Mathologer recently published a terrific video about the Golden Ratio and Infinite descent:
As usual, this video is absolutely terrific and I was excited to share it with the boys. Here are their reactions after seeing the video this morning:
My younger son thought the discussion about the Golden Spiral was interesting, so we spent the first part of the project today talking about golden rectangles, the golden ratio, and the golden spiral:
My older son was interested in ideas about irrational numbers and why the spirals were infinitely long for irrational numbers. We explored that idea for using a rectangle with aspect ration of 
Unfortunately I did a terrible job explaining the ideas here. Luckily we were reviewing ideas from Mathologer’s video rather than seeing these ideas for the first time. I’ll definitely have to revisit these ideas with the boys later.
Mike's Math Page 