NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

NCMath1–SystemsofEquationsandInequalities

NC2MLUNITBRIEF

GENERALIZINGRELATIONSHIPSACROSS6-8Since6thgrade,studentshavebeengeneralizingmathematicalrelationshipsusingvariables.Initially,studentsutilizevariablestocreateexpressionsfromverbaldescriptionsorequationsinwhichavariablerepresentsamissingvalue.Inmiddlegrades,studentsalsoseevariablesinequationswheretheycantrulyvary(likein𝒂 + 𝒃 = 𝒃 +𝒂)andvariablesdescribingrelationshipsinwhichthevalueofonedeterminesthevalueofanother(incovaryingrelationshipslike𝟑𝒙 + 𝟒𝒚 = 𝟏𝟖).

BUILDINGFROMNCMATH1UNITS1&2FollowingtheCollaborativePacingGuideforNCMath1,this5thunitonlinearsystemsissupportedbyandextendsthecontentoffunctionsandlinearrelationshipsinUnits1and2.Inthisunit,studentswillcreategraphicalrepresentationsoflinearequalitiesandinequalities,whilealsoconnectingto

thealgebraicorsymbolicrepresentations.Tasksthataskstudentstocreategraphsfromformulas,butalsoformulasfromgraphscansupportstudents’understandingofhowthesedistinctrepresentationsrelate.

BUILDINGASYSTEMFROMCONTEXTStudentswillbeaskedtocreatesystemsoflinearequationsandinequalitiesfromcontexts(NC.M1.A-CED.3)andwillcontinuethiscreationofsystemsastheyprogressthroughNCMath2andNCMath3.ThedistinctioninNCMath1isthatthefocusofisonlinearrelationshipswithinasystem.

WithinUnit5,studentswilltakeagivencontext,assignvariablestorepresentquantitiesexistinginthatcontext,andthencreatemathematicalequationsorinequalitiestorepresenthowthecontextdescribestherelationshipsthatexistbetweenthevariables.Thesystemthatiscreatedmodelsthecontextprovidedandallowsstudentstographicallyandsymbolicallyinvestigatefeasibleandoptimalvariablevalues.

Extendingtheunderstandingofasolutiontoasinglelinearequationorinequality,studentswillinvestigatesolutionstosystemsoflinearequationsorinequalities,orpointsthatsatisfyeveryrelationshipinacollectionofmorethanoneequationorinequality.

Asolutiontomanyschoolmathematicsproblemsisoftenasinglevalue,anumber.InthisunitoftheCollaborativePacingGuideforNCMath1studentsbuildontheirunderstandingofasolutiontoalinearequationoftwovariables,asanorderedpairthatsatisfiestheequationormakestheequationtrue.

AnessentialunderstandingofUnit5isthatthereexisitsaninfinitenumberofsolutionstolinearequationsandinequalities.Thus,students’workinlinearequationsandinequalitiesconnectstheAlgebraandFunctiondomains,byutilizingalgebraicandgraphicalrepresentationsofsolutions.

Tobeginthinkingaboutsolutionstoasystemconsiderthesolutionto5x+3=28.Graphtogethery=28andy=5x+3.Howdoesthesolutionofthatsystemrelatetothesolutionof5x+3=28?Theremaynotbeaninfinitienumberofsolutionstoasystemoflineareqautions,butcouldtherebe?

Agraphofalinearfunctioncanbeusedtounderstandgraphsoflinearinequalities.Thegraphofy=3x+2dividestheplaneinto3distinctparts:1. Theareawhere𝑦 > 3𝑥 + 2

2. Theareawhere𝑦 < 3𝑥 + 2

3. Thelinewhere𝑦 = 3𝑥 + 2

NorthCarolinaCollaborativeforMathematicsLearning(NC2ML)IUNCGSchoolofEducationINorthCarolinaDepartmentofPublicInstruction

StandardNC.M2.A-REI.5callsforstudentsto:Explainwhyreplacingoneequationinasystemoflinearequationsbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.

