ISSUES RELATED TO THE PROBLEM

An early objective in this problem is to get students to plot data as an early strategy for problem-solving.

The frustration they have with not being able to plot size vs. time should be remembered. For when they do discover the differencing, the plotting of differences in the size of the slick vs. the size, and the fitting of a linear relationship between a difference and a function to obtain a differential equation, they will see differencing as a valuable tool.

This project further reinforces the use of a differential equation model. The model is not given to students. They have to construct it out of data.

Students in traditional approaches in calculus for example, are not called upon to plot data often, indeed, they may not even possess graph paper for class. However, graph paper can be supplied or one can go right to a CAS or data plotting routines.

We give the students graph paper at the start of the class and encourage them to use it. In fact, often we PREFER to be in an environment without CAS so they "mess with" the data on paper.

But it is here that the students get most frustrated for they wish to plot something they have immediately vs. time, e.g. size vs. time. But the data does not indicate what the absolute time for each observation is, only relative times, e.g., for unknown times t we know size S(t) and size S(t+10) some 10 minutes later only, but we do not know at what time t this actually took place, nor do we know how long there is between observations. This is a roadblock for the students. Incidentally, the students will invariably say that the pilot should be fired as she keeps a lousy log!

It is the concept of differencing, i.e., Delta S = S(t+10) - S(t), and realizing that one can plot Delta S vs. S or Delta S/Delta t vs. S to get a relationship between Delta S/ Delta t and S and thus obtain a differential equation in S.