Modeling with Linear Functions - Your Local Advisoralgebra1withmrleon.weebly.com/uploads/3/1/4/9/31491391/...1 Adapted from: Day 11: Modeling with Linear Functions MAFS.912.A-CED.1.2,

MAFS Algebra 1

Modeling with Linear Functions

Day 11 - Student Packet

Adapted from: 1

Day 11: Modeling with Linear Functions MAFS.912.A-CED.1.2, MAFS.912.A-CED.1.3, MAFS.912.S-ID.2.6, MAFS.912.S-ID.3.8, MAFS.912.S-ID.3.9

I CAN…

identify the quantities in a real-world situation that should be represented by distinct variables

write a system of equations given a real-world situation

graph a system of equations that represents a real-world context

write constraints for a real-world context

interpret the solution of a real-world context as viable or not viable

represent data on a scatter plot

identify a linear function, a quadratic function, or an exponential function that was found using regression

use a regression equation to solve problems in the context of the data

calculate residuals

create a residual plot and determine whether a function is an appropriate fit for the data

determine the fit of a function by analyzing the correlation coefficient

distinguish between situations where correlation does not imply causation

distinguish variables that are correlated because one is the cause of another Solution Sets to Inequalities with Two Variables

Problem Set Andy’s Cab Service charges a fee plus per mile. His twin brother Randy starts a rival business where he charges per

mile but does not charge a fee.

1. Write a cost equation for each cab service in terms of the number

of miles.

2. Graph both cost equations.

3. For what trip distances should a customer use Andy’s Cab Service?

For what trip distances should a customer use Randy’s Cab Service?

Justify your answer algebraically, and show the location of the

solution on the graph.

Adapted from: 2

4. Find two numbers such that the sum of the first and three times the second is 5 and the sum of second and two times the first is

8.

5. A chemist has two solutions: a 50% methane solution and an 80% methane solution. He wants 100 ml of a 70% methane

solution. How many ml of each solution does he need to mix?

6. Pam has two part time jobs. At one job, she works as a cashier and makes $8 per hour. At the second job, she works as a tutor

and makes $12 per hour. One week she worked 30 hours and made $268. How many hours did she spend at each job?

7. A store sells Brazilian coffee for $10 per lb. and Columbian coffee for $14 per lb. If the store decides to make a 150-lb. blend of

the two and sell it for $11 per lb., how much of each type of coffee should be used?

8. The cost of a daily truck rental is $48.00, plus an additional $0.45 for every mile driven.

a. Write a function that gives the cost of the daily truck rental and use it to determine the total cost of renting the truck for a

day and driving it 60 miles.

b. How many miles can be driven to keep the cost of the rental at most $66?

9. A theater wants to take in at least $2000 for a certain matinee. Children’s tickets cost $5 each and adult tickets cost $10 each.

a. Write an inequality describing the number of tickets that will allow the theater to meet their goal of $2000.

b. If the theater has a maximum of 350 seats, write an inequality describing the number of both types of tickets the theater

can sell.

c. Find the number of children and adult tickets that can be sold so that all seats are sold and the $2000 goal is reached.

10. James is trying to expand his pastry business to include cupcakes and personal cakes. He has 40 hours available to decorate the

new items and can use no more than 22 pounds of cake mix. Each personal cake requires 2 pounds of cake mix and 2 hours to

decorate. Each cupcake order requires one pound of cake mix and 4 hours to decorate. If he can sell each personal cake for

$14.99 and each cupcake order for $16.99, how many personal cakes and cupcake orders should James make to make the most

revenue?

Adapted from: 3

11. Jim tells you he paid a total of $23,078.90 for a car, and you would like to know the price of the car before sales tax so that you

can compare the price of that model of car at various dealers. Find price of the car before sales tax if Jim bought the car in each

of the following states:

a. Arizona, where the sales tax is 6.6%.

b. New York, where the sales tax is 8.25%.

c. A state where the sales tax is %.

