Saturday, September 12, 2009

Algebra 1: Solving Equations

I am beginning the planning stages of our unit on solving equations in Algebra 1. In my past experiences, some students pick this up very quickly, no matter how you teach it, while other students struggle mightily. I want to try some alternate approaches this year, to really reach those students who have not been able to learn this skill in the past. I remembered an order of operations approach that I read about in the NCTM magazine a few years back. I can't recall the name of the article, but a little google searching found me this document that is even better than what I remembered.

Our students in Numeracy already work with bar modeling to solve word problems, so this seems like a natural extension to solving equations. I like this approach because it helps focus on the idea that the variable is a given quantity that must be determined, instead of focusing on the steps that isolate the variable. It also might help with those difficult to master "converting verbal sentences to algebraic equations" problems. Here are a few examples of how this might look. I know the diagrams are a bit confusing at first, but I think they would make more sense to students as they watch them get created and do them by themselves.

I also like the other representation discussed in the article. This is the original order of operations process that I had been searching for. I like this because it gives a very clear framework for solving equations - reversing the order of operations.

When you look at each stage, you can draw equal signs between the boxes. These would be equivalent to the intermediate statements in the traditional "do the same thing to both sides" approach.

So for the unit, I am thinking that we would spend two or three lessons on bar models to build the concept of what we are actually trying to do (find the value of the unknown amount). Then, spend a couple lessons on the order of operations representation to build an understanding of the process for isolating the variable. Finally, transition to the traditional approach, which is clearly the fastest and cleanest way to solve an equation of the three. This would take more time, of course, but the hope is that it would build a more enduring understanding.

Has anyone tried these methods with their students?

I wonder now if it would make more sense to start in with modeling sentence/word problems with the bar model method, and not start by saying that we are "solving equations". That way, more students would be engaged with the material, and we could eventually use the bar models to develop the equations.

This way, the unit doesn't start with the problem "solve (3/5)x = 45", which will stop most kids dead in their tracks, but maybe with something like "It took Sandra 45 minutes to finish 3/5 of her homework. How long will it take her to finish it all?", which kids might have more of an entry to. After we solve it, we can then discuss how to represent it as an equation.

Edit 2:
I also need to think about how to incorporate the balance idea and preserving equality... Kids don't always know what the equal sign really means. Maybe in the transition time from the box method to the traditional method?

Edit 3:
(Written on 10/27 - at the end of the unit)
On reflection, the problem was not having enough time to really devote to the two alternative methods. Both did show a lot of promise, but we weren't able to really practice either enough for it to really stick with students. The bar model method really worked to help students set up and solve word problems, so I think I will stick with that next year. Give it some more time so that it really sinks in and can be used to get a deeper understanding of fractional coefficients. I will probably save the GERMDAS method for individual tutoring with students who are not having success with the traditional balance method. Less fights... more differentiation.

All of these lessons have been added to the box widget on the left.


Alison Blank said...

This is pretty intuitive to me, so I hope it will be to your students. My concern would be that, for the students who pick up on how to solve equations quickly, this will just be obnoxious busy work and they will complain or act out. Otherwise, it seems like it's certainly worth a try! I'm looking forward to hearing how it goes down.

Dan Wekselgreene said...

The difference in abilities is a constant struggle in my class. All of the students have taken algebra 1 already in 8th grade; we make them all take algebra 1 again with us as freshmen, because even those who have passed the class in middle school tend not to actually be able to do any algebra (procedurally, let alone conceptually).

But they all remember different pieces here and there. I'm sure quite a few will already know how to solve equations like 2x + 5 = 11. And you're right, when they do know how to do something (or just think they do), they become very resistant to trying "the long way".

However, I doubt those same kids can solve 5 - (2/3)x = -3. Using these visual methods might really help them, even though they "know a faster way". So I would probably push them to try these methods anyway.

I will do a diagnostic before we start the unit, though, to determine if there are students who really have already mastered solving equations. I might have a total of 4 or 5 out of all my algebra 1 students. For those kids, I will probably come up with some alternate assignments for them to work on during this unit. Then I would give them the choice: learn the same topic in a new way, or learn a new topic independently. Both have value.

Mr. K said...

The second method is very similar to the way I teach it.

