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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