Solving-Linear-Equations-in-Algebra-I求解線性方程組的代數(shù)I-2-資料課件

WhatmustallAlgebraIstudentsknowaboutsolvingandworkingwithLinearEquations?WhatmustallAlgebraIstuden1StudentsshouldUnderstandthebigideasofequivalenceandlinearity ModelingrealsituationswithvariablesUseappropriatetoolssuchasalgebratilesandgraphingcalculators,andspreadsheetsregularlyUnderstandthatgeometricobjectscanberepresentedalgebraically(linescanbedescribedusingcoordinates),andalgebraicexpressionscanbeinterpretedgeometrically(systemsofequationsandinequalitiescanbesolvedgraphically) GoalsfromtheNewJerseyAlgebraICoreContentStandardsStudentsshouldGoalsfromthe2

EQUIVALENCE:Numbers,expressions,functions,orequationshavemanydifferent–butequivalent–forms.Theseformsdifferintheirefficacyandefficiencyininterpretingorsolvingaproblem,dependingonthecontext.Algebraextendsthepropertiesofnumberstorulesinvolvingsymbolstotransformanexpression,function,orequationintoanequivalentformandsubstituteequivalentformsforeachother.Solvingproblemsalgebraicallytypicallyinvolvestransformingoneequationtoanotherequivalentequationuntilthesolutionbecomesclear.

EQUIVALENCE:3LinearityTherelationshipbetweentwoquantitiescanoftenberepresentedgraphicallybyalinearfunction.LinearfunctionscanbeusedtoshowarelationshipbetweentwovariableswithaconstantrateofchangeLinearfunctionscanbeusedtoshowtherelationshipbetweentwoquantitiesthatvaryproportionately.Linearfunctionscanalsobeusedtomodel,describe,analyze,andcomparesetsofdata.UnderstandinglinearfunctionsshouldbeprominentintheAlgebraIcontent. Linearity4Recognize,describeandrepresentlinearrelationshipsusingwords,tables,numericalpatterns,graphsandequations.Describe,analyzeandusekeycharacteristicsoflinearfunctionsandtheirgraphs. Graphtheabsolutevalueofalinearfunctionanddetermineandanalyzeitskeycharacteristics.Recognize,expressandsolveproblemsthatcanbemodeledusinglinearfunctions.Interpretsolutionsintermsofthecontextoftheproblem.BenchmarksRelatedtoLinearRelationshipsRecognize,describeandrepres5Solvesingle-variablelinearequationsandinequalitieswithrationalcoefficients. Solveequationsinvolvingtheabsolutevalueofalinearexpression. Graphandanalyzethegraphofthesolutionsetofatwo-variablelinearinequality. Solvesystemsoflinearequationsintwovariablesusingalgebraicandgraphicprocedures. Recognize,expressandsolveproblemsthatcanbemodeledusingsingle-variablelinearequations;one-ortwo-variableinequalities;ortwo-variablesystemsoflinearequations. Solvesingle-variablelineare6BuildingUnderstandingforWritingandEvaluatingExpressionsAlgebrashouldbecomealanguagethroughwhichwecandescribevarioussituationsBuildingUnderstandingforWri7Whatisthreeplusfivetimestwo?TryenteringthisproblemonthehomescreenofthegraphingcalculatorHowmanywayscanyouenteritonthehomescreen?Isthereanorderfortheoperationswhentheproblemiswrittenhorizontally?4+3*2Whatisthreeplusfivetimes8OrderofOperationsEvaluateexpressionswithinparenthesesorothergroupingsymbols.Evaluateallpowers.Multiplyanddividefromlefttoright.Addandsubtractfromlefttoright.Howwouldyouhavetowrite

4+3?2