12. The sum of two numbers is 70 and the difference is 11. What are the numbers?

13. A rectangular field is enclosed by a fence on three sides and a wall on the fourth side. The total length of the fence is

320 yards. If the field has a total perimeter of 400 yards, what are the dimensions of the field?

14. A ray cuts a line forming two angles. The difference between the two angles is . What does each angle measure?

15. Of the two non-­‐right angles in a right triangle, one measures twice as many degrees as the other. What are the angles?

16. A phone company offers a choice of three text-messaging plans. Plan A gives you unlimited text messages for $10 a month; Plan

B gives you 60 text messages for $5 a month and then charges you $0.05 for each additional message; and Plan C has no

monthly fee but charges you $0.10 per message.

a. Write an equation for the monthly cost of each of the three plans.

b. If you send 30 messages per month, which plan is cheapest?

c. What is the cost of each of the three plans if you send 50 messages per month?

d. Determine the values for which each plan is the cheapest?

Adapted from: 4

17. Axel and his brother like to play tennis. About three months ago they decided to keep track of how many games they have each

won. As of today, Axel has won 18 out of the 30 games against his brother.

a. How many games would Axel have to win in a row in order to have a 75% winning record?

b. How many games would Axel have to win in a row in order to have a 90% winning record?

c. Is Axel ever able to reach a 100% winning record? Explain why or why not.

d. Suppose that after reaching a winning record of 90% in part (b), Axel had a losing streak. How many games in a row would

Axel have to lose in order to drop down to a winning record of 60% again?

18. Omar has $84 and Calina has $12. How much money must Omar give to Calina so that Calina will have three times as much as

Omar?

a. Solve the problem above by setting up an equation.

b. In your opinion, is this problem easier to solve using an equation or using a tape diagram? Why?

19. Patty makes $10 per hour mowing lawns and $15 per hour babysitting. She

wants to make at least $150 per week, but can work no more than 15 hours a

week. Write and graph a system of linear inequalities. List 2 possible

combinations of hours that Patty could work at each job.

20. A movie theater charges $4.50 for children and $8.00 for adults.

a. On a certain day, 1200 people enter the theater and $8375 is collected. How many children and how many adults

attended?

b. The next day, the manager announces that she wants to see them take in $10000 in tickets. If there are 240 seats in the

house and only five movie showings planned that day, is it possible to meet that goal?

c. At the same theater, a 16-ounce soda costs $3 and a 32-ounce soda costs $5. If the theater sells 12,480 ounces of soda for

$2100, how many people bought soda?

Adapted from: 5

21. The local theater in Jamie’s home town has a maximum capacity of people. Jamie shared with Venus the following graph

and said that the shaded region represented all the possible combinations of adult and child tickets that could be sold for one

show.

a. Venus objected and said there was more than one reason that Jamie’s thinking was flawed. What reasons could Venus be

thinking of?

b. Use equations, inequalities, graphs, and/or words to describe for Jamie the set of all possible combinations of adult and

child tickets that could be sold for one show.

c. The theater charges for each adult ticket and for each child ticket. The theater sold tickets for the first showing

of the new release. The total money collected from ticket sales for that show was . Write a system of equations

that could be used to find the number of child tickets and the number of adult tickets sold, and solve the system

algebraically. Summarize your findings using the context of the problem.

22. Students and adults purchased tickets for a recent basketball playoff game. All tickets were sold at the ticket booth—season

passes, discounts, etc. were not allowed.

Student tickets cost each, and adult tickets cost each. A total of was collected. tickets were sold.

a. Write a system of equations that can be used to find the number of student tickets, , and the number of adult tickets, ,

that were sold at the playoff game.

b. Assuming that the number of students and adults attending would not change, how much more money could have been

collected at the playoff game if the ticket booth charged students and adults the same price of per ticket?

c. Assuming that the number of students and adults attending would not change, how much more money could have been

collected at the playoff game if the student price was kept at per ticket and adults were charged per ticket instead

of ?

