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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

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

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

Eachyear,ScottparticipatesintheMachoMarchpromotion.The

goalofMachoMarchistoraisemoneyforcharitybyfinding

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.

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

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?

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

functionstohelpustoseesomeofthesimilaritiesanddifferencesbetweencubicfunctionsand

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)