This morning I had the boys each choose a chapter that they found interesting and we walked through the proof that Polster in that chapter.

My younger son went first – the idea he found interesting was dividing up a square in different ways. Here’s the introduction and an explanation of two of the four ideas:

Here’s how we finished up the last two proofs:

Next my older son found a neat proof relating the volume of a sphere to the volume of a cylinder and a cone. He struggled a little bit to understand the proof, but the struggle that takes place in this and the next video is a great way to see how kids learn and think about math.

Just so there’s no confusion, the formula he derives for the area of the slice of the cylinder / cone we are looking at isn’t right. He’ll discover the mistake and correct it in the next video.

Here we find the second formula that we need to show how the volume of a sphere relates to the volume of the cylinder / cone combination.

Finally, we revisited an old 3d print that we had showing the relationship between the volume of a sphere, cylinder, and a cone. The print is designed by Steve Portz and is on Thingiverse here: