SDUHSD Math 1 Module 4 SE - San Dieguito Union High ...teachers.sduhsd.net/dpillsbury/IM1/Modules/SDUHSD Math 1...2 SDUHSD Math 1 College Prep Module 4 Overview Prerequisite Concepts

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©2012MathematicsVisionProject|MVPInpartnership withtheUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense

IntegratedMath1Module4

LinearandExponentialFunctions

AdaptedFrom

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org

InpartnershipwiththeUtahStateOfficeofEducation

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SDUHSDMath1CollegePrep

Module4OverviewPrerequisiteConcepts&Skills:

Writerecursiveandexplicitrules/formulasforarithmeticandgeometricsequences Usefunctionnotationtoevaluatefunctions Modelsequenceswithatableofvaluesandgraph

SummaryoftheConcepts&SkillsinModule4:

Transitionfromarithmeticandgeometricsequencestolinearandexponentialmodels. Distinguishbetweencontinuousvdiscrete Comparelinearandexponentialmodels Applylinearandexponentialfunctiontomodelsituations(population) Solvelinearandexponentialequations Developandusesimpleandcompoundinterestformulas Analyzerateofchangeforagivencontext Representlinearequationsusingslope‐intercept,standard,andpoint‐slopeformandidentifythebenefits

andidealusesofeachformContentStandardsandStandardsofMathematicalPracticeCovered:

ContentStandards:F.IF.3,F.IF.6,F.IF.7,F.LE.1,F.LE.2,F.LE.3,F.LE.5,F.BF.1,F.BF.2,A.SSE.1,A.SSE.3,A.CED.2,A.REI.3

StandardsofMathematicalPractice:1.Makesenseofproblemsandpersevereinsolvingthem. 2.Reasonabstractlyandquantitatively. 3.Constructviableargumentsandcritiquethereasoningofothers. 4.Modelwithmathematics. 5.Useappropriatetoolsstrategically. 6.Attendtoprecision. 7.Lookforandmakeuseofstructure. 8.Lookforandexpressregularityinrepeatedreasoning.

Materials:

GraphingutilityModule4Vocabulary:

Linearmodel/equation/function Exponentialmodel/equation/function Domain DiscreteFunction ContinuousFunction ChangeFactor AveragerateofChange SecantLine TangentLine Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Point‐slopeform Simpleinterest Compoundinterest Principal

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ConceptsUsedIntheNextModule: Useacontexttographanddescribekeyfeaturesoffunctions Usetablesandgraphstointerpretkeyfeaturesoffunctions Interpretfunctionsusingnotation Combinefunctionsandanalyzecontextsusingfunctions Usegraphstosolveproblemsgiveninfunctionnotation Definefunction Identifywhetherornotarelationisafunctiongivenvariousrepresentations

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Module4–LinearandExponentialFunctions4.1Introducingcontinuouslinearandexponentialfunctions(F.IF.3)Warm‐Up:Writeafunction/graphthesituationClassroomTask:ConnectingtheDots:PiggiesandPools–ADevelopUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.14.2DefininglinearandexponentialfunctionsbaseduponthepatternofchangeandIdentifyingratesofchangeinlinearandexponentialfunctions(F.LE.1,F.LE.2)Warm‐Up:RepresentationsoflinearandexponentialfunctionsClassroomTask:

SortingOuttheChange–ASolidifyUnderstandingTask Where’sMyChange–APracticeUnderstandingTask

Ready,Set,GoHomework:LinearandExponentialFunctions4.24.3ComparingthegrowthoflinearandexponentialfunctionsandComparinglinearandexponentialmodelsofpopulation(F.BF.1,F.BF.2,F.LE.1,F.LE.2,F.LE.3,F.LE.5,F.IF.7)Warm‐Up:GraphingLinearEquationsinSlope‐InterceptFormClassroomTask:Growing,Growing,Gone–ASolidifyUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.34.4Interpretingequationsthatmodellinearandexponentialfunctions(A.SSE.1,A.CED.2,F.LE.5)ClassroomTask:MakingMyPoint–ASolidifyUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.44.5Evaluatingtheuseofvariousformsoflinearandexponentialequations(A.SSE.1,A.SSE.3,A.CED.2,F.LE.5)Warm‐Up:The4formsofalinearequationClassroomTask:EfficiencyExperts–ASolidifyUnderstandingTaskReady,GoHomework:LinearandExponentialFunctions4.54.6Understandingandinterpretingformulasforexponentialgrowthanddecay(A.SSE.1,A.CED.2,F.LE.5,F.IF.7)Warm‐Up:InterestClassroomTask:UpaLittle,DownaLittle–ASolidifyUnderstandingTaskSet,GoHomework:LinearandExponentialFunctions4.64.7Solvingexponentialandlinearequations(A.REI.3)Warm‐Up:Writingalinearequationinslope‐interceptformClassroomTask:XMarkstheSpot–APracticeUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.7&OptionalModule4ReviewHomework

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4.1WarmUpRepresentationsofexponentialfunctionsBillyhasawarrenof16rabbits.Eachmonththereare1.5timesmorerabbits.1. Completethetablebelowshowingthenumberofrabbitsasafunctionoftime.

Month #ofRabbits0 1 2 3 4

2. Drawagraphofthesituation,clearlylabelingandscalingtheaxes.

3. Writeafunctionthatmodelsthesituation.

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www.flickr.com/photos/teegardin

4.1ConnectingtheDots:PiggiesandPoolsADevelopUnderstandingTask1. Mylittlesister,Savannah,isthreeyearsold.Shehasapiggybankthatshewantsto

fill.ShestartedwithfivepenniesandeachdaywhenIcomehomefromschool,sheisexcitedwhenIgiveherthreepenniesthatareleftoverfrommylunchmoney.Createamathematicalmodelforthenumberofpenniesinthepiggybankondayn.

