingFormulastoCreateEquationsInadditiontobeingabletotranslatewordproblemsintoequations,studentsalsoneedtobeabletoidentifywhenacommonformulaisneededforthegivencontext.Thesearemostcommonlygeometricformulas(likeperimeter,area,orvolumeofvariousshapes)orphysicalformulas(suchasF=ma,p=mv,V=IR,v=d⁄t,KE=.mv2,orGPE=mgh).Itisimportanttonotethatthisassumesthatstudentsarealreadyfamiliarwiththerelevantformulasfrompreviouslearning.Studentswhoarenotalreadyfamiliarwiththeformulasneedtobesupportedinunderstandingthembeforetheywillbeabletomatchthemtocontexts.Matchingformulasmaybetreatedasavariationfromthecreatingequationsprocess.Whenattemptingtowritedowntheequalityorinequality,studentswillnoticethatthereisn’tenoughinformation.Ageneralrelationshipmightbeimplied,butnospecificequalityorinequalityisdescribed.Whateverwilltheydo?Well,studentsneedtoidentifytheformulathatdescribesthatrelationship.Theycanlookforcluessuchasappropriateunits(volume,forexample,isalwaysincubicunits).Oncethecorrectformulaismatchedtothecontextualrelationship,thestudentscancontinuesolvingtheproblemasusual.RearrangingFormulasOnceaformulaiswrittenalgebraically,studentscanmanipulateithowevertheywant(withinmathematicalreason).
MAFS.912.A-CED.1.4ContentComplexity:Level1:Recall
Theprocessforrearrangingitisidenticaltotheprocessofrearranginganyequationorsystemofequations.Itinvolvessimplifyingexpressionsandsolvingequations,whichstudentsshouldalreadyknowhowtodo.Thebestwaytorearrangequestionswhenlookingforaparticularvalueistoisolatethatvalue.Studentsshouldrearrangetheequationssothatthedesiredvalueisononesideoftheequalsign,andthere’sawholebigmessofstuffontheother.Thatway,they’llbeabletopluginthevaluestheyknowandendupwithwhatevertheywantequalssomenumber.
MAFS.912.A.REI.1.1ITEMSSPECIFICATIONS:ItemTypesEditingTaskChoice–Mayrequirechoosingthenextstepinasolutionmethod.EquationEditor–Mayrequirecreatinganexpressionorvalue.GRID–Mayrequiredragginganddroppingsteps,equations,and/orjustificationstocreateaviableargument.HotText–Mayrequirerearrangingequationsorjustifications.MultipleChoice–Mayrequireidentifyingexpressions,statements,orvalues.OpenResponse–Mayrequirecreatingawrittenresponse.ClarificationsStudentswillcompleteanalgebraicproofofsolvingalinearequation.Studentswillconstructaviableargumenttojustifyasolutionmethod.AssessmentLimitItemswillnotrequirethestudenttorecallnamesofpropertiesfrommemory.StimulusAttributesItemsshouldbesetinamathematicalcontext.Itemsmayusefunctionnotation.Itemsshouldbelinearequationsintheformofax+b=c,a(bx+c)=d,ax+b=cx+d,ora(bx+c)=d(ex+f),wherea,b,c,d,e,andfarerationalnumbers.Equationsmaybegiveninformsthatareequivalenttothese.Coefficientsmaybearationalnumberoravariablethatrepresentsanyrealnumber.Itemsshouldnotrequiremorethanfourproceduralstepstoreachasolution.ResponseAttributesItemsmayaskthestudenttocompletestepsinaviableargument.Itemsshouldnotaskthestudenttoprovidethesolution.Calculator
MAFS.912.A.REI.1.1Complexity:Level3:StrategicThinking&ComplexReasoning
No
