NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

NCMath1–QuadraticFunctions

NC2ML UNIT BRIEF

NCMATH1–UNIT4,QUADRATICFUNCTIONSInNCMath1,studentsbegintobuildanunderstandingofquadraticfunctionsbyexaminingkeyfeaturesandcoefficientstofactorquadraticexpressionsandsolvequadraticequationsincontext.InNCMath2,thestandardsextendthisworktofocuson:creatingequationsdescribingquadraticrelationships;analyzingandinterpretingequationsbymakingconnectionsacrossrepresentations;comparingandtransformingquadraticrelationships,andintroducingcomplexnumbers.

ThedistinctionbetweenQuadraticswithinthetwocoursesisnotintheirblendofAlgebraandFunction.Bothsetsofcoursestandardscallforconnectingalgebraicandgraphicalrepresentations.SincetheNCMathStandardsfirstintroducequadraticfunctionsinNCMath1(the4thunitofCollaborativePacingGuide),studentsinthiscourseareexposedtoquadraticsrequiringlesssophisticatedalgebraicmaneuveringwhencomparedtothoseinNCMath2.InNCMath1,studentsrewrite“factorable”quadraticexpressionsandsolveforrealroots,consideringthemaszerostothecorrespondingfunctions.InNCMath2studentswillengagewithmoresophisticatedalgebraictechniquesincludingcompletingthesquareandthequadraticformula,whilecontinuingtobeabletojustifytheirwork.

BUILDINGFROM8thGRADEWhendecidinghowtobeginthisunit,itmaybeusefultostartwithataskthatisaccessibleusingapproachesthatbuilduponmathematicslearningin8thgrade.Growingpatterntasksinwhichstudentsmustidentifyapattern,generalizethepattern,andgraphtherelationshipcanbe

utilizedtoaddressnumerousstandardswithinthisunit(e.g.NC.M1.A.SSE.1;F.BF.1b;A.CED.1;A.CED.2)andcanhelpbuilduponstudentsgrowingunderstandingofrateofchangeandfunctionfamilies.

https://www.illustrativemathematics.org/content-standards/tasks/75InNCMath1,studentsengagefirstwithlinearandexponentialfunctions,buildinganunderstandingoftheirrespectiveadditiveandmultiplicativeratesofchange.Thatis,linearfunctionscanbecharacterizedbyaconstantrateofchange,andexponentialfunctionsbyarateofchangethatisproportionaltothevalueofthefunction.Whilenotexplicitinthestandards,manyadvocatethatteacherscansupportstudentsinextendingtheirunderstandingofquadraticsbydesigninginstructionalopportunitiesthathighlightthatquadraticshavealinearrateofchange(NCTM,2014).Forexample,inthegrowingtowerproblemabove,byexaminingdifferentrepresentations,studentsmayrecognizethatthepatternhasarateofchange(orfirstdifference)thatchangeslinearlyandaseconddifferencethatisconstant.Researchhasshownthatstudentsareindeedabletorecognizetheaveragerateofchangeover

NC.M1.A-REI.4:Solvefortherealsolutionsofquadraticequationsinonevariablebytakingsquarerootsandfactoring.

NC.M2.A-REI.4:Understandthatthequadraticformulaisthegeneralizationofsolving𝑎𝑥# + 𝑏𝑥 + 𝑐byusingtheprocessofcompletingthesquare.

NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

equalintervalsofaquadraticfunctionaslinear(Lobatoetal.,2012).

STANDARDANDFACTOREDFORMNCMath1studentsarerequiredtoonlyengagewiththestandardandfactoredformsofaquadraticexpression.Algebraisatoolthatstudentswillusetofactorquadraticexpressionstodeterminethezerosofthecorrespondingquadraticfunction(A-SSE.3),themainpurposeforfactoringanypolynomial.Studentswillneedtobeabletojustifytheirprocessofsolvingquadraticsusingmathematicalreasoning(A.REI.1).Whilemany“tricks”havebecomepopularforteachingfactoringandmayprovideshort-termresults,itisimportanttodevelopstudentsconceptualunderstandingoffactoring.Boaler(1998)foundthatstudentswhoviewmathematicsaslearningandusingrulesstruggledtosolveproblemswhenpresentedwithnewanddifferentsituations.Usinggeometricrepresentationsofmultiplyingbinomialsandfactoringtrinomialsusingalgebratiles(realorvirtual)canhelpstudentsmakesenseoftheprocessoffactoringinordertogivethemafoundationforfactoringalgebraicexpressionsthataremorecomplicated.Anotherimportantconnectiontomakewithstudentslearningtofactoristhelinkbetweenfactoringbygroupingandthedistributiveproperty.Usingthe"boxmethod"tobothmultiplyalgebraicexpressionsandfactoralgebraicexpressionsandconnectingittothegeometricrepresentationsaffordedbyalgebratilesisonewaytodrawattentiontothisconnectionbetweenfactoringanddistributing.