2. Ourfamilyhasasmallpoolforrelaxinginthesummerthatholds1500gallonsofwater.Idecidedtofillthe

poolforthesummer.WhenIhad5gallonsofwaterinthepool,IdecidedthatIdidn’twanttostandoutsideandwatchthepoolfill,soIhadtofigureouthowlongitwouldtakesothatIcouldleave,butcomebacktoturnoffthewaterattherighttime.Icheckedtheflowonthehoseandfoundthatitwasfillingthepoolatarateof2gallonseveryminute.Createamathematicalmodelforthenumberofgallonsofwaterinthepoolattminutes.

3. I’mmoresophisticatedthanmylittlesistersoIsavemymoneyinabankaccountthatpaysme3%intereston

themoneyintheaccountattheendofeachmonth.(IfItakemymoneyoutbeforetheendofthemonth,Idon’tearnanyinterestforthemonth.)Istartedtheaccountwith$50thatIgotformybirthday.CreateamathematicalmodeloftheamountofmoneyIwillhaveintheaccountaftermmonths.

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4. Attheendofthesummer,Idecidetodraintheswimmingpool.Inoticedthatitdrainsfasterwhenthereismorewaterinthepool.Thatwasinterestingtome,soIdecidedtomeasuretherateatwhichitdrains.Ifoundthatitwasdrainingatarateof3%everyminute.Createamathematicalmodelofthegallonsofwaterinthepoolattminutes.

5. Compareproblems1and3.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?6. Compareproblems1and2.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?7. Compareproblems3and4.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?

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Name: LinearandExponentialFunctions 4.1Ready,Set,Go!ReadyTopic:RecognizingarithmeticandgeometricsequencesPredictthenext2termsinthesequence.Statewhetherthesequenceisarithmetic,geometric,orneither.Justifyyouranswer.1. 4, 20, 100, 500,⋯ 2. 3, 5, 8, 12,⋯3. 64, 4, 3,27,⋯ 4. 1.5, 0.75, 0, 0.75,⋯5. 40, 10, , , ⋯ 6. 1, 11, 111, 1111,⋯7. 3.6, 5.4, 8.1, 12.15,⋯ 8. 64, 47, 30, 13,⋯9. Createapredictablesequenceofatleast4numbersthatisNOTarithmeticorgeometric.

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SetTopic:DiscreteandcontinuousrelationshipsIdentifywhetherthefollowingstatementsrepresentadiscreteoracontinuousrelationship.10.Thehaironyourheadgrows inchpermonth.11.Foreverytonofpaperthatisrecycled,17treesaresaved.12.Approximately3.24billiongallonsofwaterflowoverNiagaraFallsdaily.13.Theaveragepersonlaughs15timesperday.14.ThecityofBuenosAiresadds6,000tonsoftrashtoitslandfillseveryday.15.DuringtheGreatDepression,stockmarketpricesfell75%.GoTopic:SlopesoflinesDeterminetheslopeofthelinethatpassesthroughthefollowingpoints.16. 15, 9 , 10, 4 17. 0.5, 4 , 3, 3.5 18. 50, 85 , 60, 80 19.

5 204 173 14

20.

1 1

0

1 2

21.

5 330 305 27

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4.2WarmUpRepresentationsofLinearandExponentialFunctionsIneachofthefollowingproblems,youaregivenoneoftherepresentationsofafunction.Completetheremaining3representations.Identifytherateofchangefortherelation.1. Equation:

Table:

CreateaContext:Youandyourfriendsgotothestatefair.Itcosts$5togetintothefairand$3eachtimeyougoonaride.

Graph

2. Equation:

Table:Time Amount1 182 533 1624 4865 14586 4374

CreateaContext:

Graph

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© 2012 www.flickr.com/photos/ 401(K) 2012

4.2SortingOuttheChangeASolidifyUnderstandingTask1A. Whichofthefollowingappliestothegiveninformation?

Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither

B. Bepreparedtodescribethepatternofchangeandtotellhowyoufoundit.1.

30 5725 4720 3715 2710 175 70 3

Typeofpatternofchange(circleone):


HowIfoundthepatternofchange:

2. 0 3, 1 ⋅




3.

3 53 103 203 153 353 50




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Usethefigurebelowtoanswerquestions4and5.Step1 Step2 Step3 Step4

4. Thepatternofchangeintheperimeterofthefiguresfromonesteptothenext.




5. Thepatternofchangeinthe

areaofthefiguresfromonesteptothenext.




6.

0 54 82 3.58 16 0.5




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7. Thealgaepopulationincreasesby3%eachyear.




8.




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© 2012 www.flickr.com/photos/JennuineC

aptures

4.2Where’sMyChange?APracticeUnderstandingTask2LookthroughtheproblemsthatyouworkedwithintheSortingOuttheChangetask.Chooseoneproblemfromyourlinearcategory(equaldifferencesoverequalintervals)andoneproblemfromyourexponentialcategory(equalfactorsoverequalintervals).Addasmanyrepresentationsasyoucantotheproblemyouselected.Forinstance,ifyouchooseproblem#1whichisatable,youshouldtrytorepresentthefunctionwithagraph,anexplicitequation,arecursiveequation,andastorycontext.Identifytherateofchangeinthefunction.Ifthefunctionislinear,identifytheconstantrateofchange.Ifthefunctionisexponential,identifythefactorofchange.Linear:Scenario/Context:RecursiveEquation: ExplicitEquation:Table: Graph:

Explainhowtherateofchangeappearsineachofyourrepresentations.

