math 1cp to 2h matrices and vectors se ... bridge ( 1 to 2...2 sduhsd math 1cp to math 2h summer...
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©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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
Module2HonorsOverview
Belowarethestandards,concepts,andvocabularyfromtheCOMPLETEMODULEinIntegratedMath1Honors
PrerequisiteConcepts&Skills:
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
ContentStandardsandStandardsofMathematicalPracticeCovered:
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
Module7HonorsOverview
Belowarethestandards,concepts,andvocabularyfromtheCOMPLETEMODULEinIntegratedMath1Honors
PrerequisiteConcepts&Skills:
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
ContentStandardsandStandardsofMathematicalPracticeCovered:
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
©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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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?
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
Giventhefollowingmatrices,performtheindicatedoperationwhenpossible.
3 15 2
7 2 16 4 3
537
0 78 35 9
4 86 2
5.
6.
7.
8.
9.
10.
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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”.
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
GoTopic:Representingvisualpatternsofchangewithequations,findingpatternsCreatetablesandequationsforeachattributeofthevisualpatternbelow.Ifyouareunabletocreateanequationthenstatethepatternyounotice.(Alltrianglesareequilateralandthesidelengthofthetriangleinstep1isoneunitinlength.)
Step1 Step2 Step3
14.ThewidthofthelargetrianglewithrespecttotheStepnumber.15.ThenumberofsmalltriangleswithsidelengthofoneinthelargetrianglewithrespecttotheStepnumber.16.TheperimeterofthelargetrianglewithrespecttotheStepnumber.17.Thenumberof60°anglesinthefigurewithrespecttotheStepnumber.18.ThenumberofwhitetrianglesinthelargetrianglewithrespecttotheStepnumber.
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SDUHSDMath1CPtoMath2HSummerBridge
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 ?
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SDUHSDMath1CPtoMath2HSummerBridge
©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.
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SDUHSDMath1CPtoMath2HSummerBridge
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
=__
__
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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?
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SDUHSDMath1CPtoMath2HSummerBridge
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:
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
©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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
5. 2 63 2 254 12
6. 3 2 8
3 2 3 154 2 3 1
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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?
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
©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
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SDUHSDMath1CPtoMath2HSummerBridge
3.4 2 1 32 1 1 13 1 2 7
4. Eachoftheabovematricesrepresentsasystemofequations.Foreachproblem,writethesystemofequations
representedbytheoriginalmatrix.Determinethesolutionforeachsystemusingtherow‐reducedmatrixyouobtained,andthencheckthesolutionsintheoriginalsystem.
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SDUHSDMath1CPtoMath2HSummerBridge
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?
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
2012 www.flickr.com/photos/an
oldent/
7.4HTheArithmeticofVectorsASolidifyUnderstandingTaskThefollowingdiagramshowsatrianglethathasbeentranslatedtoanewlocation,andthentranslatedagain.Arrows, and ,havebeenusedtoindicatethemovementofoneofthevertexpointsthrougheachtranslation.Theresultofthetwotranslationscanalsobethoughtofasasingletranslation,asshownbythethirdarrow, ,inthediagram.
Drawarrowstoshowthemovementoftheothertwoverticesthroughthesequenceoftranslations,andthendrawanarrowtorepresenttheresultantsingletranslation.Whatdoyounoticeabouteachsetofarrows?
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SDUHSDMath1CPtoMath2HSummerBridge
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?
