I love watching my younger son learn and think about geometry

My younger son and I have just started talking about some basic geometry. He has so many neat ideas which make talking about geometry with him super fun.

Today we were returning to a topic from a few days ago – angles in polygons. The specific topic for today was exterior angles. For the first part of our talk today we reviewed what happens with interior and exterior angles in a triangle:

Next up was quadrilaterals – what happens with exterior angles here? The fact that the interior and exterior angles here have the same sum was a little confusing to him – his initial reaction was that the two sums should never be the same.

After expressing some skepticism, he proceeds to calculate the sum of the exterior angles a different way and finds that the sum is indeed 360 degrees. Having done calculation two different ways, he now believes the sum is 360 degrees and even begins to wonder if you get 360 degrees as the sum of the exterior angles for any polygon.

Since he was looking to generalize, we moved on to studying pentagons. He’s getting comfortable now with the idea of chopping polygons up into triangles, so he sees that the sum of the interior angles is 540 degrees. He then does a similar calculation to what he did in the last two videos to compute that the sum of the exterior angles is 360 degrees.

Now he really believes that the sum of the exterior angles of any polygon will be 360 degrees. Fun 🙂

One thing that made me really happy about this short project was his skepticism about the sums for the quadrilateral. The answer didn’t seem right to him, and he looked for an alternate way to find the answer. When he found the same answer the second time around not only did he have the confidence to believe it, he began to think that there might be something more general happening and wanted to check the next step. Love the curiosity of kids!

A fun power of a point problem

In 2001 my wife competed in the Ironman triathlon world championships in Kona, Hawaii for the first time. Probably 20 to 25 of our friends were there to watch and vacation and on one of the days some of us were wondering how far you could see out in the ocean from different heights. Made for some fun scrawling around in the sand 🙂

Today I saw that exact idea in the review problems in the “Power of a Point” chapter in our Introduction to Geometry book. I thought it was make both a fun geometry and a fun number sense project for my older son, so I deviated from the book a little bit.

First up was the statement of the problem followed by us talking through the solution (with one false start)

Next we went to Wolfram Alpha to look at lots of different heights. I guess I hadn’t had enough coffee yet, but I kept goofing up the code while I was trying to get things set up – that’s why the code we are using is so forced. But no matter, we got to explore a little bit and also test out a bit of number sense.

So, I think both of us were a little tired for this exercise, so sorry for a few of the extra mistakes. But, this is a really neat problem and also a fun exercise to do next time you are at the beach 🙂