Followers of Systematic Initiative for Modeling Investigations and Opportunities with Differential Equations (SIMIODE) will be familiar with the use of M&M candies to model a population with deaths and immigration. A starting number of plain M&M candies (say P_0) are placed in a cup, gently shaken and rolled out onto a plate. The M&M candies with the letter “m” face up are deemed dead or emigrated and are removed from the population. Then some number (say k) M&M candies immigrate into the population, simulated by adding an additional “k” M&Ms to the plate. Finally the candies in the plate are transferred to the cup and the entire process repeats.
The goal is to see what happens to the population in the long run. In the DfEq class I asked the students to jot down their guess as to the long run behavior of the population. The two most common guesses were “stabilize at k persons” or “go to extinction.” The students then tossed M&Ms and collected data for something like fifteen iterations at which time it appeared that the population was going to be stable at about 2*k. These students, junior and senior math and physics majors were then asked to formulate an equation that would model what their simulation had revealed.
Their first result was a discrete formula, iterative, that started with P_1=P_0*1/2+k and after calculating through P_4 noticed a pattern and wrote the pattern as P_n=P_0*(1/2)^n+(2^n-1)/2^(n-1) *k. They were quite happy with this compact description of their experience and I asked them to take the limit of this expression as n->∞. Quickly they concluded the limit was 2*k, just as they had observed in the physical experiment. Finally, since these are differential equations students I asked them to set up a differential equation that matched the situation. dP/dt=-1/2 P+k and p(0)=P_0 was agreed upon and solved for P(t)=(P_0+2k) e^(- 1/2 t)+2k.
The solution was seen to model their experience with the physical M&M candies and the limit as t grew large again converged to 2k. This project went very well as I expected it to.
The success was on my mind two days later when I received a surprise request to be the program at our monthly Teachers’ Circle which was only two days away. I still had M&M candies and cups on hand and so decided to take the Teachers’ Circle through the exercise. For any not familiar with a Teachers’ Circle ours is organized by two of my colleagues and is an active outreach to area high school mathematics teachers. They meet once a month and enjoy a math themed program, discuss topics of mutual interest, and enjoy a nice meal together. And now I am the January program for the Teachers’ Circle.
When I was introduced and started passing around paper plates and cups and M&M candies everyone knew that they were going to be participating in something different. Right from the start the teachers got into the spirit of the simulation, my colleagues wanted to participate as well and we had a great time tumbling M&M candies and speculating on what the result would be. I had also asked this group to speculate on the long-term size of the population. No one in the Teachers’ Circle guessed that the limiting population size would be 2k. When I asked them to write an equation that described our exercise they quickly came up with the iterative description (see above) and with the slightest bit of help we noticed the patterns that allowed us to express the population size at iteration n without the need of developing the population at the preceding iteration.
I did not request that this group develop a differential equation, but I did ask them to remember with me and to evaluate the limit of their function as the number of iterations increased without bound and they were happy to see that their experience and their equation both predicted a long term population of size 2k.
This was a very successful experience for both my college differential equations class and for the in-service teachers in the Circle.
We all thank SIMIODE for suggesting this exercise to us.
John T. Sieben, Prof. Mathematics and Computer Science, Texas Lutheran University, Seguin, TX 78155
John Thomas Sieben Mathematics, Texas Lutheran University, Seguin TX USA