Paying for Your Wheels

In this lesson, students consider the costs of owning a car and ways to lessen those costs. In particular, highway and city mileage are considered, and optimal mileage is calculated using fuel consumption versus speed data.

Research has solidly established the importance of teaching mathematics in contexts that capture student interest and involvement. Weaving real-world financial issues into secondary mathematics instruction allows teachers to infuse personal money management into standards–based math lessons. The following lesson uses ideas taken from the book Teaching Money Applications to Make Mathematics Meaningful, Grades 7-12 written by Elizabeth Marquez and Paul Westbrook.

This lesson begins with a quick introduction to the costs associated with owning a car. The major part of the lesson, however, focuses on one specific cost — fuel consumption. In particular, students will analyze the amount of fuel used by a car at various speeds, and will attempt to identify the "optimal speed" by performing regression analysis on speed versus fuel economy data. (Note that the data provided below is fictional. To make it more applicable to your students, you may wish to obtain fuel economy information about a car that students are likely to own in your area.)

To begin, ask students to imagine that they have used their hard-earned savings to buy the car shown below. To embellish the introduction, you can tell them that you scoured local publications and dealer web sites to help them find a good deal. Tell them that you found the ad shown below from a local dealership, called to inquire about the car, and believe that it is a good car for a good price.

Now that the car belongs to them, ask students how much they think it costs, on average, to operate the car each month. Give them time to think about the question, itemize all the costs they come up with, and record their estimate.

 

2002 Aston-Marquez – $4,614 RX400 4-door sedan, 2.4L, 4-cyl 4A
Bucket Seats	Center Console
Power Brakes	Rear Window Defroster
Premium Audio System	2.4L I4 DOHC 16V FI Engine
15-Inch Wheels	4-Speed Automatic Transmission
Clock	Tachometer
Steel Wheels	Traction Control
4-Wheel ABS	Driver and Passenger Front Airbags
Front and Rear Head Airbags	Anti-Theft Alarm System
Cloth Seats	Rear Split Bench Seat
Power Steering	Remote Trunk Release
Tilt Steering Wheel	Air Conditioning
Intermittent Windshield Wipers	Automatic On/Off Headlights
Daytime Running Lights	AM/FM/CD Audio System
Original Owner, Garaged 52,138 miles

 

Have students share their thoughts with a partner, reaching a consensus on the monthly expenses that are incurred. Ask a few pairs of students to present their results to the class. Then, allow all students to access the True Cost To Own Calculator available at Edmunds.com.

Alternatively, if technology is not available to all students, share the following facts with students so they can determine the reasonableness of their estimates. (The costs shown below are based on information taken from the True Cost To Own calculator for a car similar to the Aston-Marquez described above and on using a northern New Jersey zip code.)

• Insurance: $94. Insurance costs about $1,132 annually, although it could be as high as $2,000 a year ($167 per month) depending on your driving record and the type of coverage you select.
• Maintenance and Repairs: $104. These costs include scheduled maintenance (the items defined by warranty) as well as unscheduled maintenance and repairs for batteries, brakes, headlights, hoses, turn-signal bulbs, wiper blades, tires, and so forth.
• Fuel: $101. This estimate is based on EPA mileage figures, assuming 45% highway driving and 55% city driving, automatic transmission, self-service gas prices (about $2.19 per gallon), and driving 15,000 miles per year.
• Registration/License/Inspection/Taxes: $28.
• Financing: $25. Edmunds includes financing as an expense, even if you do not need to finance the vehicle. They do this to show the "opportunity cost" of owning a car. Paying for the cost of the car ($4,614) means that you don't have the money to invest in something that appreciates (yields a return) rather than depreciates.
• Depreciation: $53. This assumes that the vehicle is in good condition, is driven approximately 15,000 miles per year, and will eventually be sold to a private party.

Students will probably be surprised that the True Cost To Own calculator estimates that the total monthly cost exceeds $400, especially since the car is "paid for." Students may want to remove depreciation and financing as considerations, since they were initially asked to consider only operating costs. Still, the remaining cost of $327 a month is probably significant for most students.

