A Fawn Nguyen inspired geometry problem

Last week Fawn Nguyen posted that she was going to so a fun Five Triangles problem with her class:

I typically love the problems posted by Five Triangles and their geometry problems, in particular, are consistently outstanding.  Too bad I’ve not really covered any geometry with either kid yet 😦

But, I have been working on fractions and decimals with my younger son and this problem had a really interesting infinite series hiding in it, so I though it would be fun to talk through with the boys even if I would have to skip over the interesting geometry.

I spent the first 5 minutes just introducing some basic concepts about triangles so that they could understand the problem.  The fact that we are dealing with an equilateral triangle here significantly simplifies the explanation because we can work with the medians rather than the angle bisectors.  Also, though we didn’t dwell on it, with an equilateral triangle it isn’t hard to believe that the medians intersect in a single point.

With that basic introduction out of the way, now we could spend a little time talking through the problem.  The first challenge is to find the radius of the second circle.  My older son had a one geometric idea that was going to be a little more difficult to work through than I was hoping for, but then my younger son noticed that we could draw in a new line segment that would make finding the radius of our new circle pretty easy.  From there we moved on to talking about the infinite series that is hiding in this problem:

From our picture in the last video we were able to see that 2/3 + 2/9 + 2/27 + . . .  = 1.  Now we try to see if there’s a way to sum up that series without appealing to the geometry.  This particular problem is pretty similar to converting repeating decimals to fractions which is what my younger son and I have been talking about for the last week.   I really loved the various ideas that the kids threw out here:

Finally, we wrap up by showing how to sum up the above series by using base 3.  We start by talking about why .9999… = 1 in base 10 and move on to show how the same argument shows that .22222…. = 1 in base 3.  Luckily the series we are looking at is easy to write in base 3.  Fun!!

So, yet another thanks to Fawn Nguyen for alerting me to a really great problem.  Though not quite the point of the problem as originally posed, I love the connection between arithmetic and geometry hiding in this problem.  It was really fun to talk through with the boys.