Math 1CP to 2H Matrices and Vectors SE ... Bridge ( 1 to 2...2 SDUHSD Math 1CP to Math 2H Summer...

1

©2012MathematicsVisionProject|MVPInpartnership withtheUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense

Math1CPtoMath2HonorsSummerBridge

MatricesandVectorsMath1HModules1H,2H,7H

Adaptedfrom:

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org

InpartnershipwiththeUtahStateOfficeofEducation

2

SDUHSDMath1CPtoMath2HSummerBridge

Module1HonorsOverview

Belowarethestandards,concepts,andvocabularyfromtheCOMPLETEMODULEinIntegratedMath1Honors

PrerequisiteConcepts&Skills:

Operationswithintegers,fractions,decimalsandvariableexpressions DistributiveProperty Solvingbasiconeandtwostepequations Solvinglinearinequalities Understandingofthenumberlineandthecoordinateplane Plottingpoints Evaluatingexpressions Orderofoperations

SummaryoftheConcepts&SkillsinModule1Honors:

Teambuildingskillsandgrouproles Communicationskills(orally&written) IntroductiontoCCStandardsofMathPracticesthroughdailytasks Writeexpressionstorepresentacontextand/orgivenavisual Solvelinearequations Solvelinearinequalities&graphingsolutionsonanumberline Solveliteralequations Solveabsolutevalueequationsandinequalities Intervalnotation Writelinearequationandinequalitiestorepresentacontext Usematricestorepresentandmanipulatedata Operationswithmatrices:multiplybyascalar,add,subtract,andmultiply Useagraphingcalculatorforcompletingoperationswithmatrices.

ContentStandardsandStandardsofMathematicalPracticeCovered:

ContentStandards:N.Q.1,A.REI.1,A.REI.3,A.REI.3.1,N.VM.6,N.VM.7,N.VM.8 StandardsofMathematicalPractice:

1. Makesenseofproblems&persevereinsolvingthem.2. Reasonabstractly&quantitatively3. Constructviablearguments&critiquethereasoningofothers4. Modelwithmathematics5. Useappropriatetoolsstrategically6. Attendtoprecision7. Lookforandmakeuseofstructure8. Lookforandexpressregularityinrepeatedreasoning

3


Module1HVocabulary: Expression Equation Inequality Lessthan/Lessthanorequalto Greaterthan/Greaterthanorequalto Distribute Solve Simplify Context Slope y‐intercept Coordinateplane Justify Evaluate Open/closeddot Matrix/matrices Scalarmultiplication Intervalnotation

ConceptsUsedIntheNextModule:

Graphlinearequations&inequalities Solvelinearequations&inequalities Definevariablesfromacontext Writeequationsfromacontext Determineifagivenpointisasolutiontoanequation,inequality,orsystemsofequations Solvesystemsoflinearequationsbygraphing,substitution,&elimination Solvesystemsoflinearinequalitiesbygraphing Graphlinesusingtechnology(i.e.graphingcalculatorsorDesmos) Representingandsolvingsystemsofequationswithmatrices

4





Operationswithintegers,fractions,decimalsandvariableexpressions Solvemultistepequationsandinequalities Graphlinearequationsinslope‐interceptandstandardform Evaluateexpressionsusingtheorderofoperations Writelinearequationsandinequalitiestorepresentacontext Arrangedatawithinanarray/matrix Arithmeticoperationswithmatrices

SummaryoftheConcepts&SkillsinModule2Honors:

Reinforcegrouprolesandcommunicationskills(orally&written) CCStandardsofMathPracticesthroughdailytasks Writelinearequationsandinequalitiestorepresentasetofconstraints Usegraphstosolvesystemsofequationsandinequalities Usetechnology(GraphingCalculators/Desmos)tographlinearfunctionsanddeterminethemost

appropriatewindowtouse. Solvesystemsofequationsalgebraically Identifytypesofsolutionsofasystemoflinearequationsincludingonesolution,nosolution,orinfinitely

manysolutions Interpretsolutionsofsystemsinthecontextofasituation. Determineifagivenpointisasolutiontoanequation,inequality,orsystemofequations Writeanobjectivefunctiontodeterminetheoptimalsolutionforasituation Identifycornerpointsofafeasibleregionofthegraphofasystemofinequalitiesalgebraicallyand

graphically Understandthattheoptimalsolutionforlinearprogrammingproblemsisalwaysontheboundaryofthe

feasibleregion Performrowreductionofmatrices Interpretsolutionsfromsolvingsystemsofequationsusingmatrices


ContentStandards:A.CED.2,A.CED.3,A.CED.4,A.REI.5,A.REI.6,A.REI.8,A.REI.9,A.REI.10,A.REI.12,A.SSE.1,N.Q.1,N.Q.2,F.LE.1b,F.LE.5

StandardsofMathematicalPractice:1. Makesenseofproblems&persevereinsolvingthem.2. Reasonabstractly&quantitatively3. Constructviablearguments&critiquethereasoningofothers4. Modelwithmathematics5. Useappropriatetoolsstrategically6. Attendtoprecision7. Lookforandmakeuseofstructure8. Lookforandexpressregularityinrepeatedreasoning

5


Module2HVocabulary: SystemofEquations/Inequalities Constraint Solutionregion Feasibleregion Objectivefunction Optimalsolution Pointofintersection Boundaryofthesolutionregion/feasibleregion Inconsistentsolutionforasystemofequations Dependentsolutionforasystemofequations Atleast Morethan/Nomorethan Solid/Dottedline Rowreductionformofamatrix Augmentedmatrix

IntheNextModule:

Sequences‐arithmetic,geometric,andother Representsequencesusingdot/tilediagrams,context,tables,graphs,andequations(recursiveand

explicit) Arithmeticandgeometricmean Identifyacommondifference/ratioofarithmeticandgeometricsequences

6





ApplyPythagoreanTheorem Graphlinearandexponentialfunctions Writelinearequationsinstandard,slope‐intercept,andpoint‐slopeform Identify/solveforslopeandx‐andy‐interceptsoflinearfunctions Solvemulti‐stepequations Identifybasicgeometricshapesandcharacteristics Usefunctionnotation

SummaryoftheConcepts&SkillsinModule7H:

Usecoordinatestofinddistancesanddeterminetheperimeterofgeometricshapes Proveslopecriteriaforparallelandperpendicularlines Usecoordinatestoalgebraicallyprovegeometrictheorems Writetheequation bycomparingparallellinesandfindingk Determinethetransformationfromonefunctiontoanother Translatelinearandexponentialfunctionsusingmultiplerepresentations Definingandoperatingwithvectorsasquantitieswithmagnitudeanddirection Propertiesofmatrixadditionandmultiplication,includingidentityandinverseproperties Findingthedeterminantofamatrixandrelatingittotheareaofaparallelogram Solvingasystemoflinearequationsusingthemultiplicativeinversematrix Usingmatrixmultiplicationtoreflectandrotatevectorsandimages Solvingproblemsinvolvingquantitiesthatcanberepresentedbyvectors


ContentStandards:G.GPE.4,G.GPE.5,G.GPE.7,F.BF.3,F.BF.1,F.IF.9 StandardsofMathematicalPractice:

1. Makesenseofproblems&persevereinsolvingthem.2. Reasonabstractly&quantitatively3. Constructviablearguments&critiquethereasoningofothers4. Modelwithmathematics5. Useappropriatetoolsstrategically6. Attendtoprecision7. Lookforandmakeuseofstructure8. Lookforandexpressregularityinrepeatedreasoning

7


Module7HVocabulary: Pythagoreantheorem Reciprocal Parallel Slopetriangle Kite Proof Rhombus Parallelogram Trapezoid Diagonal Reflection Translation Distanceformula Functionnotation Perpendicular Hypotenuse Construction Quadrilateral Equilateral Square Polygon

Rotation Transformation Lineofsymmetry Triangle Pentagon Hexagon Heptagon Octagon Similar Congruent Inscribed Translationformequation Slope‐interceptform Lineofreflection Vector Magnitude Direction Directedlinesegment

Resultantvector Scalarmultiplication Componentformofavector Associativeproperty Commutativeproperty Distributiveproperty Additiveinverse Multiplicativeinverse Matrixmultiplication Multiplicativeidentity Additiveidentity Determinant Squarematrix

ConceptsUsedIntheNextModule:

Usecontexttodescribedatadistributionsandcomparestatisticalrepresentations Describedatadistributionsandcomparetwoormoredatasets Interprettwo‐wayfrequencytables Usecontexttointerpretandwriteconditionalstatementsusingrelativefrequencytables Developanunderstandingofthevalueofthecorrelationcoefficient Estimatecorrelationoflinesofbetfitandcomparetocalculatedresultsoflinearregressionandcorrelation

coefficient Uselinearmodelsofdataandinterprettheslopeandinterceptofregressionlineswithvariousunits Useresidualplotstoanalyzethestrengthofalinearmodelofdata

8


MatricesandVectors1.3HOrganizingdataintorectangulararraysormatrices(A.REI.3.1,N.VM.6,N.VM.7,N.VM.8,N.Q.1)ClassroomTask:CafeteriaConsumptionandCost–ADevelopUnderstandingTaskReady,Set,GoHomework:MatrixMadness1.3H1.4HMultiplyingmatrices(N.VM.8,N.Q.1)WarmUp:WorkingwithMatricesClassroomTask:EatingUptheLunchroomBudget–ASolidifyUnderstandingTaskReady,Set,GoHomework:MatrixMadness1.4H1.5HPracticingthearithmeticofmatrices(N.VM.6,N.VM.7,N.VM.8,N.Q.1)WarmUp:IntrotoMatrixMultiplicationClassroomTask:TheArithmeticofMatrices–APracticeUnderstandingTaskReady,Set,GoHomework:MatrixMadness1.5H2.4HAnintroductiontosolvingsystemsoflinearequationsusingmatrices(A.REI.9)WarmUp:OperationswithMatricesClassroomTask:ToMarketwithMatrices–ASolidifyUnderstandingTaskReady,Set,GoHomework:PetSitters2.4H2.5HSolvingsystemsoflinearequationsusingmatrices(A.REI.8,A.REI.9)WarmUp:BasicMatrixOperationsClassroomTask:SolvingSystemswithMatrices–APracticeUnderstandingTaskReady,Set,GoHomework:PetSitters2.5H7.4HDefiningandoperatingwithvectorsasquantitieswithmagnitudeanddirection(N.VM.1,N.VM.2,N.VM.3,

