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Lesson 7
Lesson 7: Comparing Linear Functions and Graphs
Date: 11/19/14
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Lesson 7: Comparing Linear Functions and Graphs
Student Outcomes
Students compare the properties of two functions represented in different ways, including tables, graphs,
equations, and written descriptions.
Students use rate of change to compare functions, determining which function has a greater rate of change.
Lesson Notes
The Fluency Exercise included in this lesson will take approximately minutes and should be assigned either at the
beginning or at the end of the lesson.
Classwork
Exploratory Challenge/Exercises 1–4 (20 minutes)
Students work in small groups to complete Exercises 1–4. Groups can select a method of their choice to answer the
questions, and their methods will be a topic of discussion once the Exploratory Challenge is completed. Encourage
students to discuss the various methods (e.g., graphing, comparing rates of change, using algebra) as a group before
they begin solving.
Exercises
Exercises 1–4 provide information about functions. Use that information to help you compare the functions and answer
the questions.
1. Alan and Margot drive from City A to City B, a distance of miles. They take the same route and drive at
constant speeds. Alan begins driving at 1:40 p.m. and arrives at City B at 4:15 p.m. Margot’s trip from City A to City
B can be described with the equation , where is the distance traveled in miles and is the time in minutes
spent traveling. Who gets from City A to City B faster?
Student solutions will vary. Sample solution is provided.
It takes Alan minutes to travel the miles. Therefore, his constant rate is
miles
per minute.
Margot drives miles per hour (minutes). Therefore, her constant rate is
miles per
minute.
To determine who gets from City A to City B faster, we just need to compare their rates in
miles per minutes:
Since Margot’s rate is faster, she will get to City B faster than Alan.
Scaffolding:
Providing example language for
students to reference will be
useful. This might consist of
sentence starters, sentence
frames, or a word wall.
MP.1
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2. You have recently begun researching phone billing plans. Phone Company A charges a flat rate of a month. A
flat rate means that your bill will be each month with no additional costs. The billing plan for Phone Company B
is a linear function of the number of texts that you send that month. That is, the total cost of the bill changes each
month depending on how many texts you send. The table below represents the inputs and the corresponding
outputs that the function assigns.
Input
(number of texts)
Output
(cost of bill)
At what number of texts would the bill from each phone plan be the same? At what number of texts is Phone
Company A the better choice? At what number of texts is Phone Company B the better choice?
Student solutions will vary. Sample solution is provided.
The equation that represents the function for Phone Company A is .
To determine the equation that represents the function for Phone Company B, we need the rate of change:
The equation for Phone Company B is shown below.
Using the assignment of to ,
The equation that represents the function for Phone Company B is .
We can determine at what point the phone companies charge the same amount by solving the system:
After texts are sent, both companies would charge the same amount, . More than texts means that
the bill from Phone Company B will be higher than Phone Company A. Less than texts means the bill from
Phone Company A will be higher.
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3. A function describes the volume of water, , that flows from Faucet A in gallons for minutes. The graph below is
the graph of this linear function. Faucet B’s water flow can be described by the equation
, where is the
volume of water in gallons that flows from the faucet in minutes. Assume the flow of water from each faucet is
constant. Which faucet has a faster rate of flow of water? Each faucet is being used to fill tubs with a volume of
gallons. How long will it take each faucet to fill the tub? How do you know? The tub that is filled by Faucet A
already has gallons in it. If both faucets are turned on at the same time, which faucet will fill its tub faster?
Student solutions will vary. Sample solution is provided.
The slope of the graph of the line is
because is a point on the line that represents gallons of water that
flows in minutes. Therefore, the rate of water flow for Faucet A is
. To determine which faucet has a faster flow
of water, we can compare their rates.
Therefore, Faucet B has a faster rate of water flow.
Faucet A
It will take minutes to fill
a tub of gallons.
Faucet B
It will take minutes to fill a
tub of gallons.
The tub filled by Faucet A that already has
gallons in it
Faucet B will fill the tub first because it will
take Faucet A minutes to fill the
tub, even though it already has gallons
in it.
