Module 1: Quadratic Functions - Mathematics Vision Project

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2017 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 1

Quadratic Functions

SECONDARY

MATH TWO

An Integrated Approach

SECONDARY MATH 2 // MODULE 1

QUADRATIC FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 1 - TABLE OF CONTENTS

QUADRATIC FUNCTIONS

1.1 Something to Talk About – A Develop Understanding Task

An introduction to quadratic functions, designed to elicit representations and surface a new type of

pattern and change (F.BF.1, A.SSE.1, A.CED.2)

READY, SET, GO Homework: Quadratic Functions 1.1

1.2 I Rule – A Solidify Understanding Task

Solidification of quadratic functions begins as quadratic patterns are examined in multiple

representations and contrasted with linear relationships (F.BF.1, A.SSE.1, A.CED.2)


1.3 Scott’s Macho March – A Solidify Understanding Task

Focus specifically on the nature of change between values in a quadratic being linear (F-BF, F-LE)


1.4 Rabbit Run– A Solidify Understanding Task

Focus on maximum/minimum point as well as domain and range for quadratics (F.BF.1, A.SSE.1,

A.CED.2)


1.5 The Tortoise and the Hare– A Solidify Understanding Task

Comparing quadratic and exponential functions to clarify and distinguish between each type of growth as

well as how that growth appears in each of their representations (F.BF.1, A.SSE.1, A.CED.2, F.LE.3)


SECONDARY MATH 2 // MODULE 1

QUADRATIC FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

1.6 How Does it Grow – A Practice Understanding Task

Incorporating quadratics with the understandings of linear and exponential functions (F.LE.1, F.LE.2,

F.LE.3)


SECONDARY MATH II // MODULE 1

QUADRATIC FUNCTIONS – 1.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

1.1 Something to Talk About

A Develop Understanding Task

Cellphonesoftenindicatethestrengthofthephone’ssignalwithaseriesofbars.Thelogobelowshowshowthismightlookforvariouslevelsofservice.

1. Assumingthepatterncontinues,drawthenextfigureinthesequence.

2. Howmanyblockswillbeinthefigure10?

3. Examinethesequenceoffiguresandfindaruleorformulaforthenumberoftilesinanyfigurenumber.

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1.1 Something to Talk About – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskistosurfaceideasandrepresentationsforquadraticfunctions.The

taskisdesignedtoelicittables,graphs,andequations,bothrecursiveandexplicittodescribeagrowing

pattern.Theclassroomdiscussionwillfocusonthegrowthshowninthevariousrepresentations,

developingtheideathatquadraticfunctionsshowlinearratesofchange.

CoreStandardsFocus:

F.BF.1 Writeafunctionthatdescribesarelationshipbetweentwoquantities.*

a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfroma

context.

A.SSE.1 Interpretexpressionsthatrepresentaquantityintermsofitscontext.*

A.CED.2 Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;

graphequationsoncoordinateaxeswithlabelsandscales.

*Focusonsituationsthatexhibitaquadraticorexponentialrelationship.

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbydrawingstudents’attentiontothepatternandaskingstudentstodrawthenextfigure

inthesequence(question#1).Haveastudentshowfigure5andtalkaboutwhattheynoticedabout

thepatterntodrawfigure5.Askstudentstoworkontherestofthetask.

Explore(SmallGroup):

Asstudentsareworkingintheirgroups,payattentiontowaysthattheyareusingthepattern.Watch

forstudentsthatareusingtablestokeeptrackofthegrowthofthepatternandfindthenumberof

blocksinfigure10.Listenforstudentsthatareusingreasoningaboutthepatterntopredictfigure10.

Some





studentsmaybewritingvariousequations,baseduponthewaytheyseethepatterngrowing.Ifthey

arenoticingthepartthatisaddedontothepatterneachtime,theymaywritearecursiveequation.

Sincethisisanimportantrepresentationforthediscussion,encouragethisthinking,evenifthe

notationisn’tentirelycorrect.Ifstudentsarepayingattentiontotherelationshipbetweenthefigure

numberandthenumberofblocksinthefigure,theymaytrytowriteanexplicitequationforthe

pattern.Youmayexpectstudentstothinkaboutthe“emptyspace”orthesquaresthatcouldbeusedto

completearectangle.Theymaynoticethatifthefigureiscopiedandrotated,itcanbefituponitselfto

formarectanglelikeso:

Thisreasoningmayhelpthemtofindtheexplicitformulaforthepattern.Asstudentswork,encourage

asmanyrepresentationsaspossible.

Discuss(WholeClass):

Whenstudentshavefinished#2andhadachancetoworkon#3(somestudentsmaynothavefound

explicitequations),beginthediscussionwiththenumberofsquaresinfigure10(problem#2).Begin

byhavingagroupthatusedatabletopresenttheirwork.

Askstudentswhatpatternstheyseeinthetable.Askhowthosesamepatternsshowupinthefigures.

Oneimportantpatterntonoticeisthefirstdifference,orthechangeinoutputsoneachlineofthetable.

Figure4





Askastudentthathascreatedagraphtoshowtheirrepresentation.Anexampleisshownbelow.Have

abriefdiscussionofwhythepointsinthisgraphshouldnotbeconnectedbaseduponthiscontext.

Askstudentswhattheyknowabouttherelationbaseduponthetableandgraph.Someconclusions

thatshouldbedrawn:

• Therelationisafunction.

• Thefunctionisdiscrete.

• Thedomainandrangeofthefunctionareboththesetofnaturalnumbers.

• Thefunctionisnotlinearbecauseitdoesnothaveaconstantrateofchange.

• Thefunctionisnotexponentialbecauseitdoesnothaveaconstantratiobetweenterms.

• Thefunctionisalwaysincreasing.

• Theminimumvalueofthefunctionis(1,1).

Studentsmayrefertotherelationshipasasequence,comparingittoarithmeticandgeometric

sequences.Thisideaisappropriatebutcanbeextendedtoincludethebroadercategoriesoflinearand

exponentialfunctions.

Turnthediscussiontofindinganequationtorepresentthepattern.Askstudentstosharetheir

recursiveequations.Youmaywishtostartwithastudentthathasusedarecursiveideabuthasnot

usedfunctionnotation.Theideaisthatthefunctionstartsat1,andthenthtermcanbefoundbytaking

theprevioustermandaddingn.Askstudentstoshowhowthiscanbeseeninthetable,movingfrom

oneoutputrowtothenext.Infunctionnotation,thisisrepresentedby:

! 1 = 1, ! ! = ! ! − 1 + !





Askastudenthowtheycanusetheequationtodescribethefiguresasthepatterngoesfromfigure4to

figure5.

