Math 1 CP to Math 2 Honors Summer Bridge Functions Module 9H Bridge ( 1 to 2... · 2017. 4. 17. ·...

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Math1CPtoMath2HonorsSummerBridge

QuadraticFunctionsModule9H

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TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org

InpartnershipwiththeUtahStateOfficeofEducation

SDUHSDMath1CPtoMath2HSummerBridge

Module9HOverview

Belowarethestandards,concepts,andvocabularyfromtheCOMPLETEMODULEinIntegratedMath1Honors

PrerequisiteConcepts&Skills:

Graphingfunctions Identifyingdomain,range,andintervalsofincreaseanddecrease Usingpatternstofindrecursiveandexplicitrules Usingdifferenceswithinatableofvaluestodeterminethetypeoffunctionthatcanbeusedtorepresenta

dataset Comparingfeaturesoffunctions Comparinglinearandexponentialfunctions Findingmultiplerepresentationsforacontext

SummaryoftheConcepts&SkillsinModule9H:

Usepatternstodeterminethetypeoffunction Usemultiplerepresentationstocomparelinear,quadratic,andexponentialfunctions Determinedomain,range,maximum,andminimumvaluesforquadratics Determineaverageratesofchange

ContentStandardsandStandardsofMathematicalPracticeCovered:

ContentStandards:F.BF.1,A.SSE.1,A.CED.2,F‐BF,F‐LE,F.LE.2,F.LE.3 StandardsofMathematicalPractice:

1. Makesenseofproblems&persevereinsolvingthem2. Attendtoprecision3. Reasonabstractly&quantitatively4. Constructviablearguments&critiquethereasoningofothers5. Modelwithmathematics6. Useappropriatetoolsstrategically7. Lookfor&makeuseofstructure8. Lookfor&expressregularityinrepeatedreasoning

Module9HVocabulary:

Quadraticfunctions Maximumvalue Minimumvalue Averagerateofchange Binomial Perimeter Area Domain Range First&seconddifferences Parabola Explicitequation Recursiveequation Increasing Decreasing Intervalnotation Continuous Discrete Acceleration Initialspeed Instantaneousspeed Averagespeed Atrest Exponentialfunction

ConceptsUsedintheNextModule: Determineifacontextrepresentsalinear,exponential,orquadraticfunction Determinethemaximumorminimumvalue Findmultiplerepresentationsfromacontext Determinethetransformationsbeingappliedtoafunction Identifyfeaturesoffunctions(domain,range,intervalsofincrease/decrease)

Module9H–QuadraticFunctions9.1HSolidificationofquadraticfunctionsbeginsasquadraticpatternsareexaminedinmultiplerepresentationsandcontrastedwithlinearrelationshipsandFocusspecificallyonthenatureofchangebetweenvaluesinaquadraticbeinglinearandfocusonmaximum/minimumpointaswellasdomainandrangeforquadratics(F.BF.1,A.SSE.1,A.CED.2,F.LE,F.IF.9)WarmUp:SomethingtoTalkAbout–ADevelopUnderstandingTaskClassroomTask:RabbitRun–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.1H9.2HFocusonmaximum/minimumpointaswellasdomainandrangeforquadratics(F.BF.1,A.SSE.1,A.CED.2,F.LE)ClassroomTask:Scott’sMachoMarch–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.2H9.3HExaminingquadraticfunctionsonvarioussizedintervalstodetermineaverageratesofchange(F.BF.1,A.SSE.1,A.CED.2)ClassroomTask:LookoutBelow–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.3H

www.flickr.com/photos/robwallace/ 9.1HWarmUp–SomethingtoTalkAbout

ADevelopUnderstandingTaskCellphonesoftenindicatethestrengthofthephone’ssignalwithaseriesofbars.Thelogobelowshowshowthismightlookforvariouslevelsofservice.

Figure1 Figure2 Figure3 Figure4

1. Assumingthepatterncontinues,drawthenextfigureinthesequence.2. HowmanyblockswillbeintheFigure10logo?3. Examinethesequenceoffiguresandfindaruleorformulaforthenumberoftilesinanyfigurenumber.