STANDARDDECONSTRUCTION:Propertiesofoperationscanbeusedtochangeexpressionsoneithersideoftheequationtoequivalentexpressions.Inaddition,addingthesametermtobothsidesofanequationormultiplyingbothsidesbyanon-zeroconstantproducesanequationwiththesamesolutions.Otheroperations,suchassquaringbothsides,mayproduceequationsthathaveextraneoussolutions.Whowasitthatsaid,“Ajourneyofathousandmilesbeginswithasinglestep”?WasitSocrates?Confucius?BugsBunny?Whoeveritwas,theyprobablyweren’tthinkingaboutalgebrawhentheycoinedthatgem.Tomanystudents,solvinganalgebraicequationmayfeellikeajourneyofathousandmiles.Beforestudentsbeginwiththeirsinglestep,theyshouldprobablygetacompassorsomething.Studentsshouldbeabletofigureoutthelogicalnextstepinsolvinganequationusingthepreviousstep.Soundssimple,butittakesmorethanjustputtingonefootinfrontoftheother.Knowingthatwhateverisdonetoonesideoftheequationmustbedonetotheotherisagoodstart,butthatdoesn’ttellthemwhattodo.Ideally,studentsshouldnotbememorizingasetofrulesandproceduresandusingthemtosolveequations.Theyshouldunderstandhowthenextstepinsolvinganequationcanbelogicallyderivedfromthepreviousstep,but
thereisageneralformatforhowtogetstartedontheiralgebraicjourney.Whenwesolveanequation,it’susuallyagoodideatogetthevariablewewantononesideoftheequalsign.Then,wedowhatwecantosimplifyitasmuchaswecan.Forexample,let’ssolvetheequationx2–3x–7=2x+17.Subtracting2x+17frombothsidesgetsallourx’sononesideoftheequation.That’sagoodplacetostart.x2–3x–7–(2x+17)=2x+17–(2x+17)x2–5x–24=0Wealsotooktheopportunitytosimplifybycombiningliketerms.Sincetheequationwehaveisaquadraticequation,wecanfactoritintotheproductoftwolinearterms.(x–8)(x+3)=0Thisproductwillequal0wheneitherx–8orx+3isequalto0,sowecansolveitbysettingeachofthemto0.Asaresult,x=8and-3.Ifstudentsarereallystruggling,wesuggeststartingsimple.Givethemeasierlinearequationsandslowlyworkyourwaytomorecomplexones.Pointoutpatternsinequations,makesuretheyknowthequadraticformula,andremindthemofhelpfulfactoringtricks.Mostofall,getthemtopractice.It’sdifficulttotakeeventhefirststepofajourneyifyoucan’twalk.Afterenoughexercises,yourstudentsshouldbeabletotackleanythousand-milejourneyfasterthantheRoadRunner.
MAFS.912.A.REI.2.3AssessedwithinA-CED.1.1STANDARDDECONSTRUCTION:Withlinearequations,studentsshouldbeabletofindthesolutionorsolutionsthatmaketheequationtrue.Weusuallygetoneorseveralspecificanswerswithequations,butinequalitiessingaslightlydifferenttune.AsdifferentasUnderPressureandIceIceBaby.
MAFS.912.A.REI.2.3Complexity:Level2:BasicApplicationofSkills&Concepts
Studentsshouldfirstunderstandthedifferencebetweenanequationandinequality.Anequationusesthe=signwhileanequalitymayuse<,>,≤,or≥.Ifwefindthatx≤2,weknowthatxcanbe2oranythinglessthan2.Ifweknowthatx<2,xcannotbe2,butitcanbeanythinglessthan2.Withinequalities,studentsshouldfindthesetofnumbersthatmaketheinequalitytrue.Inequalitieswon’ttellusexactlywhichnumberxwillequal.Instead,it’llgiveusarangeofpossiblexvalues,allofwhichwillworkfortheinequality.Studentsshouldalsoknowhowtoworkwithinequalities.Algebraically,theyaren’tthatdifferentfromanequalsign.Still,multiplyinganddividingbynegativenumbersswitchesthedirectionofthesign(1>-2butmultiplyingbothsidesby-1givesus-1<2).Ifstudentsareunsure,itmightbehelpfulforthemtovisualizeinequalitiesonanumberline.Sometimes,lettersmayrepresentconstantsandcoefficientsinequations.Studentsshouldknowhowtotreattheseasnumbers.Forinstance,theanswertotheequationx+4m=2x+mwouldbewrittenasx=3m.It’sokayforoursolutiontobeintermsofmbecausemistreatedasaconstant.Isn’tthatnice,nice,baby?Weapologizeinadvanceifthatbaselineisstuckinyourheadfortherestoftheday.
