SECONDARY MATHEMATICS II // MODULE 2

STRUCTURES OF EXPRESSIONS

Mathematics Vision Project

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2.4 A SQUARE DEAL

A Solidify Understanding Task

QuadraticQuilts,Revisited

RememberOptima’squiltshop?Shebasesherdesignsonquiltsquaresthatcanvaryinsize,

soshecallsthelengthofthesideforthebasicsquarex,andtheareaofthebasicsquareisthe

functionA 𝑥 = 𝑥!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresor

smallersquares.

1.Sometimesacustomerordersmorethanonequiltblockofagivensize.Forinstance,

whenacustomerorders4blocksofthebasicsize,thecustomerservicerepresentatives

writeupanorderforA 𝑥 = 4𝑥!.Modelthisareaexpressionwithadiagram.

2.Oneofthecustomerservicerepresentativesfindsanenvelopethatcontainstheblocks

picturedbelow.Writetheorderthatshowstwoequivalentequationsfortheareaofthe

blocks.

© 2

014

ww

w.fl

icke

r.co

m/p

hoto

s/ea

myl

ove

SECONDARY MATHEMATICS II // MODULE 2

STRUCTURES OF EXPRESSIONS

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

3.Whatequationsfortheareacouldcustomerservicewriteiftheyreceivedanorderfor2

blocksthataresquaresandhavebothdimensionsincreasedby1inchincomparisontothe

basicblock?Writetheareaequationsintwoequivalentforms.Verifyyouralgebrausinga

diagram.

4.Ifcustomerservicereceivesanorderfor3blocksthatareeachsquareswithboth

dimensionsincreasedby2inchesincomparisontothebasicblock?Again,show2different

equationsfortheareaandverifyyourworkwithamodel.

5.Clementineisatitagain!Whenisthatdoggoingtolearnnottochewuptheorders?

(ShealsochewsOptima’sshoes,butthat’sastoryforanotherday.)Herearesomeofthe

ordersthathavebeenchewedupsotheyaremissingthelastterm.HelpOptimaby

completingeachofthefollowingexpressionsfortheareasothattheydescribeaperfect

square.Then,writethetwoequivalentequationsfortheareaofthesquare.

2𝑥! + 8𝑥

3x! + 24𝑥

SECONDARY MATHEMATICS II // MODULE 2

STRUCTURES OF EXPRESSIONS

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

Sometimesthequiltshopgetsanorderthatturnsoutnottobemoreorlessthanaperfect

square.Customerservicealwaystriestofillorderswithperfectsquares,oratleast,they

startthereandthenadjustasneeded.

6.Nowhere’sarealmess!Customerservicereceivedanorderforanarea

A x = 2𝑥! + 12𝑥 + 13.Helpthemtofigureoutanequivalentexpressionfortheareausing

thediagram.

7.Optimareallyneedstogetorganized.Here’sanotherscrambleddiagram.Writetwo

equivalentequationsfortheareaofthisdiagram:

SECONDARY MATHEMATICS II // MODULE 2

STRUCTURES OF EXPRESSIONS

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

8.Optimarealizesthatnoteveryoneisinneedofperfectsquaresandnotallordersare

cominginasexpressionsthatareperfectsquares.Determinewhetherornoteach

expressionbelowisaperfectsquare.Explainwhytheexpressionisorisnotaperfect

square.Ifitisnotaperfectsquare,findtheperfectsquarethatseems“closest”tothegiven

expressionandshowhowtheperfectsquarecanbeadjustedtobethegivenexpression.

A. 𝐴 𝑥 = 𝑥! + 6𝑥 + 13 B. 𝐴 𝑥 = 𝑥! − 8𝑥 + 16

C. 𝐴 𝑥 = 𝑥! − 10𝑥 − 3 D.𝐴 𝑥 = 2𝑥! + 8𝑥 + 14

E. 𝐴 𝑥 = 3𝑥! − 30𝑥 + 75 F. 𝐴 𝑥 = 2𝑥! − 22𝑥 + 11

9.Nowlet’sgeneralize.Givenanexpressionintheform𝑎𝑥! + 𝑏𝑥 + 𝑐 (𝑎 ≠ 0),describea

step-by-stepprocessforcompletingthesquare.