/photos/bishi/2314705514

9.1HRabbitRunASolidifyUnderstandingTaskMishahasanewrabbitthatshenamed“Wascal”.ShewantstobuildWascalapensothattherabbithasspacetomovearoundsafely.Mishahaspurchaseda72footrolloffencingtobuildarectangularpen.1. IfMishausesthewholerolloffencing,whataresomeofthepossibledimensionsofthepen?2. IfMishawantsapenwiththelargestpossiblearea,whatdimensionsshouldsheuseforthepen?Justifyyour

answer.3. Writeamodelfortheareaoftherectangularpenintermsofthelengthofoneside.Includebothanequation

andagraph.

4. Whatkindoffunctionisthis?Why?5. HowdoesthisfunctioncomparetothesecondtypeofblockIlogosinIRule?

Name: QuadraticFunctions 9.1HReady,Set,GoReadyTopic:AddingandmultiplyingbinomialsSimplifythefollowingexpressions.Forthepartbproblems,multiplyusingthegivenareamodel.1a. 6 1 10 b. 6 1 10

2a. 8 3 3 4 b. 8 3 3 4

3a. 5 2 7 13 b. 5 2 7 13

4a. 12 3 4 3 b. 12 3 4 3

5. 5 5

6. Comparethestyleyouranswersin#1–4(parta)toyouranswersin#1–4(partb).Lookforapatterninthe

answers.Howaretheydifferent?7. Theanswerto#5isadifferent“shape”thantheotherpartbanswers,eventhoughyouwerestillmultiplying.

Explainhowitisdifferentfromtheotherproducts.Trytoexplainwhyitisdifferent.Find2examplesofmultiplyingbinomialsthatwouldhaveasimilarsolutionas#5.

8. Tryaddingthetwobinomialsin#5. 5 5 Doesthisanswerlook

differentthanthoseinparta?Explain.

SetTopic:ComparingareaandperimeterCalculatetheperimeterandtheareaofthefiguresbelow.Youranswerswillcontainavariable.9.

a. Perimeter: b. Area:

13.Comparetheperimetertotheareaineachofproblems9‐12.Inwhatwayarethenumbersandunitsinthe

perimetersandareasdifferent?

Topic:Recognizinglinear,exponential,andquadraticequations.14. Ineachsetof3functions,onewillbelinearandonewillbeexponential.Oneofthethreewillbeanew

categoryoffunction.Statethetypeoffunctionrepresented(linear,exponential,ornewfunction).Listthecharacteristicsineachtableand/orgraphthathelpedyoutoidentifythelinearandtheexponentialfunctions.Forthegraph,placeyouraxessothatyoucanshowall5points.Identifyyourscale.Findanexplicitandrecursiveequationforeach.

Typeandcharacteristics?Explicitequation: Recursiveequation:

GoTopic:GreatestCommonFactor(GCF)FindtheGCFofthegivennumbers.16.15 and25 17. 12 and32 18. 17 and51

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

19.6 ,18 , 12 20. 49 and36 21. 11 ,33 ,and3

9.2HScott’sMachoMarchASolidifyUnderstandingTaskPart1:Afterlookinginthemirrorandfeelingflabby,Scottdecidedthathereallyneedstogetinshape.Hejoinedagymandaddedpush‐upstohisdailyexerciseroutine.Hestartedkeepingtrackofthenumberofpush‐upshecompletedeachdayinthebargraphbelow,withdayoneshowinghecompletedthreepush‐ups.Afterfourdays,Scottwascertainhecancontinuethispatternofincreasingthenumberofpush‐upsforatleastafewmonths.

1. Modelthenumberofpush‐upsScottwillcompleteonanygivenday.Includebothexplicitandrecursive

equations.Scott’sgymissponsoringa“MachoMarch”promotionduringthemonthofMarch.Thegoalof“MachoMarch”istoraisemoneyforcharitybydoingpush‐ups.Scotthasdecidedtoparticipateandhassponsorsthatwilldonatemoneytothecharityifhecandoatotalofatleast500push‐ups.Asabonus,theywilldonateanadditional$10forevery100push‐upshecandobeyondhisgoalof500push‐ups.2. Estimatethetotalnumberofpush‐upsthatScottwilldoinamonthifhecontinuestoincreasethenumberof

push‐upshedoeseachdayinthepatternshownabove.3. Howmanytotalpush‐upswillScotthavedoneafteronefullweek?

4. Modelthetotalnumberofpush‐upsthatScotthascompletedonanygivendayduring“MachoMarch”.Includebothrecursiveandexplicitequations.

