Lesson4-11:ConstantRate(Part1)

Module4:LinearEqua:onsTopicB:LinearEqua:onsinTwoVariablesandTheirGraphs

Lesson4-11:ConstantRate

PurposeforLearning:

Essen:alQues:on(s):

Vocabulary:

•  Studentsknowthedefini/onofconstantrateinvariedcontextsasexpressedusingtwo

variableswhereoneis𝑡represen/nga/meinterval.

•  Studentsgraphpointsonacoordinateplanerelatedtoconstantrateproblems.

Howcantablesandgraphsbeusedtorepresentpropor/onalrela/onships?

Averagerate,Constantrate,twovariableequa/on,coordinateplane

PART1—Example1(inyourstudentbooklet):Considerthefollowingwordproblem:

Paulinemowsalawnataconstantrate.Supposeshemowsa𝟑𝟓

squarefootlawnin𝟐.𝟓minutes.Whatarea,insquarefeet,canshe

mowin𝟏𝟎minutes?𝒕minutes?

SincePaulinemowsataconstantrate,𝐶,then35/2.5 =𝐶and 𝐴/10 =𝐶.Therefore,

𝟑𝟓/𝟐.𝟓 = 𝑨/𝟏𝟎 𝟑𝟓𝟎=𝟐.𝟓𝑨𝟑𝟓𝟎/𝟐.𝟓 = 𝟐.𝟓/𝟐.𝟓 𝑨𝟏𝟒𝟎=𝑨Paulinemows𝟏𝟒𝟎squarefeetoflawninsquarefeetoflawnin𝟏𝟎minutes.minutes.

WhatisPauline’saverageratein2.5minutes?

Pauline’saverageratein𝟐.𝟓minutesis 𝟑𝟓/𝟐.𝟓 squarefeetperminute.

WhatisPauline’saverageratein10minutes?

Let𝑨representareamowed.representareamowed.Pauline’saverageratein𝟏𝟎minutesis 𝑨/𝟏𝟎 squarefeetperminute.

Example1(con:nued):

Paulinemowsalawnataconstantrate.Supposeshemowsa𝟑𝟓

squarefootlawnin𝟐.𝟓minutes.Whatarea,insquarefeet,canshe

mowin𝟏𝟎minutes?𝒕minutes?

𝟑𝟓/𝟐.𝟓 = 𝒚/𝒕 𝟐.𝟓𝒚=𝟑𝟓𝒕𝟐.𝟓/𝟐.𝟓 𝒚= 𝟑𝟓/𝟐.𝟓 𝒕𝒚= 𝟑𝟓/𝟐.𝟓 𝒕

Whatisthemeaningof 35/2.5 intheequa/on𝑦= 35/2.5 𝑡?

Thenumber 𝟑𝟓/𝟐.𝟓 representstherateatwhichPaulinecanmowalawn.

(35squarefeetin2.5minutes)

Ifwelet𝑦representthenumberofsquarefeetPaulinecanmowin𝑡minutes,thenPauline’saverageratein𝑡minutesis𝑦/𝑡 squarefeetperminute.Writethetwovariableequa/onthatrepresentstheareaoflawn,𝑦,Paulinecanmowin𝑡minutes.

Example1(con:nued):

Paulinemowsalawnataconstantrate.Supposeshemowsa𝟑𝟓

squarefootlawnin𝟐.𝟓minutes.Whatarea,insquarefeet,can

shemowin𝟏𝟎minutes?𝒕minutes?

Wecanorganizetheinforma/oninatable:

(:meinminutes) Linearequa:on: (areainsquarefeet)

Example1(con:nued):

Paulinemowsalawnataconstantrate.Supposeshemowsa𝟑𝟓

squarefootlawnin𝟐.𝟓minutes.Whatarea,insquarefeet,can

shemowin𝟏𝟎minutes?𝒕minutes?

Wecanorganizetheinforma/oninatable:

(:meinminutes) Linearequa:on: (areainsquarefeet)

Example1(con:nued):

Paulinemowsalawnataconstantrate.Supposeshemowsa𝟑𝟓

squarefootlawnin𝟐.𝟓minutes.Whatarea,insquarefeet,can

shemowin𝟏𝟎minutes?𝒕minutes?

•  Onacoordinateplane,wewillletthe𝑥-axisrepresent/me𝑡,inminutes,

andthe𝑦-axisrepresenttheareaofmowedlawninsquarefeet.Thenwehavethefollowinggraph:

BecausePaulinemowsataconstantrate,wewouldexpectthesquarefeetofmowedlawntocon?nuetoriseasthe?meinminutesincreases.

