SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

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3.1 Getting Ready for a Pool Party A Develop Understanding Task

Sylviahasasmallpoolfullofwaterthatneedstobeemptiedandcleaned,thenrefilledforapool

party.Duringtheprocessofgettingthepoolready,Sylviadidallofthefollowingactivities,each

duringadifferenttimeinterval.

Removedwaterwithasinglebucket

Filledthepoolwithahose

(samerateasemptyingpool)

Drainedwaterwithahose

(samerateasfillingpool)

Cleanedtheemptypool

Sylviaandhertwofriendsremovedwaterwith

herthreebuckets

Tookabreak

1.Sketchapossiblegraphshowingtheheightofthewaterlevelinthepoolovertime.Besureto

includeallofactivitiesSylviadidtopreparethepoolfortheparty.Rememberthatonlyoneactivity

happenedatatime.Thinkcarefullyabouthoweachsectionofyourgraphwilllook,labelingwhere

eachactivityoccurs.

2.CreateastoryconnectingSylvia’sprocessforemptying,cleaning,andthenfillingthepooltothe

graphyouhavecreated.Doyourbesttouseappropriatemathvocabulary.

3.Doesyourgraphrepresentafunction?Whyorwhynot?Wouldallgraphscreatedforthis

situationrepresentafunction?

1

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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3 . 1 Getting Ready for a Pool Party – Teacher Notes A Develop Understanding Task

Purpose:Thistaskisdesignedtodeveloptheideasoffeaturesoffunctionsusingasituation.

Featuresoffunctionssuchasincreasing/decreasingandmaximum/minimumcanbedifficultfor

studentstounderstand,eveninagraphicalrepresentationiftheyarenotusedtoreadingagraph

fromlefttoright.Asituationusingthewaterlevelofapooloveraperiodoftimecanprovide

opportunitiesforstudentstomakeconnectionstothesefeatures.Whilesomepartsofthegraph

needtocomebeforeothers(emptyingthepoolbeforefillingthepool),othersituationscanbe

switchedaround(emptyingthewaterwithbucketsandemptyingthewaterwithahose).Thekey

featuresofthistaskinclude:

• Thesketchofthegraphisdecreasingwhenthewaterisbeingemptiedfromthepoolandthegraphisincreasingwhenthepoolisbeingfilledwithwater.

• Thesketchofthegraphshowsaheightofzeroduringaperiodoftimewherethepoolisemptyandbeingcleaned.

• Thesketchofthegraphiscontinuouswhenthehoseisused(bothforfillingandemptying)andthattherateofchangeisthesamebothwhenfillingandwhenemptying.

• Thesketchofthegraphlookslikea“stepfunction”whenusingabucket,withthewaterleveldroppingthreetimesfasterwhenSylviahasfriendsassisting.

• Studentscommunicatetheirunderstandingofgraphsinpart2• Studentsexpressthatthissituationisafunctionbyindicatingthateveryinputoftimehas

exactlyoneoutputrepresentingthedepthofwater.

CoreStandardsFocus:

F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.�

RelatedStandards:F.IF.1,A.REI.10,F.IF.5,F.IF.7

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

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StandardsforMathematicalPracticeofFocusintheTask:

SMP1–Makesenseofproblemsandpersevereinsolvingthem

SMP3–Constructviableargumentsandcritiquethereasoningofothers

TheTeachingCycle:

Launch(WholeClass):

Readtheinitialsituationandthefirstquestion.Makesurestudentsunderstandtheyaretographa

situationwhereallmethodsforemptying,cleaning,andthenfillingthepoolareusedinthe

problem.Whentheyhavecompletedtheirgraph,theyaretowriteastoryconnectingSylvia’s

processforemptying,cleaning,andthenfillingthepooltothegraphtheyhavecreated.

Explore(SmallGroup):

Yourstudentsmayalreadybefamiliarwithstrategiesforcreatinggraphsgivenasituation.They

arealsofamiliarwithslopeandrateofchangeaswellasgraphingcontinuousandnon-continuous

situations(itcanbearguedthatthegraphiscontinuousevenduringthebucketremovalasthereis

notaninstantaneousjump).Thecontextofthisproblemfocusesondecreasingandincreasing

intervalsofthegraph,rateofchange,andtheideaofafunctionbeingacontinuouslinear

relationshipvsasituationwhoseratechangesina‘stepfunction’fashion.Duringthemonitoring

phase,pressstudentswhoarenotbeingspecificenoughbyaskingquestionssuchas:

• Whatishappeningduringeachintervaloftimeontheirgraph?

• Compareandcontrastthedifferentactivities.Howshouldthegraphforthesesituations

looksimilar?Different?

Thiswillhelpbringoutthefeaturesoffunctionsdescribedinthislesson(seepurposestatement

above).Otherthanthefeaturesoffunctionslistedabove,thistaskalsosurfacestheideaofdomain

andstepfunctions.Ifyoudonothaveanystudentscreateastepfunctionrepresentationduringthe

bucketsituation,buttheydoshowadiscreterepresentation,plantousethisduringthediscussion

partofthelessontogetathowthedomainiscontinuous,evenifthegraphisnot.

Havestudentssharetheirstorywithapartner,thenhavethemdiscusswhattheyagreeaboutwith

eachother’sgraphaswellaspossibleerrorsintheirthinking.

