Erie Community College
Title III
Linear Equations Project
Interdisciplinary Course Materials
Office Management
Course: MT007/ M013
Course Outline Topic: Graphing: Cartesian coordinate system, Graphing Linear equations, slope intercept form, point-slope form and applications
Project Title: Depreciation of a Car or As the Age Goes Up, the Price Goes Down
Project description: Students will be asked to analyze some data of the age and price of two different car models. They will do this using their knowledge about graphing, slope, and simple linear equations that they should have already learned in the graphing section of Elementary Algebra.
Author: Janet Evert
Curriculum Expert: Susan Ballard
Semester Created: Fall 2007
A. Essential Question (What does this project attempt to answer?) How is the price of a car related to the age of a car? What is the rate of change of the price of a car in relation to the age of the car? Is this the same for all makes and models of cars? How can we use what we know about linear equations and slopes to make the best choice in purchasing a car?
B. Introduction
You will use your knowledge of graphing to really understand what is involved in the process of making a wise purchase when it comes to buying a car. You will analyze the data of new and used car prices for two different makes and models of cars.
After doing this, you will then be able to analyze what happens to the value of any car that you might be interested in buying. This will better prepare you to make a wise choice in purchasing any car.
C. Basic Directions: You will use the data listed for two different models of car listed in the tables below. Using what you know about slopes and lines, you will find the average rate of change for each model and find the depreciation rate of each car. You will determine a linear equation that would allow you to predict the value of a car based on age for a certain time period.
D. Things to Learn Before Starting the Project
1. Know how to create a graph and plot points.
2. Know how to find the slope going through two points.
3. Understand the concept of a slope and how it applies to a real-life situation.
4. Know how to write an equation of a line knowing the slope and a point.
5. Understand the concept of the y-intercept both graphically and algebraically.
E. The Project Assignment:
Listed below are prices of two different models of cars. One is a Honda Civic—the other is a Volvo S60.
These prices were obtained from Kelly Blue book (www.kbb.com)., All models chosen have the same options (4-door sedan, automatic, etc.) and are listed in Good Condition. For used cars, it was assumed that the mileage would increase 12, 000 miles each year. Prices chosen were for Private Party Sales.
Honda Civic EX - 4-Door Sedan
Model Year / Age / Price2008 / new / $20,145
2007 / 1 / $17,795
2006 / 2 / $16,810
2005 / 3 / $14,420
2004 / 4 / $13,000
2003 / 5 / $11,365
Volvo S60 – 4-Door Sedan
Model Year / Age / Price2008 / new / $31,472
2007 / 1 / $22,745
2006 / 2 / $20,510
2005 / 3 / $17,900
2004 / 4 / $15,220
2003 / 5 / $13,895
Part I – Create a Graph
1. Determine which value is the input value and which one is the output value for this problem. The input value is normally graphed along the horizontal axis and is typically the one that you would use to find the output value. Use P for Price of a car and A for Age of a car.
2. Using a sheet of graph paper, you are to plot the points for both of these cars on the same graph. You must begin by choosing an appropriate scale for both the age of a car and the price of a car, so look at all of the values you have on the table to do this. Label both axes clearly and be sure that the input variable is on the horizontal axes and the output value is on the vertical axes.
Plot the points for the prices of the Honda Civic and then, on the same graph, plot the points for the Volvo S60. Be sure that the scale used makes the points easy to graph and easy to read.
Part II: - Honda Civic - first year
1. Interpret the slope of the line that connects the two points for a new Civic and one that is one-year old.
a. Using the data above and looking at the graph of the points for the Honda Civic, list below the two points that will be used to find the rate of change of the price of a Honda Civic from and age of 0 to 1 year in age. Connect these two points with a line on the graph.
List below the two points: (Age, Price)
New car: ( , )
One-year-old car: ( , )
b. Find the slope of the line connecting these two points. Write the answer as a change in price over the change in age. This is also called the depreciation rate of the price of the car in the first year.
Show the work below:
The rate of change of the price of a car in the first year is ______
An increase in age of one year would result in what change in the price of a car? ______
2. Find the vertical (P) intercept of this line.
a. Write the intercept as a point: ( , )
b. What is the practical meaning of the vertical intercept for this problem?
3. Find the equation of the line that connects the two points and would describe the relationship between the age of a car and the price of that car for the first year.
Explain how you did this and show the work below.
Part III: Honda Civic – First year to third year
1. Interpret the slope of a line that connects the two points for a car that is one-year old and a car that is three-years old.
Using the data given and looking at the graph of the points, list below the two points that will be used to find the rate of change of the price of a Honda civic from and age of one to three years in age. Connect these points on the graph.
List below the two points: (Age, Price)
One-year-old car: ( , )
Three-year-old car: ( , )
2. a. Determine the rate of change in the price of a car from its one-year value to its three-year value. (Find the slope of the line connecting these two points.)
b. Is the rate of change any different than it was for the first year? Compare the two slopes.
c. What can you say about the rate of depreciation comparing the two rates.
