SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.3

Mathematics Vision Project

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

/HVVRQ�� Building the Perfect Square

A Develop Understanding Task

QuadraticQuiltsOptimahasaquiltshopwhereshesellsmanycolorfulquiltblocksforpeoplewhowanttomaketheirownquilts.Shehasquiltdesignsthataremadesothattheycanbesizedtofitanybed.Shebasesherdesignsonquiltsquaresthatcanvaryinsize,soshecallsthelengthofthesideforthebasicsquarex,andtheareaofthebasicsquareisthefunction! ! = !!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresorsmallersquares.

1. IfOptimaadds3inchestothesideofthesquare,whatistheareaofthesquare?

WhenOptimadrawsapatternforthesquareinproblem#1,itlookslikethis:

2. Useboththediagramandtheequation,! ! = (! + 3)!toexplainwhytheareaofthequiltblocksquare,! ! ,isalsoequalto!! + 6! + 9.

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+3 II ! ! ! A -- lxt35* A -_ (xt3)LXt3)

X XZ * HH A=X2t6× +3 .

SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.3

Mathematics Vision Project

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

ThecustomerservicerepresentativesatOptima’sshopworkwithcustomerordersandwriteuptheordersbasedontheareaofthefabricneededfortheorder.Asyoucanseefromproblem#2therearetwowaysthatcustomerscancallinanddescribetheareaofthequiltblock.Onewaydescribesthelengthofthesidesoftheblockandtheotherwaydescribestheareasofeachofthefoursectionsoftheblock.

Foreachofthefollowingquiltblocks,drawthediagramoftheblockandwritetwoequivalentequationsfortheareaoftheblock.

3. Blockwithsidelength:! + 2.

4. Blockwithsidelength:! + 1.

5. Whatpatternsdoyounoticewhenyourelatethediagramstothetwoexpressionsforthearea?

6. Optimalikestohaveherlittledog,Clementine,aroundtheshop.Onedaythedoggotalittlehungryandstartedtochewuptheorders.WhenOptimafoundtheorders,oneofthemwassochewedupthattherewereonlypartialexpressionsforthearearemaining.HelpOptimabycompletingeachofthefollowingexpressionsfortheareasothattheydescribeaperfectsquare.Then,writethetwoequivalentequationsfortheareaofthesquare.

a. !! + 4!

b. !! + 6!

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X +2 (Xt2)Z

X ,/⇒( xtzlcxtz)

I112+4×+4

=¥9i.

'E'I'Yates×2+2×+1

+ 4

+9

SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.3

Mathematics Vision Project

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

c. !! + 8!

d. !! + 12!

7. If !! + !" + !isaperfectsquare,whatistherelationshipbetweenbandc?Howdoyouusebtofindc,likeinproblem6?

Willthisstrategyworkifbisnegative?Whyorwhynot?

Willthestrategyworkifbisanoddnumber?Whathappenstocifbisodd?

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+ 16

+36

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negative it will becomepositive .

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yes , c -- (E)bz is a fraction f so c will

be a fraction