EXAMININGTHESTANDARDSANDINSTRUCTION

DespitetherelativelyshortlistofstandardsascomparedwithUnits1and2,theCollaborativePacingGuiderecommendsspendingampleinstructionaltimeonUnit5.Takingthisrecommendationprovidesopportunitiesforstudentstobuildflexibilityinconnectingandutilizingmulitplerepresentationsintheanalysisofasystem,whichrequiresconsideringmultipleequationsand/orinequalitiesandmultiplegraphs(NCTM,2014).Asthegraphofasystemmayplainlyshowasolutiontothesystem,“seeing”thesolutionwithintheformulasmaybemorechallenging.Studentsareexpectedtobeabletoalgebraicallymanipulatevariableswithinandacrossformulas,inordertofindsolutions.Thisskillmayseemsimpleenough,sincethesesystemswillonlyinvolvelinearrelationships,butstudentswhohavebeentaughttomanipulatevariableswithrotememorizationofstepsandwithoutopportunitiestoattachmeaningtosymbolsandrateofchangemaystruggle.SeetheNCMath1unitbriefforanexplanationofcovariationandrateofchange.

Insolvingasystem,wemakeexplicituseoftheequalitystatedwithinanequation.Forexample,consider𝒙 + 𝒚 =𝟓and𝒙 − 𝒚 = 𝟐.Thesymbol“=”guaranteesthateachpairofexpressionswithintheequationsarethesame,sothat(𝒙 + 𝒚) + (𝒙 − 𝒚) = 𝟓 + 𝟐.Whichalsomeansthatx=3.5.Further,since𝒙 + 𝒚 = 𝟓,then𝟐(𝒙 + 𝒚) = 𝟏𝟎.Theessentialunderstandingisthateachofthesemanipulationsbuildonthegivenequalities,sothatthegivenequalitiesarestilltrueandtherelationshipsdefinedbetweenthevariablesstillhold.Studentscanthengraphthesenewrelationshipstovisuallycomparehowthenewequationsrelatetotheoriginalsystem.TIMEFORTECHNOLOGYAnaffordanceofNCMath1Unitsarethemultipleopportunitiesteachershavetosupportstudentsingainingexperienceinbothgraphingfunctionsbyhandandbyusingtechnology.ThecontentofUnit5iswellsuitedtotheuseofagraphingutilityandmostutilitieshavethecapabilitytographinequalitiesaswell.Twoexamplesareprovidedbelow.

1) TIgraphingcalculator:presstheY= buttontoenterthefunctiontobegraphed,thenleft-arrowovertothe

backslashsymboltotheleftof“Y1”andpress ENTER toscrollthroughtheoptionsforchoosingtheshadingappropriatefortheinequalitydesired.

2) Desmos(desmos.com):choosetheappropriateinequalityonthekeypadorusethekeyboardasyoucreatethefunctiontoindicatetheinequality.

Researchhasshownthatstudentsmayhavedifficultiesconnectinggraphswiththephyscicalcontexttheyarerepresentingandthatstudentsneedampleopportunitiestodevelopproficiencyininterpretinggraphs(McDermottetal.,1987).Regardlessofthetechnologyutilized,providingstudentsopportunitiestousetechnologytographrelationshipswithmoreefficiencyallowsmoretimeforanalyzingsystemsincontextandcontinuetobuildingconceptualunderstandingofsystems,rateofchangeandcovariation.

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

NorthCarolinaCollaborativeforMathematicsLearningwww.nc2ml.orgRev.8/22/17

QUESTIONSTOCONSIDERConsidertheequation:

𝟏𝟎 = ½(𝒙 + 𝟒)

• Doyousee,“Halfofsomethingis10?”ordoyousee,“Ineedtodividebothsidesby½”?

• Whywoulditbeimportantforstudentstoseeboth?• Whatelsewouldyouanticipatestudentswouldseeinthisequation?

• Howcouldyouconnectastudents’guessandcheckapproachtosolvingasystemofequationstographicalapproach?Whywouldthisbeuseful?

References McDermott,L.,Rosenquist,M.L.,&vanZee,E.H.(1987).Studentsdifficulties

inconnectinggraphsandphysics:examplesfromkinematics.AmericanJournalofPhysics,55(6).

NationalCouncilofTeachersofMathematics.(2014).Principlestoactions:Ensuringmathematicssuccessforall.Reston,VA:NationalCouncilofTeachersofMathematics.