Matrices Revisited · matrices that answer the question of weekly revenue collected. Have students...

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

This work is licensed under the Creative Commons Attribution CC BY 4.0

MODULE 10 HONORS

Matrices Revisited

ALGEBRA II

An Integrated Approach

Standard Teacher Notes

ALGEBRA II // MODULE 10H

MATRICES REVISITED

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

MODULE 10H - TABLE OF CONTENTS

MATRICES REVISITED

10.1H Row by Row . . . – A Solidify Understanding Task

Extending row reduction of matrices to systems of equations in n variables (MVP Honors

Standard)

READY, SET, GO Homework: Matrices Revisited 10.1H

10.2H . . . and Row by Column – A Solidify Understanding Task

Reviewing matrix multiplication in preparation for solving systems of equations using inverse

matrices (N.VM.6, N.VM.8)


10.3H More Arithmetic of Matrices – A Solidify Understanding Task

Examining properties of matrix addition and multiplication, including identity and inverse

properties (N.VM.8, N.VM.9)


10.4H The Determinant of a Matrix – A Develop Understanding Task

Finding the determinant of a matrix and relating it to the area of a parallelogram (N.VM.10,

N.VM.12)


10.5H Solving Systems with Matrices, Revisited – A Solidify Understanding Task

Solving a system of linear equations using the multiplicative inverse matrix (A.REI.1)



MATRICES REVISITED


10.6H All Systems Go – A Practice Understanding Task

Solving systems of linear equations using matrices (A.REI.8, A.REI.9)



MATRICES REVISITED – 10.1H


10.1H “Row by Row . . .”

A Solidify Understanding Task

Carloslikestobuysuppliesforthetwin’sbusiness,CurbsideRivalry,attheAllaDollarPaint

Storewherethepriceofeveryitemisamultipleof$1.Thismakesiteasytokeeptrackofthetotalcost

ofhispurchases.ClaritaisworriedthatitemsatAllaDollarPaintStoremightcostmore,sosheis

goingovertherecordstoseehowmuchCarlosispayingfordifferentsupplies.Unfortunately,Carlos

hasonceagainforgottentowritedownthecostofeachitemhepurchased.Instead,hehasonly

recordedwhathepurchasedandthetotalcostofalloftheitems.

CarlosandClaritaaretryingtofigureoutthecostofagallonofpaint,thecostofapaintbrush,

andthecostofarollofmaskingtapebasedonthefollowingpurchases:

Week1:Carlosbought2gallonsofpaintand1rollofmaskingtapefor$30.

Week2:Carlosbought1gallonofpaintand4brushesfor$20.

Week3:Carlosbought2brushesand1rollofmaskingtapefor$10.

1. Determinethecostofeachitemusingwhateverstrategyyouwant.Showthedetailsofyourworksothatsomeoneelsecanfollowyourstrategy.

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Youprobablyrecognizedthatthisproblemcouldberepresentedasasystemofequations.In

previousmathcoursesyouhavedevelopedseveralmethodsforsolvingsystems:graphing,

substitution,elimination,androwreductionofmatrices.

2. Whichofthemethodsyouhavedevelopedpreviouslyforsolvingsystemsofequationscouldbeappliedtothissystem?Whichmethodsseemmoreproblematic?Why?

IntheMVPAlgebraItasksToMarketwithMatricesandSolvingSystemswithMatricesyou

learnedhowtosolvesystemsofequationsinvolvingtwoequationsandtwounknownquantitiesusing

rowreductionofmatrices.(Youmaywanttoreviewthosetwotasksbeforecontinuing.)

3. Modifythe“rowreductionofmatrices”strategysoyoucanuseittosolveCarlosandClarita’ssystemofthreeequationsusingrowreduction.Whatmodificationsdidyouhavetomake,andwhy?

2




4. Decideonareasonablecontextandwriteastoryfortheinformationgiveninthefollowingmatrix:

"1 2 2 162 1 3 171 2 1 13

(

5. Solvefortheunknownsinyourstorybyusingrowreductiononthegivenmatrix.Checkyourresultsinthestorycontexttomakesuretheyarecorrect.

3




10.1H “Row by Row” – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistoextendtheprocessofsolvinglinearsystemsusingrow

reductionofmatricestolinearsystemsthatincludemorethantwoequationsandmorethantwo

unknowns.Therowreductionofmatricesmethodforsolvingsystemsofequationswasintroduced

forthe2-equation,2-variablecaseintheMVPAlgebraIHonorscurriculum.Thistaskreviewstheidea

ofrepresentingasystemofequationswithanaugmentedmatrix,thenrowreducingthematrixusing

ideasthatemergedfromstudents’workwithsolvingsystemsofequationsbyelimination:

• Replaceanequation(orarowofthematrix)withamultipleofthatequation(orrow);

Example:3)* → )* .• Replaceanequation(orarowofthematrix)withthesumordifferenceofthatequation(or

row)withamultipleofanotherequation(orrow);Example:2), + )* → )*

CoreStandardsFocus:

MVPHonorsStandard:Solvesystemsoflinearequationsusingmatrices.RelatedStandards:A.REI.6

StandardsforMathematicalPractice:

SMP6–Attendtoprecision

SMP7–Lookforandmakeuseofstructure

SMP8–Lookforandexpressregularityinrepeatedreasoning

TheTeachingCycle:

Launch(WholeClass):

Priortobeginningthistaskreviewwithstudents,asneeded,thefollowingtasksfromtheMVPAlgebra

Icurriculum:

5.11HToMarketwithMatrices

5.12HSolvingSystemswithMatrices




Explore(SmallGroup):

Studentsmightsolvethesysteminquestion1usinganinformalmethodbasedonthesimilaritemsin

eachpurchase.Astheymovethroughquestions2and3,helpstudentsformalizetheirworkusingrow

reductionofmatrices.

Listenforstudentstopresentinthewholeclassdiscussionwhocreateinterestingcontextstofitthe

matrixgiveninquestion4,andcansuccessfullyapplythestrategyofrowreductiontosolvethis

system.Thismatrixismorecomplicatedthanthematrixstudentswilluseinquestion3,sincethere

arenotalotof0coefficients.Watchforstrategiestoemergethatinvolvegetting1sand0sinthe

columnsinwaysthatwillnotimpactcoefficientsinarowthatalreadyhas1sor0sintheappropriate

positionswhenthatrowandamultipleofanotherrowareaddedtogether.

Discuss(WholeClass):

Movetothewholeclassdiscussionwhenstudentshavemadeenoughprogressonquestion5thatthey

caneitherdescribeanefficientprocedureforrowreducingamatrix,ortheyareprimedandreadyfor

somehintsthatwilldemystifytheprocess.Asneeded,provideasuggestionforanextpossiblestepin

therow-reductionprocess,thenletstudentscontinuetoworkofdevelopingtheirstrategies.