Itismuchhardertousethetabletofindtheexplicitequationthantherecursiveequation.Most

studentsthataresuccessfulinwritinganexplicitformulaforthepatternusethevisualpattern,

thinkingabouttherectanglemadebycopyingandrotatingthefigure.Eveniftheyhavenotwrittenan

equation,haveastudentthatconsideredthisstrategytoshowhowtheyusedthedrawingforfigure4,

asbelow:

“Usingfigure4,Inoticethattherectangleformedhaswidthof5andaheightof4.WhenItrieditwith

otherrectanglesIsawthatthewidthwasalwaysonemorethantheheight.”

Helpstudentstoseethattheheightofarectangleformedinthisfashionisnandthewidthisn+1.The

areaoftherectangleis!(! + 1),butsincetheactualfigureisonlyhalfoftherectangle,thefunctionthatdescribesthenumberofblocksineachfigureis:! ! = !

! !(! + 1).

Concludebygoingbacktothetableandthefirstdifference.Askstudentswhattypeoffunctionappears

tobeformedbythefirstdifference.Thiswillbringuptheseconddifference,therateofchangeofthe

firstdifference.Inthiscasetheseconddifferenceisalways1,whichmeansthatthefirstdifferenceis

linear.Tellstudentsthatfunctionswithalinearrateofchangearecalledquadraticfunctions.





Inaddition,tellstudentsthatbecausethedomainisrestrictedinthiscase,weareonlyseeinghalfthe

graph.Displaythegraphof! ! = !! !(! + 1),demonstratingthetypicalquadraticgraph.Askstudents

whattheynoticeaboutthegraph.Askstudentswhytheequationwouldgiveagraphthatdecreases,

thenincreases,andissymmetrical.Studentswillworkextensivelywiththegraphsofquadratic

equationsinmodule2,butitisimportantthattheynoticemanyofthefeaturesofthegraphastheyare

developinganunderstandingofquadraticrelationshipsinmodule1.

AlignedReady,Set,Go:QuadraticFunctions1.1




Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

1.1

READY Topic:DistributivePropertySimplifythefollowingexpressions

1.3 2x + 7 2.−12 5x − 4

3.5a −3a + 13 4.9x 6x − 2

5.!"! 12x + 18 6.

!"! 10a − 25b 7.

!!"!! 121x + 22

SET Topic:RecognizingLinearExponentialandQuadraticFunctionsIneachsetof3functions,onewillbelinearandonewillbeexponential.Oneofthethreewillbeanewcategoryoffunction.Listthecharacteristicsineachtablethathelpedyoutoidentifythelinearandtheexponentialfunctions.Whataresomecharacteristicsofthenewfunction?Findanexplicitandrecursiveequationforeach.8.Linear,exponential,oranewkindoffunction?

a.

Typeandcharacteristics?

Explicitequation:

Recursiveequation:

! !(!)6 64

7 128

8 256

9 512

10 1024

b.


Explicitequation:

Recursiveequation:

! !(!)6 36

7 49

8 64

9 81

10 100

c.


Explicitequation:

Recursiveequation:

! !(!)6 11

7 13

8 15

9 17

10 19

READY, SET, GO! Name PeriodDate

2






1.1

9.Linear,exponential,oranewkindoffunction?d.


Explicitequation:

Recursiveequation:

! !(!)-2 -17

-1 -12

0 -7

1 -2

2 3

e.


Explicitequation:

Recursiveequation:

! !(!)-2 1/25

-1 1/5

0 1

1 5

2 25

f.


Explicitequation:

Recursiveequation:

! !(!)-2 9

-1 6

0 5

1 6

2 9

10.Graphthefunctionsfromthetablesin#8and#9.Addanyadditionalcharacteristicsyounoticefromthegraph.Placeyouraxessothatyoucanshowall5points.Identifyyourscale.Writeyourexplicitequationabovethegraph.a.Equation:

b.Equation:

c.Equation:

d.Equation:

e.Equation:

f.Equation:

3






1.1

GO Topic:RatesofChangeIdentifytherateofchangeineachoftherepresentationsbelow.

11.

12. 13.

x f(x)25 65

26 68

27 71

28 74

14.

f 0 = 7; f n + 1 = f n + 5

15.

16.

Slopeof!"A(-3,12)B(-11,-16)

17.Georgeisloadingfreightintoanelevator.Henoticesthattheweightlimitfortheelevatoris1000lbs.He

knowsthatheweighs210lbs.Hehasloaded15boxesintotheelevator.Eachboxweighs50lbs.Identify

therateofchangeforthissituation.

18.

Independentvariable 4 5 6 7 8

Dependentvariable 5 5.5 6 6.5 7

19.

! −4 = 24 !"# ! 6 = −36

4

3

2

1

–1

–2

–3

–4

–6 –4 –2 2 4 6

4

3

2

1

–1

–2

–3

–4

–6 –4 –2 2 4 6

4

3

2

1

–1

–2

–3

–4

–6 –4 –2 2 4 6

4





1.2 I Rule!

A Solidify Understanding Task

Marcohasstartedanewblogaboutsportsat

ImaginationHighSchool(mascot:thefightingunicorns)thathehasdecidedtocall“ISite”.

Hecreatedalogoforthewebsitethatlookslikethis:

Heisworkingoncreatingthelogoinvarioussizestobeplacedondifferentpagesonthe

website.Marcodevelopedthefollowingdesigns:

1. Howmanysquareswillbeneededtocreatethesize100logo?

2. Developamathematicalmodelforthenumberofsquaresinthelogoforsizen.

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Marcodecidestoexperimentwithmakinghislogo“blockier”sothatitlooksstronger.

Here’swhathecameupwith:

3. AssumingthatMarcocontinueswiththepatternasithasbegun,drawthenext

figure,size4,andfindthenumberofblocksinthefigure.

6





4. Developamathematicalmodelforthenumberofblocksinalogoofsizen.

5. Comparethemodelsthatyoudevelopedforthefirstsetoflogostothesecondsetoflogos.Inwhatwaysaretheysimilar?Inwhatwaysaretheydifferent?

7





1.2 I Rule! – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifystudents’understandingofquadraticfunctionsandtheirrepresentations,byprovidingbothanexampleandanon-exampleofaquadraticfunction.Thetaskprovidesanopportunityforstudentstocomparethegrowthoflinearfunctionstothegrowthofquadraticfunctions.Equations,bothrecursiveandexplicit,graphs,andtablesareusedtodescribetherelationshipbetweenthenumberofblocksandthefigurenumberinthistask.CoreStandardsFocus:A-SE.1 Interpretexpressionsthatrepresentaquantityintermsofitscontext.*A-CED.2 Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;

graphequationsoncoordinateaxeswithlabelsandscales.F-BF.1 Writeafunctionthatdescribesarelationshipbetweentwoquantities.*

a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

F-IF.9 Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,graphically,numericallyintables,orbyverbaldescriptions).*Focusonsituationsthatexhibitaquadraticorexponentialrelationship.Constructandcomparelinear,quadratic,andexponentialmodelsandsolve

problems.

