This file can be found here.

Below is a bit of mathematics that I would have discussed in a class. with the goal of graphing in parametric mode on a graphic calculator. For students not yet up to that, I would adapt this so that they can experiment with different rotational speeds, reverse directions, etc., and would not include the pre-plotted graph.

\[\begin{array}{c}
A = ( - .75,0)\\
Jack = ( - .75 + \cos ( - 2t),0 + \sin ( - 2t))\\
C = (.75,0)\\
Jill = (.75 + \cos (3t),0 + \sin (3t))\\
{\rm{red}}\,{\rm{point = }}\frac{{Jack + Jill}}{2}\\
{\rm{red}}\,{\rm{point = }}\frac{{\left( {\cos ( - 2t) + \cos (3t),\sin ( - 2t) + \sin (3t)} \right)}}{2}
\end{array}\]

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