AlignedReady,Set,Go:MatricesRevisited10.1H

ALGEBRA II // MODULE 10

MATRICES REVISITED– 10.1H

10.1H


Needhelp?Visitwww.rsgsupport.org

READY Topic:SolvingSystemsbySubstitutionandElimination

Solveeachsystemofequationsusinganalgebraicmethod.

1.! − 2$ = 63! − $ = 13

2.2! − 3$ = 9! + 5$ = −28 3.

! − $ = 04! + $ = 9

4.2! + 3$ = 2−4! − 6$ = −14

5.7! − $ = 14! + 7$ = −48

6.8! + 5$ = 9−! − 5$ = 5

7.Doanyofthesystemsinproblems1-6representparallellines?

Ifso,howdoyouknow?

8.Doanyofthesystemsinproblems1-6representperpendicularlines?

Ifso,howdoyouknow?

READY, SET, GO! Name PeriodDate

4



10.1H



SET Topic:Solvingmatricesusingrowreduction

9.Createamatrixtomatcheachstepinthesolvingofthesystemofequationsgiven.Also,writea

descriptionofwhathappenedtotheequationandthematrixbetweensteps.

SystemofEquations Description Matrix

GivenSystem 03! + 2$ = 40! − 7$ = −2 13 2

1 −7 240−23 ê −345 → 45 ê

Step1 0 3! + 2$ = 40−3! + 21$ = 6 ê 1 2

−3 2406 3 ê ê

Step2 0 3! + 2$ = 400! + 23$ = 46 ê 1 0 2403

ê ê

Step3 03! + 2$ = 400! + $ = 2 ê 1 2 3

ê ê

Step4 03! + 0$ = 360! + $ = 2 ê 1 2 3

ê ê

Step5 0! + 0$ = 120! + $ = 2 1 2 3

5



10.1H



GO Topic:Reviewinghistograms

10.Flipacoin5times.Recordthenumberoftimesthecoinlandswithheadsup.Repeatthis

process20times,eitherbyhandorbysimulationusingtechnology,eachtimerecordingyour

resultsinthetablebelow.(Everytimeyouperformthesimulation,countthenumberofheadsyouhaveandrecordtheresultintheTallycolumnbelow.Forexample,ifyouflipthecoin5timesandget3heads,putatallymarkbythe3headsor60%row.)http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

11.Createahistogramofyourresults.Describetheshapeofthehistogram(Shape,Center,Spread)

#Heads %Heads Tally0 0%

1 20%

2 40%

3 60%

4 80%

5 100%

6



10.1H



12.Flipacoin20times.Recordthenumberoftimesheadslandssideup.Repeatthisprocess20

timeseitherbyhandorbysimulationusingtechnology.

http://www.rossmanchance.com/applets/CoinTossing/CoinToss.html

Recordyourresultsinthetablebelow.

13.Createahistogramofyourresultsbelow.Describetheshapeofthehistogram(Shape,Center,

Spread)

#Heads

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Frequency #Heads

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Frequency

0 0% 11 55%

1 5% 12 60%

2 10% 13 65%

3 15% 14 70%

4 20% 15 75%

5 25% 16 80%

6 30% 17 85%

7 35% 18 90%

8 40% 19 95%

9 45% 20 100%

10 50%

7



10.1H



14.Comparetheshapecenterandspreadofeachdistribution.Whatdoyounotice?

15.Ifyourepeatedthisprocesswith500flipsinsteadof5or20,predictwhatwouldhappentothe

shape,spread,andcenterofthenewhistogram.

8




10.2H “. . . and Row by Column”


Intheprevioustaskyouchoseacontextandwroteastoryfortheinformationgiveninthe

followingmatrix:

"1 2 2 162 1 3 171 2 1 13

(

Yourstorycouldalsohavebeenrepresentedbythefollowingsystemofequations:) + 2+ + 2, = 162) + + + 3, = 17) + 2+ + , = 13

Usingrowreductiononthematrix,youfoundthatthesolutiontothissystemofequationsis:

x=2,y=4,z=3.Inalatertaskinthismoduleyouwilllearnanothermethodforsolvinglinear

systemsusingmatrices.Thisnewmethodwillusematrixmultiplication,solet’sreviewthat

operation.

Wecanverifythatx=2,y=4,z=3isasolutiontothesystemofequationsgivenaboveusing

matrixmultiplication.

1. Usethefollowingtoreviewhowmatrixmultiplicationworks.Explainhowthenumbersinthematrixontherightsideoftheequationwereobtainedastheproductofthetwomatricesontheleftsideoftheequation.

"1 2 22 1 31 2 1

( ∙ "243( = "

161713(

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2. Inacontext,eachentryinamatrixrepresentstwopiecesofinformation,dependingontherowandcolumninwhichitislocated.Organizethefollowinginformationintoamatrix,andlabeltherowsandcolumns.

• Week1:Claritapainted6curbsidelogosandCarlospainted4drivewaymascots• Week2:Claritapainted8curbsidelogosandCarlospainted3drivewaymascots• Week3:Claritapainted5curbsidelogosandCarlospainted6drivewaymascots

3. CarlosandClaritacharge$8foracurbsidelogoand$20foradrivewaymascot.Usingthisadditionalinformation,showhowyoucanusematrixmultiplicationtodeterminehowmuchrevenueCarlosandClaritacollectedduringweeks1,2and3.

10




10.2H “. . . and Row by Column” – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistoreviewmatrixmultiplication,inpreparationforsolving

systemsofequationsusingmatrixmultiplicationandinversematrices,asdevelopedinthe

subsequenttask.StudentshavebeenintroducedtomatrixmultiplicationinthefollowingAlgebraI

tasks:4.7HCafeteriaConsumptionsandCost,4.8HEatingUptheLunchroomBudget,and

4.9HArithmeticofMatrices.

CoreStandardsFocus:

N.VM.6Usematricestorepresentandmanipulatedata.

N.VM.8Add,subtract,andmultiplymatricesofappropriatedimensions.

RelatedStandards:N.Q.1


SMP2–ReasonAbstractlyandQuantitatively

SMP4–Modelwithmathematics


TheTeachingCycle:

Launch(WholeClass):

Priortobeginningthistaskreviewwithstudents,asneeded,thefollowingtasksfromtheMVP

AlgebraIcurriculum:

4.7HCafeteriaConsumptionsandCost

4.8HEatingUptheLunchroomBudget

4.9HArithmeticofMatrices





Verifythatstudentscanrecalltherowbycolumnstrategyformultiplyingtwomatricesbylisteningto

theirconversationsonquestion1.Thiswillprovideformativeassessmentastohowmuchworkwith

theprevioustaskslistedinthelaunchyouwillneedtodowithstudents.

