My older son had a neat problem about counting rectangles on his homework (from an enrichment math program). He and I talked about these types of problems for a little bit during the afternoon, but they are such good examples that I thought it would be fun to do a short project on them.
We started with an introduction to the problem and a simple case – how many rectangles (with integer coordinates) are there in a 1×6 grid?
The neat thing here is that there was an accidental connection to Kelsey Houston-Edwards’s latest “Proof” video that we did a project on a few days ago:
Kelsey Houston-Edwards’s “Proof” video is incredible
Next we moved on to a more complicated problem – how many rectangles are in a 4×3 grid? Can we use the choosing number idea from the last video to count all of the rectangles here, too?
Finally, I wanted to show a counting procedure that was a little different than the choosing number approach in the last video. Here we defined a rectangle by the corners rather than the rows and columns.
This is a more complicated approach, but I thought it would be useful to show that two seemingly different counting strategies led to the same answer.