sotheansweris14?OrderofOperationsEvaluateex9LearningtobuildmathematicalexpressionsLearningtobuildmathematical10Let’stryperformingastringofoperationstoseewhatweget.Onpaper:Startwith6.Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Startwith20.Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Startwith-4Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Let’stryperformingastring11Theseproblemsappearprettysimplebecausewearegivingallthedirectionsinshortstepsandyouareperformingthemintheorderinwhichtheyaredescribed.Theseproblemsappearprettys12Let’sseeifwecanlearntowriteexpressionsthroughasimilaractivity.Startwithachartandcompleteeachlinebasedonthedirectionsgiven.Isthereanyrelationshipbetweenthestartingnumberandtheresultinganswer?Howiswhatwedidinthefourstepsequivalenttothisonerelationship?DescriptionExpressionLet’sseeifwecanlearntow13DescriptionExpressionStartwithanumberMultiplythenumberby2Add6Divideby2Subtracttheresultfrom10DescriptionExpressionStartwit14Afterwehavewrittentheexpressionwecantestit:Onthegraphingcalculatorhomescreentype:6=>x:10-(2x+6)/2=>meansSTOGeneralFormat:YourNumber=>x:10-(2x+6)/2Confirm20and-4Afterwehavewrittentheexpr15UsingtheDescription/ExpressionTemplatePickanynumberDividethenumberby4Add7Multiplytheresultby2Subtract8Findthevalueofyourexpressionwhenx=2,-5,8Setuptheexpressionforthisproblem:UsingtheDescription/Expressi16NumberTricksEachpersonpickanynumberfrom1to25.Add9toit.Multiplytheresultby3.Subtract6fromthecurrentanswer.Dividethisanswerby3.Nowsubtractyouroriginalnumber.Compareyourresults.Willtheanswerbethesameregardlessonthenumberyoubeginwith?Whyisthis?Writeoutthealgebraicexpressionforthisnumbertrick.NumberTricksEachpersonpick17DescriptionExpressionStartwithanumberAdd9Multiplytheresultby3Subtract6fromtheresultDivideby3SubtracttheoriginalnumberThisisaprettycomplexexpression.Canweputtheseinanequationandsolveforx?DescriptionExpressionStartwit18Ifyouweretoldtheexpressionontheleftdescribesseveraloperationsthatwereperformedtoagivennumberandthattheresultequalsto7,describealltheoperationsthatwereperformedonxandwhatordertheywereperformedtoarriveattheanswer7?Ifyouweretoldtheexpressio19YoucreateatrickCreateyourowntrickthathasatleast5stages.Testitonyourcalculatorwithatleastfourdifferentnumberstomakesurealltheanswersarethesame.Whenyouthinkyourtrickworks,testitonyourothergroupmembers.YoucreateatrickCreateyour20Writeinwordsthenumbertrickthatisdescribedabove.Testthenumbertricktobesureyougetthesameresultnomatterwhatnumberyouchoose.Canyouexplainwhythisnumbertrickwork?AnalyzingaNumberTrickWriteinwordsthenumbertric21Giventheexpressionontheleft,youmightwanttothinkofsubtractionasaddingtheoppositeandre-writetheexpressionWrite,inwords,thenumbertrickthatisdescribedabove.Testthenumbertricktobesureyougetthesameresultnomatterwhatnumberyouchoose.Whichoperationsthatundopreviousoperationsmakethisnumbertrickwork?Giventheexpressiononthele22Daxun,Lacy,Claudia,andAlareworkingonanumbertrick.Herearethenumbersequencestheirnumbertrickgenerates:a.Describethestagesofthisnumbertrickinthefirstcolumn.b.CompleteClaudia’ssequence.c.WriteasequenceofexpressionsforAlinthelastcolumn.Daxun,Lacy,Claudia,andAla23Lessons2.1and2.2:reviewproportionsandintroducetheideaofundoingtosolveaproportion.Lesson2.3:derivinglinearexpressionsfrommeasurementLesson2.4:introducesdirectvariationequationsasanalternativetosolvingproportions(aspeciallinearfunction),createascatterplotofarealdataset,modelwithalinethroughthepoints,andwriteanequationintheformy=kxtodescribethatline.Lesson2.5:introducestherelatedtopicofinversevariation(notalinearfunction)Lesson2.7:rulesfororderofoperationsbyanalyzinghowthestepsinlinearexpressionsthatdescribe“numbertricks”undoeachothertoendwiththesamenumberLesson2.8:writelinearequationstorepresentsequencesofstepsandsolvethoseequationsbyundoing.InChapter2Lessons2.1and2.2:reviewp24Whatdoesitmeantosolveanequation?Isitanymorethanjustundoingtheprocedureofbuildinganequation?Whatdoesitmeantosolvean25Chooseasecretnumber.Nowchoosefourmorenon-zeronumbersandinanyrandomorderaddoneofthem,multiplybyanother,subtractanother,anddividebythefinalnumberRecordinwordswhatyoudidandyourfinalresultonthecommunicatorwithablankBuildingandEvaluatinganExpressionorEquationtemplate.(Donotrecordyoursecretnumber.)Switchcommunicatorsandhaveanotherstudentsfindyoursecretnumber.Chooseasecretnumber.26?Add2Ans+2xX+2Multiplyby5Ansx5Subtract4Ans-4Divideby8Ans÷82,5,4,8Add2,Multiplyby5,Subtract4,Anddivideby8Revealthe

resultsWhatwasmystartingnumber??Add2Ans+2xX+2Multiplyby5An27Tosomenumber,add3,multiplyby2,add18,andfinallydivideby6.a.Convertthedescriptionintoanexpression,andwriteanequationthatstatesthatthisexpressionisequalto15.b.Findthestartingnumberifthefinalresultis15.c.Testyoursolutiontopartbusingyourequationfromparta.Tosomenumber,add3,multipl28SolveEquationsisJustUndoingOperationsUsetheBuildandUndoExpressionorEquationsCharttocompletethefollowingnumbertrick.Completethefirstthreecolumnsonly.PickanumberDividethenumberby4Add7Multiplytheresultby2Subtract8SolveEquationsisJustUndoin29SolveEquationsisJustUndoingOperationsDescription/SequenceExpressionPickanumber?XDivideby4/4x4Add7+7-7Multiplyby2x2/2Subtract8-8+82836181144SolveEquationsisJustUndoin30Insteadofbuildinganequation,let’sstartwithanequationandsolveitHereisanequation.Whatisitsaying?Firstbuildtheequation,thenwe’llsolveit.PickanumberxInsteadofbuildinganequatio31TrysolvingtheseequationsPlacetheBuildingandUndoinganExpressionTemplateinyourCommunicator?.Recordanequationinthecellatthetop.Completethedescriptioncolumnusingtheorderofoperations.Completetheundocolumn.Finally,workupfromthebottomofthetabletosolvetheequation.WriteafewsentencesexplainingwhythismethodworkstosolveanequationTrysolvingtheseequationsPla32SimplifyingtheTechniqueofSolvinganEquationSimplifyingtheTechniqueofS33-3x÷4+742-735x4140+3143-3x÷4+742-735x4140+314334Anequationisastatementthatsaysthevalueofoneexpressionisequaltothevalueofanotherexpression.Solvingequationsistheprocessyouusedtodeterminethevalueoftheunknownthatmakestheequationtrue.Thisiscalledthesolution.Anequationisastatementtha35InChapter3,studentsuseequationstomodellineargrowthandgraphsofstraightlinesandlearnthebalancingmethodforsolvingequations.Thischapterbuildstowardtheconceptoffunction,whichisformalizedinChapter8.Chapter3InChapter3,studentsuseequ36Lesson3.1:developmentoflineargrowthwithrecursivesequences.Lesson3.2:linearplots.Lesson3.3:walkinginstructionstostudymotionLesson3.4:interceptformofalinewithstartingvalueandrateofchangeLesson3.5:ratesofchangeLesson3.6:balancingtechniqueforsolvingequationsLesson3.7:Modelreal-worlddatawithlinearequationsLesson3.1:developmentoflin37SolvingEquationsbyBalancingEquationsSolvingEquationsbyBalancing38Thefigureillustratesabalancedscale.Thisisbecause4yellowsquaretilesbalanceswith4squareyellowsquaretiles.Buildthisscaleinfrontofyou.Let’sdiscoversomethingswecandotobalancedscalethatwillkeepitthatkeepthescaleinfigure1balanced.Thefigureillustratesabalan39___Whatwouldhappenifyouadded2yellowsquarestilestobothsidesofthefigure?