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10 CC BY https://flic.kr/p/ayDEGr SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.3 4. 3 Solving Equations Literally A Practice Understanding Task Solveeachofthefollowingequationsforx: 1. 3x + 2 =7 5 2. 3x + 2y =7 5 3. 4x − 5 = 11 3 4. 4x − 5y = 11 3 5. 2 (x + 3) = 6 5 6. 2 (x + y) = 6 5 7. 2(3x + 4) = 4 x + 12 8. 2(3x + 4 y) = 4 x + 12y Writeaverbaldescriptionforeachstepoftheequationsolvingprocessusedtosolvethefollowing equationsforx.Yourdescriptionshouldincludestatementsabouthowyouknowwhattodonext. Forexample,youmightwrite,“FirstI__________________,because_______________________...” 9. ax + b −d=e c Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 10. r⋅ mx +s=t n SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.3 4. 3 Solving Equations Literally – Teacher Notes A Practice Understanding Task Purpose:Thistaskprovidespracticeforsolvinglinearequationsinonevariable,solvinglinear equationsintwovariablesforoneofitsvariables,andsolvingliteralequations.Theprocessfor solvingmultivariableequationsforoneofitsvariablesbecomesmoreapparentwhenjuxtaposed withsimilarly-formattedequationsinonevariable.Theonlydifferenceinthesolutionprocessis theabilitytocarryoutnumericalcomputationstosimplifytheexpressionsintheone-variable equations. CoreStandardsFocus: A.REI.1Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbers assertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution. Constructaviableargumenttojustifyasolutionmethod. A.REI.3Solvelinearequationsandinequalitiesinonevariable,includingequationswith coefficientsrepresentedbyletters. RelatedStandards: StandardsforMathematicalPracticeofFocusintheTask: SMP7–Lookandmakeuseofstructure SMP8-Lookforandexpressregularityinrepeatedreasoning AdditionalResourcesforTeachers: Ananswerkeyforthequestionsinthetaskcanbefoundasaseparatepageattheendofthese teachernotes.Itisrecommendedthatyouworkthroughthetaskyourselfbeforeconsultingthe answerkeytodevelopabettersenseofhowyourstudentsmightengageinthetask. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.3 TheTeachingCycle: Launch(WholeClass): Encouragestudentstonotethesimilaritiesintheirworkonthepairsofproblemsinquestions1-8. Alsopointoutthattheyaretowritedetailedexplanationsoftheirsolutionstrategyonproblems9 and10.Onewaytofacilitatethiswouldbetohavestudentsfoldapieceofpaperinhalflengthwise. Ontheleftsideofthepapertheywriteouttheiralgebrasteps,andontherightsidetheywriteout theirjustifications. Explore(SmallGroup): Monitorstudentswhileworkingontheseproblemsandofferappropriatefeedback,asnecessary. Someoftheproblemshavealternativestrategies,suchas#5whereyoucandistributethe2/5first, ormultiplybothsidesoftheequationsby5/2first. Helpstudentsrecognizethedifferencebetweenchangingtheformofanexpressionononesideof anequation,vs.writinganequivalentequationbyapplyingthesameoperationtobothsides. Ifstudentsarehavingdifficultieswith#9or#10,havethemwritearelatedequationinwhichthey replaceallletterswithnumbersexceptforthex.Seeiftheycansolvetherelatedequationforxand ifthatworkcanhelpthemsolvetheoriginalliteralequation.Problem10involvesasquarerootin ordertoemphasizethatoneofthekeyissuesinsolvinganequationisto“un-do”anoperationby applyingtheinverseoperationtobothsides.Helpstudentsthinkabouthowthatwouldplayoutin problem10.Thatis,howmightthey“un-do”asquareroot? Discuss(WholeClass): Havestudentspresenttheirsolutionprocessforanyproblemsthatmayhavebeendifficultfora numberofstudents.Youmightalsowanttohavestudentscritiqueeachother’sexplanationson problems9and10byhavingstudentsexchangepapers.Theyshouldfoldtheirpartner’spaperin half,sothatonlytherightsidewiththewrittenexplanationisshowing.Onaseparatesheetof papertheyshouldwrite-outthealgebrastepstheywouldtaketosolveeachproblem,basedonlyon thewordingoftheirpartners’explanation.Theyshoulddiscussanyexplanationsthatareunclear withtheirpartner. AlignedReady,Set,Go:EquationsandInequalities4.3 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 11 SECONDARY MATH I // MODULE 4 4.3 SOLVING EQUATIONS AND INEQUALITIES – 4.3 READY, SET, GO! Name PeriodDate READY Topic:SolvingInequalities. Usetheinequality−9 < 2tocompleteeachrowinthetable. Applyeachoperationtotheoriginal Result Istheresulting inequality−9 < 2 inequalitytrueorfalse? Example:Add3tobothsides -9+3<2+3→-6<5 True 1.Subtract7frombothsides. 2.Add15tobothsides. 3.Add-10tobothsides. 4.Multiplybothsidesby10. 5.Dividebothsidesby5. 6.Multiplybothsidesby-6. 7.Dividebothsidesby-3. 8.Whatoperationswhenperformedonaninequality,reversetheinequality? (Beveryspecific!) SET Topic:Solveliteralequationsthatrequiremorethanonestep. Solvefortheindicatedvariable.Showyourwork!!! 9.Solveforh. ! = 25!ℎ 10.Solveforh. ! = !! ! ℎ 11.Solveform.! = 7! + 6 12.Solveform.! = !" + ! 13.Solveforz. ! = ! + 7 3 14.Solveforz. ! = ! + 7 ! 15.Solveforx. !!! ! !!!! = 4 16.Solveforx. − 9 = 6 18.Solveforx. !! 20.Solveforx. ! ! = 4 !! 17.Solveforx. ! ! − 9! = 6 19.Solveforx. ! ! ! − 2 = 12 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org ! ! − 2! = 12 12 SECONDARY MATH I // MODULE 4 4.3 SOLVING EQUATIONS AND INEQUALITIES – 4.3 GO Topic:Identifyingx-interceptsandy-intercepts Locatethex-interceptandy-interceptinthetable.Writeeachasanorderedpair. 21. 22. 23. ! ! ! ! ! ! -4 12 0 -6 -3 10 -3 10 3 -5 -2 8 -2 8 6 -4 -1 6 -1 6 9 -3 0 4 0 4 12 -2 1 2 1 2 15 -1 2 0 2 0 18 0 3 -2 x–intercept: x–intercept: x–intercept: y–intercept: y–intercept: y–intercept: Locatethex-interceptandthey-interceptinthegraph.Writeeachasanorderedpair. 24. 25. x–intercept: x–intercept: y–intercept: y–intercept: Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 13 SECONDARY MATH I // MODULE 4 4.3 SOLVING EQUATIONS AND INEQUALITIES – 4.3 Solveeachequationforx.Providethejustificationsforeachstep.Seethefirstexampleasa reminderforthetypesofjustificationsthatmightbeused. Example: 26. 4x–10=2 Justification 3x–6=15 Justification +6+6 AdditionProperty ofequality 3x=21 DivisionProperty 33 ofequality x=7 27. 28. –16=3x+11 Justification 6–2x=10 Justification 29. 6x+3=x+18 30. Justification 3x–10=2x+12 Justification 31. 32. X(B+7)=9 Justification 12x+3y=15 Justification Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org