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The Mathematics Vision Project
Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 2
Linear & Exponential Functions
SECONDARY
MATH ONE
An Integrated Approach
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONETIAL FUNCTIONS
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
MODULE 2 - TABLE OF CONTENTS
LINEAR AND EXPONENTIAL FUNCTIONS
2.1 Piggies and Pools – A Develop Understanding Task
Introducing continuous linear and exponential functions (F.IF.3)
READY, SET, GO Homework: Linear and Exponential Functions 2.1
2.2 Shh! Please Be Discreet (Discrete!) – A Solidify Understanding Task
Connecting context with domain and distinctions between discrete and continuous functions
(F.IF.3, F.BF.1a, F.LE.1, F.LE.2)
READY, SET, GO Homework: Linear and Exponential Functions 2.2
2.3 Linear Exponential or Neither – A Practice Understanding Task
Distinguishing between linear and exponential functions using various representations (F.LE.3, F.LE.5)
READY, SET, GO Homework: Linear and Exponential Functions 2.3
2.4 Getting Down to Business – A Solidify Understanding Task
Comparing growth of linear and exponential models (F.LE.2, F.LE.3, F.LE.5, F.IF.7, F.BF.2)
READY, SET, GO Homework: Linear and Exponential Functions 2.4
2.5 Making My Point – A Solidify Understanding Task
Interpreting equations that model linear and exponential functions (A.SSE.1, A.CED.2, F.LE.5, A.SSE.6)
READY, SET, GO Homework: Linear and Exponential Functions 2.5
2.6 Form Follows Function – A Solidify Understanding Task
Building fluency and efficiency in working with linear and exponential functions in their various forms
(F.LE.2, F.LE.5, F.IF.7, A.SSE.6)
READY, SET, GO Homework: Linear and Exponential Functions 2.6
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.1 Connecting the Dots: Piggies and Pools
A Develop Understanding Task
1. Mylittlesister,Savannah,isthreeyearsold.Shehasapiggybankthatshewantstofill.She
startedwithfivepenniesandeachdaywhenIcomehomefromschool,sheisexcitedwhenI
giveherthreepenniesthatareleftoverfrommylunchmoney.Useatable,agraph,andan
equationtocreateamathematicalmodelforthenumberofpenniesinthepiggybankondayn.
2. Ourfamilyhasasmallpoolforrelaxinginthesummerthatholds1500gallonsofwater.I
decidedtofillthepoolforthesummer.WhenIhad5gallonsofwaterinthepool,Idecidedthat
Ididn’twanttostandoutsideandwatchthepoolfill,soIhadtofigureouthowlongitwould
takesothatIcouldleave,butcomebacktoturnoffthewaterattherighttime.Icheckedthe
flowonthehoseandfoundthatitwasfillingthepoolatarateof2gallonseveryminute.Usea
table,agraph,andanequationtocreateamathematicalmodelforthenumberofgallonsof
waterinthepoolattminutes.
CCBYStockm
onkeys.com
https://flic.kr/p/d8t7K
W
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SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3. I’mmoresophisticatedthanmylittlesistersoIsavemymoneyinabankaccountthatpaysme
3%interestonthemoneyintheaccountattheendofeachmonth.(IfItakemymoneyout
beforetheendofthemonth,Idon’tearnanyinterestforthemonth.)Istartedtheaccountwith
$50thatIgotformybirthday.Useatable,agraph,andanequationtocreateamathematical
modeloftheamountofmoneyIwillhaveintheaccountaftermmonths.
4. Attheendofthesummer,Idecidetodrainthe1500gallonswimmingpool.Inoticedthatit
drainsfasterwhenthereismorewaterinthepool.Thatwasinterestingtome,soIdecidedto
measuretherateatwhichitdrains.Ifoundthat3%wasdrainingoutofthepooleveryminute.
Useatable,agraph,andanequationtocreateamathematicalmodelofthegallonsofwaterin
thepoolattminutes.
2
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.1
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5. Compareproblems1and3.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?
6. Compareproblems1and2.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?
7. Compareproblems3and4.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?
3
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.1
READY Topic:RecognizingarithmeticandgeometricsequencesPredictthenext2termsinthesequence.Statewhetherthesequenceisarithmetic,geometric,orneither.Justifyyouranswer.1.4,-20,100,-500,... 2.3,5,8,12,...3.64,48,36,27,... 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. SET Topic:DiscreteandcontinuousrelationshipsIdentifywhetherthefollowingstatementsrepresentadiscreteoracontinuousrelationship.10.Thehaironyourheadgrows½inchpermonth.11.Foreverytonofpaperthatisrecycled,17treesaresaved.12.Approximately3.24billiongallonsofwaterflowoverNiagaraFallsdaily.13.Theaveragepersonlaughs15timesperday.14.ThecityofBuenosAiresadds6,000tonsoftrashtoitslandfillseveryday.15.DuringtheGreatDepression,stockmarketpricesfell75%.