ComparelinearandexponentialgrowthstudiedinSecondaryMathematicsItoquadraticgrowth.

TheTeachingCycle:

Launch(WholeClass):

Beginbyhavingstudentsreadthetaskandunderstandingthecontext.Askthemtocomparethetwosetsoflogosinthefirstandsecondpartsofthetaskandsharewhattheynoticeaboutthemathematical/geometricattributesofthefigures.Havestudentsgetstartedonquestions1and2.






Part1:Monitorstudentsastheywork,lookingfortheiruseoftables,graphs,andequations.Ifstudentsarestuck,askhowtheyseethefigureschanging.Itmaybeusefultoprovidecoloredpencilstohelpthemkeeptrackofwhatchangesandwhatstaysthesameineachfigure.Encouragestudentstouseasmanyrepresentationsaspossiblefortheirmathematicalmodel.


Part1:Startthediscussionwithstudentspresentingatableandagraph.Askstudentswhattypeoffunctionthisisandhowtheyknow.Usingthefirstdifference,highlighttheideasthattherateofchangeisalways5.

Askstudentstoshowhowtheyseethechangeof5ineachoffigures.Then,askastudenttopresentarecursiveequationandconnectittothefigures.Therecursiveformulais:

! 1 = 7, ! ! = ! ! − 1 + 5Askastudenttoshowhowtheyusedthediagramtofindanexplicitformula.Apossibleexplanationis,“Inoticedthatforn=2therewere5groupsof2blocks(showninred)andthen2moreblocksleftover.WhenItrieditontheothersizesitworkedthesamewaysoIdecidedtheequationis:! ! = 5! + 2.”

SizeNo. NumberofSquares

1 72 123 174 225 27n

SizeNo.

NumberofSquares





Atthispoint,askstudentstoworkonthesecondpartofthetaskwiththe“blockier”logos.


Asyoumonitorstudentwork,ensurethatstudentsareabletocorrectlydrawthenextfigure.Listenforhowtheyareusingthediagramtothinkabouthowthefiguresarechanging.Encouragestudentstousealltherepresentationsincludinggettingbothanexplicitandrecursiveequationiftimepermits.Discuss(WholeGroup):

Part2.BeginthediscussionmuchlikePart1.Havestudentsshowatableandgraph,makingconnectionstothefigures.Directstudentstolookatthefirstdifferencesinthetableandhowtheyseethegrowingdifferencesonthegraph.Muchasinthediscussionin“SomethingtoTalkAbout”,helpstudentstoseethatthefirstdifferencesarelinear,makingthisaquadraticfunction.





Askstudentstopresentanexplicitequationandtoshowhowtheyusedthediagramtogettheequation.Onepossiblewaytothinkaboutitisshownbelow,withastudentsaying,“Inoticedthateachtimetherearesevenlargesquares.Forn=2,thesquaresaremadeupof4smallersquares,forn=3,thesquaresaremadeupof9smallersquares,andsoon,givingtheequation:! ! = 7!!.





Askstudentstosharetheirthinkingaboutarecursiveequation.Theyhaveusedthediagramasimilarway,noticingthateachtimethebigsquaresarewrappedwithan“L”shapethatadds2! − 1squareseachtimeasshownbelow:

Thisthinkingyieldstheequation:

! 1 = 7, ! ! = ! ! − 1 + 7(2! − 1)Askstudentshowtheyseetherateofchangeinthetableshowingupintherecursiveequation.Theymaynoticethatthepartaddedontotheprevioustermisthechange,whichforaquadraticfunctionwillbealinearexpression.Endthediscussionwithacomparisonoflinearandquadraticfunctions.Leadtheclassinadiscussiontocompleteatablesuchas:

Linear Quadratic

Rateofchange Constant LinearGraph Line ParabolaEquation Highestpoweredtermisx Highestpoweredtermisx2

termTable Firstdifferenceisconstant Firstdifferenceislinear,

seconddifferentisconstant.AlignedReady,Set,Go:QuadraticFunctions1.2






1.2

READY Topic:DistributivePropertySimplify.Firstusethedistributivepropertyandthencombinetheliketerms.

Example:

!" !" + ! + ! !" + ! → !"!! + !" + !" + ! → !"!! + !" + !" + ! → !"!! + !!" + !

1.2x 5x + 3 + 7 5x + 3

2.8x x + 1 + 2 x + 1

3.6x x − 10 − 1 x − 10

4.1x 3x + 4 + 5 3x + 4

5. 3x 8x + 3 − 4 8x + 3

6.5x 2x + 6 + 2 2! + 6

7.7x −5x + 2 − 13 −5x + 2

8.−4x 12x + 3 + 3 12x + 3

SET Topic:ComparingAreaandperimeterCalculatetheareaandperimeterofeachfigurebelow.Theareamaybewrittenasaproduct.Includethecorrectunitonyouranswer.(Youranswerswillcontainavariable.)9. 10.

a.Perimeter:______________________ a.Perimeter:______________________

b.Area:____________________________ b.Area:____________________________


liketermsSimplifiedform

(x+1)in

(x+1)inxcm

xcm

8






1.2

11. 12.


b.Area:____________________________ b.Area:____________________________

13. 14.


b.Area:____________________________ b.Area:____________________________

15.Comparetheperimetertotheareaineachofproblems(9-14).

Inwhatwayarethenumbersandunitsintheperimetersandareasdifferent?

GO Topic:GreatestCommonFactorFindtheGCFforthegiventerms.

16.15abc2and25a3bc 17.12x5yand32x6y 18.17pqrand51pqr3

19.7x2and21x 20.6x2,18x,and-12 21.4x2and9x

22.11x2y2,33x2y,and3xy2 23.16a2b,24ab,and16b 24.49s2t2and36s2t2

(a+5)ft

(b+3)ft ami

bmi

(x+3)m

(x–2)m

(x+4)in

(x+1)in

9





1.3 Scott’s Macho March


After looking in the mirror and feeling flabby, Scott

decided that he really needed to get in shape. He

joined a gym and added push-ups to his daily exercise routine. He started keeping track of the

number of push-ups he completed each day in the bar graph below, with day one showing he

completed three push-ups. After four days, Scott was certain he could continue this pattern of

increasing the number of push-ups for at least a few months.

