Foundations for Algebra

Winter Student Enrichment Packet

Detroit Public Schools Community District Department of Curriculum and Instruction

Office of Mathematics

NOTE TO THE STUDENT

This Winter Student Enrichment Packet has been compiled to complement middle school mathematics classroom instruction aligned to the Michigan Standards for Mathematics. The packet is intended to be used for review and practice of previously taught and new concepts. We strongly encourage you to work diligently to complete the activities for the choice board. You may experience some difficulty with some activities in this packet, but we encourage you to think critically and creatively and complete them to the best of your ability.  

   

FoundationsforAlgebraWinterEnrichmentChoiceBoard:Three‐CourseMeal

Completethespecifiednumberofappetizers,maindishes,anddessertsasindicatedbelow.

Appetizers(Select2)

Identifyifthegivenfunctionislinearornonlinear.Classifythesolutionstolinearequationsashavingonesolution,nosolution,orinfinitelymanysolutions.Solveeachinequalityandgraphitssolution.

MainDishes(CompleteAll)

Writeequationsinslope‐interceptformgivengraphsofthelinesonthecoordinategrid.Equationsandfunctionsmixedpractice.Writeasystemofequationstomatchareal‐worldsituation.

Desserts(Select1)

PuzzleTime:Solvethesolutionofthesystemsofequationsusingmultiplicationwiththeadditionmethod(equationsarenotnecessarilyinstandardform).

CrosswordPuzzle:Solvethepuzzleinrelationtolinearfunctions.

1

2

3

1

2

1

2

3

Appetizer1:IdentifyingLinearorNonlinearFunctionsDirections:Identifyifthefunctionislinearornonlinear.Writetheletterofyouranswerintheappropriateboxbelow.

N.Thegraphshowsafunction. R.Thegraphshowsafunction.IsthefunctionIsthefunctionlinearornonlinear?linearornonlinear?O.Isthefunctiony=3x+2 Y.Isthefunctiony=–8linearornonlinear?linearornonlinear?

B.Isthefunction 6 E.Thetableshowsafunction.

linearornonlinear?Isthefunctionlinearornonlinear? R.Thetableshowsafunction.Isthefunctionlinearornonlinear?

T.Thegraphshowsafunction.Isthe

functionlinearornonlinear?

D.Isthefunction √

linearornonlinear?

Linear

Nonlinear

Appetizer2:ClassifyingSolutionstoLinearEquationsDirections: 1.Putacheck()markontheblankbeforetheitemnumberiftheequationhasonesolution.2.Putanmarkontheblankbeforetheitemnumberiftheequationhasnosolution.3.Putapound(#)markontheblankbeforetheitemnumberiftheequationhasinfinitelymanysolutions.

______1.6x–3=5x+5

______2.8x–3=8x+5

______3.3x–3=–3+3x

______4.11x–2x+15=8+7+9x

______5.3(x–14)+1=–4x+5

_______6.–3x+32–7x=–2(5x+10)

______7.21(8x+26)=13+4x

______8.18x+21=6(3x+25)

______9.8–9x=15x+7+3x

_____10.5(x+9)=5x+45

_____11.6x+1=1+6x

_____12.–2x–4=–2x+4

Appetizer3:SolvingInequalitiesandGraphingSolutions

Directions:Solveeachinequalityandgraphitssolution.Showyourwork. Graphofsolution

1.18≥3x +8 1.

2.x–2x+3>6 2.3.–4+3x –8 3.4.–4x–6 –18 4.5.28 3–5x 5.

6.32 x+7 –3 6.

MainDish1:WritingEquationsinSlope‐InterceptForm Directions:Writetheequationofthelineshownineachgraphinslope‐interceptform.Notethat

eachgridlineisequivalentto1unit.Writeeachanswerintheblank.1.______________________________2.__________________________3._____________________________

4.______________________________ 5.__________________________ 6.____________________________

MainDish2:EquationsandFunctionsMixedPractice

MultipleChoice:Circlethecorrectansweroranswersforeachproblem.1.WhichofthefollowingcouldNOTbethegraphofafunction?Selectallthatapply.

a. b. c. d.

2.Worriedaboutgoingoverhisstoragelimit,Kennethmonitoredthenumberofundeletedvoicemailmessagesstoredonhisphoneeachday.

Accordingtothegraph,whatwastherateofchangebetweenThursdayandFriday?

a. –3voicemailmessagesperday

b. –2voicemailmessagesperday

c. 3voicemailmessagesperday

d. 2voicemailmessagesperday

3.ThisgraphshowshowthetotalnumberofstampsMariahasinhercollectionisrelatedtotheamountofmoneyshespendsonadditionalstamps.

With$60tospendonnewstamps,howmanytotalstampscanMariahaveinhercollection?

a.60stamps b.20stampsc.80stamps d.50stamps

4.Cristellestartedmakingthistableofvaluestoshowtherelationshipbetweenthenumberofscarvesshesells,x,andtheprofit,y,indollars,sheearnsinherbusiness.

x 1 2 3 4 5y ? –110 –90 –70 –50

Whatisthemissingvalueinthistable? a.–100 b.–120 c.–130 d.–1505.Staceyenteredtheamountshechargedonhercreditcardeachmonthintoaspreadsheet.

