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Modeling Linear Relationships




 


Videos to help Grade 8 students learn how to determine a linear function given a verbal description of a linear relationship between two quantities.

New York State Common Core Math Grade 8, Module 6, Lesson 1.

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Common Core For Grade 8

Lesson 1 Student Outcomes

• Students determine a linear function given a verbal description of a linear relationship between two quantities.
• Students interpret linear functions based on the context of a problem.
• Students graph linear functions by constructing a table of values, plotting points, and drawing the line.

Lesson 1 Summary

A linear function can be used to model a linear relationship between two types of quantities. The graph of a linear function is a straight line.

A linear function can be constructed using a rate of change and initial value. It can be interpreted as an equation of a line in which:

  • The rate of change is the slope of the line and describes how one quantity changes with respect to another quantity.
  • The initial value is the y-intercept.

Lesson 1 Classwork

Example 1: Logging On
Lenore has just purchased a tablet computer, and she is considering purchasing an “access plan” so that she can connect to the Internet wirelessly from virtually anywhere in the world. One company offers an internet access plan so that when a person connects to the company's wireless network, the person is charged a fixed access fee for connecting, PLUS an amount for the number of minutes connected based upon a constant usage rate in dollars per minute. Lenore is considering this company’s plan, but the company’s advertisement does not state how much the fixed access fee for connecting is, nor does it state the usage rate. However, somewhere on the company’s website, it says that a 10 minute session costs $0.40, a 20 minute session costs $0.70, and a 30 minute session costs $1.00. Lenore decides that she will use these pieces of information to determine both the fixed access fee for connecting and the usage rate.

Exercises 1 - 6
1. Lenore makes a table of this information and a graph where number of minutes is represented by the horizontal axis and total session cost is represented by the vertical axis. Plot the three given points on the graph. These three points appear to lie on a line. What information about the access plan suggests that the correct model is indeed a linear relationship?

2. The rate of change describes how the total cost changes with respect to time.
a. When the number of minutes increases by 10(such as from 10 minutes to 20 minutes or from 20 minutes to 30 minutes), how much does the charge increase?
b. Another way to say this would be the “usage charge per 10 minutes of use.” Use that information to determine the increase in cost based on only ONE minute of additional usage. In other words, find the “usage charge per minute of use.”

3. The company’s pricing plan states that usage rate is constant for any number of minutes connected to the internet. In other words, the increase in cost for 10 more minutes of use (the value that you calculated above) will be the same whether you increase from 20 to 30 minutes, 30 to 40 minutes, etc. Using this information, determine the total cost for 40 minutes, 50 minutes, and 60 minutes of use. Record those values in the table, and plot the corresponding points on the graph in Exercise 1.

4. Using the table and the graph in Exercise 1, compute the hypothetical cost for zero minutes of use. What does that value represent in the context of the values that Lenore is trying to figure out?

5. On the graph in Exercise 1, draw a line through the points representing 0 to 60 minutes of use under this company’s plan. The slope of this line is equal to the rate of change, which in this case is the usage rate.

6. Using for the number of minutes and for total cost in dollars, write a function to model the linear relationship between minutes of use and total cost.

Example 2: Another Rate Plan
A second wireless access company has a similar method for computing its costs. Unlike the first company that Lenore was considering, this second company explicitly states its access fee is $0.15, and its usage rate is $0.04 per minute: Total Session Cost = $0.15 + $0.04(number of minutes)

Exercises 7 - 9
7. Let x represent the number of minutes used and y represent the total session cost. Construct a linear function that models the total session cost based on the number of minutes used.

8. Using the linear function constructed in Exercise 7, determine the total session cost for sessions of 0, 10, 20, 30, 40, 50, and 60 minutes, and fill in these values in the table below.

9. Plot these points on the original graph in Exercise 1 and draw a line through these points. In what ways does the line that represents this second company's access plan differ from the line that represented the first company's access plan?

Exercises 10 - 12
MP3 download sites are a popular forum for selling music. Different sites offer pricing that depend on whether or not you want to purchase an entire album or individual songs “a la carte.” One site offers MP3 downloads of individual songs with the following price structure: a $3 fixed fee for monthly subscription PLUS a charge of $0.25 per song.

10. Using x for the number of songs downloaded and y for the total monthly cost, construct a linear function to model the relationship between the number of songs downloaded and the total monthly cost

11. Construct a table to record the total monthly cost (in dollars) for MP3 downloads of 10 songs, 20 songs, and so on up to 100 songs.

12. Plot the 10 data points in the table on a coordinate plane. Let the x-axis represent the number of songs downloaded and the y-axis represent the total monthly cost (in dollars) for MP3 downloads.





Exercises 13 - 16
A band will be paid a flat fee for playing a concert. Additionally, the band will receive a fixed amount for every ticket sold. If 40 tickets are sold, the band will be paid $200. If 70 tickets are sold, the band will be paid $260.

13. Determine the rate of change.

14. Let x represent the number of tickets sold and y represent the amount the band will be paid. Construct a linear function to represent the relationship between the number of tickets sold and the amount the band will be paid.

15. What is the fee the band will be paid for playing the concert (not including ticket sales)?

16. How much will the band receive for each ticket sold?


Lesson 1 Exit Ticket

A rental car company offers a rental package for a mid-size car. The cost is comprised of a fixed 30 administrative fee for the cleaning and maintenance of the car plus a rental cost of 3 per day.

1. Using for the number of days and for the total cost in dollars, construct a function to model the relationship between the number of days and the total cost of renting a mid-size car.

2. The same company is advertising a deal on compact car rentals. The linear function 30 + can be used to model the relationship between the number of days ( ) and the total cost ( ) of renting a compact car.

a. What is the fixed administrative fee?

b. What is the rental cost per day?



 

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