A while back, I wrote the intro to one variable lesson. I’d meant to follow it up, and never did. I also took some pictures, but i seem to have lost most of them, and the materials used for those lessons have disappeared into boxes while I was temporarily displaced from my room.

Here then, is the half assed follow up:

The previous lesson provided some valuable foundations: undoing operations, and the order of undoing operations.

However, it didn’t provide direct understanding. So the next step is to make a bunch of those little paper footballs. It turns out that if you cut a piece of 24 lb paper into a 4“x11.5” strip, it weighs exactly as much as a post 1982 penny^{1}. You can get very nice brightly colored paper packs of 24 lb paper. For each color, you can tuck a number of pennies inside, giving you unknown weights ranging from 1 to 6 pennies (the value being one penny more than the number of pennies hidden inside)[2]. You can then use a balance (with a big equals sign taped to the middle) to make up equations with various number of unknowns and knowns on each side, and let the kids solve them with the caveat that the scale must balance after each step.

This teaches them the “do the same thing to both sides” rule. It only teaches it for nice round positive numbers. And even with a team of kids on each side writing down the expressions, and what the steps are that they’re performing on those expressions, they need more help with just the sheer mechanics.

Enter the sentence display. You’ve seen them in every english class. Some teachers use them for word walls to reinforce vocabulary. This is how I use them:

You create decks for each problem. Each card gets a term. This is important – we’ve learned to see expressions in terms of their components, but to the kids thry’re just really long strings of mixed up symbols. Breaking them up by putting each term on a card helps reinforce the idea of the basic components (i.e. terms), and using the “opposite cards”[3] to cancel things out allows them to see how the equations might be manipulated.

The only parts I don’t create for each deck are the equal signs – I always leave those there, and lined up. It is a link back to the equal sign in the middle of the balance, and reinforces the left side/right side idea of doing the same thing to both sides.

I’m still not completely happy with my success rate on this – I still only manage to wean about 85% of the kids from the previous method. But given that these are kids who are learning this for the second time around, and that *none* of them got it last time, I suppose I can count it as a success.

^{1} This actually led into an interesting exploration (for me) of how many of my kids understood conservation of mass. Not a math thing, but I let them spend some time arguing about whether a folded piece of paper was heavier than it was before it was folded. Then i walked into the neighboring science teachers’ classrooms during breaks, and their kids the same thing…

^{2} I had a great picture of this. Really. Somewhere…

^{3} They figure out pretty quickly that opposite cards always have to come in pairs.