ISSUES IN SOLUTION

Once students catch on to the idea that we 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 they know an initial condition and they can use separation of variable techniques to solve the differential equation. They use the solution to check against their data and to look at long term behavior of the growth model.

The issue is how to prompt students to plot differences Delta S/ Delta t and S without telling them. They will quite naturally difference the data given the juxtaposition of the columns in the table, but they need to be nudged beyond this. The frustration of not being able to plot against time will cause some to freeze, but in groups of three or so there are usually some individual members of the several groups who will attempt plotting the differences and then the class is off and running. It is worth taking the time to point out how the group overcame the stumbling block and to ask those who thought of the idea what made them think that way.

We have gotten students, working in groups, to discover the essence of this approach within a 50 minute period, even doing the linear fitting by hand on graph paper, but in other situations with CAS in computer lab environment.

An alternative solution came up when students started to attempt to predict the next size of the slick from difference equation approach. The approach divides the average growth rate into the actual growth between the later observation and the next initial observation to get an estimate of the time between these two observations and uses this to push time until the next initial observations. We need to modify this approach as it proves inaccurate compared to the first model solution we obtained so we take the average of the growth rates from consecutive intervals (not just over one interval, the front interval) and determine the time it takes between observation intervals.

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