5. WillScottmeethisgoalandearnthedonationforthecharity?Willhegetabonus?Ifso,howmuch?Explain.Part2:MultiplyingBinomials&UsingPatternstoDetermineRecursiveEquations:Thefollowingproblemsarefactorizationsofnumericalexpressionscalledquadratics.Giventhefactors,multiplythebinomialsusingtheareamodeltofindthequadraticexpression.Simplifytheexpressionsbycombiningliketermsandwritingthetermsindescendingorderofpowers(example: .Thendistributethecoefficient,ifoneexists.1. 5 7 2. 3. 2 9 4

4. 3 1 4 5. 2 3 5 1 6. 2 5 7 3 1

Usefirstandseconddifferencestoidentifythepatterninthetablesaslinear,quadratic,orneither.Writetherecursiveequationforthepatternsthatarelinearorquadratic.7.

3 232 171 110 51 12 3 13

a. Pattern:b. Recursiveequation:

3 42 01 20 21 02 43 10

a. Pattern:b. Recursiveequation:

Name: QuadraticFunctions 9.2HReady,Set,Go!ReadyTopic:ApplyingtheslopeformulaCalculatetheslopeofthelinebetweenthegivenpoints.Useyouranswertoindicatewhichlineisthesteepest.1. 3, 7 5, 17 2. 12, 37 4, 3 3. 11, 24 21, 40 4. 55, 75 15, 40 Steepestline: SetTopic:InvestigatingperimetersandareasAdamandhisbrotherareresponsibleforfeedingtheirhorses.Inthespringandsummer,thehorsesgrazeinanunfencedpasture.Thebrothershaveerectedaportablefencetocorralthehorsesinagrazingarea.Eachdaythehorseseatallofthegrassinsidethefence.Thentheboysmoveittoanewareawherethegrassislongandgreen.Theportablefenceconsistsof16separatepiecesoffencingeach10feetlong.Thebrothershavealwaysarrangedthefenceinalongrectanglewithonelengthoffenceoneachendand7piecesoneachsidemakingthegrazingarea700sq.ft.Adamhaslearnedinhismathclassthatarectanglecanhavethesameperimeterbutdifferentareas.Heisbeginningtowonderifhecanmakehisdailyjobeasierbyrearrangingthefencesothatthehorseshaveabiggergrazingarea.Hebeginsbymakingatableofvalues.Helistsallofthepossibleareasofarectanglewithaperimeterof160ft.,whilekeepinginmindthatheisrestrictedbythelengthsofhisfencingunits.Herealizesthatarectanglethatisorientedhorizontallyinthepasturewillcoveradifferentsectionofgrassthanonethatisorientedvertically.Soheisconsideringthetworectanglesasdifferentinhistable.Usethisinformationtoanswerquestions5–9onthenextpage.

5. FillinAdam’stablewithallofthearrangementsforthefence. Lengthin

“fencing”unitsWidthin

“fencing”unitsLengthinft. Widthinft. Perimeter Area

1unit 7units 10ft 70ft 160ft 700 a. 2units 160ftb. 3units 160ftc. 4units 160ftd. 5units 160fte. 6units 160ftf. 7units 160ft

6. DiscussAdam’sfindings.Explainhowyouwouldrearrangethesectionsoftheporta‐fencesothatAdamwillbeabletodolesswork.Explainwhat“lesswork”meansforAdamandhisbrother.

7. MakeagraphofAdam’s investigation. Let lengthbe the

independentvariableandareabethedependentvariable.Labelandscaletheaxes.

8. Whatistheshapeofyourgraph? 9. Explainwhatmakesthisfunctionquadratic.

GoTopic:Comparinglinearandexponentialratesofchange.Indicatewhichfunctionischangingfaster.10.

16a. Examinethegraphattheleftfrom0to1.Whichgraphdoyouthinkisgrowingfaster?

b. Nowlookatthegraphfrom2to3.Whichgraph

isgrowingfasterinthisinterval?

Topic:Linear,Exponential,&QuadraticSequencesDetermineifeachsequenceislinear,exponential,orquadratic.Explainhowyoucandeterminethetypeofpatternbaseduponthetable.Thenfindtherecursiveandexplicitequationsforeachpattern.17.

1 32 93 274 815 243

TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:

1 62 43 24 05 2

19.1 42 73 124 195 28

20. 1 52 83 114 145 17

1 22 83 184 325 50

22.1 12 43 164 645 256

www.flickr.com/photos/melissawall 9.3HLookOutBelow!

