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Pablo Muñoz Amira Presto Superintendent of Schools Mathematics Instructional Chair Summer Math Assignment: The Passaic High School Mathematics Department requests all students tocomplete the summer assignment. Students must show work on whitelined paper and return the assignment to their math teacher by MondaySeptember 11, 2017. Assessment on the summer assignment will beadministered the first week of school. Thank you and have a great summer.

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DIRECTIONS:Thisworksheetconsistsof9parts.ItisabriefreviewofconceptsthatarecriticaltounderstandingAlgebra1nextyear.ItistobecompletedduringthesummerBEFOREthefirstdayofschool2017.TurnthisassignmentintoyourAlgebrateacheronthefirstdayofschool.

PART1:VARIABLESANDEXPRESSIONSYoucanrepresentmathematicalphrasesandreal-worldrelationshipsusingsymbolsandoperations.Thisiscalledanalgebraicexpression.

Forexample,thephrase3plusanumberncanbeexpressedusingsymbolsandoperationsas3+n.

Whatisthephrase5minusanumberdasanalgebraicexpression?

Thephrase5minusanumberd,rewrittenasanalgebraicexpression,is5—d.Theleftsideofthetablebelowgivessomecommonphrasesusedtoexpressmathematicalrelationships,andtherightsideofthetablegivestherelatedsymbol.

Exercises:Writeanalgebraicexpressionforeachwordphrase.

1.5plusanumberd 2.theproductof5andg

3.11fewerthananumberf 4.17lessthanh

5.thequotientof20andt 6.thesumof12and4

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Writeawordphraseforeachalgebraicexpression.

7.h+6 8.m–5 9.q×10

10.35r

11.h+m 12.5n

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NOTE:Multipleoperationscanbecombinedintoasinglephrase.

Thephrase11minustheproductof3andanumberd,rewrittenasanalgebraicexpression,is11–3d.

Exercises

Writeanalgebraicexpressionforeachphrase.

13. 12lessthanthequotientof12andanumberz

14. 5greaterthantheproductof3andanumberq

15. thequotientof5+handn+3

16. thedifferenceof17and22t

Writeanalgebraicexpressionorequationtomodeltherelationshipexpressedineachsituationbelow.

17. Marielaisbuildingamodelboat.Everyinchonhermodelisequivalentto3.5feetontherealboathermodelisbasedon.Whatwouldbethemathematicalruletoexpresstherelationshipbetweenthelengthofthemodel,m,andthelengthoftheboat,b?

18. Javierisputtingawaysavingsforhiscollegeeducation.EverytimeJavierputsmoneyinhisfund,hisparentsputin$2.WhatistheexpressionfortheamountgoingintoLyn’sfundifLynputsinLdollars?

Whatisthephrase11minustheproductof3anddasanalgebraicexpression?

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PART2:ORDEROFOPERATIONSANDEVALUATINGEXPRESSIONS

Exponentsareusedtorepresentrepeatedmultiplicationofthesamenumber.Forexample,4×4×4×4×4=45.Thenumberbeingmultipliedbyitselfiscalledthebase;inthiscase,thebaseis4.Thenumberthatshowshowmanytimesthebaseappearsintheproductiscalledtheexponent;inthiscase,theexponentis5.45isreadfourtothefifthpower.

Howis6×6×6×6×6×6×6writtenusinganexponent?

Thenumber6ismultipliedbyitself7times.Thismeansthatthebaseis6andtheexponentis7.6×6×6×6×6×6×6writtenusinganexponentis67.

Exercises

Writeeachrepeatedmultiplicationusinganexponent.

1.4×4×4×4×4 2.2×2×2

3.1.1×1.1×1.1×1.1×1.1 4.3.4×3.4×3.4×3.4×3.4×3.4

5.(–7)×(–7)×(–7)×(–7) 6.11×11×11

Writeeachexpressionasrepeatedmultiplication.

7.43 8.54

9.1.5210.

427

⎛ ⎞⎜ ⎟⎝ ⎠

11.x7 12.(5n)5

13.Vanessawantstodeterminethevolumeofacubewithsidesoflengths.Writeanexpressionthatrepresentsthevolumeofthecube.

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NOTE:Theorderofoperationsisasetofguidelinesthatmakeitpossibletobesurethattwopeoplewillgetthesameresultwhenevaluatinganexpression.Withoutthisstandardorderofoperations,twopeoplemightevaluateanexpressiondifferentlyandarriveatdifferentvalues.Forexample,withouttheorderofoperations,someonemightevaluateallexpressionsfromlefttoright,whileanotherpersonperformsalladditionsandsubtractionsbeforeallmultiplicationsanddivisions.