1. Model the number of push-ups Scott will complete on any given day. Include both explicit and

recursive equations.

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Scott’s gym is sponsoring a “Macho March” promotion. The goal of “Macho March” is to raise money

for charity by doing push-ups. Scott has decided to participate and has sponsors that will donate

money to the charity if he can do a total of at least 500 push-ups, and they will donate an additional

$10 for every 100 push-ups he can do beyond that. So now Scott is going to track the total number

of push-ups done up to any given day of the month.

2. Estimate the total number of push-ups that Scott will do in a month if he continues to increase

the number of push-ups he does each day in the pattern shown above.

3. Draw the diagram that shows the total number of pushups that Scot has done in the month at

the end of each day.

4. How many push-ups will Scott have done after a week?

11





5. Model the total number of push-ups that Scott has completed on any given day during “Macho

March”. Include both recursive and explicit equations.

6. Will Scott meet his goal and earn the donation for the charity? Will he get a bonus? If so, how much? Explain.

12





1.3 Scott’s Macho March – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifystudentunderstandingofquadraticfunctionsby

givinganotheropportunitytocreateaquadraticmodelforacontext.Thistaskintroducestheidea

thatquadraticfunctionsaremodelsforthesumofalinearfunction,whichobviouslycreatesa

linearrateofchange.Again,studentshavetheopportunitytousealgebraic,numeric,andgraphical

representationstomodelastorycontextwithavisualmodel.

StandardsFocus:

F-BF.1:Writeafunctionthatdescribesarelationshipbetweentwoquantities.*


F-LE:Constructandcomparelinear,quadraticandexponentialmodelsandsolveproblems.

A-CED.1Createequationsandinequalitiesinonevariableandusethemtosolveproblems.Include

equationsarisingfromlinearandquadraticfunctions,andsimplerationalandexponential

functions.

A-CED.2Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;


F-IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesof

graphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbal

descriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionis

increasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;end

behavior;andperiodicity.

F-IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative

relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofperson-hoursittakes

toassemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainfor

thefunction.

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbyclarifyingthecontexttobesurethatstudentsunderstandthatinthefirstpartof

thetasktheyarelookingatthenumberofpushupsthatScottdoeseachdayandinthesecondpart





ofthetask,theyarelookingatthesumofthenumberofpushupsthatScotthascompletedona

particularday.Question1maybefamiliarforstudentsthatdidthe“Scott’sWorkout”taskinthe

previousyear.

Explore(SmallGroup):Monitorstudentthinkingastheywork.Sincestudentsshouldbevery

familiarwithrepresentationsforlinearfunctions,don’tallowanygrouptospendtoomuchtimeon

question#1.Encouragestudentstousetables,graphs,andrecursiveandexplicitequationsasthey

workonthetask.Listentostudentsandidentifydifferentgroupstopresentandexplaintheirwork

ononerepresentationeach.Ifstudentsarehavingdifficultywritingtheequation,askthemtobe

surethattheyhavetheotherrepresentationsfirst.Liketheprevioustasks,itwillbehelpfultouse

thevisualmodeltodeveloptheexplicitequationforthequadratic.

Discuss(WholeClass):Whenthevariousgroupsarepreparedtopresent,startthediscussion

withatableforproblem#1.Besurethatthecolumnsofthetablearelabeled.Afterstudentshave

presentedtheirtable,askstudentstoidentifythedifferencebetweenconsecutivetermsandthe

typeoffunction(linear).Havestudentspresentbothanexplicitandrecursiveequationandto

connecttheirequationstothegeometricrepresentationandthetable.Askstudentstoidentifythe

domainandrangeofthefunction.

Brieflydiscussquestions2and3,focusingonstrategiesstudentsusedtocomeupwitha

reasonableestimatein#2.Turnthediscussiontoquestion#4.Extendthetablepresentedearlier

toincludethesumofthepushups.

Recursiveequation: !(1) = 3, !(!) = !(! − 1) + 2

Explicitequation: !(!) = 3+ 2(! − 1)





Begintocompletethefirstdifferencetableandaskstudentswhattheynotice.Theyshouldseethat

thefirstdifferenceisthesameasthenumberofpushupseachday.Askstudentswhythisturnsout

tobethecase.Atthispoint,identifythatthefunctionisquadratic,basedonthelinearrateof

change.

Askstudentstopresentagraph.

Dependingonthescaleselected,theirgraphsmaylooknearlylinear.Connectthegraphtothetype

ofgrowthshowninthetabletoexplainwhythegraphisn’treallylinear.

Turnthediscussiontotherecursiveequation.Asktheclasstobepreparedtoconnecttheir

equationswiththevisualmodelintheproblem.Studentsthathavenoticedthateachfigureis

composedofthepreviousfigureplusthatday’spushupswillwriteanequationsimilarto:

! 1 = 3,! ! = !(! − 1) + (2! + 1)





Comparetheequationforthelinearandquadraticfunction.Askstudentstoidentifysimilarities

anddifferences.Theimportantthingtonoticeisthatthechangeinthefunctionistheexpression

addedtotheendoftherecursiveformula.Ifthechangeisasinglenumber,asinthelinearfunction,

thatshowsaconstantrateofchange.Ifthechangeisalinearexpression,thenthefunctionis

quadratic.

Havestudentspresenthowtheythoughtaboutthevisualrepresentationtowriteanexplicit

formula.Studentsmayuseastrategysimilarto“SomethingtoTalkAbout”bycopyingand

reflectingthefigureontopofitselftoformarectangle.Inthiscase,theywillgetanequationthatis:

! ! = ![ !!!! !!)]! ,withthenumeratorofthefractionrepresentingtheareaoftherectangleand

thendividingby2toaccountforthefactthattheactualfigureisonlyhalfoftheareaofthe

rectangle.Otherstudentsmayhaveusedotherstrategiestofindanexplicitequation,butthey

shouldallsimplifytobe! ! = !(! + 2).Askstudentsforthedomainandrange.Concludethediscussionwithasummaryofwhattheclassknowsaboutquadraticfunctionsthus

far.Thediscussionshouldincludefeaturessuchas:

• Thefirstdifferenceislinearandtheseconddifferenceisconstant.

• Theyaremodelsforthesumofthetermsinalinearfunction.

• Theymodelsituationswithchangein2dimensions(likearea)

• Theequationsaretheproductof2linearfactors.

• Theequationshaveasquaredterm.

• Thegraphscurve.(Atthispoint,studentshavenotexperiencedafullparabolagraph

becauseofthelimiteddomains.)