Accordingtothetable,whatwastherateofchangebetweenAprilandMay?

a.–$8permonthb.$8permonthc.$150permonthd.–$150permonth6.Howistheequationy=‐4x+1writteninfunctionnotation? a.f(x)=–4f+1 b.f(x)=–4x+1 c.f(x)=–4y+1 d.f(x)=–4f(x)+17.Sophiawalksataspeedof2.5milesperhour.Whichequationmodelsthisunitrate? a.x=2.5y b.x=2.5+y

c.y=2.5x d.y=2.5+x8.Whichnumberlinebelowcorrectlyshowsthesolutionsetfortheinequality 4 16x ?a. b.

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c.d.

9.TriangleMNPissimilartotriangleQRS. Whichsideshavethesameslope? a. and b. and c. and d. and

10.Whatisthesolutiontothesystemofequationsshownonthisgraph?

a.(1,4) b.(–1,4) c.(4,1) d.(4,–1)

11.Inspectthesystemofequationsshownbelow.

3 12 6 2

Howmanysolutionsdoesthissystemoflinearequationshave?a.0 b.exactly2c.exactly1 d.infinitelymany

12.Staceywantstographthissystemofequations:

3 4

2 3 6

Whichchoicebelowshowshowsheshouldrewritetheseequationsinslope‐interceptform?

a.3 4

2 b.3 4

3

c.3 4

2 d.3 4

3

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MainDish3:WritingSystemsofEquationsSystemsofequationscanbeusedtorepresentmanyreal‐worldsituations.AnexampleofsystemsofequationsappliedinaUniformMotionProblemisshownbelow.Problem:Flyingwiththewind,aplanecanfly750milesin3hours.Againstthewind,theplanecanflythesamedistancein5hours.Writeasystemofequationstorepresentthesituation.

Strategy1.Chooseonevariabletorepresenttheplaneinacalmconditionandasecondvariabletorepresent

therateofthewindorcurrent. let p=therateoftheplaneincalmair w=rateofthewindsothat p+w=rateoftheplanetravelingwiththewind p‐w=rateoftheplanetravelingagainstthewind2. Usetheexpressionsfromstep1,aswellasthetimetraveledwithandagainstthewind,toget

expressionsforthedistancestraveledwithandagainstthewind.Recall:Rate Time Distance isusedtodeterminetheexpressionforthedistancetraveled.Organizethedatainatable.

Trip Rate Time = Distance

Withthewind =

Againstthewind =

3. Writeasystemofequationsthatmodelstheproblem.

Trip Rate Time = Distance

Withthewind (p+w) 3 = 750

Againstthewind (p–w) 5 = 750

So,thesystemofequationswouldbe:3 7505 750

WritingSystemsofEquationsActivity

Directions:Writeasystemofequationstorepresenteachsituationthatcouldbeusedtodeterminetheanswertothequestionintheproblem.Donotfindthesolutiontothesystem.

1. Flyingwiththewind,anairplanemadea708‐miletripin3hours.Flyingagainstthewindtomakethesametrip,theplanetook4hours.Whatisthespeedofthewindandhowfastwouldtheplanetravelinstillair?

2. Afieldgoalis3pointsandtheextrapointafteratouchdownis1point.Tobegintheseason,akickersuccessfullymadeatotalof30kicksforatotalof56points.Howmanyfieldgoalsandhowmanyextrapointsdidthekickermake?

3. Arestauranthasonepriceforadultsandanotherpriceforchildrentoeatatitsbuffet.Twofamiliesateatthebuffet.TheTraymorefamilyhastwoadultsandthreechildrenandtheirbillwas$49.50.TheWillisfamilyhasthreeadultsandonechildandtheirbillwas$44.50.Whatisthecostforanadulttoeatatthebuffetandwhatisthecostofachildtoeatatthebuffet?

4. Astudentissellingpizzasfor$6andsubsandwichesfor$4forafundraiser.Thestudentsold13moresubsandwichesthanpizzasandcollectedatotalof$262.Howmanypizzasandhowmanysubsandwichesdidthestudentsell?

5. Thelibraryishavingabooksale.Hardcoverbooksaresellingfor$5eachandpaperbackbooksaresellingfor$3each.Mrs.Smithspent$51tobuy13books.HowmanyofeachtypeofbookdidMrs.Smithpurchase?

6. Atastore,allshirtsarepricedthesameandalljeansarepricedthesame.Amyboughtfourshirtsandthreepairsofjeansfor$150.Briannaboughtoneshirtandtwopairsofjeansfor$70.Whatisthecostofashirtatthestore?Whatisthecostforapairofjeansatthestore?

Dessert

1 PuzzleTime(AdaptedfromAlgebrawithPizzazz!)

Dessert2LinearFunctionsCrosswordPuzzle

Across:

3.The__________linetestthatallowsyoutodiscovervisuallywhetherarelationisafunction.

5.Thefirstcoordinatesinasetoforderedpairs.

7.Asetoforderedpairs.

8.Anorderedpairthatmakesanequationwithtwovariablestrue.

9.Amathematicalphrasethatusesvariables,numbers,andoperations.

10.Theratioofriseoverrun.

Down:

1.Arelationshipinwhicheachmemberofthedomainispairedwithexactlyonememberofthe

range.

2.A(n)________________equationisanequationwhosegraphisaline.

4.Amathematicalsentencewithanequalsign.

6.Aletterstandsforanumber.

7.Thesecondcoordinatesinasetoforderedpairs.