MAFS.912.A.REI.4.10AssessedwithinA-REI.4.11STANDARDDECONSTRUCTION:Studentsshouldunderstandthatequationswithtwovariablescanberepresentedgraphically.Theshapethatresultsonthecoordinateplaneisavisualrepresentationofallthesolutionstothatequation.Whatdoesthatmean?Itmeanswe’renotjustpullingrabbitsoutofhats!Theequationsactuallymeansomethingvisually.Anequationwithtwovariablescanbeanythingfromy=xtox2+y2=4to19x13=y.Somearesimplerthanothers,ofcourse,buttheyallhaveanxanday.Thatmeansinsteadofhavinganequationwithonevariable(andthereforeonesolution),wecanhavemany
MAFS.912.A.REI.4.10Complexity:Level1:Recall
differentsolutions.Graphically,wecanrepresentthesesolutionsbydrawingacurveorlinethroughallthepairsofsolutions(oneforxandonefory)thatworkforthatparticularequation.Let’staketheequation7x–18=yandseehowwecanrepresentthisgraphically.Howdoweprovetoastudentthatindeed,alineofatwo-variableequation,whengraphed,showsallofthesolutions?Let’sshowthemhowtopullthatrabbitoutofthehatthemselves.Sinceanytwopointsdefinealine,allweneedtodoisinputtwovaluesforxandseewhattheoutputyvaluesare.We’llpickthreepointsjusttobesureourgraphisalineandnotsomeweirdcurve.Let’spickthenumbers-1,0,and3forx.Plugginginthenumbersforxintoourequation7x–18=ygivesus-25,-18,and3fortheyvalues.Sothepointsinourgraphbecome(-1,-25),(0,-18),and(3,3).Ifwegraphtheseonthex-ycoordinateplane,we’llhavethis:
Ifyourstudentsdon’tbelieveyou,provetothemthattheequationandgraphcorrespondtooneanother.Takeapointonthelinethatiseasilyidentifiable,say(2,-4),andplugthevaluesintotheequation.Ifwedothat,we’llhave-4=7(2)–18,whichsimplifiesto-4=-4.Thatway,studentswillbesurethatpointsonthelineorcurvearevalidsolutionstotheequation,andviceversa.Butdon’tstopthere.It’salsoimportanttoprovetheopposite.Forexample,thecoordinate(4,1),whichisnotontheline,isalsonotasolutiontoourequation.Ifwepluginthecoordinates,wecanconfirmthis:1=7(4)–18isfalsebecause1≠10.Thismeans(4,1)isn’tasolutiontoourequationandnotapointontheline.Nowyoucanpatyourselfonthebackandprovetostudentsthatteachersaren’tjustuptosomemagictricks.Everythinginmathprettymuchworksasit’ssupposedto.Thismethodcanbeappliedtotwo-variableequationsofhigherorders.Thegenericshapesoftheseequations
(suchasquadraticsmakingaparabola)shouldbeknownandassociatedwitheachotheralready.Otherwise,studentswillneedtographseveralpointsbeforeverifyingthegraphthatcorrespondswiththeparticularequation.