1.3

READY Topic:MultiplyingtwobinomialsInthepreviousRSG,youwereaskedtousethedistributivepropertyontwodifferenttermsinthesame

problem.Example:!"#$%&#' !"# !"#$%"&' 3! 4! + 1 + 2 4! + 1 .Youmayhavenoticedthatthebinomial 4! + 1 occurredtwiceintheproblem.Hereisasimplerwaytowritethesameproblem: 3! + 2 4! + 1 .Youwillusethedistributivepropertytwice.Firstmultiply3! 4! + 1 ;thenmultiply+2 4! + 1 .Addtheliketerms.Writethex2termfirst,thex-termsecond,andtheconstanttermlast.

!" !" + ! + ! !" + ! → !"!! + !" + !" + ! → !"!! + !" + !" + ! → !"!! + !!" + !

Multiplythetwobinomials.(Youranswershouldhave3termsandbeinthisform!!! + !" + !.)1. ! + 5 ! − 7 2. ! + 8 ! + 3 3. ! − 9 ! − 4

4. ! + 1 ! − 4 5. 3! − 5 ! − 1 6. 5! − 7 3! + 1

7. 4! − 2 8! + 10 8. ! + 6 −2! + 5 9. 8! − 3 2! − 1 SET Topic:DistinguishingbetweenlinearandquadraticpatternsUsefirstandseconddifferencestoidentifythepatterninthetablesaslinear,quadratic,orneither.Writetherecursiveequationforthepatternsthatarelinearorquadratic.

10.a.Pattern:b.Recursiveequation:

! !-3 -23-2 -17-1 -110 -51 12 73 13

11.

a.Pattern:b.Recursiveequation:

! !-3 4-2 0-1 -20 -21 02 43 10


! !-3 -15-2 -10-1 -50 01 52 103 15


liketermsSimplifiedform

13






1.3


! !-3 24-2 22-1 200 181 162 143 12


! !-3 48-2 22-1 60 01 42 183 42


! !-3 4-2 1-1 00 11 42 93 16

16.a.Drawfigure5.b.Predictthenumberofsquaresinfigure30.Showwhatyoudidtogetyourprediction.GO Topic:InterpretingrecursiveequationstowriteasequenceWritethefirstfivetermsofthesequence.

17.! 0 = −5; ! ! = ! ! − 1 + 8 18.! 0 = 24; ! ! = ! ! − 1 − 5

19.! 0 = 25; ! ! = 3! ! − 1 20.! 0 = 6; ! ! = 2! ! − 1

Figure 5Figure 4Figure 3Figure 2Figure 1

14





1.4 Rabbit Run


Mishahasanewrabbitthatshenamed“Wascal”.ShewantstobuildWascalapensothat

therabbithasspacetomovearoundsafely.Mishahaspurchaseda72footrolloffencingto

buildarectangularpen.

1. IfMishausesthewholerolloffencing,whataresomeofthepossibledimensionsofthepen?

2. IfMishawantsapenwiththelargestpossiblearea,whatdimensionsshouldsheuseforthesides?Justifyyouranswer.

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3. Writeamodelfortheareaoftherectangularpenintermsofthelengthofoneside.Includebothanequationandagraph.

4. Whatkindoffunctionisthis?Howdoyouknow?

5. HowdoesthisfunctioncomparetothesecondtypeofblockIlogosinIRule?

16





1.4 Rabbit Run – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifyandextendstudentthinkingaboutquadratic

functionstoincludethosewithamaximumpoint.Studentswillusethegraphofthefunctionto

discussthedomainandrangeofacontinuousquadraticfunctioninadditiontoidentifyingthe

maximumvalueandfindingtheintervalsonwhichthefunctionisincreasinganddecreasing.

CoreStandardsFocus:

F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.*


A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.*

a.Interpretpartsofanexpression,suchastermsfactors,andcoefficients.

A.CED.2Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;



RelatedStandards:F.IF.4

Note:Graphingtechnology(eithercalculatorsorcomputersoftwaresuchasGeogebraor

Desmos)wouldbeusefulforthistask.

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbyfamiliarizingstudentswiththecontextofbuildingarectangularrabbitpenthat

canbesurroundedby72feetoffencing.Askstudentstothinkofsomepossibledimensionsforthe

pen.Besurethattheyarethinkingaboutperimeters,notareas,sothattheyknowthatarectangle

withdimensionsof8x9won’twork.Aftertheyhavesuccessfullyfoundoneortwopossible

dimensionsforthepen,askifallthedifferentrectangleswillhavethesamearea.Comparethe

areasofacoupleoftherectanglesthattheyhavefoundandtellstudentsthattheoneoftheirjobsin





thistaskwillbetofindtherectanglewiththemostareabecausethatwillgivetherabbitthemost

roomtomovearound.


Asstudentsareworking,circulatetoseethattheyaretryingvariousvaluesforthedimensionsof

therectanglethathaveaperimeterof72.Itmayhelpthemtobesystematicandorganizedin

thinkingaboutthepossibilitiesforthedimensionssotheycanusethepatternstheyobservetofind

therelationshipbetweenthelengthandwidthandwritetheequationforthearea.


Startthediscussionwithatablesuchas:

Atthispoint,itmaybeusefultodisplaytherestofthetableusingtechnologysothatstudentscan

seeapossiblemaximumvalueinthetable.Studentsshouldnoticethatoncetheyreachalengthof

18,theybegintogetthesamerectangles,butwiththelengthandwidthswitched(different

orientation).

DiscussthedomainandrangeoftheA(x).Thisisthefirstcontinuouscontext,whichstudentsmay

notimmediatelyrecognize.Theyshouldbeabletowritethedomaininintervalnotationorset

buildernotation.Askifthisisaquadraticfunctionandhowtheyknow.Studentsshouldbe

justifyingtheiranswerusingthefirstdifferenceintheirtables.Thismayraisesomecontroversy

becausethefirstdifferenceispositive,thennegative.Resolvethiscontroversybyseparatingthe

tableintotwosections(atthelineofsymmetry)andcheckingforalinearfirstdifferenceand

constantseconddifferenceineachsection.

Lengthx Width AreaA(x)1 35 352 34 683 33 994 32 1285 31 1556 30 180! 36 − ! !(36 − !)





Askastudenttopresentanequationforthefunction:! ! = ! 36 − ! . Askstudentshowthisequationisconnectedtothetableandhowitfitswhattheyknowaboutquadraticfunctions.Ask

whatisdifferentaboutthisequationcomparedtootherquadraticsthattheyhaveseen.Sinceitis

importantforthemtorecognizethatthe!! termisnegative,youmayaskthemtodistributethextoseeanotherformoftheequation.

