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14 SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.4 A Develop Understanding Task Foreachsituationyouaregivenamathematicalstatement andtwoexpressionsbeneathit. 1. Decidewhichofthetwoexpressionsisgreater,iftheexpressionsareequal,orifthe relationshipcannotbedeterminedfromthestatement. 2. Writeanequationorinequalitythatshowsyouranswer. 3. Explainwhyyouransweriscorrect. Watchout—thisgetstricky! Example: Statement:! = 8 Whichisgreater?! + 5 or 3! + 2 Answer:3x+2>x+5becauseifx=8, 3! + 2 = 26, ! + 5 = 13 and 26 > 13. Tryityourself: 1. Statement:! < ! Whichisgreater?! − ! or ! − ! 2. Statement:2! − 3 > 7 Whichisgreater?5 or ! 3. Statement:10 − 2! < 6 Whichisgreater?! or 2 4. Statement:4! ≤ 0 Whichisgreater?1 or ! Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org CCBYSydneyG 4.4 Greater Than? https://flic.kr/p/9jJwL1 15 SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.4 5. Statement:nisaninteger Whichisgreater?! or − ! 6. Statementx>y Whichisgreater?x+aory+a 7. Statement:x>y Whichisgreater?x–aory–a 8. Statement:5 > 4 Whichisgreater?5! or 4! 9. Statement:5 > 4 Whichisgreater? 5 4 or x x € € 10. Statement:0 < x< 10 and 0 < y< 12 Whichisgreater?! or ! n+2 11. Statement: 3 ≥ 27 Whichisgreater?! or 1 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.4 4.4 Greater Than? – Teacher Notes A Develop Understanding Task Purpose:Thepurposeofthistaskistochallengestudentstoreasonaboutinequalityrelationships andtodevelopanunderstandingofthepropertiesofinequalities.Eachproblemrequiresreasoning aboutnumbers,includingnegativenumbersandfractions,andthinkingmathematicallyaboutthe variouspossibilitiesinthegivenproblemsituation. CoreStandardsFocus: A.REI.1Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbers assertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution. a.Constructaviableargumenttojustifyasolutionmethod. b.Solveequationsandinequalitiesinonevariable. A.REI.3Solvelinearequationsandinequalitiesinonevariable,includingequationswith coefficientsrepresentedbyletters. MathematicsINote:Extendearlierworkwithsolvinglinearequationstosolvinglinearinequalities inonevariableandtosolvingliteralequationsthatarelinearinthevariablebeingsolvedfor. StandardsforMathematicalPracticeofFocusintheTask: SMP1–Makesenseofproblemsandpersevereinsolvingthem SMP2–Reasonabstractlyandquantitatively SMP8–Lookforandexpressregularityinrepeatedreasoning TheTeachingCycle: Launch(WholeClass): Explaintostudentsthatthistaskisabiglogicpuzzle.Alloftheproblemsrequirethinkingaboutall thedifferentpossibilitiestodecidewhichexpressionisgreater.Therearesomethatcannotbe determinedfromtheinformationgiven.Whenthathappens,studentsshouldwritedownwhat informationtheywouldneedinordertohaveadefiniteanswerforthequestion.(Youmaychoose Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org SECONDARY MATH I // MODULE 4 EQUATIONS AND INEQUALITIES – 4.4 nottotellstudentsthisinadvancesothattheyhaveanopportunitytowrestlewiththeideasandto justifytheirposition.)Startbyaskingstudentstoreadtheexamplegiven.Toconfirmthe instructions,askhowtheyseethethreerequiredpartsoftheexplanationintheanswer.Next,refer studentstoproblem#1.Givethemafewminutestoanswerandwritetheirownexplanation.Ask theclassfortheiranswersandexplanationsandmodelhowtowriteananswerwithacomplete explanation.Youmayalsowanttomodelthinkingaboutpossiblevaluesforxandy,like:“Ifxisa negativenumber,thenymustalsobeanegativenumberbecauseitislessthanx.” Explore(SmallGrouporPairs):Monitorstudentsastheywork.Encouragethemtothinkabout thevariouspossibilitiesforxandyineachcase.Besurethattheirwrittenexplanationsadequately communicatetheirlogic.Watchforproblemsthatgeneratedisagreementordifficultyfortheclass