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Exponential:Scenario/Context:RecursiveEquation: ExplicitEquation:Table: Graph:

Explainhowtherateofchangeappearsineachofyourrepresentations.

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4.2ResourcePage

LinearFunctions:EqualDifferencesOverEqualIntervals

ExponentialFunctions:

EqualFactorsOverEqualIntervals

Representation: Wecanidentifytherateofchangeby: Representation: Wecanidentifythechangefactorsby:

Tables Tables

Graphs Graphs

Equations Equations

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Name: LinearandExponentialFunctions 4.2Ready,Set,Go!ReadyTopic:RatesofchangeinlinearmodelsSaywhichsituationhasthegreatestrateofchange1. Theamountofstretchinashortbungeecordstretches6incheswhenstretchedbya3poundweight.Aslinky

stretches3feetwhenstretchedbya1poundweight.2. Asunflowerthatgrows2incheseverydayoranamaryllisthatgrows18inchesinoneweek.3. Pumping25gallonsofgasintoatruckin3minutesorfillingabathtubwith40gallonsofwaterin5minutes.4. Ridingabike10milesin1hourorjogging3milesin24minutes.Topic:Recognizingthegreaterrateofchangewhencomparinglinearfunctionsandexponentialfunctions.Identifywhethersituation“a”orsituation“b”hasthegreaterrateofchange.5. a.

10 489 438 387 33

b.

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6. a.

b.

7. a. Leehas$25withheldeachweekfromhissalarytopayforhissubwaypass.

b. Joseoweshisbrother$50.Hehaspromisedtopayhalfofwhatheoweseachweekuntilthedebtispaid.

8. a.x 6 10 14 18y 13 15 17 19

b. Thenumberofrhombiineachshape.

Figure1 Figure2 Figure3

9. a. 2 5 b. Inthechildren’sbook,TheMagicPot,everytimeyouputoneobjectintothepot,twoofthesameobjectscomeout.Imaginethatyouhave5magicpots.

10. a. Examinethegraphattheleftfrom0to1.Whichgraphdoyouthinkisgrowingfaster?

b. Nowlookatthegraphfrom2to3.Whichgraphis

growingfasterinthisinterval?

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SetTopic:linearratesofchangeDeterminetherateofchangeineachtablebelow.11.

3 131 50 13 11

12.

4 40 42 86 16

13.10 145 825 050 10

14.Completethetablesbelowtocreatetheindicatedtypeoffunction.

a. LinearFunction b. ExponentialFunction

5 5

4 4

3 3

2 2

1 1

0 0 Topic:Recognizinglinearandexponentialfunctions.Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.15.Thepopulationofatownisdecreasingatarateof

1.5%peryear.

16. Joanearnsasalaryof$30,000peryearplusa4.25%commissiononsales

17.Thenumberofgiftsreceivedeachdayof“The12DaysofVacation”asafunctionoftheday.(“Onthe4th dayofVacationmytruelovegavetome,4callingbirds,3Frenchhens,2turtledoves,andapartridgeinapeartree.”)

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GoTopic:Recursiveandexplicitequationsofgeometricsequences.Writetherecursiveandexplicitequationsforeachgeometricsequence.18.Marissahassaved$1000inajar.Sheplanstowithdrawhalfofwhat’sremaininginthejarattheendofeach

month.Recursiveequation: ExplicitEquation:

19.

Time(Days)

NumberofBacteria

1 102 1003 10004 10000

RecursiveEquation:ExplicitEquation:

10.Foldsinpaper

Numberofrectangles

0 11 22 43 8


21. 1024, 256, 64, 16, …

RecursiveEquation:

ExplicitEquation:

22. 3, 9, 27, 81, …


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Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.23.

x 1 2 3 4 5

y 2 162

24.

x 1 2 3 4 5

y 3

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4.3ResourcePageLinearModel:

1910 1930 1950 1970 1990 2010 2030

Year#

Population 92 250

ExplicitFormula:ExponentialModel:

Year Year# PercentGrowth

5% 10%

1910 92 92 92 92 92 92 92

1930

1950

1970

1990 250

2010

2030

Percenttoolowortoohigh?

ExplicitFormula:

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4.3Warm‐UpGraphinglinearequationsinslope‐interceptform.Graphthefollowingequations.1. 3 1 2. 5 4

3. 1 2 3 4. 2 4

5. – 6 6. 5 6

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©www.flickr.com/photos/arenam

ont

4.3Growing,Growing,GoneASolidifyUnderstandingTask21. TheU.S.populationin1910was92millionpeople.In1990thepopulationwas250

million.Createbothalinearandanexponentialmodelofthepopulationfrom1910to2030,withprojecteddatapointsatleastevery20years,startingin1910.

LinearModel: ExponentialModel: Equationfromdatapoints: Equationfromdatapoints:

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2. TheactualU.S.populationdata(inmillions)was:1930: 122.81950: 152.31970: 204.9

WhichmodelprovidesabetterforecastoftheU.S.populationfortheyear2030?Explainyouranswer.

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Name: LinearandExponentialFunctions 4.3Ready,Set,Go!ReadyTopic:Comparingarithmeticandgeometricsequences.The1stand5thtermsofasequencearegiven.Fillinthemissingnumbersforanarithmeticsequence.Thenfillinthenumbersforageometricsequence.1.

Arithmetic 3 48

Geometric 3 48

2.