Turnthediscussiontothegraph(usingtechnology).Askstudentsaboutthefeaturesofthegraph

thattheynotice.Theyshouldcomeupwithalistlike:

• Thefunctioniscontinuous.

• Thefunctionincreasesintheinterval(0,18)anddecreasesin(18,36).

• Thefunctionhasamaximumat(18,324).

Askhowthisgraphcompareswiththegraphsofotherquadraticfunctionsthattheyhaveseen.

Sincethisisthefirstfullparabolatheyhaveseen,youwillneedtotellthemthatbecauseofthe

limiteddomainsthatwereusedinpreviouscontexts,theyhaveonlyseenpartoftheparabolasthat

arerepresentativeofquadraticfunctions.Thisoneopensdownandhasamaximum,graphsof

quadraticfunctionscanalsoopenupwardandhaveaminimum.







1.4

READY Topic:ApplyingslopeformulaCalculatetheslopeofthelinebetweenthegivenpoints.Useyouranswertoindicatewhichlineisthesteepest.

1.A(-3,7)B(-5,17) 2.H(12,-37)K(4,-3)

3.P(-11,-24)Q(21,40) 4.R(55,-75)W(-15,-40)

SET Topic:Investigatingperimetersandareas

Adamandhisbrotherareresponsibleforfeedingtheirhorses.Inthespringandsummerthehorsesgrazeinanunfencedpasture.Thebrothershaveerectedaportablefencetocorralthehorsesinagrazingarea.Eachdaythehorseseatallofthegrassinsidethefence.Thentheboysmoveittoanewareawherethegrassislongandgreen.Theporta-fenceconsistsof16separatepiecesoffencingeach10feetlong.Thebrothershavealwaysarrangedthefenceinalongrectanglewithonelengthoffenceoneachendand7piecesoneachsidemakingthegrazingarea700sq.ft.Adamhaslearnedinhismathclassthatarectanglecanhavethesameperimeterbutdifferentareas.Heisbeginningtowonderifhecanmakehisdailyjobeasierbyrearrangingthefencesothatthehorseshaveabiggergrazingarea.Hebeginsbymakingatableofvalues.Helistsallofthepossibleareasofarectanglewithaperimeterof160ft.,whilekeepinginmindthatheisrestrictedbythelengthsofhisfencingunits.Herealizesthatarectanglethatisorientedhorizontallyinthepasturewillcoveradifferentsectionofgrassthanonethatisorientedvertically.Soheisconsideringthetworectanglesasdifferentinhistable.Usethisinformationtoanswerquestions5–9onthenextpage.


Horizontal Vertical

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1.4

5.FillinAdam’stablewithallofthearrangementsforthefence.(Thefirstoneisdoneforyou.)

Lengthin

“fencing”units

Widthin“fencing”

unitsLengthinft. Widthinft. Perimeter(ft)

Area

(ft)2

1unit 7units 10ft 70ft 160ft 700ft2

a. 2units 160ft

b. 3units 160ft

c. 4units 160ft

d. 5units 160ft

e. 6units 160ft

f. 7units 160ft

6.DiscussAdam’sfindings.Explainhowyouwouldrearrangethesectionsoftheporta-fencesothatAdamwillbeabletodolesswork.

7.MakeagraphofAdam’s

investigation.Letlengthbethe

independentvariableand

areabethedependentvariable.

Labelthescale.

8.Whatistheshapeofyourgraph?

9.Explainwhatmakesthisfunction

beaquadratic.

18






1.4

GO Topic:ComparinglinearandexponentialratesofchangeIndicatewhichfunctionischangingfaster.

10. 11. 12.

13. 14. 15.

16a.Examinethegraphattheleftfrom0to1.

WhiWhichgraphdoyouthinkisgrowingfaster?

b.Nowb.Nowlookatthegraphfrom2to3.

Whichgraphisgrowingfasterinthisinterval?

g(x)

f(x)

r(x)

s(x)

q(x)

p(x)

r(x)s(x)

w(x)

m(x)

d(x)

h(x)

g(x)f(x)

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1.5 The Tortoise and The Hare


Inthechildren’sstoryofthetortoiseandthehare,theharemocksthetortoiseforbeingslow.Thetortoisereplies,“Slowandsteadywinstherace.”Theharesays,“We’lljustseeaboutthat,”andchallengesthetortoisetoarace.Thedistancefromthestartinglineofthehareisgivenbythefunction:

! = !!(dinmetersandtinseconds)Becausethehareissoconfidentthathecanbeatthetortoise,hegivesthetortoisea1meterheadstart.Thedistancefromthestartinglineofthetortoiseincludingtheheadstartisgivenbythefunction:

! = 2!(dinmetersandtinseconds)

1. Atwhattimedoestheharecatchuptothetortoise?

2. Iftheracecourseisverylong,whowins:thetortoiseorthehare?Why?

3. Atwhattime(s)aretheytied?

4. Iftheracecoursewere15meterslongwhowins,thetortoiseorthehare?Why?

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5. Usetheproperties! = 2!and! = !!toexplainthespeedsofthetortoiseandthehareinthefollowingtimeintervals:

Interval Tortoise! = !! Hare! = !![0,2)

[2,4)

[4,∞)

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1.5 The Tortoise and The Hare – Teacher Notes A Solidify Understanding Task

SpecialNotetoTeachers:Graphingtechnologyisnecessaryforthistask.Purpose:Thepurposeofthistaskistocomparequadraticandexponentialfunctionsbyexaminingtablesandgraphsforeach.Theywillconsiderratesofchangeforeachfunctiontypeinvariousintervalsandultimately,seethatanincreasingexponentialfunctionwillexceedaquadraticfunction.CoreStandardsFocus:F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.*


A-SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.*A-CED.2Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswithlabelsandscales.*Focusonsituationsthatexhibitaquadraticorexponentialrelationship.F-LE.3Observe,usinggraphsandtables,thataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.F-IF.6Calculateandinterprettheaveragerateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval.Estimatetherateofchangefromagraph.

TheTeachingCycle:

Launch(WholeClass):

Beginthetaskbysimplymakingsurethatstudentsunderstandtheproblemsituationandthefunctionsthathavebeengiventomodelthedistancetraveledbythetortoiseandthehare.Toavoidconfusion,itmaybeusefultoclarifythattheharegavethetortoiseaonemeterheadstart,whichinthiscasemeansthatatt=0,thetortoisewasat1meter.Theequation! = 2!accountsfortheheadstartbecauseatt=0,theequationyields1.





Theotherpossiblesourceofconfusionisfoundinthetablein#5.Itisaskingstudentstodescribethespeed(thepromptdoesn’tindicateaverageorinstantaneousinhopesthatstudentsmaydescribeboth),notsimplythedistancetraveledineachinterval.