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4. Two people, Adam and Bianca, are competing to see who can save the most money in one month. Use the table
and the graph below to determine who will save more money at the end of the month. State how much money
each person had at the start of the competition.
Adam’s Savings: Bianca’s Savings:
The slope of the line that represents Adam’s savings is ; therefore, the rate at which Adam is saving money is
per day. According to the table of values for Bianca, she is also saving money at a rate of per day:
Therefore, at the end of the month, Adam and Bianca will both have saved the same amount of money.
According to the graph for Adam, the equation represents the function of money saved each day. On
day zero, he must have had .
The equation that represents the function of money saved each day for Bianca is because using the
assignment of to :
The amount of money Bianca had on day zero is .
Input
(Number of Days)
Output
(Total amount of
money)
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Discussion (5 minutes)
To encourage students to compare different methods of solving problems and to make connections between them, ask
students to describe their methods for determining the answers to Exercises 1–4. Use the following questions to guide
the discussion.
Was one method more efficient than the other? Does everyone agree? Why or why not?
How did you know which method was more efficient? Did you realize at the beginning of the problem or after
they finished?
Did you complete every problem using the same method? Why or why not?
Closing (5 minutes)
Summarize, or ask students to summarize, the main points from the lesson:
We know that functions can be expressed as equations, graphs, tables, and using verbal descriptions.
Regardless of the way that the function is expressed, we can compare it with another function.
We know that we can compare two functions using different methods. Some methods are more efficient than
others.
Exit Ticket (5 minutes)
Fluency Exercise (10 minutes): Multi-Step Equations II
RWBE: During this exercise, students will solve nine multi-step equations. Each equation should be solved in about a
minute. Consider having students work on white boards, showing you their solutions after each problem is assigned.
The nine equations and their answers are below. Refer to the Rapid White Board Exchanges section in the Module
Overview for directions to administer a RWBE.
MP.1
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Name Date
Lesson 7: Comparing Linear Functions and Graphs
Exit Ticket
Brothers, Paul and Pete, walk miles to school from home. Paul can walk to school in minutes. Pete has slept in
again and needs to run to school. Paul walks at constant rate, and Pete runs at a constant rate. The graph of the
function that represents Pete’s run is shown below.
a. Which brother is moving at a greater rate? Explain how you know.
b. If Pete leaves minutes after Paul, will he catch Paul before they get
to school?
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Exit Ticket Sample Solutions
Brothers, Paul and Pete, walk miles to school from home. Paul can walk to school in minutes. Pete has slept in
again and needs to run to school. Paul walks at constant rate, and Pete runs at a constant rate. The graph of the function
that represents Pete’s run is shown below.
a. Which brother is moving at a greater rate? Explain how you know.
Paul takes minutes to walk miles; therefore, his rate is
.
Pete can run miles in minutes; therefore, his rate is
, or
Since
, Pete is moving at a greater rate.
b. If Pete leaves minutes after Paul, will he catch Paul before they
get to school?
Student solution methods will vary. Sample answer is shown.
Since Pete slept in, we need to account for that fact. So, Pete’s
time would be decreased. The equation that would represent the
number of miles Pete walks, , walked in minutes, would be
.
The equation that would represent the number of miles Paul runs, , run in minutes, would be
To find out when they meet, solve the system of equations:
or
Pete would catch up to Paul in
minutes, which is equal to
miles. Yes, he will catch Paul before they get
to school because it is less than the total distance, two miles, to school.
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Problem Set Sample Solutions
1. The graph below represents the distance,, Car A travels in minutes. The table represents the distance, , Car B
travels in minutes. Which car is traveling at a greater speed? How do you know?
Car A
Car B:
Time in minutes
()
Distance
()
Based on the graph, Car A is traveling at a rate of miles every minutes,
. From the table, the rate that Car
B is traveling is constant, as shown below.
Since
, Car B is traveling at a greater speed.
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2. The local park needs to replace an existing fence that is feet high. Fence Company A charges for building
materials and per foot for the length of the fence. Fence Company B charges based on the length of the fence.
