## Great Comments on Modeling Scenario Offered in Comments Tab of Modeling Scenario

Bill Kerbitz, from Wayzata High School in Plymouth MN, recently commented on a Modeling Scenario, 1-001A-T-MandMDeathAndImmigration,  What is most interesting is what Bill found as unintended consequences of working with this material and we are pleased to share them with you in this entry of our Blog. Users of SIMIODE Modeling Scenarios and any SIMIODE resources are encouraged to Comment (using the Comment Tab) so all can learn from their observations and a conversation might begin on the material, improving both the material with suggestions for the author and use through observations and experiences such as Bill offers. Thank you, Bill. Incidentally, when we met up with Bill at the Joint Mathematics Meetings in January in San Diego CA he mentioned that the temperature had increased by over 100 degree Fahrenheit for him as it went from 46 below zero in MN to over 70 degrees in CA!  Yikes!!!

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Last week, as always, I started off class with a qualitative development of the logistic growth model.  The rest of the week, after some discussion of direction fields, an introduction to MATLAB, and a review of separable differential equations, I began this week with Karen Bliss' version of the M&M activity.

I have a class of 26 students - I put them in groups of 3 (well, most of them, at least) and they dug right in.  They had no trouble coming up with the exponential model, and I took the opportunity to have them learn some MATLAB syntax along the way (plotting points via t and y vectors and then creating an "ideal" set of points to compare to via a "for loop" from their proposed model).

Once they started analyzing the immigration model, a few groups predicted correctly that the population would stabilize at around 20, showing some great insight along the way ... sure enough, the simulation behaved as expected.

What I found most enjoyable about the activity was what happened afterward.  With my 85-minute class periods, (I teach in a large public high school in Plymouth, a suburb of Minneapolis) I was able to:

1. Develop the connection between the difference equation and the corresponding differential equation - there were audible admissions of understanding (oohs and aahs) as the students realized the emergence of the familiar difference quotient from their calculus classes.
2. Introduce the "integrating factor" method of solving this first-order differential equation with y(0)=50 ... as that process neared completion, they again let out audible sighs of recognition as the stable population appeared as part of the solution and the impact of the internal exponential decay dissipated with time.

The following day, I went through the method of undetermined coefficients ... there was a third opportunity for "oohs" and "aahs" as the homogeneous solution turned into none other than the transient exponential decay and the particular solution turned into the steady state (as expected) - but it was really REALLY nice to be able to point out that the "transient" part of the solution corresponding to the homogeneous equation was due to the internal growth dynamics of the population (the homogeneous equation without the immigration) and the stabilizing population (20, in this case) which popped out as the particular solution was due to the external input (immigration).  They immediately understood the correlation between those elements of the differential equation (nonhomogeneous vs. homogeneous), the resulting solutions, and their relationship to the physical implications in the simulation itself.

I've not taken this exact path through these topics before, and having now done so, I would say that I would highly recommend this sequence - the flow from M&Ms (personal experience with the model) to the integrating factor method (a typical first approach to solving 1st-order linear equations) and seeing the pieces of the solution materialize out of thin air (ok - I'm being dramatic here ... but not entirely) and then directly to undetermined coefficients just seemed to work amazingly well, and it provides a great background to which I will refer as we cover these ideas in more detail later!

Great activity - thanks for sharing!