READY, SET, GO! Name PeriodDate
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.1
Mathematics Vision Project
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2.1
GO Topic:Solvingone-stepequationsEitherfindorusetheunitrateforeachofthequestionsbelow.16.Applesareonsaleatthemarket4poundsfor$2.00.Whatistheprice(incents)foronepound?17.Threeapplesweighaboutapound.Abouthowmuchwouldoneapplecost?(Roundtothenearestcent.)18.Onedozeneggscost$1.98.Howmuchdoes1eggcost?(Roundtothenearestcent.)19.Onedozeneggscost$1.98.Ifthechargeattheregisterforonlyeggs,withouttax,was$11.88,howmanydozenwerepurchased?
20.BestBuyShoeshadabacktoschoolspecial.Thetotalbillforfourpairsofshoescameto$69.24(beforetax.)Whatwastheaveragepriceforeachpairofshoes?21.Ifyouonlypurchased1pairofshoesatBestBuyShoesinsteadofthefourdescribedinproblem20,howmuchwouldyouhavepaid,basedontheaverageprice?Solveforx.Showyourwork.22.6! = 72
23.4! = 200 24.3! = 50
25.12! = 25.80
26.!! ! = 17.31 27.4! = 69.24
28.12! = 198
29.1.98! = 11.88 30.!! ! = 2
31.Someoftheproblems22–30couldrepresenttheworkyoudidtoanswerquestions16–21.Writethe
numberoftheequationnexttothestoryitrepresents.
5
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.2 Shh! Please Be Discreet (Discrete)!
A Solidify Understanding Task
1. TheLibraryofCongressinWashingtonD.C.isconsideredthelargestlibraryintheworld.Theyoftenreceiveboxesofbookstobeaddedtotheircollection.Sincebookscanbequiteheavy,theyaren’tshippedinbigboxes.If,onaverage,eachboxcontainsabout8books,howmanybooksarereceivedbythelibraryin6boxes,10boxes,ornboxes?a. Useatable,agraph,andanequationtomodelthissituation.
b. Identifythedomainofthefunction.
2. ManyofthebooksattheLibraryofCongressareelectronic.Ifabout13e-bookscanbedownloadedontothecomputereachhour,howmanye-bookscanbeaddedtothelibraryin3hours,5hours,ornhours(assumingthatthecomputermemoryisnotlimited)?a. Useatable,agraph,andanequationtomodelthissituation.
b. Identifythedomainofthefunction.
CCBYRoche
lleHartm
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SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3. Thelibrariansworktokeepthelibraryorderlyandputbooksbackintotheirproperplacesaftertheyhavebeenused.Ifalibrariancansortandshelve3booksinaminute,howmanybooksdoesthatlibrariantakecareofin3hours,5hours,ornhours?Useatable,agraph,andanequationtomodelthissituation.
4. Woulditmakesenseinanyofthesesituationsfortheretobeatimewhen32.5bookshadbeenshipped,downloadedintothecomputerorplacedontheshelf?
5. Whichofthesesituations(inproblems1-3)representadiscretefunctionandwhichrepresentacontinuousfunction?Justifyyouranswer.
7
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. Agiantpieceofpaperiscutintothreeequalpiecesandtheneachofthoseiscutintothreeequalpiecesandsoforth.Howmanypaperswilltherebeafteraroundof10cuts?20cuts?ncuts?
ZeroCuts OneCut TwoCuts
a. Useatable,agraph,andanequationtomodelthissituation.
b. Identifythedomainofthefunction.
c. Woulditmakesensetolookforthenumberofpiecesofpaperat5.2cuts?Why?d.Woulditmakesensetolookforthenumberofcutsittakestomake53.6papers?Why?
8
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7. Medicinetakenbyapatientbreaksdowninthepatient’sbloodstreamanddissipatesoutofthepatient’ssystem.Supposeadoseof60milligramsofanti-parasitemedicineisgiventoadogandthemedicinebreaksdownsuchthat20%ofthemedicinebecomesineffectiveeveryhour.Howmuchofthe60milligramdoseisstillactiveinthedog’sbloodstreamafter3hours,after4.25hours,afternhours?a. Useatable,agraph,andanequationtomodelthissituation.
b. Identifythedomainofthefunction.
c. Woulditmakesensetolookforanamountofactivemedicineat3.8hours?Why?
d. Woulditmakesensetolookforwhenthereis35milligramsofmedicine?Why?