Oncestudentshavere-familiarizedthemselveswiththeprocessofmatrixmultiplication,theirwork

shouldfocusonhowtoorganizedataintomatricessothattherowandcolumnmultiplication

procedureprovidesmeaningfulinformationwhentryingtointerprettheresultsintheproduct

matrix.Allowstudentstimetograpplewiththisidea,perhapsbythinkingaboutthepartial

multiplicationsthatneedtobesummeduptofindtherevenueforeachweek.Watchforstudentswho

mightwanttokeeptrackofCarlosandClarita’sworkindependently,sincethatisnottheissuethe

twinsaretryingtoanswer.Labelingtherowsandcolumnsofeachfactormatrixrevealsuseful

informationforinterpretingtheresultsofthemultiplication.


Movetothewholeclassdiscussionwhenmoststudentshavesuccessfullysetupaproductof

matricesthatanswerthequestionofweeklyrevenuecollected.Havestudentspresenthowthey

thoughtaboutorganizingthetwofactormatricessothatthepartialproductsfoundbymultiplying

numbersistherowsofthefirstmatrixbythenumbersinthecolumnofthesecondmatrixwould

producemeaningfulresultsintheproductmatrix.Iftheydidnotlabeltherowsandcolumnsofeach

matrix,askthemtodoso,andtousethoselabelsintheirexplanationsoftheirwork.

Concludethediscussionbyaskingwhyitispossibletomultiplymatricesofdifferentsizes,andhow

theycantellwhenmatrixmultiplicationispossible.




10.2H



READY Topic:Addingmatrices

Addthegivenmatrices.Ifthematricescan’tbeadded,giveareasonwhynot.

1.! 5 −223 11' + !

10 8−4 19' = 2./

99 5598 6478 12

2 + ! 1 45 222 88 0' =

3./70 14 62−31 39 8566 13 −7

2 + /−70 −14 −6231 −39 −85−66 −13 7

2 = 4./6−8102 + /

−108−6

2 =

SET Topic:Multiplyingmatrices5.Recallfromtoday’slessonthatinacontext,eachentryinamatrixrepresentstwopiecesof

information,dependingontherowandcolumninwhichitislocated.Organizethe

followinginformationintoamatrix,andlabeltherowsandcolumns.

ThefollowingnumberofitemsweresoldduringthelunchrushatFriedFreddy’sCafé:

• Day1–65ordersoffriedchicken,62ordersoffish,and145ordersoffrenchfries






11



10.2H



5./1 1 22 2 31 3 4

2 ∙ /155102 = /

11111111111111111

2

6./10 20 2020 14 511 15 2

2 ∙ /5282 = /

11111111111111111

2

7./4 3 0.52 1 35 4 0.25

2 ∙ /2530202 = /

11111111111111111

2

8./5 2 44 4 48 2 5

2 ∙ /915252 = /

11111111111111111

2

GO

Topic:Findingprobabilitiesfromatwo-waytable

Thefollowingdatarepresentsarandomsampleofboysandgirlsandhowmanyprefercatsordogs.Usetheinformationtoanswerthequestionsbelow.

Cats Dogs TotalBoys 32 68 100

Girls 41 11 52

Total 73 79 152

9.5(7) = 10.5(9) = 11.5(:) = 12.5(;) =

13.5(:|9) = 14.5(:=>7) = 15.5(;|7) = 16.5(7 ∩ ;) =

17.Ifthisisarandomsamplefromaschool,whattotalpercentofboysinthisschooldoyouthink

wouldpreferdogs?

18.Whatpercentofstudentsattheschoolwouldprefercats?

19.Ifyousampledadifferent152students,wouldyougetthesamepercentages?Explain.

20.Whatwouldhappentoyourpercentagesifyouusedalargersamplesize?

12




10.3H More Arithmetic of Matrices


Inthistaskyouwillhaveanopportunitytoexaminesomeofthepropertiesofmatrix

additionandmatrixmultiplication.Wewillrestrictthisworktosquare2´2matrices.

Thetablebelowdefinesandillustratesseveralpropertiesofadditionandmultiplicationfor

realnumbersandasksyoutodetermineifthesesamepropertiesholdformatrixadditionand

matrixmultiplication.Whilethechartasksforasingleexampleforeachproperty,youshould

experimentwithmatricesuntilyouareconvincedthatthepropertyholdsoryouhavefounda

counter-exampletoshowthatthepropertydoesnothold.Canyoubaseyourjustificationonmore

thatjusttryingoutseveralexamples?

Property ExamplewithRealNumbers ExamplewithMatrices

AssociativePropertyof

Addition

(a+b)+c=a+(b+c)

AssociativePropertyof

Multiplication

(ab)c=a(bc)

CommutativePropertyof

Addition

a+b=b+a

http://commons.wikimedia.org/wiki/File:Matriz_A_por_B.png

13




CommutativePropertyof

Multiplication

ab=ba

DistributivePropertyof

MultiplicationOverAddition

a(b+c)=ab+ac

Inadditiontothepropertieslistedinthetableabove,additionandmultiplicationofreal

numbersincludepropertiesrelatedtothenumbers0and1.Forexample,thenumber0isreferred

toastheadditiveidentitybecausea+0=0+a=a,andthenumber1isreferredtoasthe

multiplicativeidentitysince .Oncetheadditiveandmultiplicativeidentitieshave

beenidentified,wecanthendefineadditiveinversesaand–asincea+-a=0,andmultiplicative

inversesaand since .Todecideifthesepropertiesholdformatrixoperations,wewill

needtodetermineifthereisamatrixthatplaystheroleof0formatrixaddition,andifthereisa

matrixthatplaystheroleof1formatrixmultiplication.

TheAdditiveIdentityMatrix

Findvaluesfora,b,canddsothatthematrixbelowthatcontainsthesevariablesplaysthe

roleof0,ortheadditiveidentitymatrix,forthefollowingmatrixaddition.Willthissamematrix

workastheadditiveidentityforall2´2matrices?

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TheMultiplicativeIdentityMatrix


roleof1,orthemultiplicativeidentitymatrix,forthefollowingmatrixmultiplication.Willthis

samematrixworkasthemultiplicativeidentityforall2´2matrices?

Nowthatwehaveidentifiedtheadditiveidentityandmultiplicativeidentityfor2´2

matrices,wecansearchfortheadditiveinversesandmultiplicativeinversesofmatrices.

FindinganAdditiveInverseMatrix


roleoftheadditiveinverseofthefirstmatrix.Willthissameprocessworkforfindingtheadditive

inverseofall2´2matrices?

