1.1 checkerboard borders - utah education network - uen.org · pdf filesecondary math i //...

SECONDARY MATH I // MODULE 1

SEQUENCES – 1.1

Mathematics Vision Project

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mathematicsvisionproject.org

1.1 Checkerboard Borders

A Develop Understanding Task

Inpreparationforbacktoschool,theschooladministrationplanstoreplacethetileinthecafeteria.Theywouldliketohaveacheckerboardpatternoftilestworowswideasasurroundforthetablesandservingcarts.Belowisanexampleoftheboarderthattheadministrationisthinkingofusingtosurroundasquare5x5setoftiles.A. Findthenumberofcoloredtilesinthecheckerboardborder.Trackyourthinkingandfinda wayofcalculatingthenumberofcoloredtilesintheborderthatisquickandefficient.Be preparedtoshareyourstrategyandjustifyyourwork.

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B. Thecontractorthatwashiredtolaythetileinthecafeteriaistryingtogeneralizeawaytocalculatethenumberofcoloredtilesneededforacheckerboardbordersurroundingasquareoftileswithanydimensions.Torepresentthisgeneralsituation,thecontractorstartedsketchingthesquarebelow.

FindanexpressionforthenumberofcoloredbordertilesneededforanyNxNsquare center.

N

N

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1 . 1 Checkerboard Borders – Teacher Notes A Develop Understanding Task

Purpose:

Thefocusofthistaskisonthegenerationofmultipleexpressionsthatconnectwiththevisuals

providedforthecheckerboardborders.Theseexpressionswillalsoprovideopportunitytodiscuss

equivalentexpressionsandreviewtheskillsstudentshavepreviouslylearnedaboutsimplifying

expressionsandusingvariables.

CoreStandardsFocus:

N.Q.2Defineappropriatequantitiesforthepurposeofdescriptivemodeling.

A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.�

a.Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

b.Interpretcomplicatedexpressionsbyviewingoneormoreoftheirpartsasasingleentity.

RelatedStandards:A.CED.2,A.REI.1

StandardsforMathematicalPracticeofFocusintheTask:

SMP1–Makesenseofproblemsandpersevereinsolvingthem.

SMP7–Lookforandmakeuseofstructure

TheTeachingCycle:

Launch(WholeClass):

Afterreadinganddiscussingthe“CheckerboardBorders”scenario,challengestudentstocomeup

withawaytoquicklycountthenumberofcoloredtilesintheborder.Havethemcreatenumeric

expressionsthatexemplifytheirprocessandrequirestudentstoconnecttheirthinkingtothe

visualrepresentationofthetiles.

Thefirstphaseofworkshouldbedoneindividually,allowingstudentsto“see”theproblemand

patternsinthetilesintheirownway.Thiswillprovideformorerepresentationstobeconsidered

later.AfterstudentsworkindividuallyforafewminutesonpartA,havethemsharewithapartner


SEQUENCES – 1.1




andbegintodevelopadditionalideasasapairorassisteachotheringeneralizingtheirstrategyfor

partB.

Explore(SmallGroup):

Forstudentswhodon’tknowwheretobegin,itmaybeusefultoasksomestarterquestionslike:

“Howmanytilesaretherealongoneside?”,“Howcanyoucountthetilesingroupsratherthanone-

by-one?”

Pressonstudentstoconnecttheirnumericrepresentationstothevisualrepresentation.Youmight

ask,“Howdoesthatfourinyournumbersentenceconnecttothevisualrepresentation?”Encourage

studentstomarkonthevisualortoredrawitsotheycandemonstratehowtheywerethinking

aboutthediagramnumerically.

Watchforstudentswhocalculatethenumberofbordertilesindifferentways.Makenoteoftheir

numericstrategiesandthedifferentgeneralizedexpressionsthatarecreated.Thediffering

strategiesandalgebraicexpressionswillbethefocusofthediscussionattheend,allowingfor

studentstoconnectbacktopriorworkfrompreviousmathematicalexperiencesandbetter

understandequivalencebetweenexpressionsandhowtoproperlysimplifyanalgebraic

expression.Promptstudentstocalculatethenumberoftilesforagivensidelengthusingtheir

expressionandthentodrawthevisualmodelandcheckforaccuracy.Requirestudentstojustify

whytheirexpressionwillworkforanysidelengthNoftheinnersquareregion.Pressthemto

generalizetheirjustificationsratherthanjustrepeattheprocesstheyhavebeenusing.Youmight

ask,“Howdoyouknowyourexpressionwillworkforanysidelength?”,or“Whatisitaboutthe

natureofthepatternthatsuggeststhiswillalwayswork?”,or“Whatwillhappenifwelookataside

lengthofsix?ten?fifty-three?”Considertheseideasbothvisuallyandintermsofthegeneral

expression.