That is, the total cost of the -foot high fence will depend on how long the fence is. The table below represents the
inputs and the corresponding outputs that the function for Fence Company B assigns.
Input
(length of fence)
Output
(cost of bill)
a. Which company charges a higher rate per foot of fencing? How do you know?
Let represent the length of the fence and represent the total cost.
The equation that represents the function for Fence Company A is . So, the rate is .
The rate of change for Fence Company B:
Fence Company B charges a higher rate per foot because when you compare the rates, .
b. At what number of the length of the fence would the cost from each fence company be the same? What will
the cost be when the companies charge the same amount? If the fence you need is feet in length, which
company would be a better choice?
Student solutions will vary. Sample solution is provided.
The equation for Fence Company B is
.
We can find out at what point the fence companies charge the same amount by solving the system:
At feet of fencing, both companies would charge the same amount (about ). Less than
feet of fencing means that the cost from Fence Company A will be more than Fence Company B. More
than feet of fencing means that the cost from Fence Company B will be more than Fence Company A.
So, for feet of fencing, Fence Company A is the better choice.
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3. The rule is used to describe the function for the number of minutes needed, , to produce toys at Toys
Plus. Another company, #1 Toys, has a similar function that assigned the values shown in the table below. Which
company produces toys at a slower rate? Explain.
Time in minutes
()
Toys Produced
()
#1 Toys produces toys at a constant rate because the data in the table increases at a constant rate, as shown below.
The rate of production for Toys Plus is and for #1 Toys is . Since , #1 Toys produces toys at a
slower rate.
4. A function describes the number of miles a train can travel, , for the number of hours, . The figure shows the
graph of this function. Assume that the train travels at a constant speed. The train is traveling from City A to City B
(a distance of miles). After hours, the train slows down to a constant speed of miles per hour.
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a. How long will it take the train to reach its destination?
Student solutions will vary. Sample solution is provided.
The equation for the graph is . If the train travels for hours at a rate of miles per hour, it will
have travelled miles. That means it has miles to get to its destination. The equation for the second
part of the journey is . Then,
This means it will take about hours
for the train to reach its destination.
b. If the train had not slowed down after hours, how long would it have taken to reach its destination?
The train would have reached its destination in about hours had it not slowed down.
c. Suppose after hours, the train increased its constant speed. How fast would the train have to travel to
complete the destination in hours?
Let represent the new constant speed of the train; then,
The train would have to increase its speed to about miles per hour to arrive at its destination hours
later.
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5.
a. A hose is used to fill up a gallon water truck at a constant rate. After minutes, there are gallons
of water in the truck. After minutes, there are gallons of water in the truck. How long will it take to fill
up the water truck?
Student solutions will vary. Sample solution is provided.
Let represent the time in minutes it takes to pump gallons of water. Then, the rate can be found as
follows:
Time in minutes ()
Amount of water pumped in gallons ()
Since the water is pumping at a constant rate, we can assume the equation is linear. Therefore, the equation
for the first hose is found by
.
If we multiply the first equation by :
The equation for the first hose is
. If the hose needs to pump gallons of water into the
truck, then
.
It would take about minutes or about hours to fill up the truck.
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b. The driver of the truck realizes that something is wrong with the hose he is using. After minutes, he shuts
off the hose and tries a different hose. The second hose has a constant rate of gallons per minute. How
long does it take the second hose to fill up the truck?
Since the first hose has been pumping for minutes, there are gallons of water already in the truck.
That means the new hose only has to fill up gallons. Since the second hose fills up the truck at a
constant rate of gallons per minute, the equation for the second hose is .
It will take the second hose minutes (or a little less than an hour) to finish the job.
c. Could there ever be a time when the first hose and the second hose filled up the same amount of water?
To see if the first hose and the second hose could have ever filled up the same amount of water, I would need
to solve for the system:
The second hose could have filled up the same amount of water as the first hose at about minutes.
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Multi-Step Equations II
1.
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