MAFS.912.N-RN.1.1AssessedWithinN-RN.1.2MAFS.912.N-RN.1.2AlsoAssessesN-RN.2.3,N-RN.1.1ITEMSPECIFICATIONSItemTypeEditingTaskChoice–Mayrequirechoosingavalue,anexpression,orastatement.EquationEditor–Mayrequirecreatingavalueoranexpression.GRID–Mayrequireidentifyingpartsofanalgebraicproof.HotText–Mayrequiredragginganddroppingvalues,expressions,orexplanations.MatchingItem–Mayrequirematchingequivalentexpressions.MultipleChoice–Mayrequireselectingavalueoranexpressionfromalist.Multiselect–Mayrequireselectingmultiplevalues.OpenResponse–Mayrequireexplainingwhytworationalexponentexpressionsareequivalentorwhytwoexpressionsareequivalent.ClarificationsStudentswillusethepropertiesofexponentstorewritearadicalexpressionasanexpressionwitharationalexponent.Studentswillusethepropertiesofexponentstorewriteanexpressionwitharationalexponentasaradicalexpression.Studentswillapplythepropertiesofoperationsofintegerexponentstoexpressionswithrationalexponents.Studentswillapplythepropertiesofoperationsofintegerexponentstoradicalexpressions.Studentswillwritealgebraicproofsthatshowthatasumorproductoftworationalnumbersisrational;thatthesumofarationalnumberandanirrationalnumberisirrational;andthattheproductofanonzerorationalnumberandanirrationalnumberisirrational.
MAFS.912.N-RN.1.1Level2:BasicApplicationofSkills&ConceptsMAFS.912.N-RN.1.2ContentComplexity:Level1:Recall
AssessmentLimitsExpressionsshouldcontainnomorethanthreevariables.ForN-RN.1.2,itemsshouldnotrequirethestudenttodomorethantwooperations.StimulusAttributesItemsshouldbesetinamathematicalcontext.ResponseAttributesItemsmayrequirethestudenttocompleteanalgebraicproof.Itemsmayrequirethestudenttodetermineequivalentexpressionsorequations.Responseswithsquarerootsshouldrequirethestudenttorewritethesquarerootsothattheradicandhasnosquarefactors.CalculatorNo
MAFS.912.N-RN.2.3AssessedWithinN-RN.1.2
MAFS.912.N-RN.2.3Level2:BasicApplicationofSkills&Concepts
MAFS.912.A-SSE.1.1AssessedwithinA-SSE.2.3STANDARDDECONSTRUCTION:Studentsshouldunderstandthevocabularyforthepartsthatmakeupthewholeexpressionandbeabletoidentifythosepartsandinterprettheirmeaningintermsofacontext.Theonlywaythatcouldbemoregeneralisifitsaid,“Dothingstothingsintermsofotherthings.”Luckily,wehavejustasmidgeonmoretoworkwith.Atitscore,thisstandardwantsstudentstostartthinkingofmathasalanguage,notapileofnumbers.Justlikeanyotherlanguage,mathcanhelpuscommunicatethoughtsandideaswitheachother,butstudentsneedtoknowthebasicsbeforetheycanreallyunderstandit.Yourstudentsprobablyalreadyhavesomeideaofwhatanexpressionisinageneralsense.Startfromthispoint.Atitssimplest,anexpressionisathoughtorideacommunicatedbylanguage.Inthesameway,amathematicalexpressioncanbeconsideredamathematicalthoughtorideacommunicatedbythelanguageofmathematics.Emphasizethatmathematicsisalanguage,justasEnglish,French,German,andPigLatinarelanguages.Studentsshouldusetheocabulary-vayofathematics-maycorrectlytobecomefluentinit.Afterall,thebestwaytolearnanewlanguageistoimmerseyourselfinit.(a.)Example.JustlikeEnglishhasnouns,verbs,andadjectives,mathematicshasterms,factors,andcoefficients.Well,sortof.Studentsshouldknowthattermsarethepiecesoftheexpressionthatareseparatedbyplusorminussigns,exceptwhenthosesignsarewithingroupingsymbolslikeparentheses,brackets,curlybraces,orabsolutevaluebars.Everymathematicalexpressionhasatleastoneterm.Forinstance,theexpression3x+2hastwoterms:3xand2.Atermthathasnovariablesisoftencalledaconstantbecauseitneverchanges.Withineachterm,therecanbetwoormorefactors,thenumbersand/orvariablesmultipliedtogether.Theterm3xhastwofactors:3andx.Therearealwaysatleasttwofactors,thoughoneofthemmaybethenumber1,whichisn’tusuallywritten.Butthat1isalwaysthere...watchingus.Finally,acoefficientisafactor(usuallynumeric)thatismultiplyingavariable.Usingtheexample,the3inthefirsttermisthecoefficientofthevariablex.
