NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

NCMath3–Polynomials

NC2MLUNITBRIEF

ALGEBRA,FUNCTION,ANDNUMBER ThethirdunitoftheNCCollaborativePacingGuideextendsstudents’priorconceptionsoffunctionandalgebratopolynomialsofhigherdegreebyfocusingon:• buildingpolynomialfunctionsincontext,oftenbasedongeometricfigures(F-BF.1,G-GMD.3,G-MG.1,F.BF.1);

• keyfeaturesandrepresentationsbycomparingandtransformingfunctions(F.IF.4,.7,.9;F.BF.3);

• connectingtheconceptsofrootsofapolynomialandzerosofafunctionbyapplyingtheFundamentalTheoremofAlgebra(N-CN.9);and

• combiningalgebraicmethodsofsolvingequationswithbroadgeneralizationsabouttheexistenceofsolutionsbyapplyingrelevanttheoremsandalgorithms(A-APR.2,.3;N-CN.9).

BUILDINGFROMNCMATH2InNCMath2,studentsengageinanin-depthlookatquadraticfunctionswithrealcoefficients.Theyconnectalgebraicsolutionstographicalintercepts,andgeneralizetheshapesofquadratics’graphsbasedonthestructureintheirdifferentforms(standard,factored,vertex).

Additionally,studentsbuildquadraticformulasfromcontext,generalizingpatternsthatinvolveaproductoftwobinomials(hencetheresultingx2term).InNCMath3,thisworkisextendedtohigherdegreepolynomials,therefore,teacherscouldbeginthisunitwithathree-dimensionalgrowingpatternormodelgeometricfiguresthatgeneralizetoacubicpolynomial(e.g.PaintedCube).

InNCMath2,studentsdevelopanunderstandingofcomplexnumbersasawaytoexpresssolutionstoquadraticequationsthatdonotproducex-interceptsforthefunction.Itisimportantinthisunitthatteacherscontinuetobuildstudents’understandingofrealnumbers-complexnumbersthatdonothaveanimaginaryfactor(e.g.rationalnumbers,p, 2)andtheirsuperset,complexnumbers–

numberswithrealand/orimaginaryfactors(e.g.𝑖, %

&, 1,

5 + 2𝑖)tosupportstudentsinundestandingandapplyingtheFundamentalTheoremofAlgebratodeterminethenumberandtypesofsolutionsforpolynomialfunctions.

THEFUNDAMENTALTHEOREMOFALGEBRAWait,wait,wait,…atheoremthatisfundamentaltoAlgebra,butwearejustnowfindingoutaboutit?Thelabelof“fundamental”,inthiscase,isnottobesynonymouswith“basic”.Thetheoremprovidestheassurancethatthenumberofcomplexrootsofanon-linearpolynomialwillequalthedegreeofthepolynomial,ifyouincludemultiplicitywhencountingroots.Thetheoremisalsoanexampleofwhat’sknownasanexistenceproof.Thatis,itguaranteestherootsexist,withouttellingyouwhattheyareorhowtofindthem.AswithTheFundamentalTheoremofArithmetic(everyintegergreaterthan1hasauniqueprimefactorization)andTheFundamentalTheoremofCalculus(ifacontinuousfunctionhasanantiderivative,theantiderivativecanbeusedtocalculateintegralvalues)studentscanusetheresultofTheFundamentalTheoremofAlgebra,withoutknowingaproofofthetheorem.DISCUSSWITHYOURCOLLEAGUESHowdoyouknowthistheoremistrue?Whatconvincesyou?Ifyouarestillaskeptic,hereisalinktoaNumberphileVideoproofasaplacetostart.

NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

INVESTIGATINGPOLYNOMIALFUNCTIONS

NCMath1,2,and3allcontainF-IF.7andF-IF.9standards,whichappearinfiveoftheeightunitsinNCMath3.Thesestandardsrequirestudentstoanalyzethecharacteristicsofdomain,range,intervalsofincreaseanddecrease,extremevalues,endbehavior,andcomparefunctionsbytheirkeyfeaturesasfoundindifferentrepresentations.

Focusingongraphicalrepresentations,studentscangeneralizethepossibleshapesofthegraphsofpolynomialfunctionsbasedsolelyonthedegreeofthepolynomialandthesignvalueoftheleadingcoefficient.Knowingtherootsofthepolynomialandtheirmultiplicitieswillhelpstudentsbetterlocatetheirsketchofthegraphofthepolynomialfunctionwithinthecoordinateplane.

