Everyone's favorite preschool activity is finding things to count. Following the general idea of finding things for parents to program with their kids, I was showing my kids some cellular automata. The first thing they wanted to do was count the black cells. They were unstoppable. Here is rule 250.
As you can see in the attached notebook, this leads naturally to Map and Total and then after a ListLinePlot they can guess the next elements and together we wrote a function for predicting the answer.
Another way to approach cellular automata, for older kids, is to use Wolfram Alpha, e.g., look at its results for the different rule numbers. http://www.wolframalpha.com/input/?i=rule+250 not to mention reading the New Kind of Science book by Stephen Wolfram.
(More advanced students and adults interested in cellular automata should consider applying to the Wolfram Science Summer School.)
(An earlier post on projects with kids was about writing letters.)
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