9
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
8. Whichofthefunctionsmodeledin#6and#7arediscreteandwhicharecontinuous?Why?
9. Whatneedstobeconsideredwhenlookingatasituationorcontextanddecidingifitfitsbestwithadiscreteorcontinuousmodel?
10. Describethedifferencesineachrepresentation(table,graph,andequation)fordiscreteandcontinuousfunctions.
11. Whichofthefunctionsmodeledabovearelinear?Whichareexponential?Why?
10
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
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2.2
READY
Topic:Comparingratesofchangeinlinearsituations.
Statewhichsituationhasthegreatestrateofchange
1.Theamountofstretchinashortbungeecordstretches6incheswhenstretchedbya3poundweight.Aslinkystretches3feetwhenstretchedbya1poundweight.2.Asunflowerthatgrows2incheseverydayoranamaryllisthatgrows18inchesinoneweek.3.Pumping25gallonsofgasintoatruckin3minutesorfillingabathtubwith40gallonsofwaterin5minutes.4.Ridingabike10milesin1hourorjogging3milesin24minutes.
SET
Topic:Discreteandcontinuousrelationships
Identifywhetherthefollowingitemsbestfitwithadiscreteoracontinuousmodel.Thendeterminewhetheritisalinear(arithmetic)orexponential(geometric)relationshipthatisbeingdescribed.5. Thefreewayconstructioncrewpours300ftofconcreteinaday.
6. Foreveryhourthatpasses,theamountofareainfectedbythebacteriadoubles.
7. Tomeetthedemandsplacedonthemthebricklayershavestartedlaying5%morebrickseachday.
8. Theaveragepersontakes10,000stepsinaday.
9. ThecityofBuenosAireshasbeenadding8%toitspopulationeveryyear.
10.AttheheadwatersoftheMississippiRiverthewaterflowsatasurfacerateof1.2milesperhour.
11.a.! ! = ! ! − 1 + 3; ! 1 = 5b. c.! ! = 2!(7)
READY, SET, GO! Name PeriodDate
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.2
Mathematics Vision Project
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2.2
GO
Topic: Solving one-step equations Solve the following equations. Remember that what you do to one side of the equation must also be done to the other side. (Show your work, even if you can do these in your head.) Example: Solve for x . 1! + 7 = 23 !"" − 7 !! !"#ℎ !"#$! !" !ℎ! !"#$%&'(.
Example: Solve for x. 9! = 63 !"#$%&#' !"#ℎ !"#$! !" !ℎ! !"#$%&'( !" !! .
Note that multiplying by !! gives the same result as dividing everything by 9.
11. 1! + 16 = 36 12. 1! − 13 = 10 13. 1! − 8 = −3 14. 8! = 56 15. −11! = 88 16. 425! = 850
17. !! ! = 10 18. − !! ! = −1 19.
!! ! = −9
12
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.3 Linear, Exponential or Neither?
A Practice Understanding Task
Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.Giveatleast2reasonsforyouranswer.
1.
Linear Exponential Neither
Why?
2. TennisTournament
Rounds 1 2 3 4 5Number
ofPlayersleft
64 32 16 8 4
Thereare4playersremainingafter5rounds
Linear Exponential Neither
Why?
CCBYJohn
Sho
rtland
https://flic.kr/p/a8u
zeA
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SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3.! = 4!
Linear Exponential Neither
Why?
4.Thisfunctionisdecreasingataconstantrate.
Linear Exponential Neither
Why?
5. Linear Exponential Neither
Why?
14
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6.Aperson'sheightasafunctionofaperson'sage(fromage0to100)
Linear Exponential Neither
Why?
7.−3! = 4! + 7
Linear Exponential Neither
Why?
8.x y-2 230 52 -134 -316 -49
Linear Exponential Neither
Why?
15
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9.HeightinInches ShoeSize
62 674 1370 967 1153 458 7
Linear Exponential Neither
Why?
10.ThenumberofcellphoneusersinCentervilleasa
functionofyears,ifthenumberofusersis
increasingby75%eachyear.
Linear Exponential Neither
Why?
11. Linear Exponential Neither
Why?
16
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
12.Thetimeittakesyoutogettoworkasafunctionthespeedatwhichyoudrive
Linear Exponential Neither
Why?
13.! = 7!!
