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)

CalcDave said...

Maybe you could start with the substitution to see how that goes. Replace x with your favorite number and then get a value. Use the order of steps to build up the function.

Also, I wonder if it would be slightly better on the board if you continued the second column at the bottom and worked up. Then the corresponding parts are next to one another rather than reversed.

I like the idea here, though!

Miss Gray said...

We use this method when we do literal equations. I was wondering if you have applies it to equations with variables on both sides (ex no solution, all real etc), if so how did you do it?

Dan Wekselgreene said...

@ CalcDave,

Using substitution is exactly what I did when students were given a constructed equation and had to figure out what the steps were. Some students were able to see the order without that crutch, and others could only see it when I had them work through the operations with a sample x.

Starting at the bottom and working up is actually how I did it several years ago when I tried this before. It makes sense to me to do it that way. But when I was reading through the FAL, they have it starting from the top. I was wondering why, but I figured they know what they are doing and probably have a reason, so I decided to go with that. I think it may be because, when they are typically solving equations, they will be expected to start at the top and work their way down, so the idea is to habituate them to the typical order of writing the steps? Maybe the trade-off is a little more confusion at the outset for less confusion later. I'm definitely not convinced either way yet...

@ Miss Gray
I haven't applied it to more complicated equations yet. And I'm not sure that this method can apply on its own, without adding in other steps, when there are variables on both sides. There is another FAL that deals with number of solutions, and I plan on looking more into that one before tackling the next phase of equation solving. How do you introduce variables on both sides?

Unknown said...

I do a variety of different things. Something that I do, which sounds similar to your current method, is that I have students analyze what is happening to the variable within the equation. They have to analyze this step by step and create a list of all of the actions that occurred. I then ask them how to undo these actions. most start undoing in the wrong order so I make the analogy of putting on clothes in the morning and taking them off at night. They typically get it and are then able to make a list of the correct actions they need to take. To help remediate students knowledge on this, particularly as I teach a lot of upper level math know, I ask a series of questions. "What do you want," "what has been done to it" and "what do you need to do to undo it." Same thing I know, but I also get them to attend to even the visual structure of equations.

You can often tell visually what things are "most closely attached to the variable." In this respect, those things are like the sock on your foot and things that are attached less tightly are the shoe. The visual structure is pretty helpful for a lot of students.

untilnextstop said...

I like it! I use a similar method called "unwrapping the onion." http://untilnextstop.blogspot.de/2012/11/visualizing-order-of-operations.html It worked very well with my 7th-graders last year (who went from having never seen variables to solving quite complex equations)!

Annie Forest said...

I love the FAL lessons. I recently used the lesson called "Solving Linear Equations in One Variable" with my 8th graders and it went great! It is a great one to use when dealing with variables on both sides and when equations can be always, never, or sometimes true.