Arithmetic 12 0.75

Geometric 12 0.75

Topic:ComparinglinearandexponentialmodelsInquestions3‐8,comparedifferentcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible. 4 3 4 3

3. Typeofgrowth

4. Whatkindofsequencecorrespondstoeachmodel?

5. Makeatableofvalues

x y x y

6. Findtherateofchange

7. Grapheachequation.

Comparethegraphs.Whatisthesame?Whatisdifferent?

8. Findthey‐interceptforeachfunction.

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9. Findthey‐interceptsforthefollowingequations a. 3

b. 3

10.Explainhowyoucanfindthey‐interceptofalinearequationandhowthatisdifferentfromfindingthe

y‐interceptofageometricequation.SetTopic:Comparinglinearandexponentialfunctions.11.Calcu‐ramahadanetincomeof5milliondollarsin2010,whileasmallcompetingcompany,Computafest,hada

netincomeof2milliondollars.ThemanagementofCalcu‐ramadevelopsabusinessplanforfuturegrowththatprojectsanincreaseinnetincomeof0.5millionperyear,whilethemanagementofComputafestdevelopsaplanaimedatincreasingitsnetincomeby15%eachyear.

a. Expresstheprojectednetincomesinthesetwobusinessplansasrecursiveformulas.

b. Writeanexplicitequationforthenetincomeasafunctionoftimeforeachcompany’sbusinessplan.

c. Compareyouranswersinaandb.Howarethetworepresentationssimilar?Howdotheydiffer?Whatrelationshipsarehighlightedineachrepresentation?

d. Explainwhyifbothcompaniesareabletomeettheirnetincomegrowthgoals,thenetincomeofComputafestwilleventuallybelargerthanthatofCalcu‐rama.InwhatyearwillthenetincomeofComputafestbelargerthanthatofCalcu‐rama?

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GoTopic:Writingexplicitequationsforlinearandexponentialmodels.Writetheexplicitequationforthetablesandgraphsbelow.12. 13.

1 4 81

0 2 3 27

1 10 2 9

2 50 1 3

14.

15.

16.

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Topic:Solvingsystemsthroughgraphing.Findthesolutiontothesystemsofequationsbygraphing.

17. 3 4

18.2 23 15

19.4

2 1 0

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© 2012 www.flickr.com/photos/teddylam

bec

4.4MakingMyPointASolidifyUnderstandingTaskZacandSionewereworkingonpredictingthenumberofquiltblocksinthispattern:

Whentheycomparedtheirresults,theyhadaninterestingdiscussion:Zac: Igot 6 1becauseInoticedthat6blockswereaddedeachtimesothepatternmusthavestartedwith

1blockat 0.Sione: Igot 6 1 7becauseInoticedthatat 1therewere7blocksandat 2therewere13,soI

usedmytabletoseethatIcouldgetthenumberofblocksbytakingonelessthanthen,multiplyingby6(becausethereare6newblocksineachfigure)andthenadding7becausethat’showmanyblocksinthefirstfigure.Here’smytable:

1 2 3 n7 13 19 6 1 7

1. WhatdoyouthinkaboutthestrategiesthatZacandSioneused?Areeitherofthemcorrect?Whyorwhynot?

Useasmanyrepresentationsasyoucantosupportyouranswer.

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ThenextproblemZacandSioneworkedonwastowritetheequationofthelineshownonthegraphbelow.

Whentheywerefinished,hereistheconversationtheyhadabouthowtheygottheirequations:Sione: Itwashardformetotellwherethegraphcrossedtheyaxis,soIfoundtwopointsthatIcouldreadeasily,

1, 3 and 9, 7 .Ifiguredoutthattheslopewas andmadeatableandcheckeditagainstthegraph.Here’smytable:

1 3 5 7 9

3 4 5 6 7 1 3

Iwassurprisedtonoticethatthepatternwastostartwiththen,subtract1,multiplybytheslopeandthensubtract3.Igottheequation 1 3.

Zac: Hey—IthinkIdidsomethingsimilar,butIusedthepoints, 7, 6 and 9, 7 .

Iendedupwiththeequation: 9 7.Oneofusmustbewrongbecauseyourssaysthatyousubtract1tothenandminesaysthatyousubtract9.Howcanwebothberight?

2. Whatdoyousay?Cantheybothberight?

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Zac: Myequationmademewonderiftherewassomethingspecialaboutthepoint 9, 7 sinceitseemedtoappearinmyequation 9 7whenIlookedatthenumberpattern.NowI’mnoticing

somethinginteresting—thesamethingseemstohappenwithyourequation, 9 2andthepoints 9, 2 .

3. DescribethepatternthatZacisnoticing.4. FindanotherpointonthelinegivenaboveandwritetheequationthatwouldcomefromZac’spattern.5. Whatwouldthepatternlooklikewiththepoint , ifyouknewthattheslopeofthelinewasm?6. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?

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ZacandSionewentbacktoworkonanextensionofthequiltproblemtheywereworkingonbefore.Nowtheyhavethispattern:

Zac: Thisoneworksalotlikethelastquiltpatterntome.Theonlydifferenceisthatthepatternisdoubling,soI

knewitwasexponential.Ithoughtthatitstartswith7blocksanddoubles,sotheequationmustbe7 2 .

Sione: Idon’tknowaboutthat.Iagreethatitisanexponentialfunction—justlookatthatgrowthpattern.But,I

madethistable:

1 2 3

7 14 28 7 2

Iusedthenumbersinthetableandgotthisequation: 7 2 .Thisseemsjustlikeallthatstuffweweredoingwiththelines,butIthinkthatthegraphsofthesetwoequationswouldbedifferent.Thereissomethingdefinitelywronghere.