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FindingaMultiplicativeInverseMatrix


roleofthemultiplicativeinverseofthefirstmatrix.Willthissameprocessworkforfindingthe

multiplicativeinverseofall2´2matrices?

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10.3H More Arithmetic of Matrices – Teacher Notes A Solidify Understanding Task

Purpose:Inpreviousmodulesstudentshavelearnedhowtoadd,subtractandmultiplymatrices.

Thepurposeofthistaskistoexaminethesimilaritiesbetweenthepropertiesofoperationswith

realnumbersandthepropertiesofoperationswithmatrices.Studentswillexaminethe

associative,distributiveandcommutativepropertiestodetermineiftheyholdformatrixaddition

andmultiplication.Theywillalsoexaminethepropertiesofadditiveandmultiplicativeidentities

andinversesformatrixoperations.Aspartofthistasktheywilldevelopastrategyforfindingthe

multiplicativeinverseofasquarematrix.Theinverseofamatrixwillbeusedinasubsequenttask

togivestudentsanalternativestrategyforsolvingsystemsoflinearequations.

CoreStandardsFocus:

N.VM.9Understandthat,unlikemultiplicationofnumbers,matrixmultiplicationforsquare

matricesisnotacommutativeoperation,butstillsatisfiestheassociativeanddistributive

properties.

N.VM.10Understandthatthezeroandidentitymatricesplayaroleinmatrixadditionand

multiplicationsimilartotheroleof0and1intherealnumbers.

RelatedStandards:N.VM.8



TheTeachingCycle:

Launch(WholeClass):

Inthistaskstudentswillbeusinglanguagelikeadditiveidentityormultiplicativeinverseorthe

associativepropertyofmultiplication.Youmightbeginthistaskbyreviewingwhattheseproperties

meanforadditionandmultiplicationofrealnumbers.Usethefirstcolumnofthechartinthetask




toreviewandillustratetheseproperties.Thenaskstudentsifvectoraddition,asdefinedinthe

previoustaskhassomeofthesesameproperties.Thatis,askquestionssuchasthefollowingand

allowstudentstimetoexpresswhytheythinkthesepropertieshold.Acceptgenericdiagramsof

vectorswithnon-specificlengthsas“representation-basedproof”oftheirclaimsaboutthese

properties.

• Isvectoradditionassociative?

• Isvectoradditioncommutative?

• Isthereavectorthatplaystherolethat0playsintheadditionofrealnumbers?Thatis,is

thereanadditiveidentityforvectoraddition?

• Dovectorshaveadditiveinverses?

• Isscalarmultiplicationdistributiveovervectoraddition?

Oncethesepropertieshavebeenreviewedintermsofoperationswithrealnumbers,andexplored

intermsofoperationswithmatrices,informstudentsthattheywillbeconsideringtheseproperties

intermsofaddingandsubtractingsquarematrices.


Studentsshouldexperimentwithsquarematricesoftheirownchoosingastheyexploreeachofthe

propertieslistedinthechartforoperationswithsquarematrices.Encouragethemtofindcounter-

examples,ortoconvincethemselveswhycounter-examplesmaynotexist.Listenfor“proof-like”

arguments,suchas“theassociativepropertyofadditionholdsbecausewearejustaddingthethree

elementsinthesamepositionofthethreematriceswhichisthesameastheassociativeproperty

forthreerealnumbers”,ratherthanjustificationbasedsolelyontryingoutspecificcases.

Theremainderofthetaskasksstudentstofindtheelementsofidentityandinversematrices.This

workshouldbefairlyeasybasedonwhatstudentsalreadyknowaboutmatrixadditionand

multiplication,withtheexceptionoffindingthemultiplicativeinverseofamatrix.Allothercases

canberesolvedbyguessandcheck.Inthecaseofthemultiplicativeinverse,suggestthatstudents

multiplyoutthematricesontheleftsideandsettheresultingexpressionsequaltotheappropriate

elementsofthematrixontherightsideoftheequation.Thisshouldleadtothefollowingfour

equations:




3a+c=1

3b+d=0

4a+2c=0

4b+2d=1

Studentscanthensolvefora,b,canddbypairingequationsthatcontainthesametwovariables

intosystemsofequations.


Havestudentssharetheirjustificationsthatmatrixadditionandmatrixmultiplicationsharethe

samepropertiesofoperationsasthecorrespondingoperationswithrealnumbers,withthe

exceptionofthecommutativepropertyofmultiplication.Havestudentspresentacounter-example

forthisexception.Makesurestudentscanfindthemultiplicativeinverseofamatrixbysettingup

thematrixequationwithvariablesastheelementsoftheinversematrix,turningthematrix

equationintotwosystemsoflinearequations,andsolvingforthevariablesthatrepresentthe

elementsoftheinversematrix.Iftechnologyisavailable,youmightshowstudentshowtoobtain

themultiplicativeinversematrixusingtechnology.Wewillusethemultiplicativeinversematrixin

alatertasktoprovideanalternativewayforsolvingsystemsusingmatrices.




10.3H



READY Topic: SET Topic::Findingtheadditiveandmultiplicativeinverseofamatrix

2.Given:Matrix

a.FindtheadditiveinverseofmatrixAb.FindthemultiplicativeinverseofmatrixA

3.Given:Matrix

a.FindtheadditiveinverseofmatrixB. b.FindthemultiplicativeinverseofmatrixB

GO Topic:Parallellines,perpendicularlines,andlengthfromacoordinategeometryperspective

Giventhefourpoints:A(2,1),B(5,2),C(4,5),andD(1,4)

4.IsABCDaparallelogram?

Provideconvincingevidenceforyouranswer.

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10.3H



5.IsABCDarectangle?

Provideconvincingevidenceforyouranswer.

6.IsABCDarhombus? Provideconvincingevidenceforyouranswer.

7.IsABCDasquare? Provideconvincingevidenceforyouranswer.

18




10.4H The Determinant of a Matrix


Intheprevioustaskwelearnedhowtofindthemultiplicativeinverseofamatrix.Usethatprocess

tofindthemultiplicativeinverseofthefollowingtwomatrices.

1.

2.

3. Wereyouabletofindthemultiplicativeinverseforbothmatrices?

Thereisanumberassociatedwitheverysquarematrixcalledthedeterminant.Ifthe

determinantisnotequaltozero,thenthematrixhasamultiplicativeinverse.