Adapted from: 6

23. A potter is making cups and plates. It takes her 6 min. to make a cup and 3 min. to make a plate. Each cup uses 3/4 lb. of clay,

and each plate uses 1 lb. of clay. She has 20 hr. available to make the cups and plates and has 250 lb. of clay.

a. What are the variables?

b. Write inequalities for the constraints.

c. Graph and shade the solution set.

d. If she makes a profit of $2 on each cup and $1.50 on each plate, how many of each should she make in order to maximize

her profit?

e. What is her maximum profit?

Using a Line to Describe a Relationship

Kendra likes to watch crime scene investigation shows on television. She watched a show where investigators used a shoe print to

help identify a suspect in a case. She questioned how possible it is to predict someone’s height is from his shoe print.

To investigate, she collected data on shoe length (in inches) and height (in inches) from 10 adult men. Her data appear in the table

and scatter plot below.

Shoe Length Height

1. Is there a relationship between shoe length and height?

2. How would you describe the relationship? Do the men with longer shoe lengths tend be taller?

When the relationship between two numerical variables 𝒙 and 𝒚 is linear, a straight line can be used to describe the relationship.

Such a line can then be used to predict the value of 𝒚 based on the value of 𝒙. When a prediction is made, the prediction error is

the difference between the actual 𝒚-value and the predicted 𝒚-value. The prediction error is called a residual, and the residual is

calculated as residual actual 𝒚-value − predicted 𝒚-value. The least squares line is the line that is used to model a linear

relationship. The least squares line is the best line in that it has a smaller sum of squared residuals than any other line.

Adapted from: 7

3. Below is a scatter plot of the data with two linear models, − and . Which of these two

models does a better job of describing how shoe length ( ) and height ( ) are related? Explain your choice.

4. One of the men in the sample has a shoe length of inches and a height of inches. Circle the point in the scatter plot in

Question 3 that represents this man.

5. Suppose that you do not know this man’s height, but do know that his shoe length is inches. If you use the model

, what would you predict his height to be? If you use the model − , what would you predict his

height to be?

6. Which model was closer to the actual height of inches? Is that model a better fit to the data? Explain your answer.

7. Is there a better way to decide which of two lines provides a better description of a relationship (rather than just comparing the

predicted value to the actual value for one data point in the sample)?

Kendra wondered if the relationship between shoe length and height might be different for men and women. To investigate, she

also collected data on shoe length (in inches) and height (in inches) for 12 women.

Shoe Length (Women) Height (Women)

8. Construct a scatter plot of these data.

9. Is there a relationship between shoe length and height for these women?

10. Find the equation of the line of best fit. (Round values to the nearest hundredth.)

Adapted from: 8

11. Suppose that these women are representative of adult women in general. Based on the least squares line, what would you

predict for the height of a woman whose shoe length is inches? What would you predict for the height of a woman whose

shoe length is inches?

12. One of the women in the sample had a shoe length of inches. Based on the regression line, what would you predict for her

height?

13. What is the value of the residual associated with the observation for the woman with the shoe length of ?

14. Below are dot plots of the shoe lengths for women and the shoe lengths for men. Suppose that you found a shoe print and that

when you measured the shoe length, you got inches. Do you think that a man or a woman left this shoe print? Explain

your choice.

15. Suppose that you find a shoe print and the shoe length for this print is inches. What would you predict for the height of the

person who left this print? Explain how you arrived at this answer.

The Relevance of the Pattern in the Residual Plot

After fitting a line, the residual plot can be constructed using a graphing calculator.

A curve or pattern in the residual plot indicates a nonlinear relationship in the original data set.

A random scatter of points in the residual plot indicates a linear relationship in the original data set.

Scatter Plot Residual Plot

Adapted from: 9

1. For each of the following residual plots, what conclusion would you reach about the relationship between the variables in the

original data set? Indicate whether the values would be better represented by a linear or a nonlinear relationship.

a.

b.

c.