Linear Exponential Neither
Why?
14.Eachpointonthegraphisexactly1/3ofthepreviouspoint.
Linear Exponential Neither
Why?
17
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
15.! 1 = 7, ! 2 = 7, ! ! = ! ! − 1 + !(! − 2)
Linear Exponential Neither
Why?
16.
! 1 = 1, ! ! = 23 !(! − 1)
Linear Exponential Neither
Why?
18
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
2.3
READY Topic:Comparingratesofchangeinbothlinearandexponentialsituations.Identifywhethersituation“a”orsituation“b”hasagreaterrateofchange.
1. x y
-10 -48
-9 -43
-8 -38
-7 -33
a.
b.
2. a.
b.
3. a.Leehas$25withheldeachweekfromhissalarytopayforhissubwaypass.
b.Joseoweshisbrother$50.Hehaspromisedtopayhalfofwhatheoweseachweekuntilthedebtispaid.
4. a.
x 6 10 14 18
y 13 15 17 19
b.Thenumberofrhombiineachshape.
Figure1Figure2Figure3
5. a.! = 2(5)! b.Inthechildren'sbook,TheMagicPot,everytimeyouputoneobjectintothepot,twoofthesameobjectcomeout.Imaginethatyouhave5magicpots.
READY, SET, GO! Name PeriodDate
-5
-5
5
5
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
Mathematics Vision Project
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2.3
SET Topic:Recognizinglinearandexponentialfunctions.Basedoneachofthegivenrepresentationsofafunctiondetermineifitislinear,exponentialorneither.6.Thepopulationofatownisdecreasingatarateof1.5%peryear.
7.Joanearnsasalaryof$30,000peryearplusa4.25%commissiononsales.
8.3x+4y=-3
9.Thenumberofgiftsreceivedeachdayof”The12DaysofChristmas”asafunctionoftheday.(“Onthe4thdayofChristmasmytruelovegavetome,4callingbirds,3Frenchhens,2turtledoves,andapartridgeinapeartree.”)
10.
11.
Sideofasquare Areaofasquare1inch 1in22inches 4in2
3inches 9in2
4inches 16in2
GO Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.
12. 13. 14.
x 1 2 3 4 5
y 2 162
x 1 2 3 4 5
y 1/9 -3
x 1 2 3 4 5
y 10 0.625
20
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.3
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2.3
15. 16. Findtherateofchange(slope)ineachoftheexercisesbelow.
17. x g(x)
-5 11
-3 4
-2 0.5
0 -6
18. t h(t)
3 13
8 23
18 43
23 53
19.
n f(n)
-7 20
-5 24
-1 32
2 38
20. (2,5)(8,29) 21. 22. (-3,7)(8,29)
x 1 2 3 4 5
y g gz4
x 1 2 3 4 5
y -3 -243
21
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.4 Getting Down to Business
A Solidify Understanding Task
Calcu-ramahadanetincomeof5milliondollarsin2010,whileasmallcompetingcompany,
Computafest,hadanetincomeof2milliondollars.ThemanagementofCalcu-ramadevelopsa
businessplanforfuturegrowththatprojectsanincreaseinnetincomeof0.5millionperyear,
whilethemanagementofComputafestdevelopsaplanaimedatincreasingitsnetincomeby15%
eachyear.
a. Createstandardmathematicalmodels(table,graphandequations)fortheprojectednet
incomeovertimeforbothcompanies.(Attendtoprecisionandbesurethateachmodelis
accurateandlabeledproperlysothatitrepresentsthesituation.)
b. Comparethetwocompanies.Howaretherepresentationsforthenetincomeofthetwo
companiessimilar?Howdotheydiffer?Whatrelationshipsarehighlightedineach
representation?
CCBYreynermed
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https://flic.kr/p/ifz6w
b
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SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.4
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c. Ifbothcompanieswereabletomeettheirnetincomegrowthgoals,whichcompanywould
youchoosetoinvestin?Why?
d. When,ifever,wouldyourprojectionssuggestthatthetwocompanieshavethesamenet
income?Howdidyoufindthis?Willtheyeverhavethesamenetincomeagain?
e. Sincewearecreatingthemodelsforthesecompanieswecanchoosetohaveadiscrete
modeloracontinuousmodel.Whataretheadvantagesordisadvantagesforeachtypeof
model?
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.4
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2.4
READY Topic:Comparingarithmeticandgeometricsequences.Thefirstandfifthtermsofasequencearegiven.Fillinthemissingnumbersifitisanarithmeticsequence.Thenfillinthenumbersifitisageometricsequence.