Fora2´2matrixthedeterminantcanbefoundusingthefollowingrule:(note:thevertical

lines,ratherthanthesquarebrackets,whichareusedtoindicatethatwearefindingthe

determinantofthematrix)

4 Usingthisrule,findthedeterminantofthetwomatricesgiveninproblems1and2above.

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19




Theabsolutevalueofthedeterminantofa2´2matrixcanbevisualizedastheareaofa

parallelogram,constructedasfollows:

• Drawonesideoftheparallelogramwithendpointsat(0,0)and(a,c).• Drawasecondsideoftheparallelogramwithendpointsat(0,0)and(b,d).• Locatethefourthvertexthatcompletestheparallelogram.

(Notethattheelementsinthecolumnsofthematrixareusedtodefinetheendpointsofthevectors

thatformtwosidesoftheparallelogram.)

5. Usethefollowingdiagramtoshowthattheareaoftheparallelogramisgivenbyad–bc.

6. Drawtheparallelogramswhoseareasrepresentthedeterminantsofthetwomatriceslistedinquestions1and2above.Howdoesazerodeterminantshowupinthesediagrams?

20




7. Createamatrixforwhichthedeterminantwillbenegative.Drawtheparallelogram

associatedwiththedeterminantofyourmatrixandfindtheareaoftheparallelogram.

Thedeterminantcanbeusedtoprovideanalternativemethodforfindingtheinverseof2´2

matrix.

8. Usetheprocessyouusedpreviouslytofindtheinverseofageneric2´2matrixwhoseelementsaregivenbythevariablesa,b,candd.Fornow,wewillrefertotheelementsoftheinversematrixasM1,M2,M3andM4asillustratedinthefollowingmatrixequation.FindexpressionsforM1,M2,M3andM4intermsoftheelementsofthefirstmatrix,a,b,candd.

M1=M2=M3=M4=Useyourworkabovetoexplainthisstrategyforfindingtheinverseofa2´2matrix:(note:the-1superscriptisusedtoindicatethatwearefindingthemultiplicativeinverseofthematrix)

wheread–bcisthedeterminantofthematrix

€

a bc d"

# $

%

& ' ⋅

M1 M2

M3 M4

"

# $

%

& ' =

1 00 1"

# $

%

& '

€

a bc d"

# $

%

& '

−1

=1

ad − bcd −b−c a"

# $

%

& '

21




10.4H The Determinant of a Matrix – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistointroducethedeterminantofasquarematrix,andto

connectthedeterminantofa2´2matrixtotheareaofaparallelogram.The“areaofa

parallelogram”representationwillbeusedtoillustratewhysomesquarematriceshavea

determinantof0.Studentswillobservethatifthedeterminantofamatrixisequaltozero,the

matrixwillnothaveamultiplicativeinverse.Inthenexttask,SolvingSystemswithMatrices,

Revisited,non-zerodeterminantswillbeusedasanindicatorthatasystemoflinearequationshasa

uniquesolution.

Inthelastpartofthetaskstudentsareintroducedtoanalternativewayforfindingtheinverseofa

2´2matrixusingthedeterminant.Thiswillsupporttheworkofthenexttaskwherestudentswill

solvesystemsoflinearequationsusinginversematrices.Oncestudentshaveaclearunderstanding

ofwhataninversematrixmeansandhavefoundafewinversesof2´2matricesusingbothofthe

methodsofthistask,youmayalsowanttointroducethemtotheuseoftechnologytofindinverse

matrices.(Athirdalgebraicmethodforfindingtheinverseofamatrixbyusingrowreductionis

notintroducedinthissequenceoftasks,butcouldbeexploredwithmoreadvancedstudents.)

CoreStandardsFocus:

N.VM.10Understandthatthedeterminantofasquarematrixisnonzeroifandonlyifthematrix

hasamultiplicativeinverse.

N.VM.12Workwith2×2matricesandinterprettheabsolutevalueofthedeterminantintermsof

area.






TheTeachingCycle:

Launch(WholeClass):

Givestudentsafewminutesindividuallytoworkonfindingtheinverseofthematricesinquestions

1and2usingthestrategydevelopedintheprevioustask.Thatis,studentsshouldlookforthe

valuesofa,b,canddthatmakethefollowingmatrixequationstrue.

Forquestion1:

Studentscanmultiplythetwomatricesontheleftsideofthisequationandsettheresulting

expressionsequaltothecorrespondingelementsinthematrixontherightsideoftheequation.

Thiswillleadtothefollowingtwosystemsthatcanbesolvedfora,b,c,andd.

and .

Forquestion2:

Asstudentsexaminetherelatedsystems,suchas ,theywillfindthatthesystemshave

nosolutionsandtherefore,thegivenmatrixdoesnothaveamultiplicativeinverse.Introducethe

determinantasanumberthatwilltellusifasquarematrixhasamultiplicativeinversebeforegoing

throughtheworkoftryingtocalculatewhattheinversematrixis,andhavestudentsapplythe

notationgivenpriortoquestion4tofindthedeterminantofthetwogivenmatrices.Thenassign

studentstoworkontheremainderofthetask.

€

5 16 2"

# $

%

& ' ⋅

a bc d"

# $

%

& ' =

1 00 1"

# $

%

& '

€

5a + c = 16a + 2c = 0" # $

€

5b + d = 06b + 2d = 1" # $

€

6 23 1"

# $

%

& ' ⋅

a bc d"

# $

%

& ' =

1 00 1"

# $

%

& '

€

6a + 2b = 13a + b = 0

" # $





Associatingthedeterminantofa2´2matrixwiththeareaofaparallelogramwillhelpstudentssee

whathappenswhenthedeterminantofamatrixiszero.Thisoccurswhentheparallelogram

degeneratesintotwooverlappinglinesegments.Inthisdrawingoftheparallelogram,thecolumns

ofthematrixaretreatedasdefiningtwovectorsthatformtwosidesofaparallelogram—thesame

parallelogramformedinthetaskTheArithmeticofVectorstoaddvectorsusingtheparallelogram

rule.Toshowthattheareaofthisparallelogramisgivenbyad–bc,studentsmaychoosetofind

theareaofthesurroundingrectangle,andthensubtracttheareaoftheextradart-shapedpieces,

leavingtheareaoftheparallelogram.

Theareaofthesurroundingrectangle

isgivenby(a+b)(c+d).Since

studentsmaynotknowhowto

multiplythesetwobinomialstogether,

theymayneedtodecomposethe

largerrectangleintosmaller

rectangles,asshowninthefollowing

diagram.

Theareaoftheextrapiecesthatneed

tobesubtractedfromthesurrounding

rectanglecanbefoundby

decomposingtheextradart-shaped

piecesintorectanglesandtriangles,as

inthefollowingdiagram.




Inquestion8studentsdevelopanalternativemethodforfindingtheinverseofa2´2matrix,ifthe

inverseexists.StudentswillfirstsetuptwosystemsofequationstosolveforM1,M2,M3andM4.