7. WhatisdifferentaboutthethinkingthatZacandSioneusedtocometodifferentequations?8. Howaretheirresultssimilartotheirresultsonthelinearquiltpatternabove?Howaretheydifferent?

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Zac: Iknow!Let’strydoingthesamethingwithyourexponentialfunctionasthelinearfunction.Whatifwetookthepoint 1, 7 andwrotetheequationthisway:

2 7

SeewhatIdid?Ididthesubtract1thingwiththexandthenaddedonthe7fromthey‐valueofthepoint.I’llbetthisisareallygoodshortcuttrick.

9. Graphthethreeexponentialfunctionstoverifytheirequivalenceortoshowthattheyarenotequivalent.

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Name: LinearandExponentialFunctions 4.4Ready,Set,Go!ReadyTopic:Writingequationsoflines.Writetheequationofalineinslope‐interceptform: ,usingthegiveninformation.1. 7, 4 2. , 3 3. 16, Writetheequationofthelineinpoint‐slopeform: ,usingthegiveninformation.4. 9, 0, 7 5. , 6, 1 6. 5, 4, 11 7. 2, 5 3, 10 8. 0, 9 3, 0 9. 4, 8 3, 1

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SetTopic:GraphinglinearandexponentialfunctionsMakeagraphofthefunctionbasedonthefollowinginformation.Addyouraxes.Chooseanappropriatescaleandlabelyourgraph.Thenwritetheequationofthefunction.10. Thebeginningvalueofthefunctionis5andits

valueis3unitssmallerateachstage.Equation:

11. Thebeginningvalueis16anditsvalueis smallerateachstage.

Equation:

12. Thebeginningvalueis1anditsvalueis10timesasbigateachstage.Equation:

13. Thebeginningvalueis 8anditsvalueis2unitslargerateachstage.

Equation:

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GoTopic:Recognizinglinearandexponentialfunctions.Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.14.3 4 3

15.ideofasquare Areaofasquare

1inch 1in22inches 4in23inches 9in24inches 16in2

16.

Topic:Slope‐interceptformRewritetheequationsinslope‐interceptform.17.2 10 6 12

18. 13 8 14

19. 11 7 2

20. 3 2 9 12

21. 2 10 25

22. 1 3

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4.5Warm‐UpThefourformsofalinearequationBelowarethe4formsofthesamelinearequation.Completethetable:

Slope‐Intercept Point‐Slope Standard RecursiveFormRateofChange x‐intercept y‐intercept

3 2 3 5 13 3 2 0 21 3

7 8 5 4 280 7

1

3 6 1 2 3 90 3

1

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©2012 www.flickr.com/photos/cannongod

4.5EfficiencyExpertsASolidifyUnderstandingTaskPart1:InvestigatingLinearFormsInourworksofar,wehaveworkedwithlinearandexponentialequationsinmanyforms.Someoftheformsofequationsandtheirnamesare:

Name Equation

SlopeInterceptForm

wheremistheslopeandbisthey‐intercept1

PointSlopeForm

wheremistheslopeand , arethecoordinatesofapointontheline

3 4

StandardForm

2 2

RecursiveForm1

Givenaninitialvalue andD=constantdifferenceinconsecutiveterms

0 1, 1

1. Createatableofvaluesforeachequationabove.Verifythatthefourequationsareequivalent.

SlopeInterceptForm: PointSlopeForm

StandardForm: RecursiveForm

2. Explainhowyouknowthatthefourequationsarelinear.

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ContextInvestigation:Whichformbesttellsthestory?Youhavebeenappointedasamathematicsefficiencyexpert.Yourjobistocomparethesefourformsofequationsfordifferentusesanddecidewhichformismostefficientandeffectiveforeachuse.Determinewhichequationwillbethemostefficientandeffectiveforeachscenario.3. Inhisjobsellingvacuums,Joemakes$500eachmonthplus$20foreachvacuumhesells.

StandardForm: Slope‐InterceptForm:20 500 20 500

4. TheTreeHuggerGranolaCompanymakestrailmixwithcandiesandnuts.Thecostofcandiesforatrailmixis

$2perpoundandthecostofthenutsis$1.50perpound.Thetotalcostofabatchoftrailmixis$540.

StandardForm: SlopeInterceptForm:2 1.5 540 360

5. GrandmaBillingsisworkingonaquiltwithblocksinthefollowingpattern.

StandardForm: RecursiveForm:6 1 1 7, 1 6

6. Apoolisfillingataconstantrateof5gallonsperminute.After13minutes,thereare500gallonsinthepool.

PointSlopeForm: StandardForm:5 13 500 5 435

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EquationInvestigation:Determinewhichofthefourformsoflinearequationstousetowriteeachequation.Thenwritetheequationusingthechosenform.7. Writetheequationofthelinewithaslopeof 2throughthepoint 2, 5 .8. Writetheequationofthelinethroughthepoints 1, 2 and 4, 1 .9. Writetheequationofthearithmeticsequencethatstartswith 7andeachtermdecreasesby3.GraphInvestigation:Graphthefollowingequations.Thendescribehoweachrepresentationofthelinearfunctionassistsincompletingthegraph.10. 5

Howdoestheslopeinterceptformassistincompletingthegraph?

11. 3 5 15

Howdoesthestandardformassistincompletingthegraph?

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12. 4 6 5

Howdoesthepointslopeformassistincompletingthegraph?

13. 0 2, 1 5

Howdoestherecursiveformassistincompletingthegraph?

EfficiencyUsingtheFourFormsofLinearEquations:14.Fillintheresourcepageattheendofthistaskwithyourfindingsforlinearfunctions.