The tradeoff that I see between the two is that your method makes the process a little bit clearer, at the expense of accessibility for the low performing kids - I believe part of the success is that they can just do this, and worry about the why it works after they have the process down. It seems possible that having them write the expressions in the tp boxes might just be more stuff to put in their head.

Mr. Sweeney said...

I've not tried that approach, but I can tell you what I do with my classes which has been pretty effective.

I start by teaching the unit to the best of my abilities, but there are always a few students who lag behind like you said. I then have a song and dance we do as a class which reinforces the steps they need to take to solve the equations. It might sound silly, but it really helps them understand that when they get stuck, there are steps they can follow.

After the unit is over, throughout the year I do a "skills drill" sheet for 10 minutes at the beginning of about two classes a week. They are 6 problems long, each of which show off some different part of solving equations. I look them over really quickly as they finish. Students that make a certain number of mistakes have to meet with me outside of class either during our study hall period or after school for a few minutes to go over it together. If a student gets through 3 of these sheets and only makes 2 mistakes between them, then they move on to a new topic for skills drill(graphing equations). If they get through all of the skills drills topics(rarely happens, but provides a carrot for them) then they can have the 10 minutes of class to work silently on something else or read a book. I've had a lot of success with these skills drills, but it does take time out of classes.

Another strategy I've used for this was having my students create wiki pages for each important topic for our exam as a group project. I intentionally made students who struggled with a particular topic responsible for explaining that topic. Students who excelled at most things were grouped together for the newest material. I was a little worried about it at first, but it worked out quite nicely in the end.

Dan Wekselgreene said...

Mr. K -

I remember that post from way back when, when I didn't think about Algebra 1. Thanks for linking to it again.

I believe that the original NCTM article that I was thinking of did it like you do. This new article got me thinking though - I like how the boxes correspond to the intermediate steps of the traditional method. I wonder if this would help the transition, especially for equations that are more than just two-steppers.

This method does have more stuff to write and keep track of, but at the same time, it is totally procedural (as long as they have mastered the order of operations) and does extend to most basic equations. So a lower skilled kid could solve an equation like -3(2 + -5x) + 30 = -36 by applying the box method.

Or maybe another way to look at the benefits is that it forces students to constantly think about the order of operations as they work, pushing them toward mastery? One can hope? :)

Dan Wekselgreene said...

Mr. Sweeney -

Thanks for the info. Your process sounds like it's pretty effective - though I don't think I have the brain space to employ something like that this year.

Any youtube clips of you guys singing and dancing? :)

Nick said...

Dan - Thanks for posting this. I really like the visual. I'll give this a try with students who struggle. My inclination would be to go with the standard approach for most kids, and reserve this as the backup for those struggling more. I would demonstrate this skill to the entire class once or twice to broaden the conceptual understanding in general. I think after it's been introduced, maybe ask kids to solve some equations using the method they like best. From there you can decide how best to get those bar approach more comfortable with the traditional approach.

David Cox said...

You are an absolute machine. I am embarassed to admit that I have never seen these the bar modeling or order of operations before. (maybe I have seen the order of ops, but it was probably on your blog :). This year I introduced equations to my 7th graders using the balance scale. Use circles for the x's and squares for the units. It was amazing how quickly kids picked up on it and transfered it to equations like:
2x + 3 = 5x -7 .

Kim Hughey said...

I have used Mr. K's method (which I stole from his blog) during summer school. I could not believed how well it worked. We spent some time with the boxes and then moved on. What the boxes helped the kids to understand was the undoing process. It seems as if many times students are not sure where to start or how to undo 16x = 32. I will find them somtimes subtracting the 16. This tells me that they don't really understand the whole order of operations concept.

Vaporizing Boxes said...

I used the order of operations boxes with my sixth graders last year as an intro to solving two step equations. I then couldn't convince my students to do the standard isolate a variable method. They loved mapping out every step. The order of operations boxes are in the NYC Impact Math curriculum for 6th grade.

Dan Wekselgreene said...

I think it's excellent if they love mapping out the steps, as it will really reinforce the order of operations for them. I think students will naturally move out of reliance on this kind of scaffolding as they get better at it (they will naturally look for short cuts) or as problems demand it.