Eachofthesevalueswillconsistofafractionwhosedenominatorcanberepresentedbyad–bc,the

valueofthedeterminant.Consequently,thevalue canbefactoredoutofthematrixasa

scalarmultiple,leadingtotherulegiveninthetaskfortheinversematrix.


Haveselectedstudentspresenttheirworkshowingthattheareaoftheparallelogramisgivenby

thevalueofthedeterminantad–bc.Inthecaseswherethedeterminantisnegative,theareaofthe

parallelogramis|ad–bc|.Thiswillcomeoutasstudentssharetheirworkonquestion7,andthe

issuewitha0determinantwillcomeoutoftheworkonquestion6.Studentsmayneedassistance

withthealgebrainquestion8.Ifso,workthroughthisderivationasawholeclass,ratherthanwith

eachindividualgroup.Thederivationstartsbyformingthefollowingtwosystems:

and

SolvingthesesystemsforM1,M2,M3andM4leadsto:

M1=

M2=

M3=

M4=

€

1ad − bc

€

aM1 + bM3 = 1cM1 + bM3 = 0" # $

€

aM2 + bM4 = 0cM2 + dM4 = 1" # $

€

dad − bc

€

−bad − bc

€

−cad − bc

€

aad − bc




Therefore,theinversematrixis or,afterfactoringoutthescalarfactor

,theinversematrixisgivenby .


€

dad − bc

−bad − bc

−cad − bc

aad − bc

#

$

% % %

&

'

( ( (

€

1ad − bc

€

1ad − bc

d −b−c a#

$ %

&

' (



10.4H



READY Topic:Solvingsystemsofequationsusingrowreduction

Giventhesystemofequations

1.Zacstartedsolvingthisproblembywriting !5 −3 32 1 10( → !1 −5 −17

2 1 10 (

DescribewhatZacdidtogetfromthematrixonthelefttothematrixontheright.

2.Leastartedsolvingthisproblembywriting!5 −3 32 1 10( → +

5 −3 31 ,

- 5.

DescribewhatLeadidtogetfromthematrixonthelefttothematrixontheright.

3.UsingeitherZac’sorLea’sfirststep,continuesolvingthesystemusingrowreduction.Showeach

matrixalongwithnotationindicatinghowyougotfromonematrixtoanother.Besuretocheck

yoursolution.

€

5x − 3y = 32x + y = 10# $ %


22



10.4H



SET Topic:Findingthedeterminantofa2X2matrix

4.Usethedeterminantofeach2´2matrixtodecidewhichmatriceshavemultiplicativeinverses,

andwhichdonot.

a. b. c.

5.Findthemultiplicativeinverseofeachofthematricesin4,providedtheinversematrixexists.

a. b. c.

6.Generallymatrixmultiplicationisnotcommutative.Thatis,ifAandBarematrices,typically

/ ∙ 1 ≠ 1 ∙ /.However,multiplicationofinversematricesiscommutative.Testthisoutbyshowingthatthepairsofinversematricesyoufoundinquestion7givethesameresultwhen

multipliedineitherorder.

€

8 −24 1#

$ %

&

' (

€

3 26 4"

# $

%

& '

€

4 23 1"

# $

%

& '

23



10.4H



GO Topic:Parallelandperpendicularlines

Determineifthefollowingpairsoflinesareparallel,perpendicularorneither.Explainhow

youarrivedatyouranswer.

7. 3x+2y=7and6x+4y=9

8. and

9. and4x+3y=3

10.Writetheequationofalinethatisparallelto andhasay-interceptat(0,4).

€

y =23x − 5

€

y = −23x + 7

€

y =34x − 2

€

y =45x − 2

24




10.5H Solving Systems with

Matrices, Revisited


Whenyousolvelinearequations,youusemanyofthepropertiesofoperationsthatwere

revisitedinthetaskMoreArithmeticofMatrices.

1. Solvethefollowingequationforxandlistthepropertiesofoperationsthatyouuseduringtheequationsolvingprocess.

Thelistofpropertiesyouusedtosolvethisequationprobablyincludedtheuseofa

multiplicativeinverseandthemultiplicativeidentityproperty.Ifyoudidn’tspecificallylistthose

properties,gobackandidentifywheretheymightshowupintheequationsolvingprocessforthis

particularequation.

Systemsoflinearequationscanberepresentedwithmatrixequationsthatcanbesolved

usingthesamepropertiesthatareusedtosolvetheaboveequation.First,weneedtorecognize

howamatrixequationcanrepresentasystemoflinearequations.

2. Writethelinearsystemofequationsthatisrepresentedbythefollowingmatrixequation.(Thinkabouttheprocedureformultiplyingmatricesyoudevelopedinprevioustasks.)

€

23x = 8

€

3 52 4"

# $

%

& ' ⋅

xy"

# $ %

& ' =

−14"

# $

%

& '

25




3. Usingtherelationshipsyounoticedinquestion3,writethematrixequationthatrepresentsthefollowingsystemofequations.

4. Therationalnumbers and aremultiplicativeinverses.Whatisthemultiplicative

inverseofthematrix ?Note:Theinversematrixisusuallydenotedby .

5. Thefollowingtableliststhestepsyoumayhaveusedtosolve andasksyoutoapply

thosesamestepstothematrixequationyouwroteinquestion4.Completethetableusing

thesesamesteps.

Originalequation

Multiplybothsidesoftheequationbythemultiplicativeinverse

Theproductofmultiplicativeinversesisthemultiplicativeidentityontheleftsideoftheequation

Performtheindicatedmultiplicationontherightsideoftheequation

€

2x + 3y = 143x + 4y = 20" # $

€

23

€

32

€

2 33 4"

# $

%

& '

€

2 33 4"

# $

%

& '

−1

€

23x = 8

€

23x = 8

€

2 33 4"

# $

%

& ' ⋅

xy"

# $ %

& ' =

1420"

# $

%

& '

€

32⋅23x =

32⋅8

€

1⋅ x =32⋅8

€

1⋅ x = 12

26




Applythepropertyofthemultiplicativeidentityontheleftsideoftheequation

6. Whatdoesthelastlineinthetableinquestion5tellyouaboutthesystemofequationsinquestion3?

7. Usetheprocessyouhavejustexaminedtosolvethefollowingsystemoflinearequations.

€

x = 12

€

3x + 5y = −12x + 4y = 4# $ %

27




10.5H Solving Systems with Matrices, Revisited – Teacher Notes


Purpose:Studentshavepreviouslysolvedsystemsusingmatricesandrowreduction—aprocess

associatedwiththeeliminationmethodforsolvingsystems.Thepurposeofthistaskistogive

studentsanalternativemethodforsolvingsystemsusingtheinverseofthecoefficientmatrix.This

methodcanbegeneralizedtosolvinglargersystemsoflinearequationsusingtechnologyto

producetheinversematrix.Youmaywishtoaugmentthistaskwithafewadditionalproblemsin

whichstudentssolvealargersystemoflinearequationsusinginversematricesproducedby

technology.Thiswillhelpthemgainanappreciationforthestrategybeingdevelopedinthistask.