N.VM.4,N.VM.5)ClassroomTask:TheArithmeticofVectors–ASolidifyUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.4H7.5HExaminingpropertiesofmatrixadditionandmultiplication,includingidentityandinverseproperties

(N.VM.8,N.VM.9)WarmUp:PropertiesofVectorsandMatricesClassroomTask:MoreArithmeticofMatrices–ASolidifyUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.5H7.6HFindingthedeterminantofamatrixandrelatingittotheareaofaparallelogram(N.VM.10,N.VM.12)WarmUp:SystemsofEquationsClassroomTask:TheDeterminantofaMatrix–ASolidifyUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.6H7.7HSolvingasystemoflinearequationsusingthemultiplicativeinversematrix(A.REI.1,A.REI.9(+))WarmUp:PropertiesofMatricesContinuedClassroomTask:SolvingSystemswithMatrices,Revisited–ASolidifyUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.7H

9


7.8HUsingmatrixmultiplicationtoreflectandrotatevectorsandimages(N.VM.11,N.VM.12)WarmUp:TransformationsClassroomTask:TransformationswithMatrices–ASolidifyUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.8H7.9HSolvingproblemsinvolvingquantitiesthatcanberepresentedbyvectors(N.VM.3,N.VM.4a,N.VM.12)WarmUp:TransformationswithMatricesClassroomTask:PlaneGeometry–APracticeUnderstandingTaskReady,Set,GoHomework:ConnectingAlgebraandGeometry7.9H,Module7HReview

10


©2012 www.flickr.com/photos/peretzp

1.3HCafeteriaConsumptionandCostsADevelopUnderstandingTaskSometimesElvirahostsafterschooleventsinthecafeteriaasclubsandteamscelebratetheiraccomplishments.Frequentlysheorderstoomuchfoodforsuchevents—andoccasionallynotenough.Forexample,shehasnoticedthatthechessclubeatslessthanthefootballteam,butmorethanthecheerleaders.Elvirahasaskedyoutohelphersortthroughherrecordsforthepastfewyears,hopingshecanbetterplanonhowmuchfoodtoorderfortheupcomingsoccerteamanddramaclubevents.Unfortunately,ElvirakeptmostofherrecordsonPost‐ItNotes,andnoweverythingisoutoforder.Fortunately,sheusedadifferentcolorofPost‐ItNoteseachyear,soyouatleasthaveaplacetostart.1. Hereistheinformationyouhaveidentifiedfromthepastthreeyearsforthesoccerteamanddramaclub

events.ThebluePost‐ItNotesarefromthreeyearsago,theyellowfromtwoyearsago,andthepinkfromlastyear’sevents.Organizethedataforeachyearinsuchawaythatitcanbecombinedwithsimilardatafromotheryears.

BluePost‐ItNoteOrdered10packagesofchipsforthesoccerteam—Waytoomuch!

YellowPost‐ItNoteOrdered6packagesofchipsforthesoccerteam—Definitelynot

enough!

BluePost‐ItNoteOrdered3dozencookiesforthe

dramaclub—Shouldhaveorderedmore

BluePost‐ItNoteOrdered4gallonsofdrinksforthesoccerteam.Theypouredsomeontheircoach!(bigmess)

PinkPost‐ItNoteOrdered8packagesofchipsforthesoccerteam—Myneighboris

ontheteam!

YellowPost‐ItNoteOrdered5dozencookiesforthedramaclub—Ireallylikethose

kids!

PinkPost‐ItNoteOrdered10packagesofchipsforthedramaclub—Theytalkeda

lotwithfakeaccents

BluePost‐ItNoteOrdered5gallonsofdrinksforthedramaclub(theytalkalotandseemtogetthirsty!)

PinkPost‐ItNoteOrdered4dozencookiesforthedramaclub—Toomuchdrama,

toolittlecharacter!

BluePost‐ItNoteOrdered8packagesofchipsforthedramaclub—Neededmore!

PinkPost‐ItNoteOrdered8dozencookiesforthesoccerteam—slippedafewextra

tomyneighbor.

PinkPost‐ItNoteOrdered4gallonsofdrinksforthesoccerteam—Watchedthe

playerslikeahawk!

PinkPost‐ItNoteOrdered4gallonsofdrinksforthedramaclub—Seemedabout

right

YellowPost‐ItNoteOrdered4gallonsofdrinksforthesoccerteam—Warnedthemnottorepeatlastyear’sprank!

YellowPost‐ItNoteOrdered3gallonsofdrinksforthedramaclub—drinksweregonelongbeforethechips

YellowPost‐ItNoteOrdered7dozencookiesforthesoccerteam—ShouldIhave

orderedmore?

YellowPost‐ItNoteOrdered12packagesofchipsforthedramaclub—Sentextrahome

withkids

BluePost‐ItNoteOrdered6dozencookiesforthe

soccerteam—Couldhaveorderedmore

11


2. YousuggesttoElvirathatforeacheventsheshouldordertheaverageamountofeachitembasedonwhatshehasorderedoverthepastthreeyears.Howmightyourepresentthisyear’sorderinaconcise,organizedway?DescribeindetailhowyoucalculatedtheamountofeachitemtobeorderedforeacheventsoElviracanfollowyourdescriptionwhenplanningforfutureevents.

3. Elvirajustinformedyouthatthesoccerteamwonthestatechampionshipandthedramaclubtookmajor

awardsattheShakespeareanFestivalcompetition.Consequently,bothgroupshavedecidedtoalloweachmemberoftheteamorclubtoinvitetwogueststoaccompanythemtotheircelebrationevents.Elviraassumesthateachoftheguestswillconsumeaboutthesameamountoffoodastheteamorclubmemberstheyaccompany.ExplaintoElvirahowtouseyourrepresentationoftheoriginalamountoffoodtoordertodeterminethenewamountoffoodtoorder.

4. ElviracanorderfoodfromeitherMainstreetMarketorGrandpa’sGrocery,andshehasgivenyoualistofthe

pricesateachstoreforeachitemtobepurchased.Shewouldlikeyoutocreatearepresentationofthetotalcostofpurchasingtherecommendedamountoffoodforeacheventfromeachstore.ElviraknowsthatforsomeeventsitmightbebesttopurchasethefoodfromMainstreetMarketandforothereventsitmaybebettertopurchasethefoodfromGrandpa’sGrocery.Shealsorealizesthatitistootimeconsumingtopurchasesomeitemsfromonestoreandsomefromanother.SinceyouwilleventuallywanttodetermineaprocedureElviracanusewhencalculatingthecostoffutureevents,youwillneedtokeeptrackofthedetailsofyourcomputationsforthetotalcostofpurchasingfoodforthesoccerteamfromeitherstore,andforpurchasingfoodforthedramaclubfromeitherstore.

MainstreetMarket Grandpa’sGroceryCostperpackageofchips $2.50 $2.00

Costperdozencookies $3.00 $4.00Costpergallonofdrink $2.00 $1.50

12


Name: MatrixMadness 1.3HReady,Set,Go!ReadyTopic:Ratios,ProportionsandmakingpredictionsThetablebelowshowshowMarcospendshistimeonatypicalday.Usethetabletoanswerthequestionsbelow.

ActivityHoursSpentper

DayHoursSpentper

WeekHoursSpentper

MonthExercise 1 WatchTV 2 Reading .75 Math 1.5

HouseholdChores 1.75 OtherSchoolwork 3 Videogames .5

TalkwithFriends 2 Eating 1.5 Sleeping 10

1. WhatfractionofadaydoesMarcospendsleeping?2. WhatpercentofthedaydoesMarcospenddoing“OtherSchoolwork”?3. WhatamountoftimewouldyoupredictMarcowouldspendonvideogamesforanentireweek?4. CompletethetablewithpredictionsforMarco’sactivitiesforanentireweek.WhatotheractivitiesmightMarco

engageinthatarenotonthetable?Whyisitpossiblethatnoteveryactivityislisted?5. CompletethetablewithpredictionsforMarco’sactivitiesforanentire30‐daymonth.Howdoyouusethe

givendatatomakepredictionsforanentireweekormonth?Explainyourreasoning.

13


SetTopic:OrganizinginformationinmatricesElvirahasbeenrunningaprivatecateringbusinesstomakeextramoney.Sheneedssomehelporganizingtheinformationinproblems6through8belowsothatshecanbetterpredictamountstopurchaseandimproveherprofits.Assistherbyorganizingtheinformationinameaningfulwaysothatshecanaveragetheyearsanddobetterforthecomingyear.6. ThelastthreeyearsElvirahascateredfamilygatheringsandcityevents.Lastyearsheprovidedthefollowing

atfamilygatheringsshecatered:5bagsofchips,6dozencookiesand4gallonsofdrink.Lastyearatcityeventssheprovidedthefollowing:16bagsofchips,19gallonsordrinkand24dozencookies.Organizethisinformation.

7. TwoyearagoElviraprovidedthefollowingatfamilyevents:5gallonsofdrink,4bagsofchipsand5dozen

cookies.Whilesheprovidedthefollowingatcityevents:20dozencookies,18gallonsofdrinkand12bagsofchips.

8. ThreeyearsagoElviraprovidedthefollowingatcityevents:14bagsofchips,20gallonsofdrinkand19dozen

cookies.Shealsoprovidedthefollowingatfamilygatherings:6bagsofchips,7dozencookiesand9gallonsordrink.

9. IfyouprovideElvirawithanaverageamounttobeorderedforthegatheringsandeventsshecatersinthe

comingyear,howmuchofeachitemwouldsheneed?Presenttheaverageinanorganizedway.

14


Topic:SolvingabsolutevalueequationsandinequalitiesSolveeachequationorinequality.Graphyoursolutiononthenumberline.Fortheinequality,writeyouranswerinbothinequalityandintervalnotations.10. | 8| 4

11. | 5| 8

Topic:Scalarmultiplicationofmatrices

12.Findthefollowing,givenmatrices: 2 04 1

, 4 37 21 9

4 6 3

a. 4 b. 2 c.

GoTopic:Creatingexpressionsandequations13. Ifcookiescost$2.50adozen,drinkis$1.75agallonandchipsare$2abagwhatwouldbethetotalcostfora

cateredeventaccordingtoyourrecommendationabove(problem9)?Showyourcalculationshere.14.Writeanexpressionbasedontheinformationabovethatwillcalculatethetotalcostforanyamountsof

cookiesc,drinkdandchipsh.15.Writeanexpressionthatwillcalculatethecostforanyamountsofcookiesc,drinkdandchipsh,ifpricesrise

tothefollowing:$2.75foradozencookies,$2.25forabagofchipsand$2foragallonofdrink.UsingthisnewexpressioncalculatethecostsforElvirainthecomingyear.