3. Find the equation of a line that connects the one-year and three-year points that would describe the relationship between the price of that car at its age.
Explain how you did this and show the work below.
4. Estimate the cost of buying a used 6-year old Honda Civic using the information from the first to the third year assuming that the slope remained the same.
a. Do this first by looking at the graph. (What is your best guess of the estimated cost?)
b. Now find the answer algebraically.
c. Do you think this equation would be a good predictor of car prices for cars that are older (ex. 10-years old)? Explain your reasoning.
Part IV: Volvo S60
Analyze the data for the Volvo S80 in the table above as you did for the Honda Civic. Follow each step as you did for the Civic and then compare the two.
1. List the points that would be used to find the rate of change of the price of a Volvo S60 for the first year:
List below the points as: (Age, Price)
New car: ( , )
One-year-old car: ( , )
2. Find the depreciation rate of the Volvo S60 for the first year (the slope of the line connecting the first two points).
3. Find the vertical (P) intercept of this line. What is the practical meaning of it for this problem?
4. Find the equation of the line that would describe the relationship between the age of a car and the price of that car for the first year.
Find the equation of the line that connects the two points. Explain how you did this and show the work below.
Part V: Volvo S60 – First year to third year
Find the depreciation rate of the Volvo S60 for the first year (the slope of the line connecting the first two points:
1. List below the two points used: (Age, Price)
One-year-old car: ( , )
Three-year-old car: ( , )
2. Determine the depreciation rate from one to three years. (Find the slope of the line connecting these two points.)
Is the slope any different for these two points than it was for the first part where you found the rate of change for the first year? Compare the two slopes.
3. Using a point and the slope (for the points of one and three years), find the equation of the line that would describe the relationship between the age of a car and the price.
Explain how you did this and show the work below.
4. Estimate the cost of buying a used Volvo S60 that is 6-years old.
a. Do this first by looking at the graph. (What is your best guess of the estimated cost?)
b. Find the answer algebraically. Using the equation that would make the most sense, predict the price of a 6-year old Volvo S60.
Part VI: Comparison
1. Looking at the two graphs, compare the rate of change in price for the first year for a Honda Civic and for a Volvo S60. Is there much of a difference? Which car does a better job of holding its value after one year?
2. Looking at the graphs from one year to three years, compare the average rate of change for the Honda and the Volvo. Is there much of a difference in the average rate of change?
3. For what age values would this problem make sense? (What range of input values would make sense for each of the two equations?) Would the equation hold true for the entire life span of the car? Why or why not? Explain your answer.
4. Which car would you prefer to buy? Explain your reasoning. Describe which car would be your best buy. Which would hold its value better?
Part VII: Do some investigating on your own
Now you are to do some investigating on your own. You will use your knowledge of graphing to really understand what is involved in the process of making a wise purchase when it comes to buying a car.
Choose a make and a model of a car that you would like to purchase. Find the new-car price, one-year price, and one that is at least two-years old. Choose models that have the same features and options, and similar conditions. Mileage should increase the same amount each year (for example, 12,000 miles per year). Use prices for private-party sales. Find the depreciation rate for a car from its new-car price to its one-year old price. Go to www.kkb.com; get the values of several different years. Enter the values in the table below.
Be sure to keep the cars similar so that you get an accurate comparison.
Make and Model of Car: ______
Model Year / Age / Price2008 / new
2007 / 1
Find each of the following and include all work.
1. The depreciation rate of the car for the first year.
2. The depreciation rate of the car between the ______year and the ______year.
3. The equation of the line that would best describe the price of a car (that is more than two-years old) based on the age of the car.
4. Predict the price of the car at an age of your choice. How much would it cost to buy a car that is ______years old? Show your work below.
5. Using Kelley Blue Book, find the price of the car and age that you chose in Step4 above. Compare your prediction with the Kelley Blue Book price. Were the prices close? If not, what do you think is the reason for the difference?
F. Student Resources (websites, books, technology, etc.) Students should refer to the Chapter on Graphing Linear Equations. In the current text, Elementary Algebra for College Students (7th edition) by Alan Angel, that would be chapter seven.
Calculators should be used for this project.
G. Faculty Resources (teacher notes, websites, books, technology, etc.)
This would make a good group project as students would need to discuss the best choices and the reasoning involved in making their decisions. This works well especially for choosing the proper scales for graphing the problem initially.
This project also provides an opportunity to discuss the fact that the linear relationship of the price of a car based on its age may hold true for only a certain range of ages. This project could be expanded to have students look at what happens to prices of cars that are older than 5 years of age.
Information on pricing can be found at www.kbb.com. Prices may also be found at similar Web sites.
H. Grading Rubric
All parts should be graded on Mathematical logic used through work shown (50%) as well as accuracy (50%)
Part I Create a Graph 10%
Part II Honda Civic – First Year 15%
Part III Honda Civic – First Year to Third Year 15%
Part IV Volvo – First Year 15%
Part V Volvo – First Year to Third Year 15%
Part VI Comparison 10%
Part VII Your Own Investigation 20%
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Erie Community College
Title III Grant