Teaching Mathematical Modeling with the Driving for Gas Problem Christine Belledin NC School of Science and Mathematics Teaching Contemporary Mathematics 2015

What is Mathematical Modeling? According to the Common Core Standards for High School: Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.

Driving for Gas problem Every driver recognizes the fluctuations in gas prices that happen almost on a daily basis. In some areas, a local radio station has a special report on the location of the gas station with the lowest price per gallon for regular gas. Of course, that station is likely to be across town from where you are driving. Is it worth the drive across town for less expensive gas? If you know the locations and the prices at all gasoline stations, at which station should you buy your gas? Does it depend upon the car you are driving, and if so, in what way? Develop a model that can be used by owners of different cars that will tell them how far they should be willing to drive based on the specifications of their car. Does it matter if you think that you are buying gallons of gas or that you are buying miles of travel? Present your results as a lookup table that can be given out at gas stations for general use with one page describing how the table was created.

Strategy #1: Start with a simpler question You drive to school every day. On the route you take from home to school, there are several gas stations. Unfortunately, the prices on your route are always high. A friend tells you she buys her gas at a station several miles off your normal route where the prices are cheaper. Would it be more economical for you to drive the extra distance for the less expensive gas than to purchase gas along your route? What did we gain from refining the question this way?

Strategy #2: Consider a specific case Suppose there is a gas station on your normal route that sells gas for $2.25 a gallon. A station 5 miles off your route sells gas for $2.10 a gallon. Should you travel the extra distance to buy gas at that station? A B $2.25 $2.10 5 miles

Questions: Why wouldn’t you drive the 5 miles to buy cheaper gas? What does this tell us about the problem?

Consider an even more specific case Station A is on your normal route and sells gas for $2.25 a gallon, while Station B is 5 miles off your route sells gas for $2.10 a gallon. Your car gets 32 miles per gallon, and your friend’s car only gets 15 miles per gallon. Should either of you travel the extra distance to buy gas at Station B? Now we can start to do some calculations!

Your Car: Paid $22.50 for 10 gallons of useable gas. $2.25/gallon Station A Station B Miles out of your way: none Cost of gasoline: 10gal • $2.25/gal) = $22.50 Paid $22.50 for 10 gallons of useable gas. $2.25/gallon Miles out of your way: 10 Cost of gasoline: 10gal • $2.10/gal) = $21.00 Gas used to/from station: 10miles/32mpg = 0.3125gal Paid $21.00 for 9.6875gal of useable gas. $2.17/gallon

Your Friend’s Car: Paid $22.50 for 10 gallons of useable gas. Station A Station B Miles out of her way: none Cost of gasoline: 10gal • $2.25/gal) = $22.50 Paid $22.50 for 10 gallons of useable gas. $2.25/gallon Miles out of her way: 10 Cost of gasoline: 10gal • $2.10/gal) = $21.00 Gas used to/from station: 10miles/15mpg = 0.67gal Paid $21.00 for 9.33gal of useable gas. $2.2508/gallon

What if your friend’s car has a larger gas tank? Suppose your friend’s car has a 25 gallon tank instead. Station A: Cost of useable gas is still $2.25 per gallon Station B: Cost of useable gas would be $50.40/23.33gal, or $2.16 per gallon Both you and your friend should buy gas at Station B!

Strategies we learned that we can apply to future modeling problems Ask an easier question first. Create a specific example. Use this to learn what is important (and what is not) about a problem. Add details or complexity step by step.

Developing an Algebraic solution Let 𝑷 ∗ represent the price per gallon at the station along our route and 𝑷 the price per gallon at the station we are considering. Let 𝑫 represent the distance in miles from the normal route that must be driven to get to and from the other station. Let 𝑴 represent the average fuel efficiency of the car you drive. Let 𝑻 represent the number of gallons of gas purchased when we buy gas. For any given station, we will compare the price per gallon of useable gas to 𝑷 ∗ .

Price per useable Gallon for other stations Cost of a tank of gas: 𝑻∙𝑷 Number of useable gallons: 𝑻− 𝟐𝑫 𝑴 Cost per useable gallon: 𝑻∙𝑷 𝑻− 𝟐𝑫 𝑴 Our cost index: 𝑰= 𝑻∙𝑷 𝑻− 𝟐𝑫 𝑴 = 𝑴∙𝑻∙𝑷 𝑴∙𝑻−𝟐𝑫 We should buy gas at the distant station if 𝑰< 𝑷 ∗ .

Other approaches and considerations What if we think about buying “miles of driving” rather than “useable gas”? Does that change our results? What if we take time into account? What else would we need to consider to do this? How can we present our model in a way that is usable for an average driver?

Things I’ve learned about teaching modeling Let students see the process of simplifying the question. Check in often. Struggle is important, but it needs to be productive struggle. Take time to debrief the process, not just the problem. Communicate with students about how they will be graded on this assignment. If this is their first experience, consider grading only on effort (or not at all). If you want to assess their ability to communicate their findings, consider giving students the chance to revise their work. You want to encourage creativity. What grade do you assign to an incorrect answer that is based on a really good idea versus a correct answer that was very straightforward

Sample of Student Work

References and Acknowledgement This problem is based on a problem created by Landy Godbold, who teaches at the Westminster Schools in Atlanta, Georgia.This version of the problem was created by Dan Teague and Dot Doyle. Doyle, Dot, and Dan Teague, “The Gas Station Problem”, Everybody’s Problems, Consortium, Number 88, COMAP, Inc. Lexington, Massachusetts, Spring/Summer, 2005.

Questions or Comments? I welcome feedback or questions! Feel free to contact me at belledin@ncssm.edu.