CoreStandardsFocus:

MVPHonorsStandard:Solvesystemsoflinearequationsusingmatrices.

A.REI.1Explaineachstepinsolvingasimpleequationasfollowingfromtheequalityofnumbers

assertedatthepreviousstep,startingfromtheassumptionthattheoriginalequationhasasolution.

Constructaviableargumenttojustifyasolutionmethod.

RelatedStandards:A.REI.6



TheTeachingCycle:

Launch(WholeClass):

Givestudentsafewminutesindividuallytoworkonquestions1and2.Whilestudentsmaysolve

theequationinquestion1bymultiplyingbothsidesoftheequationby3andthendividingboth

sidesoftheequationby2,question2encouragesthemtomultiplybothsidesoftheequationbythe




multiplicativeinverseof toget1xontheleftsideoftheequationandthentosimplify1xtox

usingthemultiplicativeidentityproperty.Lookforastudentwhohasapproachedtheproblemin

thiswaytopresent.

Workwithstudentsonusingamatrixequationtorepresentasystemofequations,assuggestedin

questions3and4.Contrastthismatrixrepresentationwiththeaugmentedmatrixrepresentation

ofasystemusedintheSystemsmodule.Youmaywanttoreviewsolvingasystembyrowreduction

andthenpointoutthatinthistaskstudentswilldevelopanalternativemethodforsolvingsystems

basedonthearithmeticofmatrices.Setstudentstoworkontheremainderofthetask.


Sincematrixmultiplicationisnotcommutative,whenstudentsmultiplybothsidesofthematrix

equationbytheinversematrix,theywillhavetoplacetheinversematrixtotheleftofthematrices

givenoneachsideoftheequation.Watchforstudentswhoarehavingdifficultymultiplying

matricesontherightsideoftheequationbecauseofthisissue.Askstudentswhythisisanissue

withthematrixequation,butnotwiththeequationthatcontainstherationalnumbercoefficient

andthemultiplicationby onbothsidesoftheequation.Likewise,rewriting1xasxisasubtle

andover-lookedideawhensolvingtheequationcontainingrationalnumbers,butitbecomesa

moreapparentissuewhenworkingwiththematrixequationandrecognizingthat can

bere-writtensimplyas becauseofthemultiplicativeidentityproperty.


Haveastudentpresenttheirworkonquestion8,showinghowtheysolvedthesystemusinga

matrixequationandaninversematrix.Whilewehavenotdevelopedaprocessforfindingthe

inverseofa3´3matrix,itmayhelpstudentsappreciatethevalueofsolvingsystemsusinginverse

matricestoconsidersolvingalargersystemusingthisstrategyversususingrowreduction,or

substitutionorelimination.Hereisa3´3systemtoconsider.

€

23

€

32

€

1 00 1"

# $

%

& ' ⋅

xy"

# $ %

& '

€

xy"

# $ %

& '




Theinverseofthecoefficientmatrixcanbeobtainedusingtechnology.Havestudentsconfirmthat

thematrixproducedbytechnologyisindeedtheinversematrixbyhavingthemcarryoutthe

multiplicationofthematrixanditsinversebyhand.Oncethesystemhasbeensolvedusingthe

inversematrix,havestudentssolvethesystemusinganothermethod,suchasrowreductionofthe

augmentedmatrix,orsubstitutionorelimination.


€

4x − 2y + z = 32x + y − z = 13x − y + 2z = 7

#

$ %

& %



10.5H



READY Topic:ReflectionsandRotations1.Thefollowingthreepointsformtheverticesofatriangle:(3,2),(6,1),(4,3)

a.Plotthesethreepointsonthecoordinategridandthenconnectthemtoformatriangle.

b.Reflecttheoriginaltriangleoverthey-axisandrecordthecoordinatesoftheverticeshere:

c.Reflecttheoriginaltriangleoverthex-axisandrecordthecoordinatesoftheverticeshere:

d.Rotatetheoriginaltriangle90°counter-clockwiseabouttheoriginandrecordthecoordinatesoftheverticeshere:

e.Rotatetheoriginaltriangle180°abouttheoriginandrecordthecoordinatesoftheverticeshere. SET Topic:SolvingSystemsUsingInverseMatricesTwoofthefollowingsystemshaveuniquesolutions(thatis,thelinesintersectatasinglepoint).2.Usethedeterminantofa2´2matrixtodecidewhichsystemshaveuniquesolutions,andwhichonedoesnot.

a. b. c.

€

8x − 2y = −24x + y = 5

# $ %

€

3x + 2y = 76x + 4y = −5# $ %

€

4x + 2y = 03x + y = 2

" # $


28



10.5H



3.Foreachofthesystemsin#2whichhaveauniquesolution,findthesolutiontothesystembysolvingamatrixequationusinganinversematrix.a. b. c.

GO Topic:ReviewingthePropertiesofArithmeticMatcheachexampleontheleftwiththenameofapropertyofarithmeticontheright.Notallanswerswillbeused.

_____4.2(x+3y)=2x+6y

_____5.(2x+3y)+4y=2x+(3y+4y)

_____6.2x+3y=3y+2x

_____7.2(3y)=(2×3)y=6y

_____8.

_____9.x+-x=0

_____10.xy=yx

a.multiplicativeinverses

b.additiveinverses

c.multiplicativeidentity

d.additiveidentity

e.commutativepropertyofaddition

f.commutativepropertyofmultiplication

g.associativepropertyofaddition

h.associativepropertyofmultiplication

i.distributivepropertyofadditionovermultiplication

€

23⋅32x = 1x

29




10.6H All Systems Go!

A Practice Understanding Task

Thismodulebeganwiththefollowingproblem:

CarlosandClaritaaretryingtofigureoutthecostofagallonofpaint,thecostofa

paintbrush,andthecostofarollofmaskingtapebasedonthefollowingpurchases:

Week1:Carlosbought2gallonsofpaintand1rollofmaskingtapefor$30.

Week2:Carlosbought1gallonofpaintand4brushesfor$20.

Week3:Carlosbought2brushesand1rollofmaskingtapefor$10.

IntheprevioussequenceoftasksMoreArithmeticofMatrices,SolvingSystemswithMatrices,

RevisitedandTheDeterminantofaMatrixyoulearnedhowtosolvesystemsusingmultiplicationof

matrices.Inthistask,wearegoingtoextendthisstrategytoincludesystemswithmorethattwo

equationsandtwovariables.

