With Spanish AP taking 20% of each class, and with all the juniors worked up over AP US stress, this was a “diminished” teaching day in the midst of a diminished teaching week. I scheduled today for “practice solving trig equations” – but I had a beautiful challenge problem, and let it and its extensions eat up two-thirds of each period:

Find x:  $x = \sqrt{2 + \sqrt{2 +\sqrt{2 +\sqrt{2 +\sqrt{2 + ...}}}}}$

Divided by missing classmates into clusters, the kids got stuck or made little progress on their own. Five minutes in the buzz changed… I could hear groups considering answers. At least two had decided to evaluate $\sqrt{2 + \sqrt{2 +\sqrt{2 +\sqrt{2 +\sqrt{2}}}}}$ or some similar partial roots, and had decided that either this thing was 2, or it was a little less than two.

They disagreed. Fairly loudly. Was x 2, or was it a little less than 2?

I took a risk. “You guys did well. It’s clearly either 2 or a little less than 2. One of them’s got to be the answer, the other’s close. Let’s go on to some trigonometry practice.”

I read them right. Nothing doing. They wanted to know. And I had them hooked for a half hour or so investigation. That’ll be Part 2.