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Part2:InvestigatingExponentialFormsDuringthecourseoftheyear,wehavealsoworkedwithformsofexponentialequations,withafewmorecomingbeforetheendofthemodule.Theformsofexponentialequationsthatwehaveseensofar:

Name Equation

ExplicitForm 10 3

RecursiveForm1 ⋅

Givenaninitialvalue r=constantratiobetweenconsecutiveterms

1 10, 1 ⋅ 3

ContextInvestigation:Whichformtellsthestorybest?Determinewhichequationwillbethemostefficientandeffectiveforeachscenario.15.GrandmaBillingshasstartedpiecingherquilttogetherandhascreatedthefollowinggrowthpattern:

Whichequationbestmodelsthenumberofsquaresineachblock?ExplicitForm: RecursiveForm:

7 2 1 7, 1 ⋅ 2

16.ThepopulationoftheresorttownofJavaHotSpringsin2003wasestimatedtobe35,000peoplewithan

annualrateofincreaseofabout2.4%.WhichequationbestmodelsthenumberofpeopleinJavaHotSprings,witht=thenumberofyearsfrom2003?

ExplicitForm: RecursiveForm:

35,000 1.024 0 35,000, 1 ⋅ 1.024

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GraphInvestigation:Graphthefollowingequations.Thendescribehoweachrepresentationofthelinearfunctionassistsincompletingthegraph.17. 2 1.8

Howdoestheexplicitformassistincompletingthegraph?

18. 0 5, 1 ⋅ 0.6

Howdoestherecursiveformassistincompletingthegraph?

EfficiencyUsingtheTwoFormsofExponentialEquations:19.Fillintheresourcepageattheendofthistaskwithyourfindingsforexponentialfunctions.

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4.5ResourcePage

Whendoyouusethisformfor…

FormofLinearEquation

Situationsincontext(i.e.storyproblems)

WritingEquations Graphing

SlopeInterceptForm

PointSlopeForm

StandardForm

RecursiveForm1 , 1

Whendoyouusethisformfor…

FormofExponentialEquation

Situationsincontext(i.e.storyproblems)

WritingEquations

Graphing

ExplicitForm⋅

RecursiveForm1 , 1 ⋅

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Name: LinearandExponentialFunctions 4.5Ready,Go!ReadyTopic:SimpleinterestWhenapersonborrowsmoney,thelenderusuallycharges“rent”onthemoney.This“rent”iscalledinterest.Simpleinterestisapercent“r”oftheoriginalamountborrowed“P”multipliedbythetime“t”,usuallyinyears.Theformulaforcalculatingtheinterestis .Calculatethesimpleinterestowedonthefollowingloans.1. $1000 11% 2 2. $6500 12.5% 5 3. $20,000 8.5% 6 4. $700 20% 6 GoTopic:Solvingmulti‐stepequationsSolvethefollowingequations5. 12 6 4 5 2 3 1

6. 5 2 4 3 5 19

7. 7 3 4 2 6 2 3 17

8. 2 1 6 3

9. Whatdoesitmeanwhenyouhavesolvedanequation?10.Explainhowalinearequationcanhavemorethanonesolution.

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Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.11.

x 1 2 3 4 5

y 10 0.625

12.

x 1 2 3 4 5

y g

13.

x 1 2 3 4 5

y 3 243

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4.6Warm‐UpInterestJuanitaborrowed$1,000andagreedtopay15%simpleinterestfor5years.Juanitadidnothavetomakeanypaymentsuntiltheendofthe5years,butthenshehadtopaybacktheamountborrowed“P”plusalloftheinterest“i”forthe5years“t.”BelowisachartthatshowshowmuchmoneyJuanitaowedthelenderattheendofeachyearoftheloan.

EndofYear

Interestowedfortheyear

Totalamountofmoneyowedtothelendertopaybacktheloan.

1 $1000 0.15 $150 A=Principal+interest=$1150

2 $1000 0.15 $150 $1300

3 $1000 0.15 $150 $1450

4 $1000 0.15 $150 $1600

5 $1000 0.15 $150 $1750

1. a. Lookforthepatternyouseeinthechartabovefortheamount(A)owedtothelender.Writeafunction

thatbestdescribesAwithrespecttotime(inyears). b. Whattypeoffunctiondoesthefunctionfrompartarepresents?2. Considerifthelendercharged15%oftheamountowedinsteadof15%oftheamountoftheoriginalloan.

Calculatetheinterestowedeachyearifthelenderrequired15%oftheamountowedattheendofeachyear.Notethattheinterestowedattheendofthefirstyearwouldstillbe$150.Completethetablebelow.EndofYear

StartingAmountfortheYear Interest

TotalAmountattheEndoftheYear

1 $1000 $1000 0.15 $150 $1150

2 $1150 $1150 0.15 $1150

3

4

5

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© 2012 www.flickr.com/photos/civisi

4.6UpaLittle,DownaLittleASolidifyUnderstandingTaskOneofthemostcommonapplicationsofexponentialgrowthiscompoundinterest.Forexample,MamaBigbucksputs$20,000inabanksavingsaccountthatpays3%interestcompoundedannually.“Compoundedannually”meansthatattheendofthefirstyear,thebankpaysMama3%of$20,000,sotheyadd$600totheaccount.Mamaleavesheroriginalmoney($20,000)andtheinterest($600)intheaccountforayear.Attheendofthesecondyearthebankwillpayinterestontheentireamount,$20,600.Sincethebankispayinginterestonapreviousinterestamount,thisiscalled“compoundinterest”.1. ModeltheamountofmoneyinMamaBigbucks’bankaccountaftertyears.