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SDUHSDMath1CPtoMath2HSummerBridge
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. ‖ ‖
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SDUHSDMath1CPtoMath2HSummerBridge
ScalarMultiplesofVectorsWecanstretchavectorbymultiplyingthevectorbyascalefactor.Forexample,2 representsthevectorthathasthesamedirectionas ,butwhosemagnitudeistwicethatof .Drawandlabeleachofthefollowingvectorsonacoordinategraph.Givethecomponentformoftheresultantvector:7. 3 8. 2 9. 3 2 10.3 2 11.3 2
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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
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SDUHSDMath1CPtoMath2HSummerBridge
SetTopic:AddingvectorsTwovectorsaredescribedincomponentforminthefollowingway:
:⟨ , ⟩and :⟨ , ⟩Onthegridsbelow,createvectordiagramstoshowthefollowing.Findthemagnitudeandcomponentformoftheresultantvector.4. 5.
magnitude: magnitude:componentform: componentform:
6. 3 7. 2
magnitude: magnitude:componentform: componentform:
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SDUHSDMath1CPtoMath2HSummerBridge
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. ⋅
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SDUHSDMath1CPtoMath2HSummerBridge
7.5HWarmUpPropertiesofVectorsandMatricesPart1:InvestigatingpropertiesofVectors:
Property Useadrawingtodetermineifthepropertyholdstrueforvectors
YesorNo
AssociativePropertyofAddition
CommutativePropertyofAddition
DistributivePropertyofMultiplicationOverAddition
where isscalar
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SDUHSDMath1CPtoMath2HSummerBridge
PartII:Investigatingpropertiesofmatrices:
Property
Usethefollowingmatricestodetermineifthepropertyholdstrueformatrices:
YesorNo
AssociativePropertyofAddition
AssociativePropertyofMultiplication
CommutativePropertyofAddition
CommutativePropertyofMultiplication
DistributivePropertyofMultiplicationOver
Addition
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SDUHSDMath1CPtoMath2HSummerBridge
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.
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SDUHSDMath1CPtoMath2HSummerBridge
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
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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.
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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
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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.
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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
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7.6HWarmUpSystemsofEquations
1. Solvethesystemofequations7 192 3 19
a. Bysubstitution: b. Byelimination:
c. Byconvertingtoamatrixandusingrowreduction.
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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.
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Theabsolutevalueofthedeterminantofa2 2matrixcanbevisualizedastheareaofaparallelogram,constructedasfollows.
• Drawonesideoftheparallelogramwithendpointsat 0, 0 and , .• Drawasecondsideoftheparallelogramwithendpointsat 0, 0 and , .• Locatethefourthvertexthatcompletestheparallelogram.•
Notethattheelementsinthecolumnsofthematrixareusedtodefinetheendpointsofthevectorsthatformtwosidesoftheparallelogram.5. Usethefollowingdiagramtoshowthattheareaoftheparallelogramisgivenby .
6. Drawtheparallelogramswhoseareasrepresentthedeterminantsofthetwomatriceslistedinquestions1and
2above.Howdoesazerodeterminantshowupinthesediagrams?
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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=
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Useyourworkabovetoexplainthisstrategyforfindingtheinverseofa2 2matrix:9. Findtheinverseofthefollowing2x2matricesusingtheformula,iftheyexist:
a. 3 15 2
b. 1 23 4
c. 4 22 1
d. 1 13 4
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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.
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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.
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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 .
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7.7HWarmUp–PropertiesofMatricesContinued
Matrix Definition‐Whatdoesitdo?Whatdowecallit?
(notation)
Whatisit?Isitalwaysthesame?
Example
AdditiveIdentityMatrix
MultiplicativeIdentityMatrix
AdditiveInverseMatrix
MultiplicativeInverseMatrix
Determinant
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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
.
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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
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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?
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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?
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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.
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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
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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.
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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
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7.8HWarmUpTransformations1. Reflect∆ overthex‐axis.Labeltheimage∆ ′ ′ ′.2. Rotate∆ counterclockwiseabouttheorigin90°.Labeltheimage∆ ′′ ′′ ′′.3. Translate∆ 5unitsleftand7unitsdown.Labeltheimage∆ ′′′ ′′′ ′′′.
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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
.
