1. Let’s get modeling!! Pair Up. Each pair take Two sets of two sheets of paper on front desk. 1 bag M&M’s (we have 25 bags) and 4 cups and 2 paper plates. From your pair’s 4 cups and 2 paper plates each person take 2 cups and 1 paper plate. First person tear open your pair’s M&M bag and pour half in one of your cups and half in one of colleague’s cup. Leave two cups empty. Second person pick which of the M&M “filled” cups and First person take remaining cup – fair division. Sit and wait for instructions – DO NOT eat M&M’s!!!

2. Modeling in Differential Equations Doing, Learning, and Teaching Society for Industrial and Applied Mathematics SIAM Student Chapter New York City College of Technology Brooklyn NY 13 September 2016 Brian Winkel, Emeritus Professor of Mathematical Science, United States Military Academy, West Point NY and Director SIMIODE

3. www.simiode.org Added area listing FREE online texts

11. Using Euler’s method in nonlinear system with step size 1.

12. Using Euler’s method in nonlinear system with step size 0.05.

15. SIMIODE provides manuscript management system for double-blind, peer-reviewed referee/edit system for online publication.

16. Start M&M Death and Immigration Mystery Modeling

17. Objective: Get students to think about change, about increment.... How long does it take an ant to build a tunnel of length x?

21. Example of “local flipping” or “flipping in the small” in this case modeling first leading to differential equation mathematics.

22. How does this suit you? What mathematics does this yield?

23. Ask questions of solved differential equations model such as “What if we double the length of the tunnel? Does this double the time?” How does this suit you? What mathematics does this yield?

28. Spread of Disease Process Experiment conducted 30 April 2013, Prepare a grid (print out M&MGridToUseForSimulations pdf file), mark 8 randomly selected cells as infected and write numbers 1 through 8 in each cell, respectively. Make a corral surrounding the grid so when we toss the M&M’s they stay on the grid. ON Web at https://www.simiode.org/resources/715

29. Place 54 M&M’s in a cup and (a)gently toss them out onto the grid, (b)if an M&M contacts an infected cell/M&M mark the cell with next number of infected, first 9, then 10, then 11, etc. and remove infected M&M’s from population, (c)when all M&M’s are marked from this toss or generation collect M&M’s and toss again until all M&M’s are infected. If you cannot perform this simulation then we have done this for you and enclose 12 generations of our simulation from which you are to gather data on the number of infected at each generation or time.

30. Equipment: 1 small bag regular M&M’s 1 small cup 1 level surface Early actions: Prepare grid with 8 randomly selected infected cells. Number them as seen here. Open bag and count out 54 M&M’ s. Place M&M’s in cup. Gently toss M&M’s onto surface. If M&M contacts an infected cell/M&M mark the cell with next number of infected, first 9, then 10, then 11, etc. Remove infected M&M’s from population. When all M&M’s are marked from this toss or generation collect M&M’s and toss again. Later action: Yummy!!!

31. Collect data for each generation on y(t), the number of M&M cells which are infected. Plot the number of infecteds as a function of the generation (time). Propose a mathematical model, using a differential equation for the rate of spread of the disease. Estimate the parameters using a sum of square errors method. Compare your model to your data, numerically and graphically. Does your model do a reasonable and reasoned job of modeling this data and hence this phenomenon?

32. Generation 1

34. Plot of data from simulation.

35. Plot of data and best fit model.

36. Videos available on SIMIODE YouTube Channel and streaming from www.simiode.org.www.simiode.org

39. Mathematical Model

40. Not all work... not all fun... building for modeling-first differential equations These images are related!

41. Many sources on falling body with resistance. Use some with real data or generate the real data with students. Source:

43. Akaike Information Criterion (AIC) k is # parameters n is # data points RSS is Sum of Square Errors

47. www.SIMIODE.org

48. Thank you. www.simiode.org