Example:
Arithmetic 4 84 164 244 324
Geometric 4 12 36 108 324
1.
Arithmetic 3 48
Geometric 3 48
2.
Arithmetic -6250 -10
Geometric -6250 -10
3.
Arithmetic -12 -0.75
Geometric -12 -0.75
SET Topic:Distinguishingspecificsbetweensequencesandlinearorexponentialfunctions.Answerthequestionsbelowwithrespecttotherelationshipbetweensequencesandthelargerfamiliesoffunctions.4.Ifarelationshipismodeledwithacontinuousfunctionwhichofthedomainchoicesisapossibility? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ ! C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ ! 5.WhichoneoftheoptionsbelowisthemathematicalwaytorepresenttheNaturalNumbers? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ !, ! ≥ 0 C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ !
READY, SET, GO! Name PeriodDate
x 3 x 3 x 3 x 3
+80 +80 +80 +80
24
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.4
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2.4
6.Onlyoneofthechoicesbelowwouldbeusedforacontinuousexponentialmodel,whichoneisit? A.! ! = ! ! − 1 ∙ 4, ! 1 = 3 B.! ! = 4!(5) C.ℎ ! = 3! − 5 D.! ! = ! ! − 1 − 5, ! 1 = 327.Onlyoneofthechoicesbelowwouldbeusedforacontinuouslinearmodel,whichoneisit? A.! ! = ! ! − 1 ∙ 4, ! 1 = 3 B.! ! = 4!(5) C.ℎ ! = 3! − 5 D.! ! = ! ! − 1 − 5, ! 1 = 328.Whatdomainchoicewouldbemostappropriateforanarithmeticorgeometricsequence? A. ! | ! ∈ !, ! ≥ 0 B. ! | ! ∈ !, ! ≥ 0 C. ! | ! ∈ !, ! ≥ 0 D. ! | ! ∈ ! 9.Whatattributeswillarithmeticorgeometricsequencesalwayshave? (Therecouldbemorethanonecorrectchoice.Circleallthatapply.) A.Continuous B.Discrete C.Domain: ! | ! ∈ ! D.Domain: ! | ! ∈ ! E.Negativex-values F.Somethingconstant G.RecursiveRule10.Whattypeofsequencefitswithlinearmathematicalmodels? Whatisthedifferencebetweenthissequencetypeandtheoverarchingumbrellaoflinear relationships?(Usewordslikediscrete,continuous,domainandsoforthinyourresponse.)11.Whattypeofsequencefitswithexponentialmathematicalmodels? Whatisthedifferencebetweenthissequencetypeandtheoverarchingumbrellaofexponential relationships?(Usewordslikediscrete,continuous,domainandsoforthinyourresponse.)
25
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LINEAR & EXPONENTIAL FUNCTIONS – 2.4
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2.4
GO Topic:Writingexplicitequationsforlinearandexponentialmodels.Writetheexplicitequationsforthetablesandgraphsbelow.Thisissomethingyoureallyneedtoknow.Persevereanddoallyoucantofigurethemout.Rememberthetoolswehaveused.(#21isbonusgiveitatry.)12. x f (x)
2
3
4
5
-4
-11
-18
-25
13. x f (x)
-1
0
1
2
2/5
2
10
50
14. x f (x)
2
3
4
5
-24
-48
-96
-192
15. x f (x)
-4
-3
-2
-1
81
27
9
3
16
17.
18
19.
20.
21.
26
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
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2.5 Making My Point
A Solidify Understanding Task
ZacandSionewereworkingonpredictingthenumberofquiltblocksinthispattern:
Whentheycomparedtheirresults,theyhadaninterestingdiscussion:
Zac:Igot! = 6! + 1 becauseInoticedthat6blockswereaddedeachtimesothepatternmusthavestartedwith1blockatn=0.
Sione:Igot! = 6 ! − 1 + 7becauseInoticedthatatn=1therewere7blocksandatn=2therewere13,soIusedmytabletoseethatIcouldgetthenumberofblocksbytakingonelessthanthen,multiplyingby6(becausethereare6newblocksineachfigure)andthenadding7becausethat’showmanyblocksinthefirstfigure.Here’smytable:
1 2 3 n7 13 19 6 ! − 1 + 7
©201
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1. WhatdoyouthinkaboutthestrategiesthatZacandSioneused?Areeitherofthemcorrect?Whyorwhynot?Useasmanyrepresentationsasyoucantosupportyouranswer.
ThenextproblemZacandSioneworkedonwastowritetheequationofthelineshownonthegraphbelow.