ASolidifyUnderstandingTaskWhathappenswhenyoudropaball?Itfallstotheground.Thatquestionsoundsassillyas“Whydidthechickencrosstheroad?”Seriously,ittookscientistsuntilthesixteenthandseventeenthcenturiestofullyunderstandthephysicsandmathematicsoffallingbodies.Wenowknowthatgravityactsontheobjectthatisfallinginawaythatcausesittoaccelerateasitfalls.Thatmeansthatifthereisnoairresistance,itfallsfasterandfaster,coveringmoredistanceineachsecondasitfalls.Ifyoucouldslowtheprocessdownsothatyoucouldseethepositionoftheobjectasitfalls,itwouldlooksomethinglikethepicturebelow.

Tobemoreprecise,objectsfallataconstantrateofaccelerationonearthofabout32feetpersecondpersecond.Thesimplestcaseoccurswhentheobjectstartsfromrest,thatis,whenitsspeediszerowhen 0.Inthiscase,theobject’sinstantaneousspeedafter1secondis32feetpersecond;after2seconds,itsinstantaneousspeedis2 32 64feetpersecond;andsoon.Otherplanetsandmoonseachhaveadifferentrateofacceleration,butthebasicprincipalremainsthesame.Iftheaccelerationonaparticularplanetisg,thentheobject’sinstantaneousspeedafter1secondisgunitspersecond;after2seconds,itsinstantaneousspeedis2 unitspersecond;andsoon.Inthistask,wewillexplorethemathematicsoffallingobjects,butbeforewestartthinkingaboutfallingobjectsweneedtobeginwithalittleworkontherelationshipbetweenspeed,time,anddistance.

Part1: AveragespeedanddistancetraveledConsideracarthatistravelingatasteadyrateof30feetpersecond.Attime 0,thedriverofthecarstartstoincreasehisspeed(accelerate)inordertopassaslowmovingvehicle.Thespeedincreasesataconstantratesothat20secondslater,thecaristravelingatarateof40feetpersecond.1. Graphthecar’sspeedasafunctionoftimeforthis20‐secondtimeinterval.

2. Calculatetheaveragespeedofthecarforthis20‐secondtimeinterval.3. Findthetotaldistancethecartravelsduringthis20‐secondtimeinterval.4. Explainhowtouseareatofindthetotaldistancethecartravelsduringthis20‐secondinterval.Thisproblemillustratesanimportantprinciple:Ifanobjectistravelingwithconstantacceleration,thenitsaveragespeedoveranytimeintervalistheaverageofitsbeginningspeedanditsfinalspeedduringthattimeinterval.Let’sapplythisideatoapennythatisdropped(initialspeedis0when 0)fromthetopoftheEmpireStateBuilding.Besuretoreferencethesecondparagraphatthestartofthetaskforinformationabouttheconstantrateofacceleration.5. Whatwillitsspeedbeafter1second?6. Graphthepenny’sspeedasafunctionoftimein1secondintervals.

7. Whatistheaveragespeedofthepennyinthe1‐secondinterval?8. Whatisthetotaldistancethatthepennyfellinthe1‐secondinterval?9. FindtheheightoftheEmpireStateBuilding.Howlongwillittakeforthepennytohittheground?10.CompletethetablebelowforthespeedsanddistancestraveledofapennydroppedfromthetopoftheEmpire

StateBuilding.

Time InstantaneousSpeed AverageSpeed DistanceTraveled

Part2: Falling,Falling,FallingWhentheastronautswenttothemoon,theyperformedGalileo’sexperimenttotesttheideathatanytwoobjects,nomattertheirmass,willfallatthesamerateifthereisnoairresistance.Becausethemoondoesn’thaveairresistance,wearegoingtopretendlikewe’retheastronautsdroppingmoonrocksandthinkingaboutwhathappens.Onthesurfaceofthemoontheconstantaccelerationincreasesthespeedofafallingobjectby6feetpersecondpersecond.Thatis,ifanobjectisdroppednearthesurfaceofthemoon(e.g.,itsinitialspeedis0when

0),thentheobject’sinstantaneousspeedafter1secondis6feetpersecond,after2seconds,itsinstantaneousspeedis12feetpersecond,andsoon.11.Usingthisinformation,completethetablebelowforthespeedsanddistancestraveledofanobjectthatis

droppedfromaheightof200feetabovethesurfaceofthemoonasafunctionoftheelapsedtime(inseconds)sinceitwasdropped.

