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.
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