15


1.4HWarmUpWorkingwithMatrices1. TheBalticSeacovers147,500squaremilesofareaandhasanaveragedepthof180feet.TheNorthSeacovers

164,900squaremilesofareaandhasanaveragedepthof308feet.TheRedSeahasanareaof174,900squaremilesandhasanaveragedepthof1,764feet.TheEastChinaSeahasanareaof256,600squaremilesandanaveragedepthof620feet.Createamatrixtodisplaythisinformationorganizedbyareaanddepthofeachsea?

2. Thismatrixshowsthecostofcellphoneserviceofferedbyseveraldifferentcompanies.

MonthlyCostfor200Minutes

CostofEachMinuteover200

MinutesCompany1Company2Company3Company4

$39.00$27.00$42.00$30.00

$0.05$0.08$0.04$0.06

Whatisthecostof320minuteswithCompany4?

16


1.4HEatingUptheLunchroomBudgetASolidifyUnderstandingTaskInCafeteriaConsumptionandCostsyoucreatedamatrixtorepresentthenumberoffooditemsElviraplannedtoorderthisyearforthesoccerteamanddramaclubcelebrations.Yourmatrixprobablylookedsomethinglikethis:(Note:labelshavebeenaddedtokeeptrackofthemeaningoftherowsandcolumns)

Chips Cookies Drinks

810 7

4 4

4

Youwerealsogiveninformationaboutthecostofpurchasingeachfooditemattwodifferentstores,MainstreetMarketandGrandpa’sGrocery.Thatinformationcouldalsoberepresentedinamatrixlikethis:

MainstreetMarket

Grandpa’sGrocery

2.503.002.00

2.004.001.50

Inquestion4oftheprevioustaskyouwereaskedtodeterminehowmucheacheventwouldcostifallofthefoodfortheeventwaspurchasedatMainstreetMarketorGrandpa’sGrocery.Thesetotalamountscouldberecordedinamatrixthatlookslikethis:

MainstreetMarket

Grandpa’sGrocery

1. Calculatethevaluesofa,b,c,anddinthematrixabove.2. Explain,indetail,howyouwouldusethenumbersinthefirsttwomatricesabovetoobtainthevaluesforthe

thirdmatrix.

17


3. Inadditiontothesoccerteamanddramaclub,Elviraplanstohosteventsforthechessclub,thecheerleadersandthefootballteam.Shegivesyouthefollowingmatrixtorepresentfooditemsthatneedtobeorderedforeachoftheevents.Canyouusematrixmultiplicationwiththecostmatrixgivenabovetodeterminethetotalcostofeacheventifitemsarepurchasedateachstore?Ifyes,showhow.Ifno,explainwhynot.

Chips Cookies Drinks

8103214

744312

44228

4. Inadditiontochips,cookiesanddrinks,Elviraplanstoaddrollsandcoldcutstotheevents’menu.Shegives

youthefollowingmatrixtorepresentallofthefooditemsthatneedtobeorderedforeachoftheevents.Canyouusematrixmultiplicationwiththecostmatrixgivenabovetodeterminethetotalcostofeacheventifitemsarepurchasedateachstore?Ifyes,showhow.Ifno,explainwhynot.

Chips Cookies Drinks Rolls ColdCuts

8103214

744312

44228

682212

452210

18


Giventhefollowingmatrices,performtheindicatedoperationwhenpossible.

3 15 2

7 2 16 4 3

537

0 78 35 9

4 86 2

5.

6.

7.

8.

9.

10.

19


Name: MatrixMadness 1.4HReady,Set,Go!ReadyTopic:EquivalentEquationsThepairsofequationsbelowareequivalent.Determinewhatwasdonetothefirstequationinordertoobtainthesecondequation.(Forexample,everythingmultipliedby5orMultiplicativePropertyofEquality)Ifmorethanoneoperationwasperformedpleaseindicatetheoperationsandtheordertheywereperformed.1. 5 3 3 15

2. 4 3 12 3

3. 6 4 20 5

Determinewhetherornotthepairsofequationsbelowareequivalent.Ifequivalentstatetheoperationsusedtocreatethesecondfromthefirst.Ifnotequivalentshowwhynot.4. 12 9 21 4 3 7

5. 2 5 10 10

6. 54 42 90 9 7 15

20


SetTopic:MatrixMultiplicationTheequipmentmanagerfortheschoolathleticsdepartmentisattemptingtorestocksomeoftheneededuniformandequipmentitemsfortheupcomingseasonsofbaseballandfootball.Ithasbeendeterminedbasedoncurrentlevelsofinventoryandthenumberofplayersthatwillbereturningthatmoresocks,pantsandhelmetswillbeneeded.Theequipmentmanagerhasorganizedtheinformationinthematrixbelow.

Socks Pants Helmets

1324 15

45 7

20

Theschoolhascontractedwithtwosupplystoresinthepastforequipmentneeds.Thematrixbelowshowshowmucheachstorechargesfortheneededitems.

BigSkySportingoods

PlayItForever

CostperpairofsocksCostperpairofpants

Costperhelmet

3.5035.0022.00

3.0040.0045.50

7. Calculatethevaluesofa,b,c,anddinthe“TotalCostsMatrix”below.

TotalCostMatrix BigSky

SportingoodsPlayItForever

BaseballFootball

8. Explain,indetail,howyouwouldusethenumbersinthefirsttwomatricesabovetoobtainthevaluesforthe

“TotalCostsMatrix”.

21


9. Alexandra,Megan,andBrittneywanttocalculatetheirfinalgradesinmathclass.Theyknowwhattheiraveragesarefortests,projects,homework,andquizzes.Theyalsoknowthattestsare40%ofthegrade,projectsare15%,homework25%,andquizzes20%.Usethefollowingmatricestocalculatetheirfinalgrades:

Tests Projects Homework Quizzes

AlexandraMegan

Brittney

927288

1008578

898085

807592

Weight

TestsProjects

HomeworkQuizzes

0.40.150.250.2

Giventhefollowingmatrices,performtheindicatedoperationwhenpossible.

A 4 2 02 4 8

1539 C

1 23 32 1

10. 11. 12. 13.

22


GoTopic:Representingvisualpatternsofchangewithequations,findingpatternsCreatetablesandequationsforeachattributeofthevisualpatternbelow.Ifyouareunabletocreateanequationthenstatethepatternyounotice.(Alltrianglesareequilateralandthesidelengthofthetriangleinstep1isoneunitinlength.)

Step1 Step2 Step3

14.ThewidthofthelargetrianglewithrespecttotheStepnumber.15.ThenumberofsmalltriangleswithsidelengthofoneinthelargetrianglewithrespecttotheStepnumber.16.TheperimeterofthelargetrianglewithrespecttotheStepnumber.17.Thenumberof60°anglesinthefigurewithrespecttotheStepnumber.18.ThenumberofwhitetrianglesinthelargetrianglewithrespecttotheStepnumber.

23


1.5HWarmUpIntrotoMatrixMultiplicationSolvethefollowingmatrixwordproblems:1. Ontwodays,astoresoldthefollowingamountsofpencils,erasers,andbinders.

Pencils Erasers Binders

Monday 4854 7

10 9

6

Tuesday

Ifthepriceforeachpencil,eraser,binder,respectively,is$0.20,$0.35,and$2.85,howmuchwasmadeeachday?

2. OldMacDonaldgrowspeaches,apricots,plums,andapplesonhisfruitfarm.Thechartbelowshowsthe

numberofboxeshesoldforeachtypeoffruit.

Peaches Apricots Plums ApplesBoxes [10 12 30 15]

Supposehesellspeachesfor$27abox,apricotsfor$15abox,plumsfor$34abox,andapplesfor$17abox.FindOldMacDonald’stotalincome.

Simplify.Write“undefined”forexpressionsthatareundefined.

3. 2 16 1

∙ 4 43 5

4. 2 66 4

∙ 5 36 2

1 22 0

5. 1 63 5

∙ 15

3032

6.

2362

4603

7. Writeanexampleofamatrixmultiplicationthatisundefined. 8. Intheexpression ⋅ ,if isa3 5matrixthenwhatcouldbethedimensionsof ?

24


©2012 www.flickr.com/photos/windysydney

1.5HTheArithmeticofMatricesAPracticeUnderstandingTaskPart1Elviralikesthewaymatricesorganizesinformationsoshecankeeptrackofseveralcomputationssimultaneously.Shedecidestoapplytheseideastoanother“sticky”situationsheoftenencountersinthelunchroom.Students’favoritedesertiscinnamonrolls—whichtheyoftenrefertoas“stickybuns”.However,notallstudentsliketheirrollswithacreamcheeseglaze,andsomepreferrollswithoutraisins.Consequently,Elvirahashercookspreparecinnamonrollsinthreedifferentvarieties.1. Organizethefollowinginformationintoamatrix.Ifhelpful,labeltherowsandcolumnstoshowwhatthe

numbersrepresent.

• Onedozenplaincinnamonrollsrequire2poundsofdough,andnoglazeandnoraisins.• Onedozenglazedcinnamonrollsrequire1.5poundsofdough,0.5poundsofcreamcheeseglaze,and0.25

poundsofraisins.• Onedozenplaincinnamonrollswithraisinsrequire1.75poundsofdough,0.25poundsofraisins,andno

glaze.2. Organizethefollowinginformationintoamatrix.Ifhelpful,labeltherowsandcolumnstoshowwhatthe

numbersrepresent.

• OnOctober31,thecooksmade20dozenplaincinnamonrolls,30dozenglazedcinnamonrollsand20dozenplaincinnamonrollswithraisins.

• OnNovember20,thecooksmade15dozenplaincinnamonrolls,40dozenglazedcinnamonrollsand10dozenplaincinnamonrollswithraisins.

3. UsetheinformationabovetofindthetotalamountofeachingredientthatwasusedonOctober31and

November20.Organizeyourworkandtheresultsintoamatrixequation.