Note:Basedonthestudentworkandthedifficultiestheymayormaynotencounter,a

determinationwillneedtobemadeastowhetheradiscussionofpartAofthetaskshouldbeheld

priortostudentsworkingonpartB.Workingwithaspecificcasemayfacilitateaccesstothe

generalcaseformorestudents.However,ifstudentsarereadyforwholeclassdiscussionoftheir


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generalrepresentations,thenstartingtherewillallowformoretimetobespentonmaking

connectionsbetweenthedifferentexpressions,andextendingthetasktomoregeneral

representations.

Asavailable,selectstudentstopresentwhofounddifferentwaysofgeneralizing.Somepossible

waysstudentsmight“see”thecoloredtilesgroupedareprovidedafterthechallengeactivity.It

wouldbeusefultohaveatleastthreedifferentviewstodiscussandpossiblymore.

Discuss(WholeClass):

Basedonthestudentworkavailable,youwillneedtodeterminetheorderofthestrategiestobe

presented.Alikelyprogressionwouldstartwithastrategythatdoesnotprovidethemost

simplifiedformoftheexpression.Thiswillpromotequestioningandunderstandingfromstudents

thatmayhavedoneitdifferentlyandallowfordiscussionaboutwhateachpieceoftheexpression

represents.Afteracoupleofdifferentstrategieshavebeenshareditmightbeusefultogetthemost

simplifiedformoftheexpressionoutonthetableandthenlookforanexplanationastohowallof

theexpressionscanbeequivalentandrepresentthesamethinginsomanydifferentways.

AlignedReady,Set,Go:GettingReady1.1


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1.1

READY

Topic:RecognizingSolutionstoEquations

Thesolutiontoanequationisthevalueofthevariablethatmakestheequationtrue.Intheequation

9! + 17 = −21, "a”isthevariable.Whena=2,9! + 17 ≠ −19, because 9 2 + 17 = 35. Thus! = 2 is NOT a solution.However,when! = −4, the equation is true 9 −4 + 17 = −19.Therefore,! = −4mustbethesolution.Identifywhichofthe3possiblenumbersisthesolutiontotheequation.

1.3! + 7 = 13 (! = −2; ! = 2; ! = 5) 2.8 − 2! = −2 (! = −3; ! = 0; ! = 5)

3.5 + 4! + 8 = 1 (! = −3;! = −1;! = 2) 4.6! − 5 + 5! = 105 (! = 4; ! = 7; ! = 10)

Someequationshavetwovariables.Youmayrecallseeinganequationwrittenlikethefollowing:

! = 5! + 2.Wecanletxequalanumberandthenworktheproblemwiththisx-valuetodeterminetheassociatedy-value.Asolutiontotheequationmustincludeboththex-valueandthey-value.Oftenthe

answeriswrittenasanorderedpair.Thex-valueisalwaysfirst.Example: !, ! .Theordermatters!

Determinethey-valueofeachorderedpairbasedonthegivenx-value.

5.! = 6! − 15; 8, , −1, , 5, 6.! = −4! + 9; −5, , 2, , 4,

7.! = 2! − 1; −4, , 0, , 7, 8.! = −! + 9; −9, , 1, , 5,

READY, SET, GO! Name PeriodDate

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1.1

SET

Topic:Usingaconstantrateofchangetocompleteatableofvalues

Fillinthetable.Thenwriteasentenceexplaininghowyoufiguredoutthevaluestoputineachcell.9.Yourunabusinessmakingbirdhouses.Youspend$600tostartyourbusiness,anditcostsyou$5.00

tomakeeachbirdhouse.

#ofbirdhouses 1 2 3 4 5 6 7

Totalcosttobuild

Explanation:

10.Youmakea$15paymentonyourloanof$500attheendofeachmonth.

#ofmonths 1 2 3 4 5 6 7

Amountofmoneyowed

Explanation:

11.Youdeposit$10inasavingsaccountattheendofeachweek.

#ofweeks 1 2 3 4 5 6 7

Amountofmoneysaved

Explanation:

12.Youaresavingforabikeandcansave$10perweek.Youhave$25whenyoubeginsaving.

#ofweeks 1 2 3 4 5 6 7

Amountofmoneysaved

Explanation:

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1.1

GO

Topic:GraphLinearEquationsGivenaTableofValues.

Graphtheorderedpairsfromthetablesonthegivengraphs.

13.

! !

0 3

2 7

3 9

5 13

14.

! !

0 14

4 10

7 7

9 5

15.

! !

2 11

4 10

6 9

8 8

16.

! !

1 4

2 7

3 10

4 13

5

1.1 checkerboard borders - utah education network - uen.org · pdf filesecondary math i //...

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