___Whatwouldhappenifyouadded1redsquaretiletobothsidesofthefigure?___Whatwouldhappenifyouadded1redsquaretotheleftsideandoneyellowsquaretiletotherightsideofthefigure?___Whatwouldhappenifyouaddeddoublethenumberoftilesonbothsidesofthefigure?___Whatwouldhappenifyouremovedoneyellowsquarefromtheleftsideandaddedoneredsquaretotherightsideofthefigure?___Whatwouldhappenifyoucutthenumberoftilesinhalfoneachsideofthefigure?___Whatwouldhappenifyoudoubledtheleftsideanddividedtherightsideby2inthefigure?___Whatwouldhappenifyouaddedoneredsquaretotheleftsideonlyinthefigure?___Whatwouldhappenifyouaddedoneyellowsquaretotherightsideonlyinthefigure?___Whatwouldhappenifyouaddedredsquaretotheleftandremovedoneyellowsquarefromtherightinthefigure?___Whatwouldhappenifyouad40Thefigureillustratesabalancedscale.Buildthisonyourscale.Howmanyredoryellowsquareswouldthegreenrectanglebeequalto?Usingoneoftheideasfromabove,wecanshowthatthegreenrectangleisequalto2yellowsquares.Showatleasttwowaysthiscanbeaccomplished.Thefigureillustratesabalan41Solving-Linear-Equations-in-Algebra-I求解線性方程組的代數(shù)I-2-資料課件42Solving-Linear-Equations-in-Algebra-I求解線性方程組的代數(shù)I-2-資料課件43x+-3=-4Makeasketchofthebalancescalethatmatcheswiththisequation.Solvetheequationbyusingthealgebratiles.2x+-3=5-2x+-3=-4+-1x-3+x=2x+1-4=2(x+2)x+4=-2x+-23x+-3=2(x1)x+-3=-4Makeasketchofth44MakingtheTransitiontosolvinganEquationAlgebraicallywithsymbolsMakingtheTransitiontosolvi45Iftheequationwas1+2x+3=7youwouldhavebuiltthebalancescaleinthefigure.Onestepyoumightdofirstiscombinetheliketerms.Thiswouldresultinthenextfigure.Thisfiguresaysthat2x+4=8.Nowyoumightthinkaboutremove4yellowsquaresfrombothsides.Thiswouldleaveyouwiththenextfigure.Thisfiguresaysthat2x=4.Thenyouwouldhavedividedbothsidesintotwoequalgroupssothegreenrectangleequals2yellowsquaresorx=2.Iftheequationwas1+2x+346Thistimeourstepswillbemorealgebraic,butbaseduponwhatwedidwiththebalancescale.Thistimeourstepswillbemo47Chapter4emphasizesslopeinthecontextoffindinglinesoffit.Chapter4Chapter4emphasizesslopein48Lesson4.1:formulafordeterminingslopeLesson4.2:usetheinterceptformtofitlinestodataLessons4.3and4.4:point-slopeformthroughapplicationLesson4.5:Usethepoint-slopeformtofitlinestodataLessons4.6and4.7:methodfordetermininglinesoffitLesson4.8:activitydayforreviewinglinesoffit.Lesson4.1:formulafordeterm49InChapter5,studentslookatsystemsoflinearequationsandconsiderlinearinequalities.Thentheyputthesetwoideastogethertothinkaboutsystemsoflinearinequalities.InChapter5,studentslookat50Lessons5.1to5.4:fivewaystosolveasystemofequations:tables,graphs,thesubstitutionmethod,theeliminationmethod,androwoperationsonmatrices.Lesson5.5:InequalitiesinonevariableareintroducedLesson5.6:graphinequalitiesintwovariablesLesson5.7:graphandsolvesystemsLessons5.1to5.4:fiveways51Students,throughusingDiscoveringAlgebraaregoingtodiscoverandlearnmuchusefulalgebraalongtheway.Learningalgebraismorethanlearningfactsandtheoriesandmemorizingproceduresandthentryingtoapplythemthroughapplicationssections.Throughthetextstudentsbeinvolvedinmathematicsandinlearning“howtodomathematics.”SuccessinalgebraisagatewaytomanyvariedcareeropportunitiesStudents,throughusingDiscov52Throughtheinvestigations,studentswillmakesenseofimportantalgebraicconcepts,learnessentialalgebraicskills,anddiscoverhowtousealgebra.Throughtheinvestigations,st53thatalgebrateachingshouldfocusonthebasicskillsoftoday,notthoseof40yearsago.Problemsolving,reasoning,justifyingideas,makingsenseofcomplexsituations,andlearningnewideasindependently—notpaper-and-pencilcomputation—arenowcriticalskillsforallAmericans.IntheInformationAgeandtheWebera,obtainingthefactsisnottheproblem;analyzingandmakingsenseofthemis.