25


Elvirawouldliketousematricestodeterminethebestplacetopurchaseingredientsfordayswhenshedecidestoservecinnamonrolls.Shehasobtainedthefollowinginformationfromthetwolocalmarkets.IfElvirashopsatMainstreetMarket,hercostsare$1.50perpoundfordough,$2.00perpoundforcreamcheeseglaze,and$5.00perpoundforraisins.AtGrandpa’sGrocery,hercostsare$1.75perpoundfordough,$4.00perpoundforraisins,and$2.25perpoundforcreamcheeseglaze.4. Usealltheinformationaboveandmatrixmultiplicationtofindthetotalcostofpurchasingtheingredientsat

eachstoreforOctober31andNovember20.Elviraisgettinggoodatmultiplyingmatrices,butrealizesthatsometimessheonlyneedsoneelementinthesumorproduct(forexample,thecostofbuyingingredientsatGrandpa’sGroceryonaspecificday)andsoshewouldliketobeabletocalculateasingleresultwithoutcompletingtherestofthematrixoperation.Forthefollowingmatrixoperations,calculatetheindicatedmissingelementsinthesumorproduct,withoutcalculatingtherestoftheindividualelementsinthesumorproductmatrix.

5. 5 2 3 67 1 4 2

+ 1 3 5 74 3 2 5

=__ __ __

__ __ __

6.

2 34 12 51 3

2 3 41 5 2

=__ __

__ __

__ __

7. 3 ∙ 2 41 5

4 ∙ 2 35 4

=__

__

26


Part2AddingandSubtractingMatrices

Thematrixfunctionscanbeaccessedbypressing 2nd [MATRIX](picturedattheright).

Toenterthematrix 2 58 11

intoyourcalculator,selectEDIT

byusingthearrowkeysandchoosingamatrix(labeled[A],[B],etc.).Select1:[A].

Nowthedimensions(rowscolumns)needtobeenteredalongwiththevaluesofeachcell.

Enterthematrix 1 03 7

asoutlinedabove,butstorethisinmatrix[B].

Now,find .Todothis,returntothehomescreenbypressing 2nd [QUIT]andthenentering 2nd [MATRIX]ENTER 2nd [MATRIX] 2 ENTER aspicturedtotheright.Recordyouranswerbelowinquestion1a.

1. Enterthefollowingexamplesintoyourhandheld.Recordtheresultsforeachexercise.

a. 2 58 11

1 03 7

b. 2 310 1

4 95 2

c. 49

1 62 8

d. 7 93 4

6 5 01 0 3

27


2. Whencanmatricesbeaddedorsubtracted?3. Howdoestheerrormessagehelpfigureouttheruletoaddandsubtractmatrices?4. Whatistheruletoaddandsubtractmatrices?

MultiplyingMatrices

Multiplytwomatricesinthesamemannerasaddingorsubtracting.Asyoucompletetheexamples,youshouldthinkaboutthedimensionsofthetwomatricesmultipliedtogetherandtheanswer.5. Enterthefollowingexamplesintoyourcalculator.Recordtheresultsforeachexercise.

a. 4 31 7

⋅ 1 38 5

b. 1 9 ⋅ 14

c. 27⋅ 3 5

d.2 5 74 11 86 0 1

⋅3 6 100 1 17 1 5

e. 2 08 4

⋅359

6. Whatwerethedimensionsofthematricesthatcouldbemultiplied?7. Ifthematricescouldbemultiplied,whatwerethedimensionsoftheresult?8. Ifanabmatrixismultipliedbyacdmatrix,whatmustbetrueinordertogetananswer?

28


Name: MatrixMadness 1.5HReady,Set,Go!ReadyTopic:SolvesystemsofequationsSolvethefollowingsystemsbygraphing.Checkthesolutionbyevaluatingbothequationsatthepointofintersection.1. 6and2 3

2. 3 4 and 0

Topic:GraphlinearinequalitiesGraphthefollowinginequalitiesonthecoordinateplane.Nameonepointthatisasolutiontotheinequalityandonepointthatisnotasolution.Showalgebraicallyandgraphicallythatyourpointsarecorrect.3. 3 5 20

Solution: Notasolution:

4. 7

Solution: Notasolution:

29


SetTopic:MatrixArithmeticPerformeachoftheoperationsindicatedonthematricesbelow.

5. 3 54 7

8 96 5

6.11 124 65 8

1 2015 92 2

7. 5 4 2 95 7 8

8. 6 7 83 5 2

4 7 2 11 2 5

9. Anice‐creamstallsellsbothgreenteaandmochaicecream.Asmallportionofeithercosts$0.75andalarge

portioncosts$1.25.Duringashortperiodoftime,thenumberoficecreamssoldisshowninthetablebelow. small largeGreenTea 3 4Mocha 6 3

a. WritedownacolumnmatrixN,representingthecostofeachportionoficecream.

b. Giventhat 3 46 3

,evaluate .

c. Explainwhatthenumbersgiveninyouranswerin(b.)signify.

30


GoTopic:EvaluatingExpressionsEvaluateeachexpressionbelowgiven: , ,and 10.

11.5 2 2

12.

13. 6 5 4

14.

15.5 6 6 2 12

Topic:SolvingabsolutevalueequationandinequalitiesSolveeachequationorinequality.Graphthesolutiononthenumberline.Fortheinequality,writetheanswerinbothinequalityandintervalnotations.16. | 2| 3

17. |5 | 8

31


2.4HWarmUpOperationswithMatricesUsematrixarithmetictosolvethefollowingequations:

1. 2 1 02 3

1 13 4

2. 1 57 65 4

⋅ 2 11 3

3. 2 4 13 0 0

0 11 40 0

3 2 34 2

32


©2012 www.flickr.com/photos/tommyhj/

2.4HToMarketwithMatricesASolidifyUnderstandingTaskCarloslearnedaboutmatriceswhenElvira,themanageroftheschoolcafeteria,wasaskedtosubstituteteachduringoneofthelastdaysofschoolbeforesummervacation.Nowthathehasworkedoutastrategyforsolvingsystemsofequationsbyeliminationofvariables,heiswonderingifmatricescanhelphimkeeptrackofhiswork.CarlosisreconsideringthefollowingscenariofromShoppingforCatsandDogs,whiletryingtorecordhisthinkingusingmatrices.

OneweekCarlospurchased6dogleashesand6catbrushesfor$45.00forClaritatousewhilepamperingthepets.Laterinthesummerhepurchased3additionaldogleashesand2catbrushesfor$19.00.Whatisthepriceofeachitem?

Carlosrealizesthathecanrepresentthisscenariousingthefollowingmatrix:

leashes brushes totalpurchase1purchase2

63 6

2 45.00

19.00

Healsorealizesthathecanrepresentthecostofeachitemwithamatrixthatlookslikethis:

leashes brushes totalpurchase1purchase2

10 0

1 4.00

3.50

So,nowheistryingtofindasequenceofmatricesthatcanfillinthegapsbetweenthefirstmatrixandthelast.Heknowsfromhispreviousworkwithsolvingsystemsofequationsthathecandoanyofthefollowingmanipulationswithequations—andherealizesthateachofthesemanipulationswouldgivehimanewrowofnumbersinacorrespondingmatrix.

• Replaceanequationinthesystemwithaconstantmultipleofthatequation• Replaceanequationinthesystemwiththesumordifferenceofthetwoequations• Replaceanequationwiththesumofthatequationandamultipleoftheother

33


1. HelpCarlosfindasequenceofmatricesthatstartswiththematrixthatrepresentstheoriginalpurchases,andendswiththematrixthatrepresentspurchasingoneleashorpurchasingonebrush.Foreachmatrixinyoursequence,writeoutthejustificationthatallowsyoutowritethatmatrixbasedonthethreemanipulationswecanperformontheequationsinasystem.

SolvebyElimination SolvebyRowReduction 6 6 45.00 3 2 19.00

63

62

45.0019.00

Multiplysecondequationby2: 6 6 45.00 6 4 38.00

→ 6

664

45.0038.00

Subtractthesecondequationfromthefirstequation: 2 7.00

→ 6

062

45.007.00

34


2. Findandjustifyasequenceofmatricesthatcouldbeusedtosolvethefollowingscenario.

OneweekCarlostriedoutcheaperbrandsofcatanddogfood.OnMondayhepurchased3smallbagsofcatfoodand5smallbagsofdogfoodfor$22.75.Becausehewentthroughthesmallbagsquitequickly,hehadtoreturntothestoreonThursdaytobuy2moresmallbagsofcatfoodand3moresmallbagsofdogfood,whichcosthim$14.25.Basedonthisinformation,canyoufigureoutthepriceofeachbagofthecheapercatanddogfood?

Createanaugmentedsequenceforthefollowingsystems.SolvethesystembyfindingasequenceofmatricesthatwillcreateamatrixinReducedRowForm.3. 4 8 24

2 64. 5 9

10 7 18

35


5. 2 63 2 254 12

6. 3 2 8

3 2 3 154 2 3 1

36


Name: PetSitters 2.4HReady,Set,Go!ReadyTopic:DeterminepatternsFindthenexttwovaluesinthepattern.Describehowyoudeterminedthesevalues.1. 3,6,9,12,______,______ Description:2. 3,6,12,24,______,______ Description:3. 24,20,16,12,______,______ Description:4. 24,12,6,3,______,______ Description:Topic:SolvingsystemsbysubstitutionandeliminationSolveeachsystemofequationsusinganyalgebraicmethod.

5.2 3 2

3 4 14

6.3 3

2 6 6

7.2 2 52 2 3

37


SetTopic:Rowreductionsinmatrices8. Createamatrixtomatcheachstepinthesolvingofthesystemofequationsgiven.Also,writeadescriptionof

whathappenedtotheequationandthematrixbetweensteps.

SystemofEquations Description Matrix

GivenSystem3 2 40

7 2 3 21 7

402

↓ ↓

Step13 2 403 21 6 ↓

23

406

↓ ↓

Step23 2 400 23 46 ↓

30

40

↓ ↓

Step33 2 400 2 ↓

↓ ↓

Step43 0 360 2 ↓

↓ ↓

Step50 12

0 2 ↓

Createasystemofequationsandsolvebyusingamatrix.9. Inoneweek,mathclubsold14calculatorsforatotalof$1140.Bluecalculatorscost$75eachandsilver

calculatorscost$85each.Howmanyofeachtypeofcalculatorweresold?10.Youaremakinggiftbaskets.Eachbasketwillcontainthreedifferenttypesofcandles:tapers,pillarsadjar

candles.Taperscost$1each,pillarscost$4each,andjarcandlescost$6each.Youput8candlescostingatotalof$24ineachbasket,andyouincludeasmanytapersaspillarsandjarcandlescombined.Howmanyofeachtypeofcandlewillbeinabasket?