Observestudentsastheywork.Thetaskdoesn’tprescribeaparticularmethodforthinkingaboutthebehaviorofeachofthetwofunctions,sotables,graphs,andequationsshouldallbeexpected.Itisgoodmathematicalthinkingtowriteanequationbysettingthetwofunctionsequaltoeachother,butstudentsmaygetstucklaboringoveranalgebraicsolution.Encouragethemtousetheirthinkingaboutequalitywithadifferentrepresentation.Youmayneedtopromptstudentstochangethewindowsontheirtographstodisplayparticularpiecesofthegraphsthatareinteresting.


Beginthediscussionofquestions#1and3withastudentthatwroteanequationthatsetsthetwofunctionsequal.Havetheclassexplainwhythisstrategymakessense,buttelltheclassthatthisisacasewherethealgebraicsolutiontotheequationisnotaccessible.Thenhaveastudentthatusedagraphtofindtheintersectionstoexplain.Havethemsharehowthegraphingstrategyrelatestotheequationwiththetwofunctionsequalthatwaswritten.Projectagraphofthetwofunctionsonthesameaxesfordiscussion,possiblybeginningwithonelikethis,whichisprobablywheremoststudentsstarted:

Brieflydiscussthedomainofthetwofunctionsinthiscontextbeginsatt=0.Onthegraphshow,thetortoiseisinred.Askstudentswhattheynoticeaboutthegraph,atwhattimetheharecaught





upwiththetortoiseandwholooksliketheywillwininthisview.Askstudentswhichcharacterwasthefastest(hadthegreatestspeedorrateofchange)intheinterval.Sincetheywereboth4metersfromthestartinglineafter2seconds,theharemusthavebeengoingfaster.Studentsshouldnoticethesteeperslopeofthehare’sgraphinthisinterval.Askastudentthatusedatabletoshowhowtoseethegreaterspeedinthetable.Askastudenttopresentthetwographsintheintervalfromt=2tot=4.

Askstudentswhattheycansayaboutthespeedoftheharevsthespeedofthetortoiseinthisinterval.Askthemtocalculatetheaveragespeedofbothcharactersandcomparethattowhattheydescribedearlier.Tellthemthedifferencebetweeninstantaneousspeedattimetandtheaveragespeedinaninterval.FindingtheinstantaneousspeedcanbedoneinCalculus(spoileralert),butitcanbeestimatedbyfindingtheaveragespeedinasmallintervalnearthepointofinterest.Weoftenusethefirstdifferenceinatableasanapproximationoftheinstantaneousrateofchange,althoughitisreallytheaveragerateofchangeintheintervalfromonerowofthetabletothenext.Nowask,whichcharacterwinstheraceandwhy.Haveastudentshowagraphsimilartotheonebelow:(Window3 < ! < 10,8 < ! < 1024)





Askstudentstodiscusstheratesofchangeinthisinterval.Whydoesthegraphofthequadraticseemsoflatcomparedtotheexponentialfunction?Makethepointthatanincreasingexponentialfunctionwilleventuallyexceedanyquadraticfunctionbecausethequadraticisincreasinglinearly,buttheexponentialisincreasingexponentially.Confirmtheideabysharingtablesforeachofthetwofunctions,lookingatthefirstdifferenceforeach.ExitTicketforStudents:Inthecaseofthetortoiseandthehare,theexponentialfunctionexceeded

thequadraticfunctionforlargevaluesoft.Iwonderifthiswouldbethecaseifthequadraticfunction

weredoubled.Compare! = 2!!and! = 2! anddecidewhichfunctiongrowsfasterforlargevaluesofx.Explainwhyyouransweriscorrect.AlignedReady,Set,Go:QuadraticFunctions1.6






1.5

READY Topic:RecognizingFunctionsIdentifywhichofthefollowingrepresentationsarefunctions.IftherepresentationisNOTafunctionstatehowyouwouldfixitsoitwas.

1.D={(4,-1)(3,-6)(2,-1)(1,2)(0,4)(2,5)} 2.Thenumberofcaloriesyouhaveburnedsincemidnightatanytimeduringtheday.

3.

4.x -12 -8 -6 -4f(x) 25 25 25 25

5.

6.

SET

Topic:Comparingratesofchangeinlinear,quadratic,andexponentialfunctionsThegraphattherightshowsatimevs.distancegraphoftwocarstravelinginthesamedirectionalongthefreeway.7.Whichcarhasthecruisecontrolon?Howdoyouknow?8.Whichcarisaccelerating?Howdoyouknow?9.Identifytheintervalinfigure1wherecarAseemstobegoingfasterthancarB.10.Forwhatintervalinfigure1doescarBseemtobegoingfasterthancarA?11.Whatinthegraphindicatesthespeedofthecars?12.AthirdcarCisnowshowninthegraph(seefigure2).All3carshavethesamedestination.Ifthedestinationisadistanceof12unitsfromtheorigin,whichcardoyoupredictwillarrivefirst?Justifyyouranswer.


12

10

8

6

4

2

5 10

B

A

Figure 1

12

10

8

6

4

2

5 10

CB

A

Figure 2

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1.5

GO Topic:IdentifyingdomainandrangefromagraphStatethedomainandrangeofeachgraph.Useintervalnotationwhereappropriate.

13a.Domain__________b.Range___________

14a.Domain__________b.Range___________

15a.Domain__________b.Range___________

16a.Domain__________b.Range___________

17a.Domain__________b.Range___________

18a.Domain__________b.Range___________

19a.Domain__________b.Range___________

20a.Domain__________b.Range___________

21.Arethedomainsof#19and#20thesame?Explain.

8

6

4

2

–2

–4

8

6

4

2

–2

–4

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1.6 How Does It Grow?

A Practice Understanding Task

Foreachrelationgiven:

a. Identifywhetherornottherelationisa

function;

b. Determineifthefunctionislinear,exponential,quadraticorneither;

c. Describethetypeofgrowth

d. Createonemorerepresentationfortherelation.

1. Aplumberchargesabasefeeof$55foraservicecallplus$35perhourforeachhour

workedduringtheservicecall.Therelationshipbetweenthetotalpriceoftheservicecalland

thenumberofhoursworked.

2.

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3.

4. ! = !! ! − 2 ! + 4

5.

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6. ! = !! ! − 2 + 4

7. Therelationshipbetweenthespeedofacarandthedistanceittakestostopwhentravelingatthatspeed.

Speed(mph)

StoppingDistance(ft)

10 12.520 5030 112.540 20050 312.560 45070 612.5

8.