“TheMathematicalMiseducationofAmerica’sYouth,”

ThePhiDeltaKappan.February,2019MichaelT.BattistaofKentStateUniversitywritesthatalgebrateachingshouldf54technology,alongwithapplications,isusedtofosteradeeperunderstandingofalgebraicideas.Theexplorationsemphasizesymbolsense,algebraicmanipulations,andconceptualunderstandings.Theinvestigativeprocessencouragestheuseofmultiplerepresentations—numerical,graphical,symbolic,andverbal—todeepenunderstandingforallstudentsandtoserveavarietyoflearningstyles.Explorationsfrommultipleperspectiveshelpstudentssimplifyandunderstandwhatformerlyweredifficultalgebraicabstractions.Investigationsactivelyengagestudentsastheymakepersonalandmeaningfulconnectionstothemathematicstheydiscover.InDiscoveringAlgebratechnology,alongwithapplica55Traditionalalgebrateachesskillsandideasbeforeexamplesandapplications.Theinvestigativeapproachworkstheotherway.Interestingquestionsandsimplehands-oninvestigationsprecedetheintroductionofformulasandsymbolicrepresentations.Byprovidingmeaningfulcontextsforstudents,theinvestigationsmotivaterelevantalgebraicconceptsandprocesses.Theinvestigationsareaccessible.Theyuseinexpensiveandreadilyavailablematerials,requirelittleprerequisitetechnicalknowledge,andfollowsimpleprocedures.Studentscanconductthemwithaminimumofdirectionandinterventionfromyou.Traditionalalgebrateachessk56TeachingwithDiscoveringAlgebradecreasesthetimestudentsspendonrotememorization,teacherexposition,andextendedperiodsofpaper-and-pencildrill.Itchangestherulesforwhatisexpectedofstudentsandwhattheyshouldexpectoftheirteacher.TeachingfromDiscoveringAlgebrarequiresnontraditionalthinkingandbehaviorandanontraditionalclassroom.Successdependsonyoursensitivity,patience,enthusiasm,anddetermination.TeachingwithDiscoveringAlge57謝謝！謝謝！58WhatmustallAlgebraIstudentsknowaboutsolvingandworkingwithLinearEquations?WhatmustallAlgebraIstuden59StudentsshouldUnderstandthebigideasofequivalenceandlinearity ModelingrealsituationswithvariablesUseappropriatetoolssuchasalgebratilesandgraphingcalculators,andspreadsheetsregularlyUnderstandthatgeometricobjectscanberepresentedalgebraically(linescanbedescribedusingcoordinates),andalgebraicexpressionscanbeinterpretedgeometrically(systemsofequationsandinequalitiescanbesolvedgraphically) GoalsfromtheNewJerseyAlgebraICoreContentStandardsStudentsshouldGoalsfromthe60

EQUIVALENCE:Numbers,expressions,functions,orequationshavemanydifferent–butequivalent–forms.Theseformsdifferintheirefficacyandefficiencyininterpretingorsolvingaproblem,dependingonthecontext.Algebraextendsthepropertiesofnumberstorulesinvolvingsymbolstotransformanexpression,function,orequationintoanequivalentformandsubstituteequivalentformsforeachother.Solvingproblemsalgebraicallytypicallyinvolvestransformingoneequationtoanotherequivalentequationuntilthesolutionbecomesclear.

