Phase IV: Output, random numbers, iterations, and simulations
Another simulation -- in response to a student question -- was to see why you divide by n - 1 instead of by n when you calculate the variance of a sample in statistics. To do this, we considered the experiment of flipping a coin over and over, getting one point for each head, and losing a point for each tail. Of course, the theoretical mean is zero points, and the standard deviation (in fact, every deviation) from the mean is 1. By taking modest-sized samples we could see that we got a better estimate of the theoretical variance by dividing by n - 1.
Finally, one of the mathematics units we studied concerned iteration schemes for solving equations. The students discovered Newton’s method for calculating nth roots by experimenting, and eventually we looked at some examples of chaotic dynamics and fractals.