Ask students which of the monthly expenses they could control. They may mention some of the following:

• Buying a car with a clean history and a brand name known for reliability, to keep maintenance costs to a minimum.
• When possible, do repairs and maintenance yourself using discounted materials.
• If you can’t afford to pay for the car in full, make as large a down payment as possible (at least 20%) and take a loan for at most 48 months (so you never owe more than the car is worth).
• Get good grades and maintain a history of safe driving to minimize your insurance costs.
• Avoid speeding, rapid acceleration, and sudden braking. In addition to putting you and others at risk, driving carelessly can decrease mpg by 5% in the city and by 33% on the highway.
• Remove excess weight — an extra 100 lbs. in your car can reduce fuel efficiency by as much as 2%.
• Use cruise control on the highway — maintaining a constant speed saves gas.
• Keep your tires filled at the proper pressure, and be sure your engine is tuned.
• When possible, drive your car at the speed at which it gets its optimal fuel economy (typically between 50 and 70 mph).

 

Analyzing Fuel Costs

Tell students that today they are going to focus on how they can drive as far as possible on the gas they buy. First, be sure students understand miles per gallon (mpg), an important ratio that gives the average number of miles they can drive, city or highway, on one gallon of gasoline. Tell students that the Aston-Marquez gets 33 mpg when driving on the highway and that the fuel tank holds 15.7 gallons. Ask them:

• How many miles will the Aston-Marquez go on a full tank if driven only on the highway? (about 518 miles)
• What is the city mpg if the car can go 440 miles on a full tank in the city? (about 28 mpg)

Now discuss the relationship between speed and gas mileage. Every car has a maximum speed at which it gets its best gas mileage. If you drive beyond that speed, the car may actually start to burn fuel faster than at slower speeds. (Note that many students are surprised by this fact.) For most cars, the optimal speed for maximum fuel efficiency is between 50 and 70 mph.

Tell students that data were collected on the speed and gas mileage of the Aston-Marquez, and display the Speed vs. Fuel Economy sheet on an overhead projector.

Speed vs. Fuel Economy Overhead

Or, rather than creating an overhead transparency, record the following information on the chalkboard or whiteboard in your classroom:

 

Speed (mph)	5	10	15	20	25	30	35	40	45	50	55	60	65
mpg	10	15	23	25	27	28	29	30	31	31.5	30	28.5	27

 

Ask students, "At what speed does the Aston-Marquez get its best gas mileage?" They should tell you it gets its best mileage at speeds near 50 mph.

Now ask them to predict the gas mileage at 75 mph. To make this prediction, they can graph speed versus miles per gallon. The lists and graph shown below were generated on a TI-83 graphing calculator:

Ask your students:

• Would a line of best fit be the best model for this data?
• If not, what model would be the best?

Students should tell you that a parabola looks like it would fit the points better than a line. They should then test their prediction to see whether a line or parabola actually does provide a better fit by performing the correlation diagnostics. (The results for linear and quadratic regression shown below were generated on a TI-83 graphing calculator.)

The correlation diagnostics of r² and R² indicate that the parabola is the better fit, since R² is closer to 1. Though r² and R² are calculated differently, their values can be compared to determine which regression fits the data better. Be sure students understand that the closer r² or R² is to 1, the better the fit.

Consequently, the parabola will provide a better prediction for the gas mileage at 75 mph. Students should calculate y(75), obtaining an answer of about 20.2 mpg.

This is worse than the fuel economy at 15 mph! Cars do not operate efficiently at very low or very high speeds. Therefore, to save money, highway driving should be done at the speed at which the car gets the most miles per gallon.

Based on this example and using current gas prices, have students calculate the money they would save by driving at 55 mph rather than 75 mph. This calculation can be completed on a per-mile basis, but note that the result may not have much impact on students because the numbers are small. It might make more of an impression on students to calculate the savings per year, as that will likely give a big number. For instance, assuming that the car is driven 15,000 miles per year, at 50 mph, the car would use 15,000 ÷ 31.5 = 476 gallons if driving at 50 mph. At 75 mph, however, the car would use 15,000 ÷ 20.2 = 742 gallons a year. If the cost of gas is $2.99 per gallon, those extra 266 gallons would cost $795.

This lesson prepared by Elizabeth Marquez.