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

border problem


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“.]



I just dug this up. It’s what I handed back students after a “pre-quiz” (i.e. a quiz at the end of the unit, but before their quiz). I had forgotten that during that first year I handed back these things with class performance percentages on them.

Anyway, the way those percentages break down is interesting to me. Is it surprising that kids had so much trouble with negative exponents in numerical context, but had such less trouble with variables?

Comment on anything that you like, but here are some prompts:

  • Do you like my questioning? (Because I didn’t. Notice that moment when I pause and try to unask my question?)
  • Describe, as best you can, the way that this kid thinks about the Distributive Property.
  • What would you say next, if you were me?
  • Does this video have implications about the way that you’d teach this topic? (And, come on, don’t give me that “they need lots of practice stuff.” Of course they do. But what else?)
  • Do you prefer having videos over images on this site? (Because I have about a half hour more footage from my work with this kid…)

Looking forward to a great bunch of comments here. Don’t let me down?


Here’s how I responded: