Summer 2014
Mathematical Modeling
“Model with mathematics” is one of the eight Standards of Mathematical Practice in the
Common Core State Standards. Modeling links classroom mathematics to everyday life, work,
and decision-making. Modeling is the process of choosing and using appropriate mathematics
to analyze empirical situations, to understand them better, and to improve decisions.
Mathematical models are not to be confused with tools (such as manipulatives or visuals for
building a pattern). Modeling with mathematics is something students do across the curriculum
as they think numerically and symbolically about the mathematics they are doing.
You can create a mathematical model to describe the
depreciation of a car at 20 percent each year. If the car loses 20
percent of its value in 1 year, then it must be worth 80 percent
of its value after a year. So, after one year, the $15,000 car is
worth:
$15,000 x 0.8
In the second year, it again loses 20 percent of its value, so it
will be worth only 80 percent of the value it had at the end of
year 1. The value at the end of year 2 would be:
($15,000 x 0.8) x 0.8
The value at the end of year 3 would be:
(($15,000 x 0.8) x 0.8) x 0.8
What pattern do you see in these representations?
So at the end of y years, the value of the car can be expressed
with this equation:
Value of the car in y years = $15,000 x 0.8^y.
The following problem provides another context appropriate for developing a mathematical
model.
Mathematical Modeling
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Summer 2014
How Many Gallons Are Left?
A car gets 23 miles per gallon of gas. It has a gas tank that holds 20 gallons.
Suppose that you were on a trip and had filled the tank at the outset. Determine a
mathematical model that describes the gallons left for any given miles traveled.
What questions might you pose to your students to help them make sense of this problem?
Let’s Empty Out the Pool
Mathematical Modeling
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Summer 2014
Suppose you use a pump to empty the water out of the pool. The
amount of water in the pool (W, measured in gallons) at any time (T,
measured in hours) is given by the following equation or model: W = 350(T – 4).
What questions might you pose to your students to help them make sense of this equation?
Work with your team to think of at least three.
Work with your team to answer your three questions like you think your students would
answer them. Make a poster to display your three questions and the work you did to answer
these questions. Be prepared to share with the whole group.
A mathematical modeling problem for further exploration
Mathematical Modeling
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Summer 2014
Pleasant’s Hardware buys widgets for $4.17 each, marks them up 35 percent over
wholesale, and sells them at that price. Create a mathematical model to relate
widgets sold (w) to profit (p). The manager asks you to determine the formula if
she were to put the widgets on sale for 25 percent off. What is your formula or
mathematical model for the sale, relating widgets sold (s) to profit (p)?
Adapted from “Teaching Student-Centered Mathematics,” Volume III, Van de Walle, BayWilliams, Lovin, and Karp; 2014.
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