Whentheywerefinished,hereistheconversationtheyhadabouthowtheygottheirequations:
Sione:Itwashardformetotellwherethegraphcrossedtheyaxis,soIfoundtwopointsthatIcouldreadeasily,(-9,2)and(-15,5).Ifiguredoutthattheslopewas-!!andmadeatableandcheckeditagainstthegraph.Here’smytable:
x -15 -13 -11 -9 n
f(x) 5 4 3 2
− 12 ! + 9 + 2
28
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Iwassurprisedtonoticethatthepatternwastostartwiththen,add9,multiplybytheslopeandthenadd2.
Igottheequation:! ! = − !! ! + 9 + 2.
Zac:Hey—IthinkIdidsomethingsimilar,butIusedthepoints,(7,-6)and(9,-7).
Iendedupwiththeequation:! ! = − !! ! − 9 − 7.Oneofusmustbewrongbecauseyours
saysthatyouadd9tothenandminesaysthatyousubtract9.Howcanwebothberight?
2. Whatdoyousay?Cantheybothberight?Showsomemathematicalworktosupportyourthinking.
Zac:Myequationmademewonderiftherewassomethingspecialaboutthepoint(9,-7)sinceitseemedtoappearinmyequation! ! = − !
! ! − 9 − 7whenIlookedatthenumberpattern.NowI’mnoticingsomethinginteresting—thesamethingseemstohappenwithyourequation,! ! = − !
! ! + 9 + 2andthepoint(-9,2)
3. DescribethepatternthatZacisnoticing.
4. FindanotherpointonthelinegivenaboveandwritetheequationthatwouldcomefromZac’spattern.
5. Whatwouldthepatternlooklikewiththepoint(a,b)ifyouknewthattheslopeofthelinewasm?
29
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
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6. Zacchallengesyoutousethepatternhenoticedtowritetheequationoflinethathasaslopeof3andcontainsthepoint(2,-1).What’syouranswer?
Showawaytochecktoseeifyourequationiscorrect.
7. Sionechallengesyoutousethepatterntowritetheequationofthelinegraphedbelow,usingthepoint(5,4).
Showawaytochecktoseeifyourequationiscorrect.
8. Zac:“I’llbetyoucan’tusethepatterntowritetheequationofthelinethroughthepoints(1,-3)and(3,-5).Tryit!”
Showawaytochecktoseeifyourequationiscorrect.
30
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9. Sione:Iwonderifwecouldusethispatterntographlines,thinkingofthestartingpointandusingtheslope.Tryitwiththeequation:! ! = −2 ! + 1 − 3.Startingpoint:Slope:
Graph:
10. Zacwonders,“Whatisitaboutlinesthatmakesthiswork?”HowwouldyouanswerZac?
11. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?
31
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
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2.5
READY Topic:Writingequationsoflines.
Writetheequationofalineinslope-interceptform:y=mx+b,usingthegiveninformation.
1.m=-7,b=4 2.m=3/8,b=-3 3.m=16,b=-1/5
Writetheequationofthelineinpoint-slopeform:y=m(x–x1)+y1,usingthegiveninformation.
4.m=9,(0.-7) 5.m=2/3,(-6,1) 6.m=-5,(4,11)
7.(2,-5)(-3,10) 8.(0,-9)(3,0) 9.(-4,8)(3,1)
READY, SET, GO! Name PeriodDate
32
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
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2.5
SET
Topic:Graphinglinearandexponentialfunctions
Makeagraphofthefunctionbasedonthefollowinginformation.Addyouraxes.Choosean
appropriatescaleandlabelyourgraph.Thenwritetheequationofthefunction.
10.Thebeginningvalueis5anditsvalueis3
unitssmallerateachstage.
Equation:
11.Thebeginningvalueis16anditsvalueis¼
smallerateachstage.
Equation:
12.Thebeginningvalueis1anditsvalueis10
timesasbigateachstage.
Equation:
13.Thebeginningvalueis-8anditsvalueis2
unitslargerateachstage.
Equation:
33
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
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2.5
GO Topic:Equivalentequations
Provethatthetwoequationsareequivalentbysimplifyingtheequationontherightsideofthe
equalsign.Thejustificationintheexampleistohelpyouunderstandthestepsforsimplifying.
YoudoNOTneedtojustifyyoursteps.
Example: Justification 2! − 4 = 8 + ! − 5! + 6 ! − 2 Add! − 5!anddistributethe6over ! − 2 = 8 − 4! + 6! − 12 Combineliketerms. = −4 + 2! 2! − 4 = 2! − 4 Commutativepropertyofaddition
14.! − 5 = 5! − 7 + 2 3! + 1 − 10!