38


GoTopic:SolvingsystemsofequationsbygraphingSolveeachsystemofequationsbygraphing.

11.2 7

3 8

12.4 73 2 8

Topic:SolvingsystemofequationswiththreevariablesSolveeachsystemofequationsusinganymethod.

13.2 4 3 373 3 3 333 3 6 48

14.6 6 406 5 6 565 2 4 35

39


2.5HWarmUpBasicMatrixOperationsSimplify.Write“undefined”forexpressionsthatareundefined.

1.451

352 2. 5 4 5 6

3.3 25 46 2

1 3 2 4. 5 34 2

⋅ 2 3 41 2 4

40


©2012 www.flickr.com/photos/dan

smath

2.5HSolvingSystemswithMatricesAPracticeUnderstandingTaskInthetask“ToMarketwithMatrices”youdevelopedastrategyforsolvingsystemsoflinearequationsusingmatrices.Anefficientandconsistentwaytocarryoutthisstrategycanbesummarizedasfollows:Torowreduceamatrix:

• Performelementaryrowoperationstoyielda"1"inthefirstrow,firstcolumn.• Createzerosinalloftheotherrowsofthefirstcolumnbyaddingthefirstrowtimesaconstanttoeach

otherrow.• Performelementaryrowoperationstoyielda"1"inthesecondrow,secondcolumn.• Createzerosinalloftheotherrowsofthesecondcolumnbyaddingthesecondrowtimesaconstantto

eachotherrow.• Performelementaryrowoperationstoyielda"1"inthethirdrow,thirdcolumn.• Createzerosinalloftheotherrowsofthethirdcolumnbyaddingthethirdrowtimesaconstanttoeach

otherrow.• Continuethisprocessuntilthefirstm×mentriesformasquarematrixwith1sinthediagonaland0s

everywhereelse.Part1–SolvingMatricesUsingReducedRowFormPracticethisstrategybycreatingasequenceofmatricesforeachofthefollowingthatbeginswiththegivenmatrixandendswiththeleftportionofthematrix(thefirstm×mentries)inReducedRowForm.Writeadescriptionofwhatyoudidtogetfromonematrixtoanotherineachstepofyoursequenceofmatrices.

1. 2 4 03 5 2

2. 4 2 21 3 11

41


3.4 2 1 32 1 1 13 1 2 7

4. Eachoftheabovematricesrepresentsasystemofequations.Foreachproblem,writethesystemofequations

representedbytheoriginalmatrix.Determinethesolutionforeachsystemusingtherow‐reducedmatrixyouobtained,andthencheckthesolutionsintheoriginalsystem.

42


5. Solvethefollowingproblembyusingamatrixtorepresentthesystemofequationsdescribedinthescenario,andthenchangingthematrixtorow‐reducedformtoobtainthesolution.

ThreeofCarlos’andClarita’sfriendsarepurchasingschoolsuppliesatthebookstore.Stanbuysanotebook,threepackagesofpencilsandtwomarkersfor$7.50.Janbuystwonotebooks,sixpackagesofpencilsandfivemarkersfor$15.50.Franbuysanotebook,twopackagesofpencilsandtwomarkersfor$6.25.Howmuchdoeseachofthesethreeitemscost?

6. Createalinearsystemthatiseitherdependent(bothequationsinthesystemrepresentthesameline)or

inconsistent(theequationsinthesystemrepresentnon‐intersectinglines).Whathappenswhenyoutrytorowreducethe2×3matrixthatrepresentsthislinearsystemofequations?

43


Part2–ReducedRowFormUsingaGraphingCalculatororOnlineMatrixCalculator7. Enterthefollowingsystemasa2 3matrix:

4 2 1410 7 25

Now,findtheReducedRowFormofthematrix.Todothis,returntothehomescreenbypressing 2nd [QUIT]andthenentering 2nd [MATRIX].MovetotheMathmenu(aspicturedtotheright)andselectB:rref(

Selectmatrix[A] ENTER .Thereducedformwillgiveyouthesolutiontoyourequation: 1, 5 .

8. Enterthefollowingexamplesintoyourhandheld.Recordtheresultsforeachexercise.

a. 3 2 25 5 10

b. 2 8 65 20 15

c. 03 2 1

3 1

d. 1.8 1.2 49 6 3

44


Name: PetSitters 2.5HReady,Set,Go!ReadyTopic:Solvingsystemsofequationsusingmatrices.1. Inanearlierassignmentyouworkedthefollowingproblem:

“Atheaterwantstotakein$2000foracertainmatinee.Children’sticketscost$5eachandadultticketscost$10each.Ifthetheaterhasamaximumof350seats,writeasystemofequationsthatcanbesolvedtodeterminethenumberofbothchildrenandadultticketsthetheatercansell.”

Setupamatrixthatgoeswiththesituationdescribedabove.

SetTopic:SolvingsystemsusingrowreducedformAssumethatthematricesbelowrepresentlinearsystemsofequations.SolvethesystembyRowReduction.

2. 3 21 2

62

3. 3 12 3

1214

4. 7 28 23

2430

5. 14 27 1

4623

45


6.1 13 55 4

127257

7. TheschooltheMartineztwinsgotoissellingticketstotheannualtalentshow.Onthefirstdayofticketsales

theschoolsold6seniorcitizenticketsand7studentticketsforatotalof$116.Theschooltookin$26ontheseconddaybyselling4seniorcitizenticketsand1studentticket.Whatisthepriceofoneseniorcitizenticketandonestudentticket?Createasystemofequationsandsolveusingmatrices.

GoTopic:SolvingsystemsofequationsSolvethefollowingsystemsofequationswithamethodofyourchoice.

8.11

2 19

9.8 163 5

10.4 9 9

3 6

11.2 4 13

3 4 2 193 2 3

46


2012 www.flickr.com/photos/an

oldent/

7.4HTheArithmeticofVectorsASolidifyUnderstandingTaskThefollowingdiagramshowsatrianglethathasbeentranslatedtoanewlocation,andthentranslatedagain.Arrows, and ,havebeenusedtoindicatethemovementofoneofthevertexpointsthrougheachtranslation.Theresultofthetwotranslationscanalsobethoughtofasasingletranslation,asshownbythethirdarrow, ,inthediagram.

Drawarrowstoshowthemovementoftheothertwoverticesthroughthesequenceoftranslations,andthendrawanarrowtorepresenttheresultantsingletranslation.Whatdoyounoticeabouteachsetofarrows?

47


Avectorisaquantitythathasbothmagnitudeanddirection.Thearrowswedrewonthediagramrepresentbothtranslationsasvectors—eachtranslationhasmagnitude(thedistancemoved)anddirection(thedirectioninwhichtheobjectismoved).Arrows,ordirectedlinesegments,areonewayofrepresentingavector.AdditionofVectors 1. Intheexampleabove,twovectors and werecombinedtoformvector .Thisiswhatismeantby

“addingvectors”.Studyeachofthefollowingmethodsforaddingvectors,thentryeachmethodtoaddvectors and giveninthediagrambelowtofind ,suchthat

2. Explainwhyeachofthesemethodsgivesthesameresult.Method1:End‐to‐EndThediagramgivenaboveillustratestheend‐to‐endstrategyofaddingtwovectorstogetaresultantvectorthatrepresentsthesumofthetwovectors.Inthiscase,theresultingvectorshowsthatasingletranslationcouldaccomplishthesamemovementasthecombinedsumofthetwoindividualtranslations,thatis .Method2:TheParallelogramRuleSincewecanrelocatethearrowrepresentingavector,drawbothvectorsstartingatacommonpoint.Oftenbothvectorsarerelocatedsotheyhavetheirtailendsattheorigin.Thesearrowsformtwosidesofaparallelogram.Drawtheothertwosides.Theresultingsumisthevectorrepresentedbythearrowdrawnfromthecommonstartingpoint(forexample,theorigin)totheoppositevertexoftheparallelogram.Question:Howcanyoudeterminewheretoputthemissingvertexpointoftheparallelogram?

48


Method3:UsingHorizontalandVerticalComponentsEachvectorconsistsofahorizontalcomponentandaverticalcomponent.Forexample,vector canbethoughtofasamovementof8unitshorizontallyand10unitsvertically.Thisisrepresentedwiththenotation⟨8, 10⟩.Vectorconsistsofamovementof7unitshorizontallyand‐5unitsvertically,representedbythenotation⟨7, 5⟩.

Question:Howcanthecomponentsoftheindividualvectorsbecombinedtodeterminethehorizontalandverticalcomponentsoftheresultingvector ?3. Examinevector giventotheright.Whilewecanrelocatethe

vector,inthediagramthetailofthevectorislocatedat 3, 2 andtheheadofthevectorislocatedat 5, 7 .Explainhowyoucandeterminethehorizontalandverticalcomponentsofavectorfromjustthecoordinatesofthepointatthetailandthepointattheheadofthevector?Thatis,howcanwefindthehorizontalandverticalcomponentsofmovementwithoutcountingacrossandupthegrid?

MagnitudeofVectorsThesymbol‖ ‖isusedtodenotethemagnitudeofthevector,inthiscasethelengthofthevector.Deviseamethodforfindingthemagnitudeofavectoranduseyourmethodtofindthefollowing.Bepreparedtodescribeyourmethodforfindingthemagnitudeofavector.4. ‖ ‖5. ‖ ‖6. ‖ ‖

49


ScalarMultiplesofVectorsWecanstretchavectorbymultiplyingthevectorbyascalefactor.Forexample,2 representsthevectorthathasthesamedirectionas ,butwhosemagnitudeistwicethatof .Drawandlabeleachofthefollowingvectorsonacoordinategraph.Givethecomponentformoftheresultantvector:7. 3 8. 2 9. 3 2 10.3 2 11.3 2

50


OtherApplicationsofVectorsWehaveillustratedtheconceptofavectorusingtranslationvectorsinwhichthemagnitudeofthevectorrepresentsthedistanceapointgetstranslated.Thereareotherquantitiesthathavemagnitudeanddirection,butthemagnitudeofthevectordoesnotalwaysrepresentlength.Forexample,acartraveling55milesperhouralongastraightstretchofhighwaycanberepresentedbyavectorsincethespeedofthecarhasmagnitude,55milesperhour,andthecaristravelinginaspecificdirection.Pushingonanobjectwith25poundsofforceisanotherexample.Avectorcanbeusedtorepresentthispushsincetheforceofthepushhasmagnitude,25poundsofforce,andthepushwouldbeinaspecificdirection.12.Aswimmerisswimmingdirectlyacrossariverwithaspeedof2ft/sec.Theriverisflowingataspeedof

10ft/sec.

a. Illustratethissituationwithavectordiagram,includingtheresultantvector.

b. Describethemeaningoftheresultantvectorthatrepresents

thesumofthetwovectorsrepresentingthemotionoftheswimmerandtheflowoftheriver.

c. Givethecomponentformoftheresultantvectorafter

1secondd. Givethecomponentformoftheresultantvectorafter5

seconds.

