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.
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?