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Success Center
Directed Learning Activity (DLA)
Linear Model
Applications
M101.1
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Directed Learning Activity: Mathematical Applications
Title: Linear Model Applications
Purpose: To deepen ones understanding of linear equations through their application to real world
situations.
Prior Knowledge: In order to be able to complete this activity, you are expected to know how to find the
slope, y-intercept, and the equation of a line if you know two points on the line. If you have not
covered these topics in your math class, then this is not the appropriate time for you to do this
activity. If you have covered these topics in class, but you do not understand how to do these
things, get help with one of the tutors.
Before you begin this activity, you should be able to do the following problem:
Line A passes through the points (3, 4) and (1, 2).
a) Find the slope of line A
b) Find the equation of line A in slope-intercept form (that is y  mx  b )
c) Give the coordinates of the y-intercept of line A.
y-intercept: ________________
Introduction: Since you are working on this DLA, presumably, you have already learned how to find the
slope, y-intercept, and the equation of a line if you know two points on the line. But honestly, how
well do you really understand the meaning of slope and y-intercept? If you are like many other
algebra students, the answer is, “not very well at all.” Through this activity, you will be challenged
to think about the meaning of the numbers that we call slope and y-intercept n the context of a reallife situation. In order to get the most out of this activity, you must complete each step in order.
Remember, the “right answer” is not the most important thing in this activity, but rather, strive to
understand the meaning of every number that you write down. The process is more important than
the answer!!
Directions: Read carefully the following problems, and answer the questions that follow each problem.
Once you complete the problems, spend some time thinking about this activity and complete the
reflection section.
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Problem 1
Sam and Mary rented the same type of car from AVIS for one day. Sam drove his car for 120 miles and
he paid $71, while Mary drove her car for 80 miles, and she paid $59. You may disregard any taxes.
a) What are the two variable quantities in this situation?
b) Which one of the variables would you classify a the independent variable and which on as the
dependent variable? Explain your answer using complete sentences.
c) Given your previous answer and the usual way in which we define x and y when we work with
linear equations, complete the following statements:
Let x = _________________________________________ and
y = _________________________________________
d) Now, write the information given in the problem as two ordered pairs: (
,
) and (
,
e) Label each axes in the coordinate plane given below. Be sure to include units
_________________________________
100
80
60
40
20
_____________________________
0
0
20
40
60
80
100
120
140 150
f) Plot the points in part d) and use them to graph the line that represents the relationship between the
distance driven and the total rental cost.
)
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g) Find the slope of the line you graphed. As you do this, think carefully about the units of the numbers
you write down. This will help you understand the meaning of slope.
h) What does this slope represent in the context of this situation? Be specific and answer using
complete sentences.
i) Write the equation of the line you graphed.
j) According to your equation, what is the y-intercept of the line?
k) Now look at your graph and locate the y-intercept on the graph? What is it? Does it agree with your
answer in part j)? If not make the appropriate corrections and briefly explain why you think the
answers did not agree.
l) What does the y-intercept represent in the context of this problem? Be specific and answer using
complete sentences.
m) Now look back at your equation and your graph. Can you see the y-intercept and the slope on the
equation? Can you see them on the graph? Label the y-intercept on the graph and think of a way to
show the slope on the graph.
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n) The equation you wrote in part i) is called a Linear Model. It is called a model because you can use
it to simulate the real-life situation. For instance, suppose that another person, say Mike, rents the
same car for one day and drives it for 195 miles. You can use your linear model to figure out how
much Mike should be charged. Do it and write your answer in a complete sentence.
o) Now suppose that another person, say Kim, wants to rent the same car for one day and does not
want to spend more than $80. Use your linear model to figure out how many miles she would be
able to drive the car.
Problem 2
When Larry became president of a service organization, he decided to start a membership campaign
with the goal of signing up 4 new members each week. He hoped that after 10 weeks the club’s total
membership would be 375.
a) What are the two variable quantities in this situation?
b) Which one of the variables would you classify a the independent variable and which on as the
dependent variable? Explain your answer using complete sentences.
c) Given your previous answer and the usual way in which we define x and y when we work with
linear equations, complete the following statements:
Let x = _________________________________________ and
y = _________________________________________
d) Now, use the given information to write an ordered pair: (
,
)
e) Assuming that Larry’s membership campaign works as planned, what is the constant rate of change
of the club’s membership? Be sure to include units.
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f) Explain how you can use your answers to parts d) and e) to find more ordered pairs
g) Label each axes in the coordinate plane given below. Be sure to include units
h) Graph the line that represents the relationship between the number of weeks since Larry became
president and the total club membership,
i) Write the equation of the line you graphed.
j) According to your equation, what is the slope of the line and what does it represent in this situation?
k) According to your equation, what is the y-intercept of the line? Now look at your graph and locate
the y-intercept on the graph? What is it? Does the answer you got from the equation match the
answer you got from the graph? If not, make the appropriate corrections and briefly explain why you
think the answers did not agree.
l) What does the y-intercept represent in the context of this problem? Be specific and answer using
complete sentences.
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m) Now look back at your equation and your graph. Can you see the y-intercept and the slope on the
equation? Can you see them on the graph? Label the y-intercept on the graph and think of a way to
show the slope on the graph.
n) The equation you wrote in part i) is called a Linear Model. It is called a model because you can use
it to simulate the real-life situation. For instance, suppose that you want to predict the number of
club members one year after Larry became president. Use your linear model to figure this out. Write
your answer in a complete sentence.
o) If Larry continues as president and the membership continues to grow at this constant rate, after how
many weeks after he first became president did the membership reach 1000?
Reflection:
a) Name one thing that you understand better about linear equations as a result of completing this
activity.
b) Name one thing about which you are still not clear related to linear equations?
c) Can you think of a way to make this activity more useful to you and other students?
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M101.1 – Linear Model Applications
__________________________________________________
PRINT STUDENT NAME
_______________________
STUDENT #
_________ Student completed the entire activity
_________ Student wrote answers using complete sentences
_________ Student demonstrated understanding of the process during the discussion of his/her work
Additional Comments:
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
___________________________________________________
PRINT INSTRUCTOR/TUTOR NAME
___________________
DATE
INSTRUCTOR/TUTOR SIGNATURE
STUDENT – DO NOT FORGET TO TURN THIS SHEET IN
AT THE FRONT DESK!
You may not get credit for completing this DLA if you fail
to leave this sheet with the front desk receptionist.