ConceptDevelopment:Inthelastlesson,welearnedaboutaveragespeedandconstantspeed.Constantspeedproblemsarejustaspecialcaseofalargervarietyofproblemsknownasconstantrateproblems.

First,wedefineAveragerate:

•  Suppose𝑉gallonsofwaterflowfromafaucetinagiven/meinterval𝑡(minutes).

•  Thentheaveragerateofwaterflowinthegiven/meintervalis𝑉/𝑡 ingallonsperminute.

𝑉 (𝑔𝑎𝑙𝑙𝑜𝑛𝑠)/𝑡 (𝑚𝑖𝑛𝑢𝑡𝑒𝑠) 

ConceptDevelopment:(con:nued)Inthelastlesson,welearnedaboutaveragespeedandconstantspeed.Constantspeedproblemsarejustaspecialcaseofalargervarietyofproblemsknownasconstantrateproblems.

Then,wedefineConstantrate:

•  Supposetheaveragerateofwaterflowisthesameconstant𝐶foranygiven/meinterval.Thenwesaythatthewaterisflowingataconstantrate,𝐶.

Example:Suppose𝑨squarefeetoflawnaremowedinagiven/meinterval𝒕(minutes).Thentheaveragerateoflawnmowinginthegiven:meintervalis 𝑨/𝒕 squarefeetperminute.Ifweassumethattheaveragerateoflawnmowingisthesameconstant𝐶foranygiven/meinterval,thenwesaythatthelawnismowedataconstantrate,𝐶.

ConceptDevelopment:

•  Whatisthedifferencebetweenaveragerateandconstantrate?

– Averagerateistherateinwhichsomethingcanbedoneoveraspecific?meinterval.Constantrateassumesthattheaveragerateisthesameoverany?meinterval.***Thewaywedefineaveragerateandconstantrateforagivensitua/onisverysimilar.Ineachcase,atranscrip/onofthegiveninforma/onleadstoanexpressionintwovariables.

Example2(inyourstudentbooklet):Waterflowsataconstantrateoutofafaucet.Supposethevolume

ofwaterthatcomesoutinthreeminutesis𝟏𝟎.𝟓gallons.How

manygallonsofwatercomesoutofthefaucetin𝒕minutes?

Writethelinearequa/onthatrepresentsthevolumeofwater,𝑉,thatcomesoutin𝑡minutes.Let𝑪representtheconstantrepresenttheconstantrateofwaterflow.

𝟏𝟎.𝟓/𝟑 =𝑪and𝑽/𝒕 =𝑪,and𝑽/𝒕 =𝑪,

then 𝟏𝟎.𝟓/𝟑 = 𝑽/𝒕 .

𝟏𝟎.𝟓/𝟑 = 𝑽/𝒕 𝟑𝑽=𝟏𝟎.𝟓𝒕𝟑/𝟑 𝑽= 𝟏𝟎.𝟓/𝟑 𝒕𝑽= 𝟏𝟎.𝟓/𝟑 𝒕

Example2(inyourstudentbooklet):

Whatisthemeaningofthenumber 10.5/3 intheequa/on𝑉= 10.5/3 𝑡?

Usingthelinearequa/on𝑉= 10.5/3 𝑡,completethetable:

(:meinminutes) Linearequa:on: (ingallons)

Thenumber 𝟏𝟎.𝟓/𝟑 representstherateatwhichwaterflowsfromafaucet.

Example2(inyourstudentbooklet):

Whatisthemeaningofthenumber 10.5/3 intheequa/on𝑉= 10.5/3 𝑡?

Thenumber 𝟏𝟎.𝟓/𝟑 representstherateatwhichwaterflowsfromafaucet.

Usingthelinearequa/on𝑉= 10.5/3 𝑡,completethetable:

(:meinminutes) Linearequa:on: (ingallons)

Example2(con:nued):Waterflowsataconstantrateoutofafaucet.Supposethevolume

ofwaterthatcomesoutinthreeminutesis𝟏𝟎.𝟓gallons.•  Onacoordinateplane,wewill

letthe𝑥-axisrepresent/me

𝑡inminutesandthe𝑦-axisrepresentthevolumeofwater.Graphthedatafromthetable.

Example2(con:nued):Waterflowsataconstantrateoutofafaucet.Supposethevolume

ofwaterthatcomesoutinthreeminutesis𝟏𝟎.𝟓gallons.

Usingthegraph,abouthowmanygallonsofwaterdoyouthinkwouldflowaXer11/2 minutes?Explain.