1. Multiplythefollowpairsofmatrices:

a.

€

1 0 00 1 00 0 1

"

#

$ $ $

%

&

' ' '

⋅

2 0 11 4 00 2 1

"

#

$ $ $

%

&

' ' '

CCBYNASA

https://flic.kr/p/9y3p6p

30




b.

2. Whatpropertyisillustratedbythemultiplicationinquestion1a?3. Whatpropertyisillustratedbythemultiplicationinquestion1b?4. Rewritethefollowingsystemofequations,whichrepresentsCarlosandClarita’sproblem,

asamatrixequationintheformAX=BwhereA,XandBareallmatrices.

5. Solveyourmatrixequationbyusingmultiplicationofmatrices.Showthedetailsofyourworksothatsomeoneelsecanfollowit.

€

0.4 0.2 −0.4−0.1 0.2 0.10.2 −0.4 0.8

#

$

% % %

&

'

( ( (

⋅

2 0 11 4 00 2 1

#

$

% % %

&

'

( ( (

€

2g + 0b +1t = 301g + 4b + 0t = 200g + 2b +1t = 10

31




Youwereabletosolvethisequationusingmatrixmultiplicationbecauseyouweregiventhe

inverseofmatrixA.Unlike2´2matrices,wheretheinversematrixcaneasilybefoundbyhand

usingthemethodsdescribedinMoreArithmeticofMatrices,theinversesofann´nmatrixin

generalcanbedifficulttofindbyhand.Insuchcases,wewillusetechnologytofindtheinverse

matrixsothatthismethodcanbeappliedtoalllinearsystemsinvolvingnequationsandn

unknownquantities.Hereisoneonlineresourceyoumightuse:https://matrixcalc.org/en/

6. Solvethefollowingsystemusingamatrixequationandinversematrices.Althoughyoumay

usetechnologytofindtheinversematrix,makesureyourecordallofyourworkinthespacebelow,includingyourinversematrix.

! + 2$ + 2% = 162! + $ + 3% = 17! + 2$ + % = 13

32




10.6H All Systems Go! – Teacher Notes A Practice Understanding Task

Purpose:Thepurposeofthistaskistoextendtheprocessofsolvinglinearsystemsusingmatrix

equationsandinversematricestolinearsystemsthatincludemorethantwoequationsandmore

thantwounknowns.Thistaskextendstheideaofrepresentingasystemwithamatrixequation

thatincludesavectorvariable,andthensolvingthematrixequationforthevectorvariableby

multiplyingbyaninversematrixtohigher-orderedn´nmatrices.Theinversesforn´nmatrices

arefoundusingtechnology.

CoreStandardsFocus:

A.REI.8Representasystemoflinearequationsasasinglematrixequationinavectorvariable.

A.REI.9Findtheinverseofamatrixifitexistsanduseittosolvesystemsoflinearequations(using

technologyformatricesofdimension3×3orgreater).



TheTeachingCycle:

Launch(WholeClass):

Posetheessentialquestionlistedabovetostudents,andhavethemconsiderthedifferentways

theyhavelearnedtosolvesystemsofequations,andwheneachmethodmightbeuseful:graphing,

substitution,elimination,rowreductionofmatrices,usinginversematrices.Studentsmightnote

thelimitationsofgraphingtosituationsthatinvolve2equationsand2unknown,andthat

substitution,eliminationandreductionofmatricesgetsmorecomplicatedortediousasthenumber

ofequationsandunknownsincreases.Atthispoint,studentswillprobablystatethatusinginverse

matricesseemsproblematic,sincetheyonlyknowhowtofindtheinverseofa2´2matrix.Once

thisissuehassurfaced,letstudentsstartonthetask.





Basedontheprevioussequencesoftasks,studentsshouldbeabletoworkthroughproblems1-5

independently,althoughtheymayneedtohaveapartner(ortheteacher)pointoutthatoneofthe

matricesinquestion1bisthematrixthatrepresentsCarlosandClarita’ssystemofequations,and

thattheyhavedemonstratedthattheothermatrixisitsinverse.

Youwillneedtodecidewhattechnologicalresourcestudentsmightusetofindtheinverseofa

matrix,agraphingcalculatorortheonlinematrixcalculatorreferencedinthetask.Provideample

supportforstudentsastheyarelearninghowtousethetechnology.

Listenforstudentswhoaresuccessfullyapplyingthestrategyofusingamatrixequationmultiplied

byaninversematrixtopresentduringthewholeclassdiscussion.


Movetothewholeclassdiscussionwhenyouhavestudentswhocanpresenthowtheyused

technologytofindtheinversematrix,andhowtheyappliedthistosolvethegivensystemof

equations.Ifnecessary,showstudentshowtofindinversematricesusingavailabletechnology,

thenletstudentsworkthroughtheproblemofsolvingthematrixequationusingtechnology.




10.6H



READY Topic:ReviewingrationalexponentsandmethodsforsolvingquadraticsWriteeachexponentialexpressioninradicalform.1. 2. 3.

10#$ %

&' 3)

&#

4. 5. 6.

6$+ 7

'# -

.'

Writeeachradicalexpressioninexponentialform.

7. 8. 9.

√30 1√723 4' 5%#10. 11. 12.

5)'6 1√)7 48 √)9:

Explaineachstrategyforsolvingquadraticequationsandexplainthecircumstancesinwhichthestrategyismostefficient.13.Graphing 14.Factoring 15.Completingthesquare 16.Whatotherstrategiesdoyouknowforsolvingquadraticequations?Whenwouldyouusethem?


33



10.6H



SET Topic:Solvingsystemswiththreeunknowns.Solvethesystemofequationsusingmatrices.Createamatrixequationforthesystemofequationsthatcanbeusedtofindthesolution.Thenfindtheinversematrixanduseittosolvethesystem.

17.;2% − 4? + A = 05% − 4? − 5A = 124% + 4? + A = 24

18.;% + 2? + 5A = −15% + ? − 4A = 12% − 6? + 4A = −12

19.;4D + E − 2F = 5

−3D − 3E − 4F = −164D − 4E + 4F = −4

20.;−6% − 4? + A = −20−3% − ? − 3A = −8−5% + 3? + 6A = −4

GO Topic:SolvingquadraticequationsSolveeachoftheequationsbelowusinganappropriateandefficientmethod.21. 22. 23.

%$ − 5% = −6 3%$ − 5 = 0 5%$ − 10 = 0

24. 25. 26.%$ + 1% − 30 = 0 %$ + 2% = 48 %$ − 3% = 0

34

Matrices Revisited · matrices that answer the question of weekly revenue collected. Have students...

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