15.6 − 13! = 24 − 10 2! + 8 + 62 + 7!
16.14! + 2 = 2! − 3 −4! − 5 − 13
17. ! + 3 = 6 ! + 3 − 5 ! + 3
18.4 = 7 2! + 1 − 5! − 3 3! + 1
19.! = 12 + 8! − 3 ! + 4 − 4!
20.Writeanexpressionthatequals ! − 13 . Itmusthaveatleasttwosetsofparenthesesandoneminussign.Verifythatitisequalto ! − 13 .
34
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.6 Form Follows Function
A Practice Understanding Task
Inourworksofar,wehaveworkedwithlinearand
exponentialequationsinmanyforms.Someofthe
formsofequationsandtheirnamesare:
LinearFunctionsEquation Name
! = 12 ! + 1
SlopeInterceptForm! = !" + !,wheremistheslopeandbisthe
y-intercept
! = 12 ! − 4 + 3
PointSlopeForm! = ! ! − !! + !!,wheremistheslopeand(x1,y1)thecoordinatesofapointontheline
! 0 = 1, ! ! = ! ! − 1 + 12
RecursionFormula! ! = ! ! − 1 +D,
Givenaninitialvaluef(a)D=constantdifferenceinconsecutiveterms
(usedonlyfordiscretefunctions)
ExponentialFunctionsEquation Name
! = 10(3)! ExplicitForm! = !(!)!
! 0 = 10, ! ! + 1 = 3! !
RecursionFormula! ! + 1 = !" !
Givenaninitialvaluef(a)R=constantratiobetweenconsecutiveterms
(usedonlyfordiscretefunctions)
CCBYPaulDow
ney
https://flic.kr/p/ta4a
35
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Knowinganumberofdifferentformsforwritingandgraphingequationsislikehavinga
mathematicaltoolbox.Youcanselectthetoolyouneedforthejob,orinthiscase,theformofthe
equationthatmakesthejobeasier.Anymasterbuilderwilltellyouthatthemoretoolsyouhave
thebetter.Inthistask,we’llworkwithourmathematicaltoolstobesurethatweknowhowtouse
themallefficiently.Asyoumodelthesituationsinthefollowingproblems,thinkaboutthe
importantinformationintheproblemandtheconclusionsthatcanbedrawnfromit.Isthe
functionlinearorexponential?Doestheproblemgiveyoutheslope,apoint,aratio,ay-intercept?
Isthefunctiondiscreteorcontinuous?Thisinformationhelpsyoutoidentifythebesttoolsandget
towork!
1. Inhisjobsellingvacuums,Joemakes$500eachmonthplus$20foreachvacuumhesells.WritetheequationthatdescribesJoe’smonthlyincomeIasafunctionofthen,thenumberofvacuumssold.
Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
2. Writetheequationofthelinewithaslopeof-1throughthepoint(-2,5)Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
36
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3.Writetheequationofthegeometricsequencewithaconstantratioof5andafirsttermof-3.
Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
3. Writetheequationofthefunctiongraphedbelow:
Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
4. ThepopulationoftheresorttownofJavaHotSpringsin2003wasestimatedtobe35,000
peoplewithanannualrateofincreaseofabout2.4%.Writetheequationthatmodelsthe
numberofpeopleinJavaHotSprings,witht=thenumberofyearsfrom2003?
Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
37
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
5. Yessica’ssciencefairprojectinvolvedgrowingsomeseedstoseewhatfertilizermadethe
seedsgrowfastest.Oneideashehadwastouseanenergydrinktofertilizetheplant.(She
thoughtthatiftheymakepeopleperky,theymighthavethesameeffectonplants.)Thisis
thedatathatshowsthegrowthoftheseedeachweekoftheproject.
Week 1 2 3 4 5
Height(cm) 1.7 2.9 4.1 5.3 6.5
Writetheequationthatmodelsthegrowthoftheplantovertime.
Nametheformoftheequationyouwroteandwhyyouchosetousethatform.
Thisfunctionis: linear exponential neither (chooseone)
Thisfunctionis: continuous discrete neither (chooseone)
Anequationgivesusinformationthatwecanusetographthefunction.Pickouttheimportant
informationgivenineachofthefollowingequationsandusetheinformationtographthefunction.
6. ! = !! ! − 5
Whatdoyouknowfromtheequationthathelpsyoutographthefunction?
38
SECONDARY MATH 1 // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
7. ! = 2!