Usingagraphingutilityisagreatwaytoefficientlycreatemultiplegraphssothatstudentscanlookforpatternsinordertogeneralizeshape.TheonlinecalculatorDesmoshostsaTeacherDesmossiteofferinglessonsthatareclassroomready(visittheTeacherDesmossiteandsearchforpolynomials).

Researchhasshownthatwhenexamininggraphicalrepresentations,studentsmayhavedifficultyconnectinggraphstothecontexttheyrepresent(Piez&Voxman,1997).Additionally,studentsmayovergeneralizecharacteristicsofparametersacrossfunctionfamilieswhenengagingwithsymbolicrepresentationsandhavedifficultyconnectingkeyfeaturesoffunctionsrecognizedintablestootherrepresentations(Wilson,1994).Therefore,ininstruction,itisimportanttoattendtomultiplerepresentationstosupportstudentsindevelopingstrongconnectionsacrossrepresentationsandtheirrelationshipstothecontextoftheproblem.

CONNECTINGTOSTUDENTS’PRIORUNDERSTANDING

Thedevelopmentofstandardalgorithmsforarithmeticareusedbasedontheirefficiencyanditisagoalthatbytheendof6thgrade,NorthCarolinastudentscanfluentlydivideusingthestandardalgorithmfordivision.Researchhasshownthatwhenstudentsengageinarithmeticusingonlystandardalgorithmstheymaylacktheunderlyingconceptsofbase-10numbersandrelationshipsbetweenmultiplicationanddivision(Ambroseetal.,2003).

It’simportanttorememberthatstudentsmayenterourclassroominNCMath3withunrefinedconceptionsofdivisionthatmayaffectthewaysinwhichtheyengagewithpolynomialdivisionorsyntheticdivision(A-APR.2,.3).Teacherscansupportstudentsinbuildingtowardanunderstandingofwhatpolynomialdivisionisbycarefullysequencingwell-craftedsimpleexamplesthatconnecttothinkingaboutfractions.

Teacherscanalsocraftexamplesthatbuilduponstudents’understandingoftheinverserelationshipbetweentheoperationsofmultiplicationanddivision.

References Ambrose,R.,Baek,J.-M.&Carpenter,T.P.(2003).Children'sinventionof

multiplicationanddivisionalgorithms.InA.Baroody&A.Dowker(Eds.),Thedevelopmentofarithmeticconceptsandskills:Recentresearchandtheory.Mahwah,NJ:Erlbaum.

Piez,C.M.,&Voxman,M.H.(1997).Multiplerepresentations—Usingdifferentperspectivestoformaclearerpicture.TheMathematicsTeacher,90(2),164-166.

Wilson,M.R.(1994).Onepreservicesecondaryteacher'sunderstandingoffunction:Theimpactofacourseintegratingmathematicalcontentandpedagogy.JournalforResearchinMathematicsEducation,346-370.

DISCUSSWITHYOURCOLLEAGUES• Howcouldyouuseequivalenciesto+

,,%,, 𝑎𝑛𝑑 +

+

tomakesenseof0120,032

0 , 𝑎𝑛𝑑 0

4120412

?

• Beforewedivide,whichofthesequotientsshouldbegreaterthan1?Howdoyouknow?

𝑥 + 2𝑥 − 1

𝑥7 − 1𝑥7

𝑥7 + 4𝑥 + 4(𝑥 + 2)7

𝑥: − 1𝑥 − 1

DISCUSSWITHYOURCOLLEAGUES• Knowingthat𝑥7 − 𝑥 − 6 = (𝑥 − 3)(𝑥 + 2),what

isthequotientof04303>017

?

• Since7043?032

= 2𝑥 + 2,thentheproduct

(𝑥 − 1)(2𝑥 + 2)wouldbeequalto_________.

• Whatinstructionaldecisionscanyoumakeinthisunittoensurestudentsdevelopbothaproceduralandconceptualunderstandingofpolynomials?

• Howdotheexamplesprovidedinthisbriefsupportyouinunderstandingwaysyoucansupportyourstudents?

LEARNMOREJoinusaswejourneytogethertosupportteachersandleadersinimplementingmathematicsinstructionthatmeetsneedsofNorthCarolinastudents.

NC2MLMATHEMATICSONLINEFormoreinformationandresourcespleasevisittheNCDPImathwikiforinstructionsonaccessingourCanvaspagecreatedinpartnershipwiththeNorthCarolinaDepartmentofPublicInstructionbyhttp://maccss.ncdpi.wikispaces.net/

NorthCarolinaCollaborativeforMathematicsLearningwww.nc2ml.orgRev.6/15/17