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

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Discuss(WholeClass):

Choosestudentstosharewhohavethefollowingaspartoftheirgraph.(Besurestudentswhoare

sharingcanalsoshowtheirgraphstoallstudentswhileexplainingthefeaturesoftheirgraphs):

• Student1:haveastudentsharethathaslabeledtheiraxesandhasclearideasaboutwhere

thegraphshouldbeincreasing/decreasing.Thisstudentisnotsharingtheirstory,but

highlightingfeaturesoftheirfunction.

• Student2:haveastudentsharethathasrepresentedthebucketpartofthegraphtobe

‘discrete’innature(notacontinuousdecreasinglineargraph).Highlightthedifference

betweenacontinuousconstantrateofchangeversusthe‘jump’intheamountofwaterin

thepoolwhenusingabucket.Atthispoint,wearestillfocusingonincreasing/decreasing,

comparingratesandtheideathatafunctioncanhavedifferentcomponentswithinthe

function(i.e.continuousanddiscrete).Iftheconversationnaturallycomesupatthistime

abouthowthediscreteportionofthegraphshouldlookdifferent(morelikeastep

function),thendiscussthisnowandwhereappropriatediscussdiscontinuousversus

discrete.Otherwisewaituntilthenextstudentsharesandthenbringitalltogether.Again,

thisstudentisnotsharingtheirstory,buthighlightingfeaturesoftheirfunction.

• Student3:selectastudenttosharetheirstorywhileshowingeachpartonthegraph.Be

suretochoosesomeonewithanaccuratestorytohighlightthefeaturesfocusedinthistask.

Besurethediscussionincludesthesefeatures:increasing/decreasing,they-intercept,

labelingtheaxesandinterpretingwhatthismeans,andtheratesofchange.Atthistime,if

the‘stepfunction’conversationhasnotoccurred,useagraphthatshowsthebucket

situationasbeingdiscrete,thenaskstudentsquestionssuchas:

• Doesthegraphtellacompletestory?

• Pointingtoanintervaloftimethatiscontinuous,askstudentstodescribewhatis

happeningateachmoment.

• Pointtothediscretepartofthegraphandaskhowmuchwaterisinthepoolbetween

thetwodiscretepoints.

WhilestudentsdonotneedtoknowhowtographstepfunctionsinSecondaryI,thepurposeofthis

conversationistohavestudentsconnectthateverypointonagraphisasolution(A.REI.10)and

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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thatsincetimeiscontinuous,everyinputvalue(thedomain)existsfromthebeginningtotheendofemptying,cleaning,thenfillingthepool(F.IF.5).AlignedReady,Set,Go:Features3.1

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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3.1

READY

Topic:GraphingLinearandExponentialFunctions

Grapheachofthefunctions.Name3pointsthatlieoneachgraph.Chooseascaleforyourgraphthatwillmakeitpossibletographyour3chosenpoints.

1.! ! = −2! + 5 2.! ! = 4 − 3! 3.ℎ ! = 5 3 !

3points: 3points: 3points:

4.! ! = 4 2 ! 5.! ! = 2.5! − 4 6.! ! = 8 3 !

3points: 3points: 3points:

READY, SET, GO! Name PeriodDate

2

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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3.1

SET Topic:Describingattributesofafunctionsbasedongraphicalrepresentation

Foreachgraphgivenmatchittothecontextualdescriptionthatfitsbest.Thenlabeltheindependentanddependentaxiswiththepropervariables.

Graphs ContextualDescriptions

7.

a.Theamountofmoneyinasavingsaccount

whereregulardepositsandsomewithdrawals

aremade.

8.

b.Thetemperatureoftheovenonadaythat

mombakesseveralbatchesofcookies.

9.

c.Theamountofgasolineonhandatthegas

stationbeforeatankertruckdeliversmore.

10.

d.Watermelonsaredeliveredtoafarmer’s

marketeverySaturdaymorning.Thenumber

ofwatermelonsavailableforsaleonThursday.

11.

e.Theamountofmileagerecordedonthe

odometerofadeliverytruckoveratime

period.

3

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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3.1

Giventhepairofgraphsoneachcoordinategrid,createalistofsimilaritiesthetwographsshareandalistofdifferences.(Considerattributeslike,continuous,discrete,increasing,decreasing,linear,exponential,restrictionsondomainorrange,etc.)12.

Similarities:

Similarities: Differences:

13.

Similarities:

Similarities: Differences:

4

SECONDARY MATH I // MODULE 3

FEATURES OF FUNCTIONS – 3.1

Mathematics Vision Project

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3.1

GO Topic:Solvingequations

Foreachequationfindthevalueofxthatmakesittrue.(Hintfor#20and#22:whensolvingalinearequation,youneedtogetthetermcontainingthevariablealoneononeside.Whensolvinganexponentialequation,youalsoneedtogetthetermcontainingthevariablealoneononeside.)

14.10! = 100,000 15.3! + 7 = 5! − 21 16.−6! − 15 = 4! + 35

17.5! − 8 = 37 18.3! = 81 19.3! − 12 = −4! + 23

20.10 = 2! − 22 21.243 = 8! + 3 22.5! − 7 = 118

5