EQUIVALENCE:61LinearityTherelationshipbetweentwoquantitiescanoftenberepresentedgraphicallybyalinearfunction.LinearfunctionscanbeusedtoshowarelationshipbetweentwovariableswithaconstantrateofchangeLinearfunctionscanbeusedtoshowtherelationshipbetweentwoquantitiesthatvaryproportionately.Linearfunctionscanalsobeusedtomodel,describe,analyze,andcomparesetsofdata.UnderstandinglinearfunctionsshouldbeprominentintheAlgebraIcontent. Linearity62Recognize,describeandrepresentlinearrelationshipsusingwords,tables,numericalpatterns,graphsandequations.Describe,analyzeandusekeycharacteristicsoflinearfunctionsandtheirgraphs. Graphtheabsolutevalueofalinearfunctionanddetermineandanalyzeitskeycharacteristics.Recognize,expressandsolveproblemsthatcanbemodeledusinglinearfunctions.Interpretsolutionsintermsofthecontextoftheproblem.BenchmarksRelatedtoLinearRelationshipsRecognize,describeandrepres63Solvesingle-variablelinearequationsandinequalitieswithrationalcoefficients. Solveequationsinvolvingtheabsolutevalueofalinearexpression. Graphandanalyzethegraphofthesolutionsetofatwo-variablelinearinequality. Solvesystemsoflinearequationsintwovariablesusingalgebraicandgraphicprocedures. Recognize,expressandsolveproblemsthatcanbemodeledusingsingle-variablelinearequations;one-ortwo-variableinequalities;ortwo-variablesystemsoflinearequations. Solvesingle-variablelineare64BuildingUnderstandingforWritingandEvaluatingExpressionsAlgebrashouldbecomealanguagethroughwhichwecandescribevarioussituationsBuildingUnderstandingforWri65Whatisthreeplusfivetimestwo?TryenteringthisproblemonthehomescreenofthegraphingcalculatorHowmanywayscanyouenteritonthehomescreen?Isthereanorderfortheoperationswhentheproblemiswrittenhorizontally?4+3*2Whatisthreeplusfivetimes66OrderofOperationsEvaluateexpressionswithinparenthesesorothergroupingsymbols.Evaluateallpowers.Multiplyanddividefromlefttoright.Addandsubtractfromlefttoright.Howwouldyouhavetowrite

4+3?2

sotheansweris14?OrderofOperationsEvaluateex67LearningtobuildmathematicalexpressionsLearningtobuildmathematical68Let’stryperformingastringofoperationstoseewhatweget.Onpaper:Startwith6.Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Startwith20.Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Startwith-4Multiply2timesastartingnumber,thenadd6,dividethisresultby2,andthensubtractyouranswerfrom10.Let’stryperformingastring69Theseproblemsappearprettysimplebecausewearegivingallthedirectionsinshortstepsandyouareperformingthemintheorderinwhichtheyaredescribed.Theseproblemsappearprettys70Let’sseeifwecanlearntowriteexpressionsthroughasimilaractivity.Startwithachartandcompleteeachlinebasedonthedirectionsgiven.Isthereanyrelationshipbetweenthestartingnumberandtheresultinganswer?Howiswhatwedidinthefourstepsequivalenttothisonerelationship?DescriptionExpressionLet’sseeifwecanlearntow71DescriptionExpressionStartwithanumberMultiplythenumberby