# Inventing Triangles

The advantage of multiple classes is you can refine, compare and contrast. Loving it.

This kind of lesson requires, from the teacher:

• Mathematical agility, to be able to quickly check if a triangle is worth exploring, or if a conjecture is worth exploring
• General knowledge, to link in unexpected ways (see below)
• Love of chaos – got no idea where we were going to travel

One group decided the outer diagonals of the triangle would be 1,3,5,7,9… and then used a Pascal-like rule to construct the rest. They abstracted to just odds and evens, and started noticed nested patterns of triangles.

Another group started with rt(2) as the diagonal, and multiplied rather than added. They too found the nested triangles…

I got very excited at the parallels with Sierpinski’s triangle. The whole class crowded around, awed into silence, to compare and contrast the two. (One made by continuing down an infinite number of rows, the other chops a finite thing infinitely many times).

Later that afternoon I noticed precisely the same pattern in a textbook. After yelling about being the Neil Armstrong of triangles, boldly going to new mathematical lands, I was disappointed to see that someone got there before the kids. Does it matter? Not really. It does matter that I was ignorant of it existing during the lesson – the excitement would have been less genuine.