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3. Findthematrixthatwillreflect overthex‐axis.4. Findthematrixthatwillrotate 90°counterclockwiseabouttheorigin.5. Findthematrixthatwillrotate 180°counterclockwiseabouttheorigin.6. Findthematrixthatwillrotate 270°counterclockwiseabouttheorigin.
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7. Isthereanotherwaytoobtainarotationof270°counterclockwiseabouttheoriginotherthanusingthematrixfoundinquestion6?Ifso,how?
Wecanrepresentpolygonsintheplanebylistingthecoordinatesofitsverticesascolumnsofamatrix.For
example,thetrianglebelowcanberepresentedbythematrix .
8. Multiplythismatrix,whichrepresentstheverticesofΔABC,bythematrixfoundinquestion2.Interpretthe
productmatrixasrepresentingthecoordinatesoftheverticesofanothertriangleintheplane.Plotthesepointsandsketchthetriangle.Howisthisnewtrianglerelatedtotheoriginaltriangle?
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9. HowmightyoufindthecoordinatesofthetrianglethatisformedafterΔABCisrotated90°counterclockwiseabouttheoriginusingmatrixmultiplication?Findthecoordinatesoftherotatedtriangle.
10.HowmightyoufindthecoordinatesofthetrianglethatisformedafterΔABCisreflectedoverthex‐axisusing
matrixmultiplication?Findthecoordinatesofthereflectedtriangle.
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Name: ConnectingAlgebraandGeometry 7.8HReady,Set,Go!ReadyTopic:AddingvectorsGivenvectors :⟨ , ⟩and :⟨ , ⟩,findthefollowingusingtheparallelogramrule:1. 2.
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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?
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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?
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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
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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.
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©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.
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3. Jondrewaparallelogramtodeterminethegroundspeedanddirectionoftheplane.Ifyouhavenotalreadydoneso,drawJon’sparallelogramandexplainhowitrepresentstheoriginalproblemsituationaswellastheanswerstothequestionsaskedinproblem2.
4. Writeamatrixequationthatwillreflecttheparallelogramyoudrewinproblem3overthey‐axis.Usethe
solutiontothematrixequationtodrawtheresultingparallelogram.5. Provethattheresultantfigureofthereflectionperformedinproblem4isaparallelogram.Thatis,explainhow
youknowoppositesidesoftheresultingquadrilateralareparallel.6. Findtheareaoftheparallelogramdrawninproblem3.Explainyourmethodfordeterminingthearea.
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Name: ConnectingAlgebraandGeometry 7.9HReady,Set,Go!ReadyTopic:ScatterplotsandtrendlinesExamineeachofthescatterplotsshownbelow.Ifpossible,makeastatementaboutrelationshipsbetweenthetwoquantitiesdepictedinthescatterplot.1.
2.
3.
4. Foreachscatterplot,writetheequationofatrendlinethatyouthinkbestfitsthedata.
a. Trendline#1b. Trendline#2c. Trendline#3
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SetTopic:ApplicationsofvectorsGiven: :⟨ , ⟩, :⟨ , ⟩, :⟨ , ⟩.Eachofthesethreevectorsrepresentsaforcepullingonanobject—suchasinathree‐waytugofwar—withforceexertedineachdirectionbeingmeasuredinpounds.5. Findthemagnitudeofeachvector.Thatis,howmanypoundsofforcearebeingexertedontheobjectbyeach
tug?Roundtothenearesthundredth.a. ‖ ‖ b. ‖ ‖ c. ‖ ‖
6. Findthemagnitudeofthesumofthethreeforcesontheobject. ‖ ‖ 7. Drawavectordiagramshowingtheresultantdirectionandmagnitudeofthemotionresultingfromthisthree‐
waytugofwar.
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GoTopic:Solvingsystems
Given:4 4 76 8 9
8. Solvethegivensystemineachofthefollowingways.
a. Bysubstitution b. Byelimination
c. Usingmatrixrowreduction d. Usinganinversematrix
top related