Therelationshipbetweenthenumberofdotsinthefigureandthetime,t.

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9. Therateatwhichcaffeineiseliminatedfromthebloodstreamofanadultisabout15%

perhour.Therelationshipbetweentheamountofcaffeineinthebloodstreamandthenumber

ofhoursfromthetimetheadultdrinksthecaffeinatedbeverageiftheinitialamountof

caffeineinthebloodstreamis500mg.

10.

.

11. ! = (4! + 3)(! − 6)

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12. MaryContrarywantstobuildarectangularflowergardensurroundedbyawalkway4meterswide.Theflowergardenwillbe6meterslongerthanitiswide.a. Therelationshipbetweenthewidthofthegardenandtheperimeterofthewalkway.

b. Therelationshipbetweenthewidthofthegardenandareaofthewalkway.

13. ! = !!!!!

+ 4

14.

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1.6 How Does It Grow? – Teacher Notes A Practice Understanding Task

Purpose:Thepurposeofthistaskistorefinestudentunderstandingofquadraticfunctionsby

distinguishingbetweenrelationshipsthatarequadratic,linear,exponentialorneither.Examples

includerelationshipsgivenwithtables,graphs,equations,visuals,andstorycontext.Studentsare

askedtodrawupontheirunderstandingofrepresentationstodeterminethetypeofchangeshown

andtocreateasecondrepresentationfortherelationshipsgiven.

CoreStandardsFocus:

F-IF.9Comparepropertiesoftwofunctionseachrepresentedinadifferentway(algebraically,

graphically,numericallyintables,orbyverbaldescriptions).


F-LEConstructandcomparelinear,quadratic,andexponentialmodelsandsolveproblems.

SpecialNotetoTeachers:Graphingtechnologyisrecommendedforthistask.

TheTeachingCycle:

Launch(WholeClass):

Toactivatestudentbackgroundknowledge,askstudentstotakethefirst3minutestowritedown

whattheyknowaboutlinear,quadratic,andexponentialfunctions.Askstudentstoshareafewof

theirideaswiththeclasswithoutrecordingtheiranswers.Goovertheinstructionsforthetaskto

ensurethatstudentsknowwhattheyaretodo.


Monitorstudentsastheywork,lookingforproblemsthataregeneratingdiscussion,student

misconceptions,orimportantideasforthediscussion.Watchfordiscussionsofwhetherornotall

thevariousfunctiontypescanbecontinuousordiscrete.Listenforhowtheydealwithproblem#5,

whichisaparabola,butnotafunction.(Suchcaseswillbeexploredinalatermodule.)Sincemost

ofthequadraticsthatstudentshaveworkedwithsofarhavedomainsthatbeginat0,studentsmay





notimmediatelythatgraphslike#14arequadratic.(Graphingquadraticsisamajortopicin

module2).Manyoftheequationsinthistaskareinunfamiliarforms,whichwillpushstudentsto

buildtablesorgraphstoidentifythetypeofchange.Graphingtechnologyshouldbeavailablefor

studentuseingraphingfunctions.Thefocusforthisworkshouldbeonratesofchangeand

representations.


Beginthediscussionbyaskingwhichrelationshipstheyclassifiedasquadraticfunctions.(Answer:

2,4,7,8,11,12b,and14).Foreachofthese,haveastudentshowasecondrepresentationandtalk

abouthowtheyidentifiedthetypeofchangeintheirtworepresentations.Whenthisiscomplete,

askastudenttoidentifyoneexponentialfunctionandsharethesecondrepresentationandtheir

justificationindecidingthefunctionisexponential.(Problems3,9,and13areexponential.)Repeat

theprocesswiththelinearfunctions.Closetheclassbyaskingstudentstowritefor3more

minutestoaddtotheirlistofthingstheyknowaboutlinear,quadratic,andexponentialfunctions.







1.6

READY Topic:Transforminglines1.Graphthefollowinglinearequationsonthegrid.Theequationy=xhasbeengraphedforyou.Foreachnewequationexplainwhatthenumber3doestothegraphofy=x.Payattentiontothe

y-intercept,thex-intercept,andtheslope.Identifywhatchangesinthegraphandwhatstaysthesame.a. y=x+3

b. y=x–3

c. y=3x

2.Thegraphofy=xisgiven.(Seefigure2.)Foreachequationpredictwhatyouthinkthenumber-2willdotothegraph.Thengraphtheequation.

a. y=x+(-2)

Prediction:

b. y=x–(-2)Prediction:

c. y=-2xPrediction:


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1.6

SET

Topic:Distinguishbetweenlinear,exponentialandquadraticfunctionsForeachrelationgiven:

a. Identifywhetherornottherelationisafunction.(Ifit’snotafunction,skipb–d.)b. DetermineifthefunctionisLinear,Exponential,QuadraticorNeither.c. Describethetypeofgrowth.d. Expresstherelationintheindicatedform.

3.Ihad81frecklesonmynosebeforeIbeganusingvanishingcream.AfterthefirstweekIhad27,thenextweek9,then3...a.Function?b.Linear,Exponential,QuadraticorNeitherc.Howdoesitgrow?d.Makeagraph.LabelyouraxesandthescaleShowall4points.

4.

x y0 811 80!!

2 80!!3 804 79!!

a.Function?b.Linear,Exponential,QuadraticorNeitherc.Howdoesitgrow?d.Writetheexplicitequation.

5.

a.Function?b.Linear,Exponential,QuadraticorNeitherc.Howdoesitgrow?d.Createatable

6.Speedinmphofabaseballvs.distanceinft.a. Function?b. Linear,Exponential,QuadraticorNeitherc. Howdoesitgrow?d. Predictthedistancethebaseballflies,ifitleavesthebatataspeedof115mph.

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1.6

GO Topic:MatchingfunctionrepresentationsMatchthefunctionontheleftwiththeequivalentfunctionontheright.

______7. f x( ) = −2x + 5 a. f x( ) = 5 2( )x ______8.

b.

______9.Iput$7000inasavingsaccountthatpays3%interestcompoundedannually.Iplantoleaveitinthebankfor20years.TheamountIwillhavethen.

c. f (1) = 2; f (n+1) = f (n)+ 2n+ 2

______10.Theareaofthetrianglesbelow.

d.

______11. f 0( ) = 5; f n( ) = 2∗ f n −1( ) e.y+x=0______12. f 0( ) = 5; f n( ) = f n −1( )− 2

f.! = ! − 1 ! + 3

______13.x -7.75 -¼ ½ 11.6f(x) 7.75 ¼ -½ -11.6

g.A=7000(1.03)20

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Module 1: Quadratic Functions - Mathematics Vision Project

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