An accident happened to me while teaching Algebra 2 this year. A very good accident. I was teaching solving equations, and I always tell my learners that solving equations is about looking for 1’s and 0’s. I stress this fact, and point out the 0’s and the 1 every time we solve something.

And of course, before we can do equations, inequalities, or absolute value equations, we do expressions. And EVERY single person in the class can rattle off “Please Excuse My Dear Aunt Sally For Looking Too Rough” (did you forget the “from left to right too?).

I taught the lesson, I had SADMEP written on the board, and as I looked at it, I realized that if I wrote the 0 and the 1 down in the right places, it would help. Then I taught Inverse and Identity, and I realized that I had the Inverse and Identity right there!

But then I had an IEP learner in my room later, and he asked me to explain it again. He was confused on when to use SADMEP and when to use PEMDAS. He has been told since elementary school that PEMDAS was how to do problems, and in high school it was all different. He really struggled in Alg 1(I had him then also), and his struggles have not abated. So, I wrote down something else on his paper. It looked like this:

Link: http://www.box.net/shared/2msix7air5

And the learner looked at this and said “AHA! I understand now! We are doing different things.”

The whole issue of Expressions and Equations became clear to him, and I realized I had just written down something that made sense. It have him a HOOK to attach his learning. PEMDAS was a hook as well, but it no longer works for solving, and putting them side by side made sense to him.

When the learner looks at it, she can see that if you have addition, you need a subtraction. If you have division, you need a multiplication. The Identity is written right there, and the inverse is the opposite! They groupings GIVE the learner a HOOK to LADDER what they know to solving equations.

My learners tell me it makes sense to them. I realize it is a ladder that allows them to practice what they know in a new situation and it gives them the same structure that PEMDAS did for expressions. It does not give them answers, it gives them KNOWLEDGE about how to solve equations.

I like it.

The notion of using inverses is the insight I labor to get most of my students to see. My attempt to describe both inverses and some of the reasoning behind the order of operations is at:

http://mathmaine.wordpress.com/2010/01/05/arithmetic-operation-pairs/