Fawn Nguyen recently published an incredible blog post about a project related to the Euclidean Algorithm that she did with her students:
Fawn Nguyen’s “Euclid’s Algorithm
[edit] The above link is broken, but Fawn’s post is here:
https://www.fawnnguyen.com/teach/euclids-algorithm
Fawn’s projects are usually very easy to do right out of the box, and this one is especially easy since you can just start with her pictures. So, we just dove in.
You’ll see from the comments my kids had that Fawn really has made using this blog post effortless:
Next I asked them to make their own shapes. They built the shapes off camera and then we talked about them.
At the end I asked them when they thought a shape would require 1x1x1 cubes.
After hearing their thoughts about relatively prime numbers at the end of the last video I asked them to make a shape that wouldn’t require 1x1x1 cubes to finish. Here’s what they made and why they thought it would work:
Such a fun project. Fawn’s work is so amazing. I love using her posts with my kids.
Mike's Math Page 
I think the last video is from yesterday.
Fixed – thanks
It’s a treat to hear your boys think through these, Mike. They are getting older, I can tell from their voices! I forgot to add to my post, which I did with my kids the next day, that given a sketch, can they figure out the two numbers. We also practiced writing out the algorithm, and the kids saw the pattern. Pretty cool. Thank you for sharing!