SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

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

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A Develop Understanding Task

OptimaPrimeisdesigningarobotquiltforhernewgrandson.Sheplansfortherobottohaveasquareface.Theamountoffabricthatsheneedsforthefacewilldependontheareaoftheface,soOptimadecidestomodeltheareaoftherobot’sfacemathematically.SheknowsthattheareaAofasquarewithsidelengthxunits(whichcanbeinchesorcentimeters)ismodeledbythefunction, ! ! = !!squareunits.1. Whatisthedomainofthefunction! ! inthiscontext?

2. Matcheachstatementabouttheareatothefunctionthatmodelsit:MatchingEquation

(A,B,C,orD)

Statement FunctionEquation

Thelengthofeachsideisincreasedby5units.

A) AB) !(!) = 5!!

Thelengthofeachsideismultipliedby5units.

C) BD) ! = (! + 5)!

Theareaofasquareisincreasedby5squareunits.

E) CF) ! = (5!)!

Theareaofasquareismultipliedby5. G) DH) ! = !! + 5

Optimastartedthinkingaboutthegraphof! = !!(inthedomainofallrealnumbers)andwonderingabouthowchangestotheequationofthefunctionlikeadding5ormultiplyingby5affectthegraph.Shedecidedtomakepredictionsabouttheeffectsandthencheckthemout.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

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

3. Predicthowthegraphsofeachofthefollowingequationswillbethesameordifferentfromthegraphof! = !!.

Similaritiestothegraphof! = !!

Differencesfromthegraphof! = !!

! = 5!!

! = (! + 5)!

! = (5!)!

! = !! + 5

4. Optimadecidedtotestherideasusingtechnology.Shethinksthatitisalwaysagoodideatostartsimple,soshedecidestogowith! = !! + 5.Shegraphsitalongwith! = !!inthesamewindow.Testityourselfanddescribewhatyoufind.

5. Knowingthatthingsmakealotmoresensewithmorerepresentations,Optimatriesafewmoreexampleslike! = !! + 2 and! = !! − 3,lookingatbothatableandagraphforeach.Whatconclusionwouldyoudrawabouttheeffectofaddingorsubtractinganumberto! = !!?Carefullyrecordthetablesandgraphsoftheseexamplesinyournotebookandexplainwhyyourconclusionwouldbetrueforanyvalueofk,given,! = !! + !.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

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

6. Afterheramazingsuccesswithadditioninthelastproblem,Optimadecidedtolookatwhathappenswithadditionandsubtractioninsidetheparentheses,orasshesaysit,“addingtothe!beforeitgetssquared”.Usingyourtechnology,decidetheeffectofhintheequations:! = (! + ℎ)!

and! = (! − ℎ)!.(Choosesomespecificnumbersforh.)Recordafewexamples(bothtablesandgraphs)inyournotebookandexplainwhythiseffectonthegraphoccurs.

7. Optimathoughtthat#6wasverytrickyandhopedthatmultiplicationwasgoingtobemorestraightforward.Shedecidestostartsimpleandmultiplyby-1,soshebeginswith! = −!!.Predictwhattheeffectisonthegraphandthentestit.Whydoesithavethiseffect?

8. Optimaisencouragedbecausethatonewaseasy.Shedecidestoendherinvestigationforthedaybydeterminingtheeffectofamultiplier,a,intheequation:! = !!!.Usingbothpositiveandnegativenumbers,fractionsandintegers,createatleast4tablesandmatchinggraphstodeterminetheeffectofamultiplier.

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