MAFS.912.A-SSE.1.1Complexity:Level2:BasicApplicationofSkills&Concepts
Theorderordegreeofamathematicalexpressionisthelargestsumoftheexponentsofthevariableswhentheexpressioniswrittenasasumofterms.Fortheexample3x+2,theorderis1,sincethevariablexinthefirsttermhasanexponentof1andtherearenoothertermswithvariables.Theexpression5x2–3x+2hasorder2,whereastheexpression3xy+5x2y3–7x+32y4hasorder5,becausetheexponentsofxandyinthesecondtermare2and3,respectively,and2+3=5.Noothertermhasahigherexponentsum.Nowthatwehaveourwords,wecanstartputtingthemtogetherandmakeexpressions.Agoodwaytoseeifstudentsreallyunderstandanexpressionlike3x+2istohavethemtranslatemathematicalexpressionsintoEnglishandviceversa.Forinstance,theexpression3x+2couldalsobewrittenas,“thesumof3timesanumberand2,”or,“2morethanthreetimesanumber.”Clearly,it’smucheasiertowritethemathematicalexpressionthantowriteitinEnglish(nottomentionPigLatin).Thetwoaredirectlyrelatedtoeachother,however,andstudentsshouldbeabletotranslatebackandforth.Atfirst,studentsmightwanttomakeuseofa“dictionary”likethetablebelowtohelpthemgofromonelanguagetotheother.
b).Example.Let’sconsideramorecomplexexpression:5x–(2–4y).InEnglish,thiscouldbestatedas“thedifference
between5timesanumberandthequantity‘4timesanothernumberlessthan2’.”That’samouthful,andthisexpressionisn’teventhatcomplex!Itshouldbeobviouswhywedomathinsymbolnotationnow:it’smucheasiertowrite.NoticehowtheEnglishexpressionmentioned“anumber”and“anothernumber.”Thisisacluethattwodifferentvariablesmustbeusedinthemathematicalexpression.Thesetwovariablesmightrepresenttwodifferentphysicalquantitiesinsomesituation,andtheexpressionshowshoweachquantitycontributestotheoverallbehavior.Howmanytermsareinthatexpression?Yourstudentswillprobablysaythree,butthereareonlytwothewaytheexpressioniswritten.Thetwotermsinsidetheparenthesesaretreatedasasinglething,sothefirsttermis5xandthesecondtermis-(2–4y).Sincethereisnonumberimmediatelyfollowingtheminussigninthesecondterm,weassumethenumberisactually1.Sothesecondtermcouldbewrittenas-1(2–4y).Thisshouldbeinterpretedasa-1multiplyingeverythinginsidetheparentheses.Ofcourse,wecanhaveexpressions,whichhaveevenmorevariables,iftherearemorechangingorunknownquantitiesinvolved.Forexample,thecompoundinterestexpressionP(1+r)nhasthreevariables:P,r,andn,eachrepresentingadifferentphysicalquantity.Aswritten,thisexpressionhasonlyoneterm,consistingoftwofactors,Pand(1+r)n.ThefirstfactordependsonlyonP,whiletheseconddependsonrandn.
MAFS.912.A-SSE.1.2AssessedWithinA-SSE.2.3
MAFS.912.A-SSE.1.2Level2:BasicApplicationofSkills&Concepts
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