Lightning in a Bottle

This is the kind of thing that teachers love - students getting excited about something you are teaching and just running away with it.  I was teaching them how to find the derivative of x^x (do you use power rule?  is it exponential?  aaah!).  After we found the first derivative, I showed them how taking the second derivative was easier.  Then some of them thought they noticed a pattern and away we went - taking derivatives that ended up looking like this -

- trying to predict what the next one would be.  Then someone noticed a pattern when you evaluate the derivatives at x = 1 and we started trying to figure that one out as well.  We were all disappointed when the bell rang and we hadn't come to any conclusions.  By the time we return after spring break the interest level will be gone but for a while there we were really DOING math.  Not just learning about it, not looking for patterns that we already know exist, not proving things that we already know to be true, but just playing with something because it looks interesting.  It was awesome and unfortunately there's just no way to plan for that kind of thing.  I can throw stuff out there but the best investigations happen when they question something I would have never considered.