What learning math sometimes looks like -> Michael Pershan’s Square

Saw this interesting problem posted on twitter today by Michael Pershan:

As I mention at the beginning of the first video, we are not studying this type of problem right now, but I thought it would be interesting to see my son try to work through it. His solution isn’t fast or necessarily a direct straight line to the answer, but it was indeed interesting. This is what learning math looks like.

In the first part I introduce the problem and my son begins to work toward the solution. His instincts are to find the length of the square. He manages to find that length at the end of this video:


After finding the side length of the square, my son is pretty puzzled as to what to do next. He spends a lot of time looking at the picture and even draws in an extra triangle. Eventually he wonders a little bit about symmetry and draws in a few other triangles. Without knowing it, yet, he’s inches away from solving the problem.


At the end of the last video my son thought that finding what A + B in his picture was, that would be a good step towards the solution. Funny enough, he’s got the answer written down on the board, but he doesn’t see the geometry right away and begins to doubt that he’s on the right path. Interesting he changes the approach to trying to find the area of the triangles and then the area of the square – this thought will help him out tremendously just a few minutes later.

Suddenly – at 2:16 in the video – he sees the connection and the whole solution to the problem falls into place.


Finally, we finish up the problem. I had him work through the quadratic formula off camera. Once he has those solutions, he’s able to write down the coordinates of the other corners of the square right away.

We wrap up this project by talking about a few other symmetries in the problem that could have allowed us to see the solution in a slightly different way.


Overall a fun little project. Sorry for the quick write up – only had 30 minutes to put the blog post together before having to run out the door.

A fun discussion about inscribed circles for kids studying geometry

I’ve been studying angles and circles lately with my older son. One of his homework problems from our Introduction to Geometry book was to prove a formula for the radius of the circle inscribed in a right triangle. The problem gave him a bit of difficulty, but talking through it led to a great discussion about tangent lines and angles.

First up, the original problem:


The prior discussion extends quite naturally to give a nice (but different) formula for the radius of the inscribed circle in a general triangle. Really the only difference in the argument is that we are looking at areas this time rather than at lengths:


Finally, we’ve just found two different formulas for the radius of the inscribed circle. For a right triangle, these two formulas ought to be the same. They don’t look the same, though, what’s going on? Oh look, the Pythagorean theorem!


So a fun geometry discussion starting from a nice little homework problem. I remember learning these formulas in high school through prepping for math contests, though I honestly don’t remember how much we dug into the ideas behind the formulas back then. It sure is fun to share these ideas with my son now. It is amazing how some simple ideas about angles in circles lead to such beautiful results.