Today I wanted to boys to explore a bit more. The plan was to explore one basic property together and then for them to play a bit on the computer individually.

Here’s the first part -> Looking at what happens when you compute the continued fraction for a rational number:

Next I had the boys go the computer and just play around.

Here’s what my younger son found. One thing that made me very happy was that he stumbled on to the Fibonacci numbers!

Here’s what my older son found. The neat thing for me was that he decided to explore what continued fractions looked like when you looked at multiples of a specific number.

So, a fun project overall. Continued fractions, I think, are a terrific advanced math topic to share with kids.

I didn’t watch your videos in this post, but if you’re playing with continued fractions, you might try starting with the golden ratio, and looking back at the ratios you can find when making zometool shapes (especially pentagon with inscribed pentagrams), and seeing how you can make scaled-up versions of a given zometool shape by using the appropriate combination of the two shortest strut sizes. It’ll get you back to seeing how the convergents are Fibonacci numbers, and so on.

While we’re at it, this is not necessarily the best produced animation, but it’s an interesting way of thinking about continued fractions geometrically:

Geometrically, a continued fraction is what you get if you take the part less than 1 (here, 0.14159…), make squares of that width, and fit as many as you can within a height of 1. Then if you take the remaining gap at the bottom, make squares of that height, and fill as many as you can into the available width, etc.

I think this is a nice geometrical interpretation (in terms of area) of what a deeply nested fraction means, which I had not seen before. Just written out symbolically, I find the nested fractions harder to reason about. YMMV.

## Comments

I didn’t watch your videos in this post, but if you’re playing with continued fractions, you might try starting with the golden ratio, and looking back at the ratios you can find when making zometool shapes (especially pentagon with inscribed pentagrams), and seeing how you can make scaled-up versions of a given zometool shape by using the appropriate combination of the two shortest strut sizes. It’ll get you back to seeing how the convergents are Fibonacci numbers, and so on.

https://www.wolframalpha.com/input/?i=Convergents%5B1,+1,+1,+1,+1,+1,+1,+1,+1,+1%5D

While we’re at it, this is not necessarily the best produced animation, but it’s an interesting way of thinking about continued fractions geometrically:

Geometrically, a continued fraction is what you get if you take the part less than 1 (here, 0.14159…), make squares of that width, and fit as many as you can within a height of 1. Then if you take the remaining gap at the bottom, make squares of that height, and fill as many as you can into the available width, etc.

I think this is a nice geometrical interpretation (in terms of area) of what a deeply nested fraction means, which I had not seen before. Just written out symbolically, I find the nested fractions harder to reason about. YMMV.