 # Linear Models

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Examples, solutions, worksheets, videos, and lessons to help Grade 8 students learn how to identify situations where it is reasonable to use a linear function to model the relationship between two numerical variables.
Students interpret slope and the initial value in a data context.

### New York State Common Core Math Grade 8, Module 6, Lesson 10

Lesson 10 Student Outcomes

• Students identify situations where it is reasonable to use a linear function to model the relationship between two numerical variables.
• Students interpret slope and the initial value in a data context.

Lesson 10 Summary
• A linear functional relationship between a dependent and independent numerical variable has the form y = mx + c or y = a + bx
• In statistics, a dependent variable is one that is predicted and an independent variable is the one that is used to make the prediction.
• The graph of a linear function describing the relationship between two variables is a line.

Lesson 10 Classwork

Example 1
Predicting the value of a numerical dependent (response) variable based on the value of a given numerical independent variable has many applications in statistics. The first step in the process is to be able to identify the dependent variable (predicted variable) and the independent variable (predictor).
There may be several independent variables that might be used to predict a given dependent variable. For example, suppose you want to predict how well you are going to do on an upcoming statistics quiz. One of the possible independent variables is how much time you put into studying for the quiz. What are some other possible numerical independent variables that could relate to how well you are going to do on the quiz?

Exercise 1
1. For each of the following dependent (response) variables, identify two possible numerical independent (explanatory) variables that might be used to predict the value of the dependent variable.

Exercise 2
2. Now, reverse your thinking. For each of the following numerical independent variables, write a possible numerical dependent variable.

Example 2
A cell phone company is offering the following basic cell phone plan to its customers: A customer pays a monthly fee of \$40.00. In addition, the customer pays \$0.15 per text message sent from the cell phone. There is no limit to the number of text messages per month, and there is no charge for receiving text messages.

Exercises 3–9
3. Determine the following:
a. Justin never sends a text message. What would be his total monthly cost?
b. During a typical month, Abbey sends 25 text messages. What is her total cost for a typical month?
c. Robert sends at least 250 text messages a month. What would be an estimate of the least his total monthly cost is likely to be?

4. Write a linear model describing the relationship between the number of text messages sent and the total monthly cost using descriptive words.

5. Is the relationship between the number of text messages sent and the total monthly cost linear? Explain your answer.

6. Let x represent the independent variable and represent the dependent variable. Write the function representing the relationship you indicated in Exercise 4 using the variables x and y.

7. Explain what \$0.15 represents in this relationship.

8. Explain what \$40.00 represents in this relationship.

9. Sketch a graph of this relationship on the following coordinate grid. Clearly label the axes and include units in the labels.

Exercise 10
10. LaMoyne needs four more pieces of lumber for his scout project. The pieces can be cut from one large piece of lumber according to the following pattern.
The lumberyard will make the cuts for LaMoyne at a fixed cost of \$2.25 plus an additional cost of 25 cents per cut. One cut is free.
a. What is the functional relationship between the total cost of cutting a piece of lumber and the number of cuts required? What is the equation of this function? Be sure to define the variables in the context of this problem.
b. Use the equation to determine LaMoyne’s total cost for cutting.
c. Interpret the slope of the equation in words in the context of this problem.
d. Interpret the intercept of your equation in words in the context of this problem. Does interpreting the intercept make sense in this problem? Explain.

Exercise 11
11. Omar and Olivia were curious about the size of coins. They measured the diameter and circumference of several coins and found the following data.
a. Wondering if there was any relationship between diameter and circumference, they thought about drawing a picture. Draw a scatter plot that displays circumference in terms of diameter.
b. Do you think that circumference and diameter are related? Explain.
c. Find the equation of the function relating circumference to the diameter of a coin.
d. The value of the slope is approximately equal to the value of π. Explain why this makes sense.
e. What is the value of the intercept? Explain why this makes sense.

Lesson 10 Exit Ticket

Suppose that a cell phone monthly rate plan costs the user 5 cents per minute beyond a fixed monthly fee of \$20. This implies that the relationship between monthly cost and monthly number of minutes is linear.
1. Write an equation in words that relates total monthly cost to monthly minutes used. Explain how you found your answer.
2. Write an equation in symbols that relates the total month cost (y ) to monthly minutes used (x ).
3. What would be the cost for a month in which minutes were used? Express your answer in words in the context of this problem.

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