1. Lesson 32: Multi-Step Problems in the Real World
Date: 5/4/15 338
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
𝟒𝟖
𝟒𝟎
𝟑𝟐
𝟐𝟒
𝟏𝟔
𝟖
Number ofPackages
Beverages for theParty
𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔
TotalNumberofBeverages
Lesson 32: Multi-Step Problemsin the Real World
Student Outcomes
 Students analyzean equation in two variables,choosean independent variableand a dependent variable,
make a table and make a graph for the equation by plotting the points in the table. For the graph,the
independent variableis usually represented by the horizontal axisand the dependent variableis usually
represented by the vertical axis.
Classwork
OpeningExercise (5 minutes)
Opening Exercise
Xin isbuying beveragesfor aparty which are individually packagedand comein packsof 𝟖. Let 𝒑 be the number of
packagesXin buysand 𝒕be the total number ofbeverages. The equation 𝒕 = 𝟖𝒑can be usedto calculatethetotal
number ofbeverageswhenthe number ofpackagesisknown. Determine the independent and dependent variablein
this scenario. Then make atable using wholenumbervaluesof 𝒑 less than 𝟔.
The independent variableis thenumber ofpackages. I know this becauseI can usetheequation todeterminethenumber
ofbeverages. Therefore, thenumber ofindividual beverages depends on thenumber ofpackages purchased.
The dependent variableis thetotal numberofbeverages becausethis is affected by thenumber of packages changing.
# ofPackages(𝒑) Total Number of Beverages(𝒕 = 𝟖𝒑)
𝟎 𝟎
𝟏 𝟖
𝟐 𝟏𝟔
𝟑 𝟐𝟒
𝟒 𝟑𝟐
𝟓 𝟒𝟎
Example 1 (7 minutes)
Example 1
Make agraph for the table in the Opening Exercise.

2. Lesson 32: Multi-Step Problems in the Real World
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
𝟐𝟓𝟎
𝟐𝟎𝟎
𝟏𝟓𝟎
𝟓𝟎
𝟏𝟎𝟎
# ofHours
𝟎 𝟏 𝟐 𝟑 𝟒 𝟓
NumberofMiles
Road Trip
 To make a graph,we must determine which variableis measured alongthe horizontal axisand which variable
is measured alongthe vertical axis.
 Generally,the independent variable is measured along the 𝑥-axis. Which axis isthe 𝑥-axis?
 The 𝑥-axis is the horizontal axis.
 Where would you put the dependent variable?
 On the 𝑦-axis, it travels vertically, or up and down.
 We want to show how the number of beverages changes when the number of packages changes. To check
that you have set up your graph correctly,try making a sentence out of the labels on the axis. Writeyour
sentence usingthe label from the 𝑦-axis firstfollowed by the label from the 𝑥-axis. “The number of beverages
depends on the number of packages purchased.”
Example 2 (3 minutes)
Example 2
Use the graph to determinewhichvariableis the independent variableand whichisthe dependent variable. Then state
the relationshipbetween thequantitiesrepresented by thevariables.
The number ofmiles driven depends on how many hours they drive. Therefore, thenumber ofmiles is thedependent
variableand thenumber ofhours is theindependentvariable. This graph shows thatthey cantravel 𝟓𝟎miles every hour.
So, thetotal number ofmiles driven increases by 𝟓𝟎every timethenumber ofhours increases by 𝟏.
MP.2