discussion.Alsolookforstudents’explanationsthatdemonstratesoundmathematicallogicor goodcommunicationtobehighlightedinthediscussion.Ifyounoticeacommonmisconception occurringduringtheexploration,plantoraiseitasanissueinthediscussion. Discuss(WholeClass):Startthediscussionwithproblems6-9.Askpreviously-selectedstudents togivetheirexplanationsforeachoftheseproblems.Besurethattheexplanationsinclude exampleofbothpositiveandnegativenumbers.Highlightfortheclassthatthesethreeproblems areaskingthemtojustifythepropertiesofinequalities.Generally,studentswillhavetested specificnumbersandmadegeneralizationsaboutallnumbersbasedupontheselectedexamples. Asktheclassiftheycancreateanargumentastowhyeachpropertycanbegeneralizedtoallreal numbers.Writeeachofthepropertiesofinequalities(addition,subtraction,multiplication,and division),andaskstudentstostatethemintheirownwords.Aftergoingthrougheachofthese, turnthediscussiontoanymisconceptionsorprovocativeproblemsthatwereselectedduringthe explorationphase. AlignedReady,Set,GoHomework:GettingReady4.4 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 16 SECONDARY MATH I // MODULE 4 4.4 SOLVING EQUATIONS AND INEQUALITIES – 4.4 READY, SET, GO! Name PeriodDate READY Topic:Writeanequationfromacontext.Interpretnotationforinequalities. Writeanequationthatdescribesthestory.Thenanswerthequestionaskedbythestory. 1.Virginia’sPaintingServicecharges$10perjoband$0.20persquarefoot.IfVirginiaearned$50 forpaintingonejob,howmanysquarefeetdidshepaintatthejob? 2.Rentingtheice-skatingrinkforapartycosts$200plus$4perperson.Ifthefinalchargefor Dane’sbirthdaypartywas$324,howmanypeopleattendedhisbirthdayparty? Indicateifthefollowingstatementsaretrueorfalse.Explainyourthinking. 3.Thenotation12 < ! meansthesamethingas! < 12.Itworksjustlike12 = ! !"# ! = 12. 4.Theinequality−2 ! + 10 ≥ 75saysthesamethingas−2! − 20 ≥ 75.Icanmultiplyby-2on theleftsidewithoutreversingtheinequalitysymbol. 5.Whensolvingtheinequality10! + 22 < 2,thesecondstepshouldsay10! > −20becauseI added-22tobothsidesandIgotanegativenumberontheright. 6.Whensolvingtheinequality−5! ≥ 45,theansweris! ≤ −9becauseIdividedbothsidesofthe inequalitybyanegativenumber. 7.Thewordsthatdescribetheinequality! ≥ 100are“xisgreaterthanorequalto100.” SET Topic:Solveinequalities.Verifythatgivennumbersareelementsofthesolutionset. Solveforx.(Showyourwork.)Indicateifthegivenvalueofxisanelementofthesolutionset. 8.2! − 9 < 3 9.4! + 25 > 13 Isthisvaluepart! = 6; !"#? ofthesolutionset? !"? Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org Isthisvaluepart! = −5; !"#? ofthesolutionset? !"? 17 SECONDARY MATH I // MODULE 4 4.4 SOLVING EQUATIONS AND INEQUALITIES – 4.4 10.6! − 4 ≤ −28 Isthisvaluepart! = −10; !"#? ofthesolutionset? 11.3! − 5 ≥ −5 !"? Isthisvaluepart! = 1; !"#? ofthesolutionset? Solveeachinequalityandgraphthesolutiononthenumberline. 12.! + 9 ≤ 7 – 10 –5 0 !"? 5 10 5 10 13.−3! − 4 > 2 – 10 14.3! < −6 ! ! ! !" 15. > − 16.−10! > 150 17. ! !! ≥ −5 – 25 –5 – 10 –5 0 5 10 – 10 –5 0 5 10 – 20 19. !(!!!) !" Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org – 15 10 0 Solveeachmulti-stepinequality. 18.! − 5 > 2! + 3 0 – 10 20 ≤ !! ! –5 0 30 20.2 ! − 3 ≤ 3! − 2 40 18 SECONDARY MATH I // MODULE 4 4.4 SOLVING EQUATIONS AND INEQUALITIES – 4.4 GO Topic:Usesubstitutiontosolvelinearsystems Solveeachsystemofequationsbyusingsubstitution. Example: ! = 12 2! − ! = 14 Thefirstequationstatesthat! = 12.Thatinformationcanbeusedinthesecondequationtofindthe valueofxbyreplacingywith12.Thesecondequationnowsays!" − !" = !".Solvethisnewequation byadding12tobothsidesandthendividingby2.Theresultisx=13. 21. !=5 −! + ! = 1 22. !=8 5! + 2! = 0 23. 2! = 10 4! − 2! = 50 24. 3! = 12 4! − ! = 5 25. ! = 2! − 5 !=!+8 26. Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org 3! = 9 5! + ! = −5