3.5 Pooling it Together - Utah Education Network · 3.5 Pooling it Together ... Write the equation...
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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 A Solidify Understanding Task Aly and Dayne work at a water park and have to drain the water at the end of each month for the ride they supervise. Each uses a pump to remove the water from the small pool at the bottom of their ride. The graph below represents the amount of water in Aly’s pool, a(x), and Dayne’s pool, d(x), over time. In this scenario, they decided to work together to drain their pools and created the equation: g(x) = a(x) +d(x). Answer the following questions about g(x). 1. What does g(x) represent? 2. Create the graph of g(x) on a new set of axes using the graphs of a(x) and d(x). Identify g(x) and label (scale, axes). Amount of water (gallons) 4,000 5 CC BY Hanumann https://flic.kr/p/7tZHkq 1 21
Transcript of 3.5 Pooling it Together - Utah Education Network · 3.5 Pooling it Together ... Write the equation...
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.
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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
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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