2. Suppose that after fitting a line, a data set produces the residual plot shown below.

An incomplete scatter plot of the original data set is shown below. The least squares line is shown, but the points in the scatter

plot have been erased. Estimate the locations of the original points and create an approximation of the scatter plot below.

Adapted from: 10

Interpreting Correlation

The Correlation Coefficient

The correlation coefficient is a number between -1 and +1 (including -1 and +1) that measures the strength and

direction of a linear relationship. The correlation coefficient is denoted by the letter r.

𝒓 𝟏 𝟎𝟎 𝒓 𝟎 𝟕𝟏

𝒓 𝟎 𝟑𝟐 𝒓 −𝟎 𝟏𝟎

𝒓 −𝟎 𝟔𝟑 𝒓 −𝟏 𝟎𝟎

Linear relationships are often described in terms of strength and direction.

The correlation coefficient is a measure of the strength and direction of a linear relationship.

The closer the value of the correlation coefficient is to +1 or -1, the stronger the linear relationship.

Just because there is a strong correlation between the two variables does not mean there is a cause-and-effect

relationship.

Adapted from: 11

1. Which of the three scatter plots below shows the strongest linear relationship? Which shows the weakest linear relationship?

Scatter Plot 1

Scatter Plot 2

Scatter Plot 3

2. Consumer Reports published data on the price (in dollars) and quality rating (on a scale of to ) for different brands of

men’s athletic shoes.

Price ( ) Quality

Rating

a. Construct a scatter plot of these data using the grid provided.

Correlation Does Not Mean There is a Cause-and-Effect Relationship Between Variables

It is sometimes tempting to conclude that if there is a strong linear relationship between two variables that one variable is

causing the value of the other variable to increase or decrease. But you should avoid making this mistake. When there is a

strong linear relationship, it means that the two variables tend to vary together in a predictable way, which might be due to

something other than a cause-and-effect relationship.

For example, the value of the correlation coefficient between sodium content and number of calories for the fast food items

in the previous example was r=0.79, indicating a strong positive relationship. This means that the items with higher sodium

content tend to have a higher number of calories. But the high number of calories is not caused by the high sodium content.

In fact sodium does not have any calories. What may be happening is that food items with high sodium content also may be

the items that are high in sugar or fat, and this is the reason for the higher number of calories in these items.

Similarly, there is a strong positive correlation between shoe size and reading ability in children. But it would be silly to think

that having big feet causes children to read better. It just means that the two variables vary together in a predictable way.

Can you think of a reason that might explain why children with larger feet also tend to score higher on reading tests?

Adapted from: 12

b. Calculate the value of the correlation coefficient between price and quality rating and interpret this value. Round to the

nearest hundredth.

c. Does it surprise you that the value of the correlation coefficient is negative? Explain why or why not.

d. Is it reasonable to conclude that higher priced shoes are higher quality? Explain.

e. The correlation between price and quality rating is negative. Is it reasonable to conclude that increasing the price causes a

decrease in quality rating? Explain.

3. The Princeton Review publishes information about colleges and universities. The data below are for six public 4-year colleges in

New York. Graduation rate is the percentage of students who graduate within six years. Student-to-faculty ratio is the number

of students per full-time faculty member.

School Number of Full-Time

Students

Student-to-Faculty

Ratio Graduation Rate

CUNY Bernard M Baruch College

CUNY Brooklyn College

CUNY City College

SUNY at Albany

SUNY at Binghamton

SUNY College at Buffalo

a. Calculate the value of the correlation coefficient between the number of full-time students and graduation rate. Round to

the nearest hundredth.

b. Is the linear relationship between graduation rate and number of full-time students weak, moderate or strong? On what

did you base your decision?

c. True or False? Based on the value of the correlation coefficient, it is reasonable to conclude that having a larger number of

students at a school is the cause of a higher graduation rate.

d. Calculate the value of the correlation coefficient between the student-to-faculty ratio and graduation rate. Round to the

nearest hundredth.

e. Which linear relationship is stronger: graduation rate and number of full-time students or graduation rate and student-to-

faculty ratio? Justify your choice.