YoucanusetheacronymP.E.M.A.(Parentheses,Exponents,MultiplicationandDivision,andAdditionandSubtraction)tohelpyouremembertheorderofoperations.

Howdoyouevaluatetheexpression3+4×2-10÷5?

3+8–10÷5=3+8–2

Therearenoparenthesesorexponents,sofirst,doanymultiplicationordivisionfromlefttoright.

=11–2=9

Doanyadditionorsubtractionfromlefttoright.

Exercises

Simplifyeachexpression.14.(5+3)2 15.(8–5)(14–6)

16.(15–3)÷4 17. 22 35+⎛ ⎞

⎜ ⎟⎝ ⎠

18.40–15÷3 19.20+12÷2–5

20.(42+52)2 21.4×5–32×2÷6

Writeandsimplifyanexpressiontomodeltherelationshipexpressedinthesituationbelow.

22.Manuelahastwoboxes.Thelargerofthetwoboxeshasdimensionsof15cmby25cmby20cm.Thesmallerofthetwoboxesisacubewithsidesthatare10cmlong.Ifsheweretoputthesmallerboxinsidethelarger,whatwouldbetheremainingvolumeofthelargerbox?

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PART3:REALNUMBERSANDTHENUMBERLINE

Anumberthatistheproductofsomeothernumberwithitself,oranumbertothesecondpower,suchas9=3×3=32,iscalledaperfectsquare.Thenumberthatisraisedtothesecondpoweriscalledthesquarerootoftheproduct.Inthiscase,3isthesquarerootof9.Thisiswritteninsymbolsas Sometimessquarerootsarewholenumbers,butinothercases,theycanbeestimated.

Whatisanestimateforthesquarerootof150?

Thereisnowholenumberthatcanbemultipliedbyitselftogivetheproductof150.

10×10=10011×11=12112×12=14413×13=169Youcannotfindtheexactvalueof 150 ,butyoucanestimateitbycomparing150toperfectsquaresthatarecloseto150.

150isbetween144and169,so 150 isbetween 144 and 169 .

144 150 169< <

12 150 13< <

Thesquarerootof150isbetween12and13.Because150iscloserto144thanitisto169,wecanestimatethatthesquarerootof150isslightlygreaterthan12.

Exercises

Findthesquarerootofeachnumber.Ifthenumberisnotaperfectsquare,estimatethesquareroottothenearestinteger.

1.100 2.49 3.9

4.25 5.81 6.169

7.15 8.24 9.40

10.Asquaremathasanareaof225cm2.Whatisthelengthofeachsideofthemat?

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PART4:PROPERTIESOFREALNUMBERS

CertainpropertiesofrealnumbersleadtothecreationofEQUALexpressions.

CommutativeProperties

Thecommutativepropertiesofadditionandmultiplication:changingtheorderofthenumbersinanadditionormultiplicationproblemdoesnotchangetheanswer

Addition:a+b=b+a Multiplication:a·b=b·a

Dothefollowingequationsillustratecommutativeproperties?a.3+4=4+3 b.(5×3)×2=5×(3×2) c.1–3=3–1

A:3+4and4+3bothsimplifyto7,sothetwosidesoftheequationinpart(a)areequal.Sincebothsideshavethesametwoaddendsbutinadifferentorder,thisequationillustratestheCommutativePropertyofAddition.

B:Theexpressiononeachsideoftheequationinpart(b)simplifiesto30.Bothsidescontainthesame3factors.However,thisequationdoesnotillustratetheCommutativePropertybecausethetermsareinthesameorderoneachsideoftheequation.

C:1–3and3–1donothavethesamevalue,sotheequationinpart(c)isnottrue.ThereisNOTacommutativepropertyforsubtractionORdivision.

AssociativePropertiesTheassociativepropertiesofadditionandmultiplication:changingthegroupingofnumbersinanadditionormultiplicationproblemdoesnotchangetheanswer

Addition:(a+b)+c=a+(b+c) Multiplication:(a·b)·c=a·(b·c)

Dothefollowingequationsillustrateassociativeproperties?

a. (1+5)+4=1+(5+4)b. 4×(2×7)=4×(7×2)

(1+5)+4and1+(5+4)bothsimplifyto10,sothetwosidesoftheequationinpart(a)areequal.Sincebothsideshavethesame#sinthesameorderbutgroupeddifferently,thisequationillustratestheAssociativePropertyofAddition.

