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

    2018 Mathematics Vision Project Original work 2013 in partnership with the Utah State Office of Education

    This work is licensed under the Creative Commons Attribution CC BY 4.0

    MODULE 3

    Polynomial Functions

    SECONDARY

    MATH THREE

    An Integrated Approach

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS

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

    MODULE 3 - TABLE OF CONTENTS

    POLYNOMIAL FUNCTIONS

    3.1 Scotts March Madness A Develop Understanding Task Introduce polynomial functions and their rates of change (F.BF.1, F.LE.3, A.CED.2) Ready, Set, Go Homework: Polynomial Functions 3.1

    3.2 You-mix Cubes A Solidify Understanding Task Graph ! = #$ with transformations and compare to ! = #%. (F.BF.3, F.IF.4, F.IF.5, F.IF.7) Ready, Set, Go Homework: Polynomial Functions 3.2

    3.3 It All Adds Up A Develop Understanding Task Add and subtract polynomials. (A.APR.1, F.BF.1, F.IF.7, F.BF.1b) Ready, Set, Go Homework: Polynomial Functions 3.3

    3.4 Pascals Pride A Solidify Understanding Task Multiply polynomials and use Pascals to expand binomials. (A.APR.1, A.APR.5) Ready, Set, Go Homework: Polynomial Functions 3.4

    3.5 Divide and Conquer A Solidify Understanding Task Divide polynomials and write equivalent expressions using the Polynomial Remainder Theorem. (A.APR.1, A.APR.2) Ready, Set, Go Homework: Polynomial Functions 3.5

    3.6 Sorry, Were Closed A Practice Understanding Task Compare polynomials and integers and determine closure under given operations. (A.APR.1, F.BF.1b) Ready, Set, Go Homework: Polynomial Functions 3.6

    3.7 Building Strong Roots A Solidify Understanding Task Understand the Fundamental Theorem of Algebra and apply it to cubic functions to find roots. (A.SSE.1, A.APR.3, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 3.7

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS

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

    3.8 Getting to the Root of the Problem A Solidify Understanding Task Find the roots of polynomials and write polynomial equations in factored form. (A.APR.3, N.CN.8, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 3.8

    3.9 Is This the End? A Solidify Understanding Task Examine the end behavior of polynomials and determine whether they are even or odd. (F.LE.3, A.SSE.1, F.IF.4, F.BF.3) Ready, Set, Go Homework: Polynomial Functions 3.9

    3.10 Puzzling Over Polynomials A Practice Understanding Task Analyze polynomials, determine roots, end behavior, and write equations (A.APR.3, N.CN.8, N.CN.9, A.CED.2) Ready, Set, Go Homework: Polynomial Functions 3.10

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.1

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

    3.1 Scotts March Madness

    A Develop Understanding Task

    Eachyear,ScottparticipatesintheMachoMarchpromotion.The

    goalofMachoMarchistoraisemoneyforcharitybyfinding

    sponsorstodonatebasedonthenumberofpush-upscompleted

    withinthemonth.Lastyear,Scottwasproudofthemoneyhe

    raised,butwasalsodeterminedtoincreasethenumberofpush-

    upshewouldcompletethisyear.

    PartI:RevisitingthePast

    BelowisthebargraphandtableScottusedlastyeartokeeptrackofthenumberofpush-upshe

    completedeachday,showinghecompletedthreepush-upsondayoneandfivepush-ups(fora

    combinedtotalofeightpush-ups)ondaytwo.Scott

    continuedthispatternthroughoutthemonth.

    1 2 3 4

    1. Writetherecursiveandexplicitequationsforthenumberofpush-upsScottcompletedonany

    givendaylastyear.Explainhowyourequationsconnecttothebargraphandthetableabove.

    ?Days A(?)

    Push-ups

    eachday

    E(?)

    Totalnumberof

    pushupsinthe

    month

    1 3 32 5 83 7 154 9 245 11 35 ?

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  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.1

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

    2. Writetherecursiveandexplicitequationfortheaccumulatedtotalnumberofpush-ups

    ScottcompletedbyanygivendayduringtheMachoMarchpromotionlastyear.

    PartII:MarchMadness

    Thisyear,Scottsplanistolookatthetotalnumberofpush-upshecompletedforthemonthlastyear

    (g(n))anddothatmanypush-upseachday(m(n)).

    3. Howmanypush-upswillScottcompleteondayfour?Howdidyoucomeupwiththisnumber?

    Writetherecursiveequationtorepresentthetotalnumberofpush-upsScottwillcompletefor

    themonthonanygivenday.

