Earlier in the week I wrote about a neat geometry / algebra problem Dan Meyer had posted on his blog. That post (including the link to Dan’s blog) is here:
It had been both fun and interesting to see all of the comments that have been posted to Dan’s blog about the problem during the week. This evening Dan posted some updates to his original post including the following commentary from Keith Devlin:
I immediately drew a simple sketch – divide the interval, fold a square from one segment, wrap a circle from the other, and then dive straight into the algebraic formulas for the areas to yield the quadratic. I was hoping that the quadratic or its solution (by the formula) would give me a clue about some neat geometric solution, but both looked a mess. No reason to assume there is a neat solution. The square has a rational area, the circle irrational, relative to the break point.
So in the end I just computed. I got an answer but no insight. I guess that reveals something of a mathematician’s meta cognitive arsenal. You can compute without insight, so when you don’t have initial insight, do the computation and see if that leads to any insight.
Having personally found some of the math hiding in the problem to be pretty neat, and in particular fun to talk through with a kid learning math , I wanted to take an extra few minutes to take sort of the opposite side of Devlin’s argument and advocate for something neat relating to math I saw in the problem.
First, a quick (50 second) review of the problem:
Next, a quick solution (again, under a minute). This isn’t meant to be instructive, but rather just a quick review of one possible solution where something interesting (I think anyway) comes up:
Finally, about 3 minutes of advocating for what I thought was particularly interesting – how does the second solution we got from the quadratic equation enter into this problem?
So that’s why I wanted to advocate a little for this problem. Thinking through some of the geometry that comes up from the two solutions to the quadratic equations can be an important (and fun) exercise for students. The significance of the first solution is clear – it is right on the line after all. The meaning of the second solution is less clear and easy to ignore. However, extending some of the geometry videos / apps posted in the comments on Dan’s blog would show that locating the point P at this second location also produces a square and circle having the same area. This second solution naturally leads to asking what other solutions might exist with P not on the original line segment. It turns out there’s a really cool geometric surprise hiding in those solutions. How fun!
All in all, a really neat problem.