Theexpressiononeachsideoftheequationinpart(b)simplifiesto56.Bothsidescontainthesame3factors.However,thesamefactorsthatweregroupedtogetherontheleftsidehavebeengroupedtogetherontherightside;onlytheorderhaschanged.ThisequationdoesNOTillustrate

Commutative = commute = to MOVE EX: Your teacher COMMUTES to work from Jersey City

Associative = associate = to talk to EX: You ASSOCIATE with your group of friends

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theAssociativePropertyofMultiplication.

Otherpropertiesofrealnumbersinclude:a.Identitypropertyofaddition: a+0=0 12+0=12b.Identitypropertyofmultiplication: a·1=a 32·1=32c.Zeropropertyofmultiplication: a·0=0 6·0=0d.Multiplicativepropertyofnegativeone: –1·a=–a –1·7=–7

Exercises

Whatpropertyisillustratedbyeachstatement?

1.(m+7.3)+4.1=m+(7.3+4.1) 2.5p·1=5p

3.12x+4y+0=12x+4y 4.(3r)(2s)=(2s)(3r)

5.17+(–2)=(–2)+17 6.–(–3)=3

PART5:ADDINGANDSUBTRACTINGREALNUMBERS

Youcanaddrealnumbersusinganumberlineorusingthefollowingrules.

Rule1:Toaddtwonumberswiththesamesign,addtheirabsolutevalues.Thesumhasthesamesignastheaddends.

Whatisthesumof–7and–4?

Useanumberline.

Thesumis-11

Usetherule.

–7+(–4) Theaddendsarebothnegative.

|–7|+|–4| Addtheabsolutevaluesoftheaddends.

7+4=11 |–7|=7and|–4|=4.

Start at zero. Move 7 spaces to the left to represent -7. Move another 4 spaces to the left to represent -4.

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–7+(–4)=–11 Thesumhasthesamesignastheaddends.

Rule2:Toaddtwonumberswithdifferentsigns,subtracttheirabsolutevalues.Thesumhasthesamesignastheaddendwiththegreaterabsolutevalue.

Whatisthesumof–6and9?

Usetherule.9+(–6) Theaddendshavedifferentsigns.

|9|–|–6| Subtracttheabsolutevaluesoftheaddends.

9–6=3 |9|=9and|–6|=6.

9+(–6)=3 Thepositiveaddendhasthegreaterabsolutevalue.

Findeachsum. 1.–4+–12 2.–3+15 3.–9+1

4.13+(–7) 5.8+(–14) 6.–11+(–5)

7.4.5+(–1.1) 8.–5.1+8.3 9.6.4+9.8

NOTE:Additionandsubtractionareinverseoperations.Tosubtractarealnumber,additsopposite.

Whatisthedifference–5–(–8)?

–5–(–8)=–5+8 Theoppositeof–8is8.

=3 UseRule2.

Thedifference–5–(–8)is3.

Exercises

Findeachdifference.

10.8–20 11.6–(–12) 12.–4–9

13.–8–(–14) 14.–11–(–4) 15.17–25

16.3.6–(–2.4) 17.–1.5–(–1.5) 18.–1.7–5.4

Problem

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19. Thetemperaturewas50C.Fivehourslater,thetemperaturehaddropped100C.Whatisthenewtemperature?

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PART6:MULTIPLYINGANDDIVIDINGREALNUMBERS

Youneedtoremembertwosimpleruleswhenmultiplyingordividingrealnumbers.

1. Theproductorquotientoftwonumberswiththesamesignispositive.

2. Theproductorquotientoftwonumberswithdifferentsignsisnegative.

Whatistheproduct–6(–30)?

–6(–30)=180 –6and–30havethesamesignsotheproductispositive.

Whatisthequotient72÷(–6)?

72÷(–6)=–12 72and–6havedifferentsignssothequotientisnegative.

Exercises:Findeachproductorquotient(refertotheexampleproblemsonthepreviouspage!)1.–5(–6) 2.7(–20) 3.–3×22

4.44÷2 5.81÷(–9) 6.–55÷(–11)

7.–62÷2 8.25·(–4) 9.(–6)2

10.–9.9÷3 11.–7.7÷(–11) 12.–1.4(–2)

13. 1 12 3

− × 14. 2 33 5⎛ ⎞

− −⎜ ⎟⎝ ⎠ 15. 3 1.

4 3⎛ ⎞−⎜ ⎟⎝ ⎠

16.Thetemperaturedropped2°Feachhourfor6hours.Whatwasthetotalchangeintemperature?

17.ReasoningSince52=25and(–5)2=25,whatarethetwovaluesforthesquarerootof25?