    4. Howmanytotalpush-upswillScottcompleteforthemonthondayfour?

    ?Days A(?)

    Push-upseach

    daylastyear

    E(?)

    Totalnumber

    ofpushupsin

    themonth

    N(?)

    Push-upseach

    daythisyear

    T(n)

    Totalpush-ups

    completedfor

    themonth

    1 3 3 3 2 5 8 8 3 7 15 15 4 9

    24 5 ?

    2

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.1

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

    5. Withoutfindingtheexplicitequation,makeaconjectureastothetypeoffunctionthatwould

    representtheexplicitequationforthetotalnumberofpush-upsScottwouldcompleteonany

    givendayforthisyearspromotion.

    6. Howdoestherateofchangeforthisexplicitequationcomparetotheratesofchangeforthe

    explicitequationsinquestions1and2?

    7. Testyourconjecturefromquestion5andjustifythatitwillalwaysbetrue(seeifyoucanmove

    toageneralizationforallpolynomialfunctions).

    3

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.1

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

    3.1

    Needhelp?Visitwww.rsgsupport.org

    READY Topic:Completinginequalitystatements

    Foreachproblem,placetheappropriateinequalitysymbolbetweenthetwoexpressionstomakethestatementtrue.If! > #, %'(: *+- > 10, %'(: *+0 < - < 1

    1. 3!____3#

    4.-3____25

    7.-____-3

    2. # !____! #

    5.-____-3

    8.-____-

    3.! + -____# + - 6.-3____-9 9.-____3-

    SET Topic:Classifyingfunctions

    Identifythetypeoffunctionforeachproblem.Explainhowyouknow.10

    - +(-) 1 3 2 6 3 12 4 24 5 48

    11.- +(-) 1 3 2 6 3 9 4 12 5 15

    12.- +(-) 1 3 2 9 3 18 4 30 5 45

    13.- +(-)1 72 93 134 215 37

    14.- +(-)1 -262 -193 04 375 98

    15.- +(-)1 -42 33 184 415 72

    16.WhichoftheabovefunctionsareNOTpolynomials?

    READY, SET, GO! Name PeriodDate

    4

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.1

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

    3.1

    Needhelp?Visitwww.rsgsupport.org

    GO Topic:Recallinglongdivisionandthemeaningofafactor

    Findthequotientwithoutusingacalculator.Ifyouhavearemainder,writetheremainderas

    awholenumber.Example: remainder217.

    18.

    19.Is30afactorof510?Howdoyouknow?

    20.Is13afactorof8359?Howdoyouknow?

    21.

    22.

    23.Is22afactorof14587?Howdoyouknow?

    24.Is952afactorof40936?Howdoyouknow?

    25.

    26.

    27.Is92afactorof3405? 28.Is27afactorof3564?

    21 1497

    30 510 13 8359

    22 14857 952 40936

    92 3405 27 3564

    5

  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.2

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

    3.2 You-mix Cubes A Solidify Understanding Task

    InScottsMarchMadness,thefunctionthatwasgeneratedby

    thesumoftermsinaquadraticfunctionwascalledacubicfunction.Linearfunctions,

    quadraticfunctions,andcubicfunctionsareallinthefamilyoffunctionscalledpolynomials,

    whichincludefunctionsofhigherpowerstoo.Inthistask,wewillexploremoreaboutcubic

    functionstohelpustoseesomeofthesimilaritiesanddifferencesbetweencubicfunctionsand

    quadraticfunctions.

    Tobegin,letstakealookatthemostbasiccubicfunction,!(#) = #& .Itistechnicallyadegree3polynomialbecausethehighestexponentis3,butitscalledacubicfunctionbecausethese

    functionsareoftenusedtomodelvolume.Thisislikequadraticfunctionswhicharedegree2

    polynomialsbutarecalledquadraticaftertheLatinwordforsquare.ScottsMarchMadness

    showedthatlinearfunctionshaveaconstantrateofchange,quadraticfunctionshavealinearrate

    ofchange,andcubicfunctionshaveaquadraticrateofchange.

    1. Useatabletoverifythat!(#) = #&hasaquadraticrateofchange.

    2. Graph!(#) = #& .

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  • SECONDARY MATH III // MODULE 3

    POLYNOMIAL FUNCTIONS 3.2

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

    3. Describethefeaturesof!(#) = #&includingintercepts,intervalsofincreaseordecrease,domain,range,etc.4. Usingyourknowledgeoftransformations,grapheachofthefollowingwithoutusingtechnology.

    a) !(#) = #& 3 b)

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