–  AEer𝟏 𝟏/𝟐 minutes,between𝟑 𝟏/𝟐 and𝟕gallonsofwaterwillflow.gallonsofwaterwillflow.Sincethewaterisflowingataconstantrate,wecanexpectthevolumeofwatertorisebetween𝟏and𝟐minutes.Thenumberofand𝟐minutes.Thenumberofminutes.ThenumberofgallonsthatflowsaEer𝟏 𝟏/𝟐  minutesthenwouldhavetobebetweenthenumberofgallonsthatflowsoutaEer𝟏minuteand𝟐minuteand𝟐minutes.

Usingthegraph,abouthowlongwouldittakefor15gallonsofwatertoflowoutofthefaucet?Explain.

–  Itwouldtakebetween𝟒and𝟓minutesfor𝟏𝟓gallonsofwatertoand𝟓minutesfor𝟏𝟓gallonsofwatertominutesfor𝟏𝟓gallonsofwatertogallonsofwatertoflowoutofthefaucet.Ittakes𝟒minutesfor𝟏𝟒gallonstoflow;minutesfor𝟏𝟒gallonstoflow;gallonstoflow;therefore,itmusttakemorethan𝟒minutesfor𝟏𝟓gallonstocomeminutesfor𝟏𝟓gallonstocomegallonstocomeout.Itmusttakelessthan𝟓minutesbecause𝟑 𝟏/𝟐 gallonsflowoutminutesbecause𝟑 𝟏/𝟐 gallonsflowouteveryminute.

(PART1)Lesson4-11:ConstantRateSUMMARY:•  Graphingpropor/onalrela/onshipsliketheselasttwo

constantrateproblemsprovideusmoreinforma/onthansimplysolvinganequa/onandcalcula/ngonevalue.Thegraphprovidesinforma/onthatisnotsoobviousinanequa/on.

Lesson4-11:ConstantRate(Part2)

Module4:LinearEqua:onsTopicB:LinearEqua:onsinTwoVariablesandTheirGraphs

Lesson4-11:ConstantRate

PurposeforLearning:

Essen:alQues:on(s):

Vocabulary:

•  Studentsknowthedefini/onofconstantrateinvariedcontextsasexpressedusingtwo

variableswhereoneis𝑡represen/nga/meinterval.

•  Studentsgraphpointsonacoordinateplanerelatedtoconstantrateproblems.

Howcantablesandgraphsbeusedtorepresentpropor/onalrela/onships?

Averagerate,Constantrate,twovariableequa/on,coordinateplane

(PART2)--Exercise1(inyourstudentbooklet):

Jesustypesataconstantrate.Hecantypeafullpageoftextin𝟑 𝟏/𝟐 minutes.Wewanttoknowhowmanypages,𝒑,Jesuscan

typeaZer𝒕minutes.

a)  Writethelinearequa:onin

twovariablesthatrepresentsthenumberofpagesJesustypesinanygiven:meinterval.

Let𝑪representtheconstantraterepresenttheconstantratethatJesustypes.Then,𝟏/𝟑.𝟓 =𝑪and𝒑/𝒕 =𝑪,therefore and𝒑/𝒕 =𝑪,therefore ,therefore

𝟏/𝟑.𝟓 = 𝒑/𝒕 .𝟏/𝟑.𝟓 = 𝒑/𝒕 𝟑.𝟓𝒑=𝒕𝟑.𝟓/𝟑.𝟓 𝒑= 𝟏/𝟑.𝟓 𝒕𝒑= 𝟏/𝟑.𝟓 𝒕

Exercise1(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes) Linearequa:on: (pagestyped)

Exercise1(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes) Linearequa:on: (pagestyped)

Exercise 1 (continued): c.)Graphthedataonacoordinateplane.

Exercise 1 (continued): d.)AbouthowlongwouldittakeJesustotypea𝟓-pagepaper?Explain.

•  Itwouldtakehimbetween𝟏𝟓and𝟐𝟎minutes.AEer𝟏𝟓and𝟐𝟎minutes.AEer𝟏𝟓minutes.AEer𝟏𝟓

minuteshewillhavetyped𝟒.𝟑pages.In𝟐𝟎minuteshecanpages.In𝟐𝟎minuteshecanminuteshecan

type𝟓.𝟕pages.Since𝟓pagesisbetween𝟒.𝟑and𝟓.𝟕,thenpages.Since𝟓pagesisbetween𝟒.𝟑and𝟓.𝟕,thenpagesisbetween𝟒.𝟑and𝟓.𝟕,thenand𝟓.𝟕,then,then

itwilltakehimbetween𝟏𝟓and𝟐𝟎minutes.and𝟐𝟎minutes.minutes.

