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.
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 r^{2} and R^{2} indicate that the parabola is the better fit, since R^{2} is closer to 1. Though r^{2} and R^{2}
are calculated differently, their values can be compared to determine
which regression fits the data better. Be sure students understand that
the closer r^{2} or R^{2} 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
1. 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.]
2. 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.]
3. 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.]
4. 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.]
5. 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.]
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?