13.Twoteamsareparticipatinginatug‐of‐war.Oneteamexertsacombinedforceof200poundsinonedirection

whiletheotherteamexertsacombinedforceof150poundsintheotherdirection.

a. Illustratethissituationwithavectordiagram.b. Describethemeaningofthevectorthatrepresentsthesumof

thevectorsrepresentingtheeffortsofthetwoteams.c. Givethecomponentformoftheresultantvector.

51


Name: ConnectingAlgebraandGeometry 7.4HReady,Set,Go!ReadyTopic: Solvingequationsusingpropertiesofarithmetic1. HerearethestepsZacusedtosolvethefollowingequation.Stateordescribethepropertiesofarithmeticor

thepropertiesofequalityheisusingineachstep.2 5 7 4 15 9 4 4 5 4 i.

2 10 7 4 15 thedistributiveproperty 9 4 4 5 4 j.

2 10 7 4 15 a. 5 4 5 4 k.

2 7 10 4 15 b. 5 4 4 5 l.

2 7 10 4 15 c. 5 0 5 m.

2 7 10 4 15 d. 5 5 n.

9 10 4 15 e. ⋅ 5 ⋅ 5 o.

9 10 10 4 1510

f. 1 1 p.

9 0 4 5 g. 1 q.

9 4 5 h.

Solveeachofthefollowingequationsforx,carefullyrecordeachstep.Thenstateordescribethepropertiesofarithmetic(ex:thedistributiveproperty,theassociativepropertyofmultiplication,etc.)orpropertiesofequality(ex:theadditionpropertyofequality)thatjustifyeachstep.2. 2 3 5 4 2 1 3. 3 2 1

52


SetTopic:AddingvectorsTwovectorsaredescribedincomponentforminthefollowingway:

:⟨ , ⟩and :⟨ , ⟩Onthegridsbelow,createvectordiagramstoshowthefollowing.Findthemagnitudeandcomponentformoftheresultantvector.4. 5.

magnitude: magnitude:componentform: componentform:

6. 3 7. 2

magnitude: magnitude:componentform: componentform:

53


8. 3 2 9. Showhowtofind usingtheparallelogramrule

magnitude: componentform:

GoTopic:Thearithmeticofmatrices

2 31 5

, 2 53 2

,and 4 2 15 2 3

Findthefollowingsums,differences,orproducts.Ifthesum,difference,orproductisundefined,explainwhy.10. 11. 12.2 – 13. ⋅ 14. ⋅ 15. ⋅ 16. ⋅

54


7.5HWarmUpPropertiesofVectorsandMatricesPart1:InvestigatingpropertiesofVectors:

Property Useadrawingtodetermineifthepropertyholdstrueforvectors

YesorNo

AssociativePropertyofAddition

CommutativePropertyofAddition

DistributivePropertyofMultiplicationOverAddition

where isscalar

55


PartII:Investigatingpropertiesofmatrices:

Property

Usethefollowingmatricestodetermineifthepropertyholdstrueformatrices:

YesorNo

AssociativePropertyofAddition

AssociativePropertyofMultiplication

CommutativePropertyofAddition

CommutativePropertyofMultiplication

DistributivePropertyofMultiplicationOver

Addition

56


http://com

mons.wikimedia.org/w

iki/File:Matriz_A_por_B.png

7.5HMoreArithmeticofMatricesASolidifyUnderstandingTaskInadditiontothepropertiesyouexploredintheWarmUp,additionandmultiplicationofrealnumbersincludepropertiesrelatedtothenumbers0and1.Forexample,thenumber0isreferredtoastheadditiveidentitybecause

0 0 ,andthenumber1isreferredtoasthemultiplicativeidentitysince ⋅ 1 1 ⋅ .Oncetheadditiveandmultiplicativeidentitieshavebeenidentified,wecanthendefineadditiveinversesaand since

0,andmultiplicativeinversesaand since ⋅ 1.Todecideifthesepropertiesholdformatrixoperations,wewillneedtodetermineifthereisamatrixthatplaystheroleof0formatrixaddition,andifthereisamatrixthatplaystheroleof1formatrixmultiplication.TheAdditiveIdentityMatrix1. Findvaluesfora,b,canddsothatthematrixcontainingthesevariablesplaystheroleof0,ortheadditive

identitymatrix,forthefollowingmatrixaddition.Willthissamematrixworkastheadditiveidentityforall2 2matrices?

3 14 2

3 14 2

TheMultiplicativeIdentityMatrix2. Findvaluesfora,b,canddsothatthematrixcontainingthesevariablesplaystheroleof1,orthe

multiplicativeidentitymatrix,forthefollowingmatrixmultiplication.Willthissamematrixworkasthemultiplicativeidentityforall2 2matrices?

3 14 2

⋅ 3 14 2

Nowthatwehaveidentifiedtheadditiveidentityandmultiplicativeidentityfor2×2matrices,wecansearchforadditiveinversesandmultiplicativeinversesofgivenmatrices.

57


FindinganAdditiveInverseMatrix3. Findvaluesfora,b,canddsothatthematrixcontainingthesevariablesplaystheroleoftheadditiveinverseof

thefirstmatrix.Willthissameprocessworkforfindingtheadditiveinverseofall2 2matrices?

3 14 2

0 00 0

FindingaMultiplicativeInverseMatrix4. Findvaluesfora,b,canddsothatthematrixcontainingthesevariablesplaystheroleofthemultiplicative

inverseofthefirstmatrix.Willthissameprocessworkforfindingthemultiplicativeinverseofall2 2matrices?

3 14 2

⋅ 1 00 1

5. Findthemultiplicativeinverseforthefollowingmatrices,ifitexits:

a. 3 14 2

58


b. 3 86 16

c. 3 67 2

6. Findingthemultiplicativeinverseonagraphingcalculator:

Enterthefollowing2x2matrxbyselecting 2 MATRIX ,andscrollto

edit.: 1 32 5

Press 2 QUIT toreturntothehomescreen

Findtheinverseoftheyourmatrixbyfirstselectingyourmatrixin2 MATRIX andthenpressing ENTER

7. Practicefindingtheinversematrixonyourcalculatorbycheckingyouranswerstoquestion#5.

59


Name: ConnectingAlgebraandGeometry 7.5HReady,Set,Go!ReadyTopic:Solvingsystemsoflinearequations

1. Solvethesystemofequations5 3 32 10

a. Bygraphing: b. Bysubstitution:

c. Byelimination:SetTopic:Inversematrices

2. Given: Matrix 5 23 1

a. FindtheadditiveinverseofmatrixA b. FindthemultiplicativeinverseofmatrixA

60


3. Given: Matrix 4 23 2

a. FindtheadditiveinverseofmatrixB b. FindthemultiplicativeinverseofmatrixBGoTopic:Parallellines,perpendicularlines,andlengthfromacoordinategeometryperspectiveGiventhefourpoints:A , ,B , ,C , ,andD , 4. IsABCDaparallelogram?Provideconvincingevidenceforyour

answer. 5. IsABCDarectangle?Provideconvincingevidenceforyour

answer.

6. IsABCDarhombus?Provideconvincingevidenceforyouranswer. 7. IsABCDasquare?Provideconvincingevidenceforyouranswer.

61


Topic:ArithmeticofVectorsandMatricesFindthecomponentformof .Thenfindthemagnitudeof 8. 2, 4 , 1, 3 9. 3, 6 , 8, 1 Let ⟨2, 1⟩and ⟨ 3,5⟩.Finduandsketchthevectoroperationsgeometrically.10. 11. – 3

Simplifyorwrite“undefined.”

12. 4 22 3

2 61 2

∙ 55

62


7.6HWarmUpSystemsofEquations

1. Solvethesystemofequations7 192 3 19

a. Bysubstitution: b. Byelimination:

c. Byconvertingtoamatrixandusingrowreduction.

63


7.6HTheDeterminantofaMatrixASolidifyUnderstandingTaskIntheprevioustaskwelearnedhowtofindthemultiplicativeinverseofamatrix.Usethatprocesstofindthemultiplicativeinverseofthefollowingtwomatrices.

1. 5 16 2

2. 6 23 1

3. Wereyouabletofindthemultiplicativeinverseforbothmatrices?Thereisanumberassociatedwitheverysquarematrixcalledthedeterminant.Ifthedeterminantisnotequaltozero,thenthematrixhasamultiplicativeinverse.Fora2 2matrix,thedeterminantcanbefoundusingthefollowingrule:

(Note:theverticallines,ratherthanthesquarebrackets,areusedtoindicatethatwearefindingthedeterminantofthematrix)4. Usingthisrule,findthedeterminantofthetwomatricesgiveninproblems1and2above.

64


Theabsolutevalueofthedeterminantofa2 2matrixcanbevisualizedastheareaofaparallelogram,constructedasfollows.

• Drawonesideoftheparallelogramwithendpointsat 0, 0 and , .• Drawasecondsideoftheparallelogramwithendpointsat 0, 0 and , .• Locatethefourthvertexthatcompletestheparallelogram.•

Notethattheelementsinthecolumnsofthematrixareusedtodefinetheendpointsofthevectorsthatformtwosidesoftheparallelogram.5. Usethefollowingdiagramtoshowthattheareaoftheparallelogramisgivenby .

6. Drawtheparallelogramswhoseareasrepresentthedeterminantsofthetwomatriceslistedinquestions1and

2above.Howdoesazerodeterminantshowupinthesediagrams?

65


7. Createamatrixforwhichthedeterminantwillbenegative.Drawtheparallelogramassociatedwiththedeterminantofyourmatrixandfindtheareaoftheparallelogram.