#ofYears Money

Equation:

2. Isthefunctionlinearofexponential?Supportyouranswerwithfeaturesfromthetableandgraph.3. UseyourmodeltofindtheamountofmoneythatMamahasinheraccountafter20years.

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Aformulathatisoftenusedforcalculatingtheamountofmoneyinanaccountthatiscompoundedannuallyis:1

Where:A=amountofmoneyintheaccountaftertyearsP=principal,theoriginalamountoftheinvestmentr=theannualinterestratet=thetimeinyears4. ApplythisformulatoMama’sbankaccountandcomparetheresulttothemodelthatyoucreated.5. Basedupontheworkthatyoudidincreatingyourmodel,explainthe 1 partoftheformula.Anothercommonapplicationofexponentialfunctionsisdepreciation.Whenthevalueofsomethingyoubuygoesdownacertainpercenteachyear,itiscalleddepreciation.Forexample,MamaBigbucksbuysacarfor$20,000anditdepreciatesatarateof3%peryear.Attheendofthefirstyear,thecarloses3%ofitsoriginalvalue,soitisnowworth$19,400.6. ModelthevalueofMama’scaraftertyears.

#ofYears Money

Equation:

7. Isthefunctionlinearofexponential?Supportyouranswerwithfeaturesfromthetableandgraph.

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8. UseyourmodeltofindhowmanyyearswillittakeforMama’scartobeworthlessthan$500?9. HowisthesituationofMama’scarsimilartoMama’sbankaccount?10.Whatdifferencesdoyouseeinthetwosituations?11.ConsideryourmodelforthevalueofMama’scaranddevelopageneralformulafordepreciation.

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Name: LinearandExponentialFunctions 4.6Set,Go!SetTopic:Evaluateusingtheformulasforsimpleinterestorcompoundinterest.Giventheformulaforsimpleinterest: ,calculatethesimpleinterestpaid.(Remember,i=interest,P=theprincipal,r=theinterestrateperyearasadecimal,t=timeinyears)1. Findthesimpleinterestyouwillpayona5yearloanof$7,000at11%peryear.2. Howmuchinterestwillyoupayin2yearsonaloanof$1500at4.5%peryear?Use tocompletethetable.Allinterestratesareannual.

i = P r t

3. $11,275 12% 3years

4. $1,428 $5,100 4%

5. $93.75 $1,250 6months

6. $54 8% 9monthsGiventheformulaforcompoundinterest: 1 ,writeacompoundinterestfunctiontomodeleachsituation.Thencalculatethebalanceafterthegivennumberofyears.(Remember:A=thebalanceaftertyears,P=theprincipal,t=thetimeinyears,r=theannualinterestrateexpressedasadecimal)7. $22,000investedatarateof3.5%compoundedannuallyfor6years.8. $4300investedatarateof2.8%compoundedannuallyfor15years.

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9. Supposethatwhenyouare15yearsold,amagicgeniegivesyouthechoiceofinvesting$10,000atarateof7%or$5,000atarateof12%.Eitherchoicewillbecompoundedannually.Themoneywillbeyourswhenyouare65yearsold.Whichinvestmentwouldbethebest?Justifyyouranswer.

GoTopic:UsingorderofoperationswhenevaluatingexpressionsEvaluatetheexpressionsforthegivenvaluesofthevariables.10. 6 10;when 7and 3 11. ;when 2,and 612. 1 ;when 5,and 3 13. 9 4 ;when 4,and 514. ;when 7,and 2 15. 7 2 ;when 2,and 4

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4.7Warm‐UpWritingalinearequationinslope‐interceptformWritetheequationofthelineinslope‐interceptformgiventhefollowinginformation.(PandQarepointsontheline)1. 0 6, 1 2. 4, : 5, 8 3. 14 2 9 0 4. : 17, 4 , : 5, 26 5. 9 6 6. 11, 8 , : 1, 8

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©2012 www.flickr.com/photos/bfurlong

4.7XMarkstheSpotAPracticeUnderstandingTaskTablePuzzles1. Usethetablestofindthemissingvaluesofx:

a. .

2 4.4

10 10

8.6

4 0.2

1.2

b.

10 10

3 6

5

0

10

c. Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwo

tablesabove?

d.

5 243

81

3

2 9

e.

5 32

8

1

2

f. Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwo

tablesabove?

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2. Whatstrategydidyouusetofindthesolutionstoequations(fromquestion1c)generatedbythetablesthatcontainedlinearfunctions?

3. Whatstrategydidyouusetofindthesolutionstoequations(fromquestion1f)generatedbythetablesthat

containedexponentialfunctions?GraphPuzzles4. Thegraphof 3isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.

a. 5 3 __________ LabelthesolutionwithanAonthegraphb. 3 1 __________ LabelthesolutionwithaBonthegraphc. 0.5 3 1 __________ LabelthesolutionwithaConthegraph

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5. Thegraphof 3 isgivenbelow.Usethegraphtosolvetheequationforxandlabelthesolutions.

a. 3 __________ LabelthesolutionwithanAonthegraphb. 3 9 __________ LabelthesolutionwithaBonthegraphc. 3√3 3 __________ LabelthesolutionwithaConthegraphd. 1 3 __________ LabelthesolutionwithaDonthegraphe. 6 3 __________ LabelthesolutionwithanEonthegraph

6. Howdoesthegraphhelptofindsolutionsforx?EquationPuzzles:Solveeachequationforx:7. 5 125 8. 7 6 9 9. 10 10,000

10.2.5 0.9 1.3 11.6

12. 16

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Name: LinearandExponentialFunctions 4.7Ready,Set,Go!ReadyTopic:Discretevs.continuous1. Giveacontextualexamplethatcanbemodeledbyadiscretefunction.2. Giveacontextualexamplethatcanbemodeledbyacontinuousfunction.Topic:Arithmeticandgeometricmeans3. Thefirstand5thtermsofasequencearegiven.Fillinthemissingnumbersforanarithmeticsequence.Then

fillinthenumbersforageometricsequence.

