For the last couple of weeks I’ve been teaching geometry to both kids. Sort of a coincidence since Art of Problem Solving’s *Prealgebra* does a little intro geometry at the end, but it is definitely a fun coincidence.

I’m enjoying watching my younger son think through some introductory ideas and enjoying watching my older son start to think about some ideas in 3-dimensional geometry. Today they each made great progress working through two challenging problems.

For my younger son the problem was a fairly complicated area problem. His approach to geometry has not involved a lot of calculations before, so I was surprised when he started calculating right from the beginning here:

Although I was impressed with his ability to work through the calculation from start to finish, I also wanted to show a more geometric approach. Fortunately we had some Magna Tiles laying around that were the exactly the right shape for this problem. These tiles allow you to see the solution without doing nearly as much calculating:

Hopefully this extra exercise with the Magna Tiles helps him build on his geometric intuition. I certainly leaned heavily on calculating rather than intuition as a kid, so I feel that I need to be extra diligent about showing ideas that help both kids build intuition.

The problem that my older son was working on this morning involved finding the distance between the centers of two faces of a tetrahedron. This problem is pretty challenging and requires you to visualize some difficulty 3D geometry. Unluckily (or maybe luckily, I’m not sure) we didn’t have our Zometool set laying around to help with the visualization – it all had to be in his head or on the board.

We probably talked about this problem for 30 minutes leading up to the video. One great moment was when he realized that the angles between the faces of the tetrahedron might not be 60 degrees even though all of the faces are equilateral triangles. Thinking about how to determine that angle led him to the solution.

After we finished our conversation I asked him to do a little recap of the problem and then threw in one extra question at the end just to mix it up a little:

So, a great morning. It is nice to watch my younger son make progress thinking through some of the ideas from elementary geometry, but also a little surprising to see him approach the problem by calculating. It is also interesting to watch my older son think through some of the ideas from 3 dimensional geometry. It isn’t easy to see all of the angles in 3 dimensional geometry, but hopefully today was a productive struggle for him.