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DividingFractions:Theproductof7and17is1.Twonumberswhoseproductis1are

calledreciprocals.Todivideanumberbyafraction,multiplybyitsreciprocal.

Whatisthequotient2 5 ?3 7

⎛ ⎞÷ −⎜ ⎟⎝ ⎠

2 5 2 73 7 3 5

1415

⎛ ⎞ ⎛ ⎞÷ − = × −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= −

Todividebyafraction,multiplybyitsreciprocal.Thesignsaredifferentsotheanswerisnegative.

Exercises:Findeachquotient.

18 1 12 3÷ 19. 26

3− ÷ 20. 2 2

5 3⎛ ⎞

− ÷ −⎜ ⎟⎝ ⎠

21. 1 12 4

⎛ ⎞÷ −⎜ ⎟⎝ ⎠

22. 5 17 2

⎛ ⎞ ⎛ ⎞− ÷ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

23. 2 13 4

− ÷

24.WritingAnotherwayofwritingabisa÷bExplainhowyoucouldevaluate

1216

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PART7:THEDISTRIBUTIVEPROPERTY

TheDistributivePropertystatesthattheproductofasumandanotherfactorcanberewrittenasthesumoftwoproducts,eachterminthesummultipliedbytheotherfactor.Forexample,theDistributivePropertycanbeusedtorewritetheproduct3(x+y)asthesum3x+3y.Eachterminthesumx+yismultipliedby3;thenthenewproductsareadded.

Whatisthesimplifiedformofeachexpression?

a.4(x+5)=4(x)+4(5)DistributiveProperty=4x+20Simplify.

b.(2x–3)(–3)=2x(–3)–3(–3)DistributiveProperty=–6x+9Simplify.

TheDistributivePropertycanbeusedwhetherthefactorbeingmultipliedbyasumordifferenceisontheleftorright.

TheDistributivePropertyissometimesreferredtoastheDistributivePropertyofMultiplicationoverAddition.Itmaybehelpfultothinkofthislongernamefortheproperty,asitmayremindyouofthewayinwhichtheoperationsofmultiplicationandadditionarerelatedbytheproperty.

Exercises

UsetheDistributivePropertytosimplifyeachexpression.

1.6(z+4)2.2(–2–k) 3.(5x+1)4 4.(7–11n)10

5.(3–8w)4.56.(4p+5)2.6 7.4(y+4) 8.6(q–2)

Writeeachfractionasasumordifference.

9. 2 59m − 10. 8 7

11z+ 11. 24 15

9f + 12. 12 16

6d −

Simplifyeachexpression.

13.–(6+j) 14.–(–9h–4) 15.–(–n+11) 16.–(6–8f)

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ThepreviousproblemsshowedhowtowriteaproductasasumusingtheDistributiveProperty.Thepropertycanalsobeusedtogointheotherorder,toconvertasumintoaproduct.

Howcanthesumofliketerms15x+6xbesimplifiedusingtheDistributiveProperty?

Eachtermof15x+6xhasafactorofx.Rewrite15x+6xas15(x)+6(x).NowusetheDistributivePropertyinreversetowrite15(x)+6(x)as(15+6)x,whichsimplifiesto21x.

Exercises

Simplifyeachexpressionbycombiningliketerms.

17.16x+12x 18.25n–17n 19.–4p+6p

20.–15a–9a 21.–9k2–5k2 22.12t2–20t2

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Bythinkingoforrewritingnumbersassumsordifferencesofothernumbersthatareeasiertouseinmultiplication,theDistributivePropertycanbeusedtomakecalculationseasier.

Howcanyoumultiply78by101usingtheDistributivePropertyandmentalmath?78×101 Writetheproduct.78×(100+1) Rewrite101assumoftwonumbersthatareeasytousein

multiplication.78(100)+78(1) UsetheDistributivePropertytowritetheproductasasum.7800+78 Multiply.

7878 Simplify.

Exercises

Usementalmathtofindeachproduct.

23.5.1×7 24.24.9×4 25.999×11 26.12×95

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PART8:INTRODUCTIONTOEQUATIONS

Anequationisamathematicalsentencewithanequalsign.Anequationcanbetrue,false,oropen.Anequationistrueiftheexpressionsonbothsidesoftheequalsignareequal,forexample2+5=4+3.Anequationisfalseiftheexpressionsonbothsidesoftheequalsignarenotequal,forexample2+5=4+2.