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
# ofweeks
#ofbooks
Exercises(20 minutes)
Students work individually.
Exercises
1. Each week Quentin earns $𝟑𝟎. Ifhe savesthismoney, create agraph that shows the total amountof money
Quentin hassaved from week 𝟏throughweek 𝟖. Writean equation that representstherelationship between the
number ofweeksthat Quentinhassaved hismoney (𝒘)and the total amountofmoney hehassaved (𝒔). Then
name the independentand dependentvariable. Writeasentence that showsthisrelationship.
𝒔 = 𝟑𝟎𝒘
𝒔is thedependent variableand 𝒘is theindependent variable.
# ofWeeks Total Saved ($)
𝟏 𝟑𝟎
𝟐 𝟔𝟎
𝟑 𝟗𝟎
𝟒 𝟏𝟐𝟎
𝟓 𝟏𝟓𝟎
𝟔 𝟏𝟖𝟎
𝟕 𝟐𝟏𝟎
𝟖 𝟐𝟒𝟎
2. Zoe iscollecting booksto donate. She started with 𝟑and collectstwo more each week. She isusing theequation
𝒃 = 𝟐𝒘+ 𝟑, where 𝒃isthe total number ofbookscollected and 𝒘isthe number ofweeksshe hasbeencollecting.
Name the independent and dependent variables. Then createa graph to representhow many booksZoe has
collected when 𝒘is 𝟓or less.
The number ofweeks is independent. Thenumber ofbooks collected is dependent.
# ofWeeks # ofBooks Collected
𝟎 𝟑
𝟏 𝟓
𝟐 𝟕
𝟑 𝟗
𝟒 𝟏𝟏
𝟓 𝟏𝟑
𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗
# ofWeeks
𝟐𝟒𝟎
𝟐𝟏𝟎
𝟏𝟖𝟎
𝟏𝟓𝟎
𝟏𝟐𝟎
𝟗𝟎
𝟔𝟎
𝟑𝟎
Money Saved
AmountofMoney($)

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
Total Cost of Ride
NumberofMiles
3. Elianaplansto visit the fair. She must pay $𝟓to enter the fair groundsand an additional $𝟑perride. Writean
equation to show the relationship between 𝒓, the number ofrides, and 𝒕, thetotalcost. State which variable is
dependentand whichisindependent. Then createagraph that modelsthe equation.
𝒕 = 𝟑𝒓+ 𝟓
The number ofrides is theindependent variableand thetotalcost is thedependent variable.
# ofRides Total Cost (in dollars)
𝟎 𝟓
𝟏 𝟖
𝟐 𝟏𝟏
𝟑 𝟏𝟒
𝟒 𝟏𝟕
Closing(5 minutes)
 Imaginethat you are helpinga friend with his math work. Here is the problem he was solving:
Henry is takinga taxi cab home. The cab company charges an initial feeof $5 plus $2 for each additional mile.
Henry uses the equation 𝑡 = 2𝑚 + 5 to calculatethe costof the ride, where 𝑡 is the total cost and 𝑚 is the
number of miles.
Your friend states that 𝑡 is the dependent variableand 𝑚 is theindependent variable. Then the friend starts to
sketch a graph.
 What would you tell your friend when lookingover his/her
work?
 I would tell my friend that the dependent variable
should go on the vertical axis. Then your graph will
show that the total cost of the ride depends on how
many miles you travel in the taxi cab.
Exit Ticket (5 minutes)

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
Name Date
Lesson 32: Multistep Problemsin the Real World
Exit Ticket
Determine which variableis theindependent variableand the dependent variable. Writean equation, make a tableand
plot the points from the table on the graph.
Enoch can type 40 words per minute. Let 𝑤 be the number of words and 𝑚 be the number of minutes.
Independent variable
Dependent variable
Equation