Time InstantaneousSpeed AverageSpeed DistanceTraveled

12.Howdidyoufindthedistancethattheobjecthastraveled?13.Writeanequationforthedistancetheobjecthasfallenasafunctionofelapsedtimet.14.Approximatelyhowlongwillittakefortheobjecttohitthesurfaceofthemoon?15.Writeanequationfortheheightoftheobjectabovethesurfaceofthemoonasafunctionofelapsedtimet.

Reminder:theobjectisdroppedfromaheightof200feetabovethesurfaceofthemoon.

16.Supposetheobjectwasnotdropped,butwasthrowndownwardfromaheightof250feetabovethesurfaceofthemoonwithaninitialspeedof10feetpersecond.Rewriteyourequationfortheheightoftheobjectabovethesurfaceofthemoonasafunctionofelapsedtimettotakeintoaccountthisinitialspeed.

17.Howisyourworkonthesefallingobjectsproblemsrelatedtoyourworkwiththerabbitrunsintheprevious

task?18.Whyarethe“distancefallen”and“heightabovetheground”functionsquadratic?

Name: QuadraticFunctions 9.3HReady,Set,Go!ReadyTopic: EvaluatingexponentialfunctionsFindtheindicatedvalueofthefunctionforeachvalueofx. , , , , ,

1. 3 2. 3.

SetTheSearsTowerinChicagois1730feettall.Ifapennywereletgofromthetopofthetower,thepositionabovethegrounds(t)ofthepennyatanygiventimetwouldbe 16 1730.4. Fillinthemissingpositionsinthechartbelow.Thenaddtogetthedistancefallen.

Distancefromgrounda. 0sec

b. 1sec

c. 2sec

d. 3sec

e. 4sec

f. 5sec

g. 6sec

h. 7sec

i. 8sec

j. 9sec

k. 10sec

5. Howfarabove thegroundisthepennywhen7secondshavepassed?

6. Howfarhasitfallenwhen7secondshavepassed? 7. Hasthepennyhitthegroundat10seconds?Justify

youranswer.

Theaveragerateofchangeofanobjectisgivenbytheformula ,whereristherateofchange,disthedistancetraveled,andtisthetimeittooktotravelthegivendistance.Weoftenusesomeformofthisformulawhenwearetryingtocalculatehowlongatripmaytake.8. Ifourdestinationis225milesawayandwecanaverage75mph,thenweshouldarrivein3hours.

3 Inthiscaseyouwouldberearrangingtheformulasothat .However,ifyourmother

findsoutthatthetriponlytook2½hours,shewillbeupset.Usetherateformulatoexplainwhy.9. Howistheslopeformula liketheformulaforrate?

Forthefollowingquestions,referbacktotheSear’sTowerproblem(questions6–9).10. Findtheaveragerateofchangeforthepennyontheinterval[0,1]seconds.11. Findtheaveragerateofchangeforthepennyontheinterval[6,7]seconds.12. Explainwhythepenny’saveragespeedisdifferentfrom0to1secondthanbetweenthe6thand7thseconds.13. Whatistheaveragespeedofthepennyfrom[0,10]seconds?14. Whatistheaveragespeedofthepennyfrom[9,10]seconds?15. Findthefirstdifferencesonthetablewhereyourecordedthepositionofthepennyateachsecond.Whatdo

thesedifferencestellyou?16. Takethedifferenceofthefirstdifferences.Thisiscalledthe2nddifference.Didyouranswersurpriseyou?

Whatdoyouthinkthismeans?

GoTopic: Evaluatingfunctions17.Find 9 giventhat 10. 18.Find 3 giventhat 2 4.

19.Find 1 giventhat 5 3 9. 20.Find giventhat .

22. Find 2 giventhat 7 11 .Topic:Linear,Exponential,orQuadraticSequencesDetermineifeachsequenceislinear,exponential,orquadratic.Explainhowyoucandeterminethetypeofpatternbaseduponthetable.Thenfindtherecursiveandexplicitequationsforeachpattern.23.

1 02 33 84 155 24

1 62 123 244 485 96

25. 1 32 123 274 485 75

1 32 73 114 155 19

Math 1 CP to Math 2 Honors Summer Bridge Functions Module 9H Bridge ( 1 to 2... · 2017. 4. 17. ·...

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