TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Our solution is not so much a solution as a set of leads the teacher may use to get the class moving and keep them going in a group modeling mode.

The process of mathematical modeling consists of a number of steps. These include accepting a situation, making some assumptions, converting the assumptions into mathematical statements, solving the mathematics, and interpreting the results in light of the situation.

We discuss a modeling activity which has proven to be of interest to regular college level calculus classes as well as high school groups. We have used it at various times and for a number of different purposes: to motivate the definition of the derivative in practice, to show the modeling process and to get students involved in that process, to confirm the notion of the antiderivative, and to permit students to see that mathematics can be applied to other areas of interest, in this case, entomology.

Students in civil engineering are intrigued with time it takes to build a tunnel and can appreciate the analysis and the conclusions drawn by this model for the time it takes to build a tunnel, especially with regard to the time it takes to build a tunnel of length 2x as compared to the time it takes to build a tunnel of length x.

Prerequisites

Definition of derivative, notion of antiderivative.

Time allotment - time management

We have finished this problem with the class participating in one 50 minute period.

Expectations

Students in course evaluations point to this exercise as useful, for it shows them the modeling process and it shows how their mathematics can be used. Since we only use this as an interjection, we do not build upon this model, but rather return to the course material.

Future payoffs

The activity of listing a large list of assumptions and concerns and then reducing to the small list necessary to formulate a mathematical model is worth practicing early and repeatedly.

Extensions

One could ease up the assumptions, e.g. the ant travels more slowly, say 75% as fast when carrying material or the tunnel is built at an angle causing down to be easer than up.

Ask how much longer to build a tunnel which is n times as long.

References and Sources

Ants, Tunnels, and Calculus: An Exercise in Mathematical Modeling, Brian J. Winkel, The Mathematics Teacher, April 1994, 87(4): 284-287.