Sharing a “visual pattern” triangular number identity with kids

Saw a fun tweet last night from Matt Enlow:

Here’s the underlying tweet since it doesn’t show up in wordpress:

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

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