A neat geometry problem I saw from Catriona Shearer

I saw this tweet from Catriona Shearer last week:

What fraction of the square is covered by these 5 identical rectangles? pic.twitter.com/x0VWYg076V

— Catriona Shearer (@Cshearer41) October 14, 2018

It was a fun problem to work through, and I ended up 3d printing the rectangles that made the shape:

2/2 it is also a great introductory 3D printing problem – here are the 5 pieces pic.twitter.com/T20Ub208wY

— Mike Lawler (@mikeandallie) October 14, 2018

Today I managed to get around to discussing the problem with the boys. First I put the pieces on our whiteboard and explained the problem. Before diving into the solution, I asked them what they thought we’d need to do to solve it:

Next we move on to solving the problem. My older son had the idea of reducing the problem to 1 variable by calling the short side of one of the rectangles 1 and the long side x.

Then the boys found a nice way to solve for x. The algebra was a little confusing to my younger son, but he was able to understand it when my older son walked through it. I liked their solution a lot.

Now that we’d solved for the length of the long side, we went back and solved the original problem -> what portion of the original square is shaded. The final step is a nice exercise in algebra / arithmetic with irrational numbers.

Definitely a fun problem – thanks to Catriona Shearer for sharing it!