SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3.5 Pooling it Together A Solidify Understanding Task
AlyandDayneworkatawaterparkandhavetodrainthewaterattheendofeachmonthfortheridetheysupervise.Eachusesapumptoremovethewaterfromthesmallpoolatthebottomoftheirride.ThegraphbelowrepresentstheamountofwaterinAly’spool,a(x),andDayne’spool,d(x),overtime.Inthisscenario,theydecidedtoworktogethertodraintheirpoolsandcreatedtheequation:
g(x)=a(x)+d(x).
Answerthefollowingquestionsaboutg(x).
1. Whatdoesg(x)represent?
2. Createthegraphofg(x)onanewsetofaxesusingthegraphsofa(x)andd(x).Identifyg(x)andlabel(scale,axes).
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SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3. Writetheequationforthefunctiong(x)usingthegraphyoucreated.Comparethisequationtothealgebraicrepresentationoffindingthesumoftheequationsfora(x)andd(x).(Theequationswerecreatedinthelasttask,“TheWaterPark”task).
4. Shouldthealgebraicequationofg(x)bethesameasthealgebraicfunctioncreatedfromthegraph?Whyorwhynot?
5. Useboththegraphicalaswellasthealgebraicrepresentationtodescribecharacteristicsofg(x)andexplainwhateachcharacteristicmeans(eachintercept,domainandrangeforthissituationandfortheequation,maximaandminima,whetherornotg(x)isafunction,etc.)
6. Explainwhyaddingthetwovaluesofthey-interceptstogetherina(x)andd(x)canbeusedtofindthey-intercepting(x).
7. Canasimilarmethodbeusedtofindthex-intercepts?Explain.
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SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3 . 5 Pooling it Together – Teacher Notes A Solidify Understanding Task
Purpose:Thepurposeofthistaskisforstudentstocombinefunctions,makesenseoffunctionnotation,andconnectmultiplerepresentations(context,equations,andgraphs).Studentswillalsoaddressfeaturesoffunctionsastheysolveproblemsthatarisefromthiscontext.CoreStandardsFocus:F.BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.�b.Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthatmodelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,andrelatethesefunctionstothemodel.
F.IF.2Usefunctionnotation,evaluatefunctionsforinputsintheirdomains,andinterpretstatementsthatusefunctionnotationintermsofacontext.F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;endbehavior;andperiodicity.�
F.IF.5Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitativerelationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofperson-hoursittakestoassemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainforthe
function.�
F.IF.7Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandinsimplecasesandusingtechnologyformorecomplicatedcases.�a.Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.e.Graphexponentialandlogarithmicfunctions,showinginterceptsandendbehavior,andtrigonometricfunctions,showingperiod,midline,andamplitude.
A.REI.11Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy=g(x)intersectarethesolutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologytographthefunctions,maketablesofvalues,orfindsuccessiveapproximations.
SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
Includecaseswheref(x)and/org(x)arelinear,polynomial,rational,absolutevalue,exponential,andlogarithmicfunctions.�A.CED.3Representconstraintsbyequationsorinequalities,andbysystemsofequationsand/orinequalities,andinterpretsolutionsasviableornon-viableoptionsinamodelingcontext.Forexample,representinequalitiesdescribingnutritionalandcostconstraintsoncombinationsofdifferentfoods.RelatedStandards:F.IF.1,A.REI.10,A.REI.11,N.Q.1,A.CED.2
StandardsforMathematicalPracticeofFocusintheTask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP2–Reasonabstractlyandquantitatively
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP4–Modelwithmathematics
SMP7–Lookandmakeuseofstructure
TheTeachingCycle:
Launch(WholeClass):
Readtheintroductionandremindstudentsofthetask“TheWaterPark”whereAlyandDaynedrainedthewaterfromthepoolstheysuperviseinawaterpark.Askthewholegroup“Whatdoesg(x)=a(x)+d(x)meanincontext?”Afterashortdiscussionthatthatg(x)representsthecombinedeffortsofdrainingthetwopools,havestudentsmovetoquestion2,whichhasthemgraphg(x)bygraphicallyaddingthetwoindividualfunctions.
Explore(SmallGroup):For‘stuck’students,promptthemwithquestionssuchas“WhatinformationdoyouknowaboutAlyandDayne’spools?”and“Howcanyourepresentororganizetheinformationyouknowabouta(x)andd(x)sothatyoucanmakesenseofg(x)?”(Studentscanfindsolutionstosolveforg(x)bycreatingatable,agraph,orlookingatequations,althoughthegoalisthattheyaddthegraphstogetheratthispointtoseevisuallythatyouareaddingoutputs.Equationsoften‘hide’thisobservation).
SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
Lookforstudentswhousedifferentrepresentationstoanswerthequestionsfromthetask.Makenoteofthisasforthewholegroupdiscussion,youmaywishtoselectstudentswhousedifferentmethodsforsolvingthefirsttwoquestions.Acommonmisconceptionwillbethatstudentswhouseinterceptswilleitheraddorfindtheaverageofthex-interceptstofindthe‘new’x-intercept.Thisisagreatopportunitytodistinguishwhyitisappropriatetoaddthey-interceptstofindthenewy-intercept(theyaretheoutputvalueshencearethevaluesofa(x)+d(x)andrepresenttheamountofwaterinthepool)butwhyyoudonotaddthex-interceptstofindthenewx-intercept(theyaretheinputvaluesandrepresenttheamountoftimeittakesforeachpooltodrainseparately).Discuss(WholeClass):
Thegoalofthistaskistomakesurestudentshaveadeeperunderstandingofkeyfeaturesoffunctions,tosurfaceideasaboutbuildingfunctions,andtodeepenunderstandingaboutfunctionnotation.Thewholegroupdiscussionshouldcoverwhateachpartofafunctionrepresentsandhowthisplaysoutwhenusingfunctionnotation.Itismosteffectivewhenstudentsseethisgraphically,numerically,andwithequationsandmakeconnectionswiththefeaturesofthefunction.Therearemanywaysthewholegroupdiscussioncanaccomplishthesegoals.Belowisasuggestionforhowtofacilitatethewholegroupdiscussionusingstudenterrorthatisalsoacommonmisconception.Youmaywishtostartthewholegroupdiscussionbyhavingtwostudentsposttheirgraphsofg(x),onebeingcorrectandtheotherbeingthecommonmisconception(onlydothisifyoufeelyourclasshasasafeenvironmentandstudentsbelievethatpartofthelearningprocessistolearnfrommistakes).Starttheconversationwithhowthesearethetwomostcommongraphsthroughouttheroomandthatmanypeoplehaveeitheroneortheotherontheirpaper.Askthewholegroupwhatissimilarandwhatisdifferent(bothgroupswillhavethesamey-intercept).Thenchooseastudentwhohascreatedatableshowingthesumoftheoutputvalueswhoagreeswiththecorrectgraph(choosethisstudentinadvance).Alsohaveastudentshowhowtheequationofa(x)+d(x)showsupinthecorrectgraph.Besurethatstudentswhoshareareexplicitintheconnectionsshowinghowtheequationsa(x),d(x),andg(x)relatetothegraphsofa(x),d(x),andg(x).Afterallstudentsseetheconnectionsbetweenthecorrectgraphandotherrepresentations,askwhatthecommonmisconceptionwasinthe‘incorrectgraph’.Intheend,studentsshouldleavewithhowxistheinputvalueandthatg(x)isthesolutiontothevalueatx.AlignedReady,Set,Go:Features3.5
SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS - RSG 3.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3.5
READY
Topic:Interpretingfunctionnotationtofindtheoutputorinputbasedonwhatisgiven
Foreachfunction,findtheindicatedvalues.
1. !"#$%: ℎ(!) = 2! – 5a. ℎ(−4) = _____b.ℎ ! = 23, ! = ______ c.ℎ(13) = ______ d.ℎ ! = −33, ! = ______
2.
3.
a. ! 2 = ______
b. ! ! = 3 , ! = ______
c.! 0 = _______d.Writetheexplicitruleforg(x).
a.! −1 = ______b.! ! = 4, ! = ______c.! 2 = ______d.Writetheexplicitruleforr(x).
SET Topic:AddingfunctionsTwofunctionsaregraphed.Graphanewfunctiononthesamegridbyaddingthetwogivenfunctions.
READY, SET, GO! Name PeriodDate
g(x) r(x)
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SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS - RSG 3.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3.5
4.ℎ ! = ! ! + ! !
5.! ! = ! ! + ! !
5.Usethegraphtoanswerthefollowingquestions.
a.Wheredoes! ! = ! ! ?b.Whatis! 4 + ! 4 ?c.Whatis! −2 − ! −2 ?d.Statetheintervalwhere! ! > ! ! .
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SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS - RSG 3.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3.5
6.Usethegraphtoanswerthefollowingquestions.a.Whereis! ! > ℎ ! ?b.Whatis! 1 − ℎ 1 ?c.Whatis! 0 + ℎ 0 ?d.Writeanexplicitrulefor
! ! and for ℎ ! .e.Sketch! ! − ℎ ! onthegraph.
GO Topic:DistinguishingbetweendiscreteandcontinuousfunctionsForeachcontextorrepresentationdeterminewhetheritisdiscreteorcontinuousorcouldbemodeledbestinadiscreteorcontinuousway.Justifyyouranswer.8.Susanputsexactly$5aweekinherpiggybank.9. 10.
10.
25
SECONDARY MATH I // MODULE 3
FEATURES OF FUNCTIONS - RSG 3.5
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3.5
11.Marshaltracksthenumberofhitshegetseachbaseballgameandisrecordinghistotalnumber
ofhitsfortheseasoninatable.12.Thedistanceyouhavetraveledsincethedaybegan.
13.Numberofgumballs Cost
5 10¢10 20¢15 30¢20 40¢
14.Stephendeposited$1,000inasavingsaccountatthebankwhenheturned21.Hedeposits
$100eachmonth.Heplanstoneverwithdrawanymoneyuntilthebalanceis$150,000.
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