Thedeterminantcanbeusedtoprovideanalternativemethodforfindingtheinverseof2 2matrix.8. Usetheprocessyouusedpreviouslytofindtheinverseofageneric2 2matrixwhoseelementsaregivenby

thevariablesa,b,candd.Fornow,wewillrefertotheelementsoftheinversematrixasM1,M2,M3andM4asillustratedinthefollowingmatrixequation.FindexpressionsforM1,M2,M3andM4intermsoftheelementsofthefirstmatrix,a,b,candd.

⋅ 1 00 1

M1= M2= M3= M4=

66


Useyourworkabovetoexplainthisstrategyforfindingtheinverseofa2 2matrix:9. Findtheinverseofthefollowing2x2matricesusingtheformula,iftheyexist:

a. 3 15 2

b. 1 23 4

c. 4 22 1

d. 1 13 4

67


Name: ConnectingAlgebraandGeometry 7.6HReady,Set,Go!ReadyTopic:Solvingsystemsoflinearequationsusingrowreduction

Giventhesystemofequations

1. Zacstartedsolvingthisproblembywriting 5 3 32 1 10

→ 1 5 172 1 10

.DescribewhatZacdidtogetfrom

thematrixonthelefttothematrixontheright.

2. Leastartedsolvingthisproblembywriting 5 3 32 1 10

→5 3 31 5 .DescribewhatLeadidtogetfrom

thematrixonthelefttothematrixontheright.3. UsingeitherZac’sorLea’sfirststep,continuesolvingthesystemusingrowreduction.Showeachmatrixalong

withnotationindicatinghowyougotfromonematrixtoanother.Besuretocheckyoursolution.

68


SetTopic:Thedeterminantofa2 2matrix4. Usethedeterminantofeach2 2matrixtodecidewhichmatriceshavemultiplicativeinverses,andwhichdo

not.

a. 8 24 1

b. 3 26 4

c. 4 23 1

5. Findthemultiplicativeinverseofeachofthematricesin4,providedtheinversematrixexists.

a.

b.

c.

6. Generallymatrixmultiplicationisnotcommutative.Thatis,ifAandBarematrices,typically ⋅ ⋅ .

However,multiplicationofinversematricesiscommutative.Testthisoutbyshowingthatthepairsofinversematricesyoufoundinquestion7givethesameresultwhenmultipliedineitherorder.

69


GoTopic:ParallelandperpendicularlinesDetermineifthefollowingpairsoflinesareparallel,perpendicularorneither.Explainhowyouarrivedatyouranswer.7. 3 2 7 and 6 4 98. 5 and 79. 2 and 4 3 310.Writetheequationofalinethatisparallelto 2andhasay‐interceptat 0, 4 .11.Writetheequationofalinethatisperpendicularto 3andpassesthroughthepoint 2, 5 .12.Writetheequationofalinethatisparallelto 3andpassesthroughthepoint 2, 5 .

70


7.7HWarmUp–PropertiesofMatricesContinued

Matrix Definition‐Whatdoesitdo?Whatdowecallit?

(notation)

Whatisit?Isitalwaysthesame?

Example

AdditiveIdentityMatrix

MultiplicativeIdentityMatrix

AdditiveInverseMatrix

MultiplicativeInverseMatrix

Determinant

71


7.7HSolvingSystemswithMatrices,RevisitedASolidifyUnderstandingTaskPart1Whenyousolvelinearequations,youusemanyofthepropertiesofoperationsthatwererevisitedinthetaskMoreArithmeticofMatrices.1. Solvethefollowingequationforxandlistthepropertiesofoperationsthatyouuseduringtheequationsolving

process.

82. Thelistofpropertiesyouusedtosolvethisequationprobablyincludedtheuseofamultiplicativeinverseand

themultiplicativeidentityproperty.Ifyoudidn’tspecificallylistthoseproperties,gobackandidentifywheretheymightshowupintheequationsolvingprocessforthisparticularequation.

Systemsoflinearequationscanberepresentedwithmatrixequationsthatcanbesolvedusingthesamepropertiesthatareusedtosolvetheaboveequation.First,weneedtorecognizehowamatrixequationcanrepresentasystemoflinearequations.3. Writethelinearsystemofequationsthatisrepresentedbythefollowingmatrixequation.(Thinkaboutthe

procedureformultiplyingmatricesyoudevelopedinprevioustasks.)

3 52 4

⋅ 14

4. Usingtherelationshipsyounoticedinquestion3,writethematrixequationthatrepresentsthefollowing

systemofequations.

2 3 143 4 20

5. Therationalnumbers and aremultiplicativeinverses.Whatisthemultiplicativeinverseofthematrix

2 33 4

?Note:Theinversematrixisusuallydenotedby 2 33 4

.

72


6. Thefollowingtableliststhestepsyoumayhaveusedtosolve 8andasksyoutoapplythosesamestepstothematrixequationyouwroteinquestion4.Completethetableusingthesesamesteps.

Originalequation 8 2 33 4

⋅ 1420

Multiplybothsidesoftheequationbythemultiplicativeinverse

⋅ ⋅ 8

Theproductofmultiplicativeinversesisthemultiplicativeidentityontheleftsideoftheequation

1 ⋅ ⋅ 8

Performtheindicatedmultiplicationontherightsideoftheequation

1 ⋅ 12

Applythepropertyofthemultiplicativeidentityontheleftsideoftheequation

12

7. Whatdoesthelastlineinthetableinquestion6tellyouaboutthesystemofequationsinquestion4?8. Usetheprocessyouhavejustexaminedtosolvethefollowingsystemoflinearequations.

3 5 12 4 4

73


Part2CarloslikestobuysuppliesforCurbsideRivalryattheAllaDollarPaintStorewherethepriceofeveryitemisamultipleof$1.Thismakesiteasytokeeptrackofthetotalcostofhispurchases.ClaritaisworriedthatitemsatAllaDollarPaintStoremightcostmore,sosheisgoingovertherecordstoseehowmuchCarlosispayingfordifferentsupplies.Unfortunately,Carloshasonceagainforgottentowritedownthecostofeachitemhepurchased.Instead,hehasonlyrecordedwhathepurchasedandthetotalcostofalloftheitems.CarlosandClaritaaretryingtofigureoutthecostofagallonofpaint,thecostofapaintbrush,andthecostofarollofmaskingtapebasedonthefollowingpurchases:

Week1: Carlosbought2gallonsofpaintand1rollofmaskingtapefor$30.Week2: Carlosbought1gallonofpaintand4brushesfor$20.Week3: Carlosbought2brushesand1rollofmaskingtapefor$10.

9. Determinethecostofeachitemusingwhateverstrategyyouwant.Showthedetailsofyourworksothat

someoneelsecanfollowyourstrategy.Youprobablyrecognizedthatthisproblemcouldberepresentedasasystemofequations.Earlierinthiscourse,youhavedevelopedseveralmethodsforsolvingsystems.10.Whichofthemethodsforsolvingsystemsofequationscouldbeappliedtothissystem?Whichmethodsseem

moreproblematic?Why?

74


IntheModule2Htasks,ToMarketwithMatricesandSolvingSystemswithMatrices,youlearnedhowtosolvesystemsofequationsinvolvingtwoequationsandtwounknownquantitiesusingrowreductionofmatrices.Youmaywanttoreviewthosetwotasksbeforecontinuing.11.Modifythe“rowreductionofmatrices”strategysoyoucanuseittosolveCarlosandClarita’ssystemof

equationsusingrowreduction.Whatmodificationsdidyouhavetomake,andwhy?InthetasksMoreArithmeticofMatrices,SolvingSystemswithMatricesRevisited,andTheDeterminantofaMatrix,youlearnedhowtosolvethesesametypesofsystemsusingthemultiplicationofmatrices.Youmaywanttoreviewthosetasksbeforecontinuing.12.Multiplythefollowingpairsofmatrices:

a.1 0 00 1 00 0 1

⋅2 0 11 4 00 2 1

Whatpropertyisillustratedbythemultiplicationinquestion4a?

b.0.4 0.2 0.40.1 0.2 0.10.2 0.4 0.8

⋅2 0 11 4 00 2 1

Whatpropertyisillustratedbythemultiplicationinquestion4b?

75


13.Rewritethefollowingsystemofequations,whichrepresentsCarlosandClarita’sproblem,asamatrixequationintheform where , and areallmatrices.

2 0 1 30 1 4 0 20 0 2 1 1014.Solveyourmatrixequationbyusingmultiplicationofmatrices.Showthedetailsofyourworksothatsomeone

elsecanfollowit.

15.Howdidyoudeterminewhichmatrixtomultiplytheequationby?Youwereabletosolvethisequationusingmatrixmultiplicationbecauseyouweregiventheinverseofmatrix .Unlike2×2matrices,wheretheinversematrixcanbeeasilyfoundbyhandusingthemethodsdescribedinMoreArithmeticofMatrices,theinversesof ingeneralcanbedifficulttofindbyhand.Insuchcases,wewillusetechnologytofindtheinversematrixsothatthismethodcanbeappliedtoalllinearsystemsinvolvingnequationsandnunknownquantities.

76


16.Solvingsystemsofequationsusingtheinverseonagraphingcalculator:

4 2 6 385 4 18

3 7 38

Entermatrix[A]andmatrix[B]:4 2 65 4 11 3 7

and381838

Press 2 QUIT toreturntothehomescreen

Multiplyintheinverseofmatrix[A]bymatrix[B] Thesolutiontothesystemofequationsistheorderedtriple:

3, 2, 5

17.Practiceusingtechnologytosolvesystemsofequationsusingtheinversewiththefollowingproblems: a. 6 2 3 17 b. 4 5 7 5 72 3 3 2 22 2x+8y+3z=‐21 2 3

77


Name: ConnectingAlgebraandGeometry 7.7HReady,Set,Go!ReadyTopic:Reflectionsandrotations1. Thefollowingthreepointsformtheverticesofatriangle: 3, 2 , 6, 1 , 4, 3 a. Plotthesethreepointsonthecoordinategridandconnect

themtoformatriangle.b. Reflecttheoriginaltriangleoverthey‐axisandrecordthe

coordinatesoftheverticeshere: c. Reflecttheoriginaltriangleoverthex‐axisandrecordthe

coordinatesoftheverticeshere: d. Rotatetheoriginaltriangle90°counter‐clockwiseabout

theoriginandrecordthecoordinatesoftheverticeshere:

e. Rotatetheoriginaltriangle180°abouttheoriginandrecordthecoordinatesoftheverticeshere:SetTopic:SolvingsystemsusinginversematricesTwoofthefollowingsystemshaveuniquesolutions(i.e.:thelinesintersectatasinglepoint).2. Usethedeterminantofa2 2matrixtodecidewhichsystemshaveuniquesolutions,andwhichonedoesnot.

a.8 2 24 5

b.3 2 76 4 5

c.4 2 03 2

3. Foreachofthesystemsinquestion#2above,findthesolutiontothesystembysolvingamatrixequationusing

aninversematrix. a. b. c.