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
Exit Ticket Sample Solutions
Determinewhichvariableisthe independent variableand the dependentvariable. Writean equation, make atable and
plot the pointsfrom thetableon the graph.
Enoch can type 𝟒𝟎wordsper minutes. Let 𝒘be thenumber ofwordsand 𝒎 be the number of minutes.
The independent variableis thenumber ofminutes. Thedependent variableis thenumber ofwords.
The equation is 𝒘 = 𝟒𝟎𝒎.
# ofMinutes # ofWords
𝟎 𝟎
𝟏 𝟒𝟎
𝟐 𝟖𝟎
𝟑 𝟏𝟐𝟎
𝟒 𝟏𝟔𝟎
𝟓 𝟐𝟎𝟎
Problem Set Sample Solutions
1. Caleb started saving money in acookiejar. He startedwith $𝟐𝟓. He adds $𝟏𝟎to the accounteach week. Write an
equation where 𝒘isthenumber ofweeksand 𝒕isthe total amount inthe account. Determinewhich variableis
independent and which isdependent. Then graph thetotalamount inthe account for 𝒘being lessthan 𝟔weeks.
𝒕 = 𝟏𝟎𝒘+ 𝟐𝟓
𝒕is thedependent variable.
𝒘is theindependent variable.
# ofWeeks Total Amount in
Account (in dollars)
𝟎 𝟐𝟓
𝟏 𝟑𝟓
𝟐 𝟒𝟓
𝟑 𝟓𝟓
𝟒 𝟔𝟓
𝟓 𝟕𝟓
𝟒𝟎
𝟖𝟎
𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔
# ofMinutes
𝟏𝟐𝟎
𝟏𝟔𝟎
𝟐𝟎𝟎
𝟐𝟒𝟎
Words typed per Minute
#ofWordsTyped
TotalAmountinAccount(indollars)
# ofweeks
Money in Savings Account

7. Lesson 32: Multi-Step Problems in the Real World
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
# ofweeks
TotalAmount($)
Total Amount Saved
2. Kevin istaking ataxi from the airport to hishome. There isa $𝟔flat fee for riding in thetaxi. In addition,Kevinmust
also pay $𝟏per mile. Write an equationwhere 𝒎 isthe number ofmilesand 𝒕isthe total cost ofthe taxiride.
Determinewhichvariableisindependent and which isdependent. Then graph thetotal cost for 𝒎 being lessthan 𝟔
miles.
𝒕 = 𝟏𝒎 + 𝟔
𝒕is thedependent variable.
𝒎 is theindependent variable.
# ofMiles Total Cost
𝟎 𝟔
𝟏 𝟕
𝟐 𝟖
𝟑 𝟗
𝟒 𝟏𝟎
𝟓 𝟏𝟏
3. Annastarted with $𝟏𝟎. She saved an additional $𝟓eachweek. Writean equation that can beused to determinethe
total amount saved, 𝒕, after agiven number ofweeks, 𝒘. Determine which variable isindependentand whichis
dependent. Then graph thetotal amount saved for first 𝟖weeks.
𝒕 = 𝟓𝒘 + 𝟏𝟎
𝒕 is thedependent variable.
𝒘is theindependent variable.
# ofWeeks Total Amount ($)
𝟎 𝟏𝟎
𝟏 𝟏𝟓
𝟐 𝟐𝟎
𝟑 𝟐𝟓
𝟒 𝟑𝟎
𝟓 𝟑𝟓
𝟔 𝟒𝟎
𝟕 𝟒𝟓
𝟖 𝟓𝟎
TotalCost($)
Total Cost ofa Taxi Ride
Number ofMiles

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 32
4. Aliyah ispurchasing produce at thefarmers’ market. She plansto buy $𝟏𝟎worth ofpotatoesand some apples. The
applescost $𝟏.𝟓𝟎 per pound. Writean equation to show thetotalcost ofthe produce,where 𝑻 isthetotal cost and
𝒂 is the number ofpoundsofapples. Determine which variable isdependent and independent. Then graph the
equation on the coordinate plane.
𝑻 = 𝟏.𝟓𝟎𝒂+ 𝟏𝟎
The total cost is thedependentvariable. Thenumber ofpounds ofapples is theindependent variable.
# ofPounds ofApples Total Cost ($)
0 10
𝟏 𝟏𝟏.𝟓𝟎
𝟐 𝟏𝟑
𝟑 𝟏𝟒.𝟓𝟎
𝟒 𝟏𝟔
𝟓 𝟏𝟕.𝟓𝟎
# oflb. ofapples
Total Cost at theFarmers’ Market
TotalCost($)