2Add6Divideby2Subtracttheresultfrom10DescriptionExpressionStartwit72Afterwehavewrittentheexpressionwecantestit:Onthegraphingcalculatorhomescreentype:6=>x:10-(2x+6)/2=>meansSTOGeneralFormat:YourNumber=>x:10-(2x+6)/2Confirm20and-4Afterwehavewrittentheexpr73UsingtheDescription/ExpressionTemplatePickanynumberDividethenumberby4Add7Multiplytheresultby2Subtract8Findthevalueofyourexpressionwhenx=2,-5,8Setuptheexpressionforthisproblem:UsingtheDescription/Expressi74NumberTricksEachpersonpickanynumberfrom1to25.Add9toit.Multiplytheresultby3.Subtract6fromthecurrentanswer.Dividethisanswerby3.Nowsubtractyouroriginalnumber.Compareyourresults.Willtheanswerbethesameregardlessonthenumberyoubeginwith?Whyisthis?Writeoutthealgebraicexpressionforthisnumbertrick.NumberTricksEachpersonpick75DescriptionExpressionStartwithanumberAdd9Multiplytheresultby3Subtract6fromtheresultDivideby3SubtracttheoriginalnumberThisisaprettycomplexexpression.Canweputtheseinanequationandsolveforx?DescriptionExpressionStartwit76Ifyouweretoldtheexpressionontheleftdescribesseveraloperationsthatwereperformedtoagivennumberandthattheresultequalsto7,describealltheoperationsthatwereperformedonxandwhatordertheywereperformedtoarriveattheanswer7?Ifyouweretoldtheexpressio77YoucreateatrickCreateyourowntrickthathasatleast5stages.Testitonyourcalculatorwithatleastfourdifferentnumberstomakesurealltheanswersarethesame.Whenyouthinkyourtrickworks,testitonyourothergroupmembers.YoucreateatrickCreateyour78Writeinwordsthenumbertrickthatisdescribedabove.Testthenumbertricktobesureyougetthesameresultnomatterwhatnumberyouchoose.Canyouexplainwhythisnumbertrickwork?AnalyzingaNumberTrickWriteinwordsthenumbertric79Giventheexpressionontheleft,youmightwanttothinkofsubtractionasaddingtheoppositeandre-writetheexpressionWrite,inwords,thenumbertrickthatisdescribedabove.Testthenumbertricktobesureyougetthesameresultnomatterwhatnumberyouchoose.Whichoperationsthatundopreviousoperationsmakethisnumbertrickwork?Giventheexpressiononthele80Daxun,Lacy,Claudia,andAlareworkingonanumbertrick.Herearethenumbersequencestheirnumbertrickgenerates:a.Describethestagesofthisnumbertrickinthefirstcolumn.b.CompleteClaudia’ssequence.c.WriteasequenceofexpressionsforAlinthelastcolumn.Daxun,Lacy,Claudia,andAla81Lessons2.1and2.2:reviewproportionsandintroducetheideaofundoingtosolveaproportion.Lesson2.3:derivinglinearexpressionsfrommeasurementLesson2.4:introducesdirectvariationequationsasanalternativetosolvingproportions(aspeciallinearfunction),createascatterplotofarealdataset,modelwithalinethroughthepoints,andwriteanequationintheformy=kxtodescribethatline.Lesson2.5:introducestherelatedtopicofinversevariation(notalinearfunction)Lesson2.7:rulesfororderofoperationsbyanalyzinghowthestepsinlinearexpressionsthatdescribe“numbertricks”undoeachothertoendwiththesamenumberLesson2.8:writelinearequationstorepresentsequencesofstepsandsolvethos