78


Topic:Solvingsystemswiththreeunknowns.Solvethesystemofequationsusingmatrices.Createamatrixequationforthesystemofequationsthatcanbeusedtofindthesolution.Thenfindtheinversematrixanduseittosolvethesystem.

4.2 4 05 4 5 124 4 24

5.2 5 15

4 126 4 12

6.4 2 53 3 4 164 4 4 4

7.6 4 203 3 85 3 6 4

GoTopic:PropertiesofarithmeticMatcheachexampleontheleftwiththenameofapropertyofarithmeticontheright.Notallanswerswillbeused._______8. 2 3 2 6 a. multiplicativeinverses

_______9. 2 3 4 2 3 4 b. additiveinverses

_______10. 2 3 3 2 c. multiplicativeidentity

_______11. 2 3 2 ⋅ 3 6 d. additiveidentity

_______12. ⋅ 1 e. commutativepropertyofaddition

_______13. 0 f. commutativepropertyofmultiplication

_______14. g. associativepropertyofaddition

h. associativepropertyofmultiplication

i. distributivepropertyofmultiplicationoveraddition

79


7.8HWarmUpTransformations1. Reflect∆ overthex‐axis.Labeltheimage∆ ′ ′ ′.2. Rotate∆ counterclockwiseabouttheorigin90°.Labeltheimage∆ ′′ ′′ ′′.3. Translate∆ 5unitsleftand7unitsdown.Labeltheimage∆ ′′′ ′′′ ′′′.

80


2012 www.flickr.com/photos/wilfbuck

7.8HTransformationswithMatricesASolidifyUnderstandingTaskVariousnotationsareusedtodenotevectors:bold‐facedtype,v;avariablewrittenwithaharpoonoverit, ;orlistingthehorizontalandverticalcomponentsofthevector,⟨ , ⟩.Inthistaskwewillrepresentvectorsbylistingtheirhorizontalandverticalcomponentsin

amatrixwithasinglecolumn, .

1. Representthevectorlabeledvinthediagrambelowasamatrixwithonecolumn.

Matrixmultiplicationcanbeusedtotransformvectorsandimagesinaplane.Supposewewanttoreflect overthey‐axis.Wecanrepresent withthe

matrix 23,andthereflectedvectorwiththematrix 2

3.

2. Findthe2 2matrixthatwecanmultiplythematrixrepresentingtheoriginalvectorbyinordertoobtainthe

matrixthatrepresentsthereflectedvector.Thatis,finda,b,canddsuchthat ⋅ 23

23

.

81


3. Findthematrixthatwillreflect overthex‐axis.4. Findthematrixthatwillrotate 90°counterclockwiseabouttheorigin.5. Findthematrixthatwillrotate 180°counterclockwiseabouttheorigin.6. Findthematrixthatwillrotate 270°counterclockwiseabouttheorigin.

82


7. Isthereanotherwaytoobtainarotationof270°counterclockwiseabouttheoriginotherthanusingthematrixfoundinquestion6?Ifso,how?

Wecanrepresentpolygonsintheplanebylistingthecoordinatesofitsverticesascolumnsofamatrix.For

example,thetrianglebelowcanberepresentedbythematrix .

8. Multiplythismatrix,whichrepresentstheverticesofΔABC,bythematrixfoundinquestion2.Interpretthe

productmatrixasrepresentingthecoordinatesoftheverticesofanothertriangleintheplane.Plotthesepointsandsketchthetriangle.Howisthisnewtrianglerelatedtotheoriginaltriangle?

83


9. HowmightyoufindthecoordinatesofthetrianglethatisformedafterΔABCisrotated90°counterclockwiseabouttheoriginusingmatrixmultiplication?Findthecoordinatesoftherotatedtriangle.

10.HowmightyoufindthecoordinatesofthetrianglethatisformedafterΔABCisreflectedoverthex‐axisusing

matrixmultiplication?Findthecoordinatesofthereflectedtriangle.

84


Name: ConnectingAlgebraandGeometry 7.8HReady,Set,Go!ReadyTopic:AddingvectorsGivenvectors :⟨ , ⟩and :⟨ , ⟩,findthefollowingusingtheparallelogramrule:1. 2.

85


SetTopic:Matricesandtransformationsoftheplane3. Listthecoordinatesofthefourverticesoftheparallelogramyoudrewinquestion1asamatrix.Usecolumnsto

representeachcoordinate(x‐valuesacrossthetoprowwithcorrespondingy‐valuesacrossthebottomrow).

Point1

Point2

Point3

Point4

x‐valuesy‐values

4. Multiplythematrixyouwroteinquestion3bythefollowingmatrix: 0 11 0

5. Plottheoriginalparallelogramformedbytheorderedpairsfromyouranswerinquestion3.Thenplotthe

parallelogramusingthepointsfromthematrixinnumber4.

Whattransformationoccurredbetweenyouroriginalparallelogramandthenewone?

86


6. Listthecoordinatesofthefourverticesoftheparallelogramyoudrewinquestion2asamatrix.Usecolumnstorepresenteachcoordinate(x‐valuesdownthefirstcolumnwithcorrespondingy‐valuesdownthesecondcolumn).

x‐values

y‐values

Point1Point2Point3Point4

7. Multiplythematrixyouwroteinquestion6bythefollowingmatrix: 1 00 1

8. Howdidtheorientationofyourmultiplicationinquestion7differfromquestion4?Why?9. Plottheoriginalparallelogramformedbytheorderedpairsfromyouranswerinquestion3.Thenplotthe

parallelogramusingthepointsfromthematrixinnumber4.Whattransformationoccurredbetweenyouroriginalparallelogramandthenewone?

Whattransformationoccurredbetweenyouroriginalparallelogramandthenewone?

87


GoTopic:TransformationsoffunctionsFunction isdefinedbythefollowingtablebelow:

2 4 6 8 10 12 14 16

8 3 2 7 12 17 22 27

10.Writeanequationfor .11.a. Fillinthevalues,inthetableabove,for assumingthat 3

b. Writeanequationfor .12.a. Fillinthevalues,inthetableabove,for assumingthat 2

b. Writeanequationfor .Topic:Findtheinverseofthefollowingmatrices:

13. 11 52 1

14. 0 21 9

88


7.9HWarmUpTransformationswithMatricesWecanrepresentpolygonsintheplanebylistingthecoordinatesofitsverticesascolumnsofamatrix.For

example,atrianglecanberepresentedbythematrix 2 5 63 7 4

(coordinatesarethecolumns)

1. Plotthetriangleinthegridbelow.

2. 1 00 1

2 5 63 7 4

representsatranslationoftheabovetriangle.Describethetransformationinthespace

below.

89


©2012www.flickr.com

/photos/49024304@N00/3611513165/

7.9HPlaneGeometryAPracticeUnderstandingTaskJon’sfatherisapilotandheisusingvectordiagramstoexplainsomeprinciplesofflighttoJon.Hisfatherhasdrawnthefollowingdiagramtorepresentaplanethatisbeingblownoffcoursebyastrongwind.Theplaneisheadingnortheastasrepresentedby andthewindisblowingtowardsthesoutheastasrepresentedby .1. Basedonthisdiagram,whatistheplane’sspeedandwhatisthewind’sspeed?Thevectordiagram

representsthespeedoftheplaneinstillair.

2. Usethisdiagramtofindthegroundspeedoftheplane,whichwillresultfromacombinationoftheplane’s

speedandthewind’sspeed.Also,indicateonthediagramthedirectionofmotionoftheplanerelativetotheground.

90


3. Jondrewaparallelogramtodeterminethegroundspeedanddirectionoftheplane.Ifyouhavenotalreadydoneso,drawJon’sparallelogramandexplainhowitrepresentstheoriginalproblemsituationaswellastheanswerstothequestionsaskedinproblem2.

4. Writeamatrixequationthatwillreflecttheparallelogramyoudrewinproblem3overthey‐axis.Usethe

solutiontothematrixequationtodrawtheresultingparallelogram.5. Provethattheresultantfigureofthereflectionperformedinproblem4isaparallelogram.Thatis,explainhow

youknowoppositesidesoftheresultingquadrilateralareparallel.6. Findtheareaoftheparallelogramdrawninproblem3.Explainyourmethodfordeterminingthearea.

91


Name: ConnectingAlgebraandGeometry 7.9HReady,Set,Go!ReadyTopic:ScatterplotsandtrendlinesExamineeachofthescatterplotsshownbelow.Ifpossible,makeastatementaboutrelationshipsbetweenthetwoquantitiesdepictedinthescatterplot.1.

2.

3.

4. Foreachscatterplot,writetheequationofatrendlinethatyouthinkbestfitsthedata.

a. Trendline#1b. Trendline#2c. Trendline#3

92


SetTopic:ApplicationsofvectorsGiven: :⟨ , ⟩, :⟨ , ⟩, :⟨ , ⟩.Eachofthesethreevectorsrepresentsaforcepullingonanobject—suchasinathree‐waytugofwar—withforceexertedineachdirectionbeingmeasuredinpounds.5. Findthemagnitudeofeachvector.Thatis,howmanypoundsofforcearebeingexertedontheobjectbyeach

tug?Roundtothenearesthundredth.a. ‖ ‖ b. ‖ ‖ c. ‖ ‖

6. Findthemagnitudeofthesumofthethreeforcesontheobject. ‖ ‖ 7. Drawavectordiagramshowingtheresultantdirectionandmagnitudeofthemotionresultingfromthisthree‐

waytugofwar.

93


GoTopic:Solvingsystems

Given:4 4 76 8 9

8. Solvethegivensystemineachofthefollowingways.

a. Bysubstitution b. Byelimination

c. Usingmatrixrowreduction d. Usinganinversematrix

Math 1CP to 2H Matrices and Vectors SE ... Bridge ( 1 to 2...2 SDUHSD Math 1CP to Math 2H Summer...

Documents

Transcript of Math 1CP to 2H Matrices and Vectors SE ... Bridge ( 1 to 2...2 SDUHSD Math 1CP to Math 2H Summer...