8. ! = −2 ! + 6 + 8
9. ! 1 = −5, ! ! = ! ! − 1 + 1
Whatdoyouknowfromtheequationthathelpsyoutographthefunction?
Whatdoyouknowfromtheequationthathelpsyoutographthefunction?
Whatdoyouknowfromtheequationthathelpsyoutographthefunction?
39
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
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2.6
READY Topic:Comparinglinearandexponentialmodels.Comparingdifferentcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible.
y=4+3x y=4(3x)
1.Typeofgrowth
2.Whatkindofsequencecorrespondstoeachmodel?
3.Makeatableofvalues
x y
x y
4.Findtherateofchange
5.Grapheachequation.
Comparethegraphs.
Whatisthesame?
Whatisdifferent?
6.Findthey-interceptfor
eachfunction.
20
18
16
14
12
10
8
6
4
2
5 10 15 20
20
18
16
14
12
10
8
6
4
2
5 10 15 20
READY, SET, GO! Name PeriodDate
40
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
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2.6
7.Findthey-interceptsforthefollowingequations
a)y=3x
b)y=3x
8.Explainhowyoucanfindthey-interceptofalinearequationandhowthatisdifferentfromfinding
they-interceptofageometricequation.
SET Topic:Efficiencywithdifferentformsoflinearandexponentialfunctions.Foreachexerciseorproblembelowusethegiveninformationtodeterminewhichoftheformswouldbethemostefficienttouseforwhatisneeded.(Seetask2.6,Linear:slope-intercept,point-slopeform,recursive,Exponential:explicitandrecursiveforms)
9. Jasmine has been working to save money and wants to have an equation to model the amount of
money in her bank account. She has been depositing $175 a month consistently, she doesn’t remember
how much money she deposited initially, however on her last statement she saw that her account has
been open for 10 months and currently has $2475 in it. Create an equation for Jasmine.
Whichequationformdoyouchose? Writetheequation.
10.
The table below shows the number of rectangles created every time there is a fold made through the
center of a paper. Use this table for each question.
Folds Rectangles
1 2
2 4
3 8
4 16
A. Find the number or rectangles created with 5 folds.
Whichequationformdoyouchose?Writetheequation.
B.Findthenumberofrectanglescreatedwith14folds.
Whichequationformdoyouchose?Writetheequation.
41
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
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2.6
11.UsinganewappthatIjustdownloadedIwanttocutbackonmycalorieintakesothatIcanloseweight.Icurrentlyweigh90kilograms,myplanistolose1.2kilogramsaweekuntilIreachmygoal.HowcanImakeanequationtomodelmyweightlossforthenextseveralweeks.Whichequationformdoyouchose?Writetheequation.12.SinceScottstarteddoinghisworkoutplanJanethasbeeninspiredtosetherselfagoaltodomoreexerciseandwalkalittlemoreeachday.Shehasdecidedtowalk10metersmoreeveryday.Ontheday20shewalked800meters.Howmanymeterswillshewalkonday21?Onday60?Whichequationformdoyouchose?Writetheequation.Foreachequationprovidedstatewhatinformationyouseeintheequationthatwillhelpyougraphit,thengraphit.Also,usetheequationtofillinanyfourcoordinatesonthetable.13. ! = !
!!8
14. ! = 5 ! − 2 − 6
Whatdoyouknowfromtheequation
thathelpsyoutographthefunction?
Whatdoyouknowfromtheequation
thathelpsyoutographthefunction?
42
SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.6
Mathematics Vision Project
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2.6
GO Topic:Solvingone-stepequationswithjustification.Recallthetwopropertiesthathelpussolveequations.TheAdditivepropertyofequalitystates:Youcanaddanynumbertobothsidesofanequationandtheequationwillstillbetrue.TheMultiplicativepropertyofequalitystates:Youcanmultiplyanynumbertobothsidesofanequationandtheequationwillstillbetrue.Solveeachequation.Justifyyouranswerbyidentifyingtheproperty(s)youusedtogetit.Example1: ! − 13 = 7Justification+13+13additivepropertyofequality ! + 0 = 20addition ! = 20additiveidentity(Youadded0andgotx.)Example2:5! = 35Justification !! ! = !"! multiplicativepropertyofequality(multipliedby
!!)
1! = 7multiplicativeidentity(Anumbermultipliedbyitsreciprocal=1)15. 3! = 15Justification
16. ! − 10 = 2Justification
17. −16 = ! + 11Justification
18. 6 + ! = 10Justification
19. 6! = 18Justification
20. −3! = 2Justification
43