Anequationisconsideredopenifitcontainsoneormorevariables,forexamplex+2=8.Whenavalueissubstitutedforthevariable,youcanthendecidewhethertheequationistrueorfalseforthatparticularvalue.Ifanopensentenceistrueforavalueofthevariable,thatvalueiscalledasolutionoftheequation.Forx+2=8,6isasolutionbecausewhen6issubstitutedintheequationforx,theequationistrue:6+2=8.

Istheequationtrue,false,oropen?Explain.a.15+21=30+6 Theequationistrue,becausebothexpressionsequal36.b.24÷8=2·2 Theequationisfalse,because24÷8=3and2·2=4;3≠4.c.2n+4=12 Theequationisopen,becausethereisavariableinthe

expression ontheleftside.

Tellwhethereachequationistrue,false,oropen.Explain.

1.2(12)–3(6)–12 2.3x+12=–19 3.14–19=–5

4.2(–8)+4=12 5.7–9+3=x 6.(28+12)÷–2=–20

7.14–(–8)–14=8 8.(13–16)÷3=1 9.42÷7+3=9

Isx=–3asolutionoftheequation4x+5=–7?

4x+5=–7

4(–3)+5=–7 Substitute–3forx.–7=–7 Simplify.

Since–7=–7,–3isasolutionoftheequation4x+5=–7.

Tellwhetherthegivennumberisasolutionofeachequation.

10.4x–1=–27;–7 11.18–2n=14;2 12.21=3p–5;9

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13.k=(–6)(–8)–14;–62 14.20v+36=–156;–6 15.8y+13=21;1

16.–24–17t=–58;217.

126 5; 73m− = + − 18.

1 38 ;384 2g − =

Writeanequationforeachsentence.

19. 13timesthesumofanumberand5is91.

20. Negative8timesanumberminus15isequalto30.

21. Jaredreceives$23foreachlawnhemows.WhatisanequationthatrelatesthenumberoflawnswthatJaredmowsandhispayp?

22. Shariffhasbeenworkingforacompany2yearslongerthanPatsy.WhatisanequationthatrelatestheyearsofemploymentofShariffSandtheyearsofemploymentofPatsyP?

Usementalmathtofindthesolutionofeachequation.

23.h+6=13 24.–11=n+2 25.6–k=14 26.5=–8+t

27. 25z= −

28 126j=

−

29.8c=–48 30.–15a=–45

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PART9:PATTERNS,EQUATIONSANDGRAPHS

Tables,equations,andgraphsaresomeofthewaysthatarelationshipbetweentwoquantitiescanberepresented.Youcanusetheinformationprovidedbyonerepresentationtoproduceoneoftheotherrepresentations;forexample,youcanusedatafromatabletoproduceagraph.Youcanalsouseanyoftherepresentationstodrawconclusionsabouttherelationship.

Are(2,11)and(5,3)solutionsoftheequationy=3x+5?

Foreachorderedpair,youcansubstitutethex-andy-coordinatesintotheequationforxandyandthensimplifytoseeifthevaluessatisfytheequation.For(2,11): For(5,3):11=3(2)+5 Substituteforxand

y.3=3(5)+5

11=11 Multiplyandthenadd.

3≠20

Sincebothsidesoftheequationhavethesamevalue,theorderedpair(2,11)isasolutionoftheequationy=3x+5.Sincethetwosidesoftheequationhavedifferentvalues,theorderedpair(5,3)isnotasolutionoftheequationy=3x+5.

ThetableshowstherelationshipbetweenthenumberofhoursKayaworksatherjobandtheamountofpayshereceives.Extendthepattern.HowmuchmoneywouldKayaearnifsheworked40hours?

Method1:Writeanequation.

y=12.50x Kayaearns$12.50perhour.

=12.50(40) Substitute40forx.=500 Simplify.

Shewouldearn$500in40hours.

Method2:Drawagraph.

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Shewouldearn$500in40hours.

Exercises

Tellwhethertheequationhasthegivenorderedpairasasolution.

1.y=x–7;(2,–5) 2.y=x+6;(–5,11) 3.y=–x+1;(–1,0)

4.y=–5x;(–3,–15) 5.y=x–8;(7,–1)6.

y = x+ 3

4;(−1,−

14

)

Useatable,anequation,andagraphtorepresenteachrelationship.

7.Ticketstothefaircost$17. 8.Brianis5yearsolderthanSam.

Usethetabletodrawagraphandanswerthequestion.

9.ThetableshowsJake’searningsforthenumberofcakeshebaked.Whatarehisearningsforbaking75cakes?

Usethetabletowriteanequationandanswerthequestion.10.ThetableshowsthenumberofmilesthatKaterunsonaweeklybasiswhiletrainingforarace.Howmanytotalmileswillshehaverunafter15weeks?

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