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.

### Learning Objectives

 Students will: Use proportions to calculate distance and miles per gallon (mpg) Analyze data using a graphing utility Determine a curve of best fit Extrapolate data

### Materials

 Speed vs. Fuel Economy Overhead Graphing calculator with projector

### Instructional Plan

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 – \$4,614RX400 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, Garaged52,138 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.
• 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.

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:

• 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.

### Questions for Students

 How could you calculate average miles per gallon for a car? Fill up the tank and "zero out" the trip meter. The next time you get gas, fill up again, and record the number of gallons purchased and the reading on the trip meter. Divide that reading by the gallons purchased. Why is highway driving more efficient than city driving? Highway driving allows for the car to attain the speed at which it gets its best mileage. City driving requires many starts and stops, each of which use more gas than driving at a constant speed. Determine the speed at which the Aston-Marquez obtains the greatest fuel economy, according to the quadratic regression graph. How does that value compare to the maximum mpg and the speed at which it occurs according to the data points given in the table of values? Explain the difference. According to the graph of the parabola, the best mileage is about 31.4 mpg, which occurs at 45 mph. However, the table shows the best mileage is 31.5 mpg, which occurs at 50 mph. These values are different because the parabola is a line of best fit — it does not have to pass through all or even any of the data points because it is a best-fit curve. It closely approximates the data, but it doesn’t necessarily match it point for point. Why do cars get optimal mileage at speeds less than 60 mph? At speeds of 60 mph and above, the power required to overcome the car's aerodynamic drag — pushing its way through the air — has a major impact. What features do you think should be included in the design of a car to minimize the aerodynamic drag and maximize the gas mileage? Answers will vary, but students will likely mention such factors as weight, shape, and type of engine.

### Assessment Options

 Ask students to calculate the fuel cost for f driving from New York City to Los Angeles in the Aston-Marquez. Use 2,800 miles as the distance, and assume that  20% of the trip is city driving. Since the price of gas varies across the country, have students use the average cost per gallon, which can be found at fueleconomy.gov. Ask students to use the quadratic regression model to predict the fuel economy for 80, 85, and 90 mph.

### Extensions

 Ask students to devise and apply a method for determining the average miles per gallon that their car (or their family’s car) gets. After they have had time to apply their method, have them present their method and results to the class. One simple method is to fill up the tank and "zero out" the trip meter When they next get gas, they should fill up the tank again, record the number of gallons purchased, and note the reading on the trip meter. The reading on the trip meter divided by the number of gallons is the miles per gallon. (Note that many cars today have the ability to calculate the fuel economy and can display it on a dashboard screen. However, these readings are often approximations, and the method described above gives a more accurate result.) Have students research what NASCAR is doing to improve fuel economy. Have students research how the French team of "hypermilers" won the 2006 Shell Eco-Marathon by achieving a fuel consumption rate of 6,786 mpg.

### Teacher Reflection

 What are the algebra and calculator skills the students need to successfully complete this lesson? Do you think students learned something about math as well as the realities of owning a car? Were clear expectations discussed so that students knew what was expected of them? If not, how can you make expectations more clear in the future? Were students focused and on task throughout the lesson? If not, what improvements could be made the next time you use this lesson? This lesson is very open-ended. Students are given a scenario and expected to make progress by themselves, with very little guidance. How might you structure this lesson to minimize frustration for low-ability students? How did students demonstrate that they were actively learning? How did students show that they had achieved the objectives of the lesson? How well were students able to connect the displayed data in the scatter plot to the appropriate class of equations to model the data? How well did students communicate their understanding of the costs of owning a car and, specifically, what they can do to minimize fuel costs?

### NCTM Standards and Expectations

 Algebra 9-12Draw reasonable conclusions about a situation being modeled. Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships. Data Analysis & Probability 9-12For bivariate measurement data, be able to display a scatterplot, describe its shape, and determine regression coefficients, regression equations, and correlation coefficients using technological tools. Number & Operations 9-12Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases.

### References

 Marquez, E., and P. Westbrook. 2007. Teaching Money Applications to Make Mathematics Meaningful. Thousand Oaks, CA: Corwin Press.
 This lesson prepared by Elizabeth Marquez.

1 period

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