Arithmetic 6250 10Geometric 6250 10

Topic:Comparingratesofchange4. Comparetherateofchangeinthepairoffunctionsinthe

graphbyidentifyingtheintervalwhereitappearsthatischangingfasterandtheintervalwhereitappears

that ischangingfaster.Verifyyourconclusionsbymakingatableofvaluesforeachfunctionandexploringtheratesofchangeinyourtables.

Topic:Determiningifasequencesislinear,exponential,orneither.5. Identifythefollowingsequencesaslinear,exponential,orneither.

a. 23, 6, 11, 28, … b. 49, 36, 25, 16, … c. 5125, 1025, 205, 41,…d. 2, 6, 24, 120,… e. 0.12, 0.36, 1.08, 3.24, … f. 21, 24.5, 28, 31.5, …

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SetTopic:ComparinglinearandexponentialfunctionsusingmultiplerepresentationsDescribethedefiningcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible.

6 5 6 5

6. Typeofgrowth

7. Whatkindofsequencecorrespondstoeachmodel?

8. Makeatableofvalues

x y

x y

9. Findtherateofchange

10.Grapheachequation.Comparethegraphs.Whatisthesame?Whatisdifferent?

11.Findthey‐interceptforeachfunction.

12.Writetherecursiveformofeachequation.

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Therewere2girlsinmygrandmother’sfamily,mymotherandmyaunt.Theyeachhad3daughters.Mytwosisters,3cousins,andIeachhad3daughters.Eachoneofour3daughtershavehad3daughters.13. Ifthepatternofeachgirlhaving3daughterscontinuesfor2moregenerations(mymomandauntbeingthe1st

generation,Iwanttoknowaboutthe5thgeneration),howmanydaughterswillbebornthen? 14.Writetheexplicitequationforthispattern. 15.Createatableandagraphdescribingthispattern.Isthissituationdiscreteorcontinuous?

GoTopic:Solvingmulti‐stepequationsSolvethefollowingequations.16.5 3 2 6

17. 6 12 10 2 3 6

18.13 12

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Recallthefollowingformulas: Simpleinterest Compoundinterest Usingtheformulasforsimpleinterestorcompoundinterest,calculatethefollowing.19.Thesimpleinterestonaloanof$12,000ataninterestrateof17%for6years.20.Thesimpleinterestonaloanof$20,000ataninterestrateof11%for5years.21.Theamountowedonaloanof$20,000,at11%,compoundedannuallyfor5years.22.Comparetheinterestpaidin#26totheinterestpaidin#27.Whichkindofinterestdoyouwantifyouhaveto

takeoutaloan?23.Theamountinyoursavingsaccountattheendof30years,ifyoubeganwith$2500andearnedaninterestrate

of7%compoundedannually.

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Module4ReviewHomeworkCharacteristicsofFunctions:Foreachofthefunctionsfindthefollowinginformation.1.

x‐intercept:____________________y‐intercept:____________________Rateofchangebetween 1and 2:____

2.

x‐intercept:_____________________y‐intercept:_____________________Rateofchangebetween 1and 2:____

3. Discussandcomparethefunctionsbyanalyzingtheratesofchange,intercepts,andwhereonefunctionis

greaterorlessthantheother.: rateofchange:

x‐intercept:y‐intercept:

: rateofchange:

x‐intercept:y‐intercept:

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LinearandExponentialModels:4. Writeanexplicitformulatomodelthenumberofdotsperday.Day1 Day2 Day3

Modelthefunctionusingthetableprovided: Useyourtabletocreateagraphofthefunction:

Day NumberofDots

5. Sherryhasahugedollcollectionof80dolls.Hermomtellsherthatsheneedstogetridof5peryeartogetit

downtoadecentnumberbeforeleavingforcollege.Writeanexplicitformulatomodelthenumberofdollsperyear.Ifsheis12,howmanywillshehaveleftwhensheis18?

6. YouboughtaToyotaCorollain2004for$12,500.Thecar’svaluedepreciatesby7%ayear.Howmuchisthe

carworthnow?Howmuchisitworthin2020?

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7. Thepopulationofalargecityincreasesbyarateof3%ayear.Whenthe2000censuswastaken,thepopulationwas1.2million.a. Writeamodelforthispopulationgrowth.

b. Whatshouldthepopulationbenow?Whatistheprojectedpopulationfor2020?

8. BankPlans:Supposeyouworkedmowinglawnsallsummerandearned$50.Twosavingsinstitutions,LinearLuckandExponentialExperimentwantyoutoletthem“holdontoyourmoney”forawhile.

LinearLuck: Thissavingsplanwilladd$100toyourbalanceforeverymonththatyouleaveyourmoneyin

theaccount.ExponentialExperiment: Thissavingsplanwillmultiplyyourbalanceby2everymonththatyouleaveyour

moneyintheiraccount.Analyzetheplans:Writetheexplicitfunctionforeachaccount,anddecidewhichaccountisbestatwhattimes.

SDUHSD Math 1 Module 4 SE - San Dieguito Union High ...teachers.sduhsd.net/dpillsbury/IM1/Modules/SDUHSD Math 1...2 SDUHSD Math 1 College Prep Module 4 Overview Prerequisite Concepts

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