Driving for Gas Pres - North Carolina School of Science and

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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?
$2.25
5 miles
$2.10
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:
STATION A
STATION B
Miles out of your way:
none
Miles out of your way: 10
Cost of gasoline:
10gal • $2.10/gal) = $21.00
10gal • $2.25/gal) = $22.50
Gas used to/from station:
Cost of gasoline:
10miles/32mpg = 0.3125gal
Paid $22.50 for 10
gallons of useable gas.
Paid $21.00 for 9.6875gal of
useable gas.
$2.25/gallon
$2.17/gallon
YOUR FRIEND’S CAR:
STATION A
STATION B
Miles out of her way:
none
Miles out of her way: 10
Cost of gasoline:
10gal • $2.10/gal) = $21.00
10gal • $2.25/gal) = $22.50
Gas used to/from station:
Cost of gasoline:
10miles/15mpg = 0.67gal
Paid $22.50 for 10
gallons of useable gas.
Paid $21.00 for 9.33gal of
useable gas.
$2.25/gallon
$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
𝑻−
Number of useable gallons:
Cost per useable gallon:
𝟐𝑫
𝑴
𝑻∙𝑷
𝟐𝑫
𝑴
𝑻−
Our cost index:
𝑻∙𝑷
𝑴∙𝑻∙𝑷
𝑰=
=
𝟐𝑫
𝑴 ∙ 𝑻 − 𝟐𝑫
𝑻−
𝑴
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
[email protected]
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