Wednesday, October 02, 2013

Constructing and Deconstructing Equations

I am ready to move my algebra 1 students into solving equations, now that they have done such a good job with patterns.  I looked through the different FALs available on the MAP site and Building and Solving Equations 1 caught my eye, as I experimented with using this method to teach equations in the past.  I don't think I did a great job of it last time, and I think I can definitely improve and hopefully make it work for the students.

The most common problem I see with students who struggle with solving equations is the order to do the steps in.  Usually, they can figure out that, in a 2-step equation, you always get rid of the subtraction/addition part first, but multi-step equations are a real issue.  And, when they get the dreaded "reverse style" equation, like 8 - 2x = 2, then everything can fall apart.

The idea of the deconstruction method is to really have students focus on the skill they do know - order of operations - to figure out what steps to take to solve.  And, it goes beyond just saying "reverse PEMDAS", to really concretely having students walk forward and backwards through all the steps.

Yesterday, I had students practice simply constructing equations.  I gave them x = 6, as suggested in the FAL, and we built one-step equations off of that using different operations.  Then, we took those new equations (like x + 5 = 11) and added a second step with a different operation.  Eventually, we built through 4-step equations.  And, we practiced checking by substitution.  The FAL includes sample student work and we analyzed that.  One student's work is correct, but can be improved by clearly showing the steps.  The other student's work is incorrect, and checking by substitution at each step reveals where the error is.  (The error has to do with fraction addition, so I didn't really discuss it with most students, since that would have been a bit too overwhelming in this lesson.)

At the end of the lesson, I could see that students were starting to get it, but still not totally clear.  So today, as a class, we built two more equations.  Then, I began the process of deconstructing them.  Here is how the board looked after those two problems:

After this, I continued with the activity in the FAL.  Each student picks two solutions, and constructs a four-step equation around them.  Then, they exchange problems, and have to deconstruct each others' equations to see if they could find the original numbers.

Students were working really hard on it, but were definitely struggling with a variety of issues - order of operations, basic calculations, how to organize their work, and so forth.  But, on the positive side, most students knew which operations they should be undoing at each step.  So, progress!  I'm going to keep working with them on this for a few more lessons.  Then, I'm going to give them a set of two-step equations, and my hypothesis is that they will be able to blaze through them, like a batter taking the donuts off the bat.  Hopefully, pushing them really hard at the outset will pay off later.  I'd appreciate hearing other teachers' thoughts on this method for teaching equations.

Here are the handouts we're going to practice with tomorrow.   (doc)   (pdf)