A presentation of the following article was given at the Joint Mathematics
Meetings, January 1013, 1996, Orlando, Florida.
Report for Mth 390, Mathematical Modeling
Taught in the Fall of 1994
by Thomas Kelley
Department of Mathematical Sciences
Metropolitan
State College of Denver
Mth 390, Mathematical Modeling, was conceived with the intention of having
the students learn mathematics as they worked through extended
mathematical modeling problems in a group setting. The mathematical
modeling projects were chosen based on the area of mathematics that the
instructor wanted the students to explore and use. A chronological
list of the projects, their mathematical content, and some remarks on
the process and outcomes follow. Note that the statement of the
project is short which requires the students to spend a considerable
amount of time talking among themselves and with their instructor(s),
who sometimes assume the role of a "client", in deciding just what
the problem is. This problem construction facet of the modeling
process proves to be indispensable in all areas involving problem
solving, not just in mathematics.
Project #1:
"Build a mathematical model of how you plan to read the
textbook."
First build an individual model and then get together with your other
two group members to build the group's model. This activity was to
introduce the students to the ideas involved in math modeling and to
disrupt their usual notions of what went on in a "math class".
This activity also demonstrated to students that mathematical models
could be developed in "non mathematical" contexts. Finally, students
were introduced to the interaction and negotiation that must go on
in constructing a group consensus.
Project #2:
"Develop and explain a method to measure the height of an
inaccessible object.
This includes the construction of a device from
readily available materials in order to measure distances and/or
angles. Use that method to measure the height of two assigned
objects, one of which was the roller coaster under construction at
Elitch's. Your written report should enable a person, unexperienced
in trigonometry, to obtain the height by following your
instructions."
The groups quickly realized that this was "just a trig problem" that
they had seen before. The purpose of this project was to force the
students to notice that just knowing the mathematics involved does
not mean that the problem is solved. As a matter of fact, the
groups' selfassessment comments often expressed amazement as to how
much time the problem took after they had solved the mathematics.
Each group learned just how difficult it was to actually use the
mathematics, to obtain accurate measurements, and to account for
errors in measurement that gave erroneous results. In fact, this was a
good exercise in spotting when a result was reasonable or not. The
writing of clear instructions for a "novice user" to implement their
methods was also a difficult task. When running this model in future
classes I plan to have the students spend more time on the error
analysis and the ways in which errors can be adjusted for and
ultimately controlled.
Project #3:
"You are a retailer of coffee. Determine the best
container for serving your product."
The mathematical intent of this project was to have the students
experimentally determine Newton's Law of Cooling. Along the way the
groups learned about defining the problem so that they knew just what
they had to model. Initially the groups were upset that the problem
was so "vague", but were mollified somewhat when told that part of the
purpose of doing this course was to disrupt traditional notions of
math courses. Each group developed their own definition of "best"
through interviews with local businesses who served coffee. They
discovered that many factors were considered (and not considered) by the
companies who made and used the various coffee containers in the
Denver area. They also learned about data collection and curve
fitting as well as customer surveys and what constituted a good
question on a survey. In addition, at this point in the course
students were starting to become more comfortable with the structure
of the course. We had 9 students, 6 males(including 1 minority ) and
3 females, and one significant choice I made as instructor was to
not have a team with only 1 female student, which more often than not
leads to the isolation of the single female student. With this in
mind, the groups for the projects were composed of three students and
one group was all female. This "3 groups of 3" structure held
throughout the course until the last project where we had two groups of
four with a female "consultant" (who was in fact a practicing high
school math teacher).
Project #4:
"The conic sections. Each group was assigned a
different conic and required to present a definition, at least one
method of construction, development of the equations for their group's
conic, and a demonstration of a physical manifestation of the conic
section."
This project differed from the others in that we started with the
mathematics and went in search of the application. The intent here
was to have the students take a topic that they were accustomed to
dealing with in terms of equations and graphing, but had lost touch
with them on a concrete level. They were required to go back to the
definition and applications of the conics so that they could see
beyond the abstractions. Each team did a very good job of showing
that they understood the definition through their handson construction
of the particular conic, complete with detailed instructions that a
person who was just learning about conics could follow. The teams
also did physical demonstrations which showed that they had a grasp
of the reflection properties of each of the conics.
Project #5:
"For a game of chance (craps, roulette, coin toss onto a
grid), do an analysis of your chances of winning if you are the "house"
. What if you are the "customer?" Your analysis should include
both physical trials (i.e. actual plays of the game) and a computer
simulation. The presentation will be a "Casino Day" where the other
teams will try to beat the game which you have "rigged" as the house."
This was our foray into probability. Students learned about
analyzing games of chance through analysis, through experiment and
through computer simulation. Simulation was a new experience for
most of the students and getting a computer or programmable
calculator to simulate playing a game helped their writing skills.
The different ways in which the students could approach the problem
helped them see that mathematics gives one flexibility in solving
problems.They also gained an appreciation for the gaming industry and
what is involved in setting up a "saleable" game. Having the
presentations in the context of a "Casino Day" resulted in a very lively
class. No reports of students visiting Las Vegas and winning big
followed this project.
Project #6:
"What relationships exist between mathematics and music?
Use a monochord to find out what the harmonic sequence sounds like.
What do musical sounds look like mathematically?" How do you
decide where to place frets on a guitar and why is a grand piano
have the shape it does? Along the way you will find out about the
Fibonacci numbers and obtain physical objects that are natural
occurrences of Fibonacci numbers. What is the Golden Ratio  how do
you construct it with numbers and does it occur in nature?"
Starting with simple notes each group proceeded to explore the
relationship between mathematics and music. This led them through
discussions of rational and irrational numbers, exponentials and
logarithms, Fibonacci numbers, and the Golden ratio. Each group had
the use of a monochord with movable frets and a ruler to see what
fractions "sounded" better. Their project was to complete a series
of worksheets taken from the book , Algebra in Everyday Life. This
setup was different in that the main "product" was not a group
report, but rather a collection of worksheets. In the future I plan to
use the worksheets to construct a project which would hang together
better than it did this time. Many of the student comments mentioned
that the material, though interesting, seemed to be disjointed.
Project #7:
"A company wants to design an automatic cab following
system for a new monorail. They are trying to decide whether a "Safe
Distance Algorithm" or a "Speed of Approach Algorithm" is the best
approach. Use a computer simulation and analysis to determine which is
better."
Students are put into two groups of 4 (instead of the usual 3 groups
of 3) with one student (who is actually an high school teacher)
serving as a technical expert for both groups (How do you divide 9
in half?). This problem was chosen so that the groups learned about
the cost benefits of simulation over building scale models. This
project took the students' calculus knowledge and extended it while
trying to solve a real problem (Denver was just putting in it's light
rail line and th enew airport was having significant problems with it's
automatic baggage system. They saw an application of the third derivative
of the position function, namely, jerk. They did not solve DIA's
baggage problems.
For Comments and Questions please contact:
Thomas Kelley
Department of Mathematical Sciences
Campus Box 038
P. O. Box 173362
Denver, CO 802173362
kelleyt@mscd.edu