Exercise2(inyourstudentbooklet):

Emilypaintsataconstantrate.Shecanpaint𝟑𝟐squarefeetin

fiveminutes.Whatarea,𝑨,canshepaintin𝒕minutes?

a)  Writethelinearequa:onin

twovariablesthatrepresentsthenumberofsquarefeetEmilycanpaintinanygiven:meinterval.

Let𝑪betheconstantratethatEmilybetheconstantratethatEmilypaints.Then,

𝟑𝟐/𝟓 =𝑪and𝑨/𝒕 =𝑪,and𝑨/𝒕 =𝑪,

therefore𝟑𝟐/𝟓 = 𝑨/𝒕 .

𝟑𝟐/𝟓 = 𝑨/𝒕 𝟓𝑨=𝟑𝟐𝒕𝟓/𝟓 𝑨= 𝟑𝟐/𝟓 𝒕𝑨= 𝟑𝟐/𝟓 𝒕

Exercise2(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes) Linearequa:on:

(areapaintedinsquarefeet)

Exercise2(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes) Linearequa:on:

(areapaintedinsquarefeet)

Exercise 2 (continued): c.)Graphthedataonacoordinateplane.

Exercise 2 (continued): d.)AbouthowmanysquarefeetcanEmilypaintin𝟐 𝟏/𝟐 minutes?Explain.

•  Emilycanpaintbetween𝟏𝟐.𝟖and𝟏𝟗.𝟐squarefeetin𝟐 and𝟏𝟗.𝟐squarefeetin𝟐 squarefeetin𝟐 𝟏/𝟐 minutes.AEer𝟐minutesshepaints𝟏𝟐.𝟖squarefeetminutesshepaints𝟏𝟐.𝟖squarefeetsquarefeet

andat𝟑minutesshewillhavepainted𝟏𝟗.𝟐squarefeet.minutesshewillhavepainted𝟏𝟗.𝟐squarefeet.squarefeet.

Exercise3(inyourstudentbooklet):Josephwalksataconstantspeed.Hewalkedtothestore,one-half

mileaway,in𝟔minutes.Howmanymiles,𝒎,canhewalkin𝒕minutes?

a)  Writethelinearequa:onin

twovariablesthatrepresentsthenumberofmilesJosephcanwalkinanygiven:meinterval,𝒕.

Let𝑪betheconstantratethatJosephbetheconstantratethatJosephwalks.Then,

𝟎.𝟓/𝟔 =𝑪and𝒎/𝒕 =𝑪,and𝒎/𝒕 =𝑪,,therefore𝟎.𝟓/𝟔 = 𝒎/𝒕 .

𝟎.𝟓/𝟔 = 𝒎/𝒕 𝟔𝒎=𝟎.𝟓𝒕𝟔/𝟔 𝒎= 𝟎.𝟓/𝟔 𝒕𝒎= 𝟎.𝟓/𝟔 𝒕

Exercise3(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes)Linearequa:on:

(distanceinmiles)

Exercise3(con:nued):b.)Completethetablebelow.Useacalculatorandroundanswerstothetenthsplace.

(:meinminutes)Linearequa:on:

(distanceinmiles)

Exercise 3 (continued): c.)Graphthedataonacoordinateplane.

Exercise 3 (continued): d.)Joseph’s friend lives 𝟒 miles away from him. About how long would it take Joseph to walk to his friend’s house? Explain.

•  ItwilltakeJosephaliLlelessthananhourtowalktohisfriend’s

house.Sinceittakes𝟑𝟎minutesforhimtowalk𝟐.𝟓miles,andminutesforhimtowalk𝟐.𝟓miles,andmiles,and

𝟔𝟎minutestowalk𝟓miles,and𝟒iscloserto𝟓than𝟐.𝟓,itminutestowalk𝟓miles,and𝟒iscloserto𝟓than𝟐.𝟓,itmiles,and𝟒iscloserto𝟓than𝟐.𝟓,itiscloserto𝟓than𝟐.𝟓,itthan𝟐.𝟓,it,it

willtakeJosephlessthananhourtowalkthe𝟒miles.miles.

LessonSummary:

•  Constantrateproblemsappearinavarietyofcontextslikepain/ngahouse,typing,walking,waterflow,etc.

***Whenconstantrateisstatedforagivenproblem,thenyoucanexpressthesitua:onasatwovariableequa:on.Theequa:oncanbeusedtocompleteatableofvaluesthatcanthenbegraphedonacoordinateplane.