In this 7th grade class, they’re studying interesting ways of counting up the boxes in the border of a square grid.

They originally start with a 10 by 10 grid, but soon after they expand to a 6 by 6 gride and a 15 by 15 grid and use their techniques to count the squares in the new border. Then, in class, they introduce the notion of variable and set about shortening one of their verbal explanations using variables.

On the board are two expressions: 6 + 6 + 4 + 4 and 15 + 15 + 13 + 13. The teacher tasks the students with writing an algebraic expression that represents the number of squares on the border. The following is a recorded interaction among students during their group work:

Sharmeen: s + s + (s-2) + (s-2). Though that’s kind of complicated. Is there any other way to put it?

Antony: What is it?

Sharmeen: Uh, mine? It was s + s + (s-2) + (s-2)

Kim: No we had to, like, um, how about we write a variable for…make a variable for thirteen.

Sharmeen: Yeah, oops. Oh, m equals… OK, so it’s s + s + m + m.

What make you of this?

[All of this is lifted from “Connecting Mathematical Ideas“.]