UTILIZINGTECHNOLOGYMajorworkofthefirst5unitsinNCMath1involvesconnectingvariousrepresentationsofmathematicalrelationships:verbaldescriptions,tablesoforderedpairs,algebraicexpressions,andgraphs(thiscanbeseenintherepetitionofstandardsacrosstheseunitsasthefunctiontypeschange).Technologyallowsstudentstoquicklyproducegraphicalrepresentations,makingtheworkofF-IF.7,8,and9lessdaunting.Withinthesestandardsstudentswillgraphsimplecasesbyhandbutthenusetechnologytoanalyzekeygraphicalfeaturesandcomparethosefeatureswithindifferentquadratics.TeacherDesmosoffersseveralvarietiesofgraphingtasks,includingaquadraticsbundleof8tasksthatbeginswithanintroductiontoquadraticsandendswiththeusefulnessofvariousquadraticforms.KokluandTopcu(2012)summarizedseveralstudiesofstudentthinkingaboutquadraticsbyidentifyingpartialunderstandingsstudents’sometimeshaveaboutquadraticfunctions.Forexample,students’maynotrealizethatthecoefficientcaffectsthegraphof𝑎𝑥# + 𝑏𝑥 + 𝑐.Also,somestudentsmaythinkoneoftheparametersofa,b,orc,isthe“slope”oftheparabola.Theyfoundthatusingdynamicprograms(likeGeogebraorDesmos)canbeapowerfulwaytocreateopportunitiesforstudentstovisualizechangesin

quadraticfunctionssincesuchsoftwareallowsstudentstoexplorehowtheparametersaffectthegraph(Koklu&Topcu,2012).

SOLUTIONS–ROOTS,ZEROS,&COMPLEXNUMBERSArootofapolynomialisavaluethatmakesthepolynomialexpressionequalto0.Azeroofapolynomialfunctionistheinputthatoutputsafunctionvalueof0,thusazeroofafunctionisthex-coordinateofanx-interceptonthegraphofthefunction.Verysimilar,yetnotexactlythesamething!Everyquadraticexpressionhasatmosttworoots.Butit’snotthecasethateveryquadraticfunctionhaszeros.Explorationsintonon-realrootsisnotcalledforinthiscourse,butknowingthisdifferenceisimportant.StudentswilllearnaboutthecomplexnumbersystemandquadraticfunctionsthatdonothaverealnumbersolutionsinNCMath2.

LEARN MORE Join us as we journey together to support teachers and leaders in implementing mathematics instruction that meets needs of North Carolina students. NC2ML MATHEMATICS ONLINE For more information and resources please visit the NC DPI math wiki for instructions on accessing our Canvas page created in partnership with the North Carolina Department of Public Instruction by http://maccss.ncdpi.wikispaces.net/

North Carolina Collaborative for Mathematics Learning www.nc2ml.org

References Boaler, J. (1998). Open and closed mathematics: Student experiences and

understandings. Journal for research in mathematics education, 41-62. Koklu, O., & Topcu, A. (2012). Effect of Cabri-assisted instruction on

secondary school students’ misconceptions about graphs of quadratic functions. International Journal of Mathematical Education in Science and Technology, 43(8), 999-1011.

Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Mathematical Thinking and Learning, 14(2), 85-119.

National Council of Teachers of Mathematics (NCTM). (2014). Putting Essential Understanding of Functions into Practice 9-12. Reston, VA: Author.

QUESTIONSTOCONSIDER• Whatwordsdoweusewhendescribingx-intercepts?Are

weprecisewhenweinterchangethevocabularybetweenx-intercepts(whentalkingaboutagraphedparabola),roots,solutions&zeroswhentalkingaboutanequation?

• Factoringisknownasthealgebraicskillthatjustwon'tstickinhighschool.Whydoyouthinkthatis?Howmightweavoidthatconsequence?

• Onekeyexampleofquadraticsisprojectilemotion.Whatwoulda,bandcrepresentinthatcontext?

Rev. 8/22/2017