November 12, 2015

Bar Models in Algebra

At ATMNE a couple weeks ago I attended Kate Marin's session "Bar Modeling: A Model for All" and remembered this sheet (below) I had used in years past which was perfect for our current unit of equations in one variable. I've seen bar modeling before but attending Kate's session reminded me of some important ideas. One is the banter that should be involved in solving any problem but particularly a problem in context. She modeled the back and forth between teacher and students of asking who the problem is about, what the problem is about and drawing a model as they tell her pieces of information. I gave students some of the banter questions in the handout and also did it with the class for the first problem. Kate also told us that the model doesn't replace the equation, the model is an aid in writing the equation. The equation is still part of the goal. However, there are many possible equations, encourage students to share as many as they can to validate everyone's way of seeing the problem (we used 72-48=3B, 3B+48=72, 72=3B+48, B+B+B+48=72...). And finally she reminded us that the final goal isn't a number, it's an answer in context, so the answer should be in a complete sentence. I modified my sheet to make it clear that every problem needs an equation and a sentence. My kids with large handwriting didn't have enough space to work but they could always draw the model and solve the equation on separate paper and record just the equation and sentence on the handout.

You might notice that none of the problems have a question. Last year some students noticed at one point, but this year no one even remarked on it! A couple times I had to encourage students to find all the missing information (the price of the sandwich as well as the fries or the amount spent on books rather than the amount left) but generally students want to fill in everything they can on their model.

I did not require students to draw a model, but I refused to discuss an incorrect equation with them until they had a model. Kids would tell me "I don't know how to do fractions or percents" but when I told them to draw a bar, and then draw 4/5, they could do that without assistance. The percents took a bit more banter, one conversation went like this:
What do you know?
"There's a 20% discount"
What does 20% mean?
"I told you I can't do percents."
What percent do you have if you have everything?
So what does 20% mean?
"20 out of 100. No way! I'm not drawing that!"
I agree, let's not.
"20, 40, 60, 80..." 
No joke, I said nothing leading and he started counting by 20 under his breath! He trailed off so I said yes! that! and he drew his bar model. Who can't do percents? Not this kid!

I'm not going to tell you how long we spent on this because you'll think less of me or my students (who are in 9th grade) or both(!) but suffice it to say, there was a lot of struggle, most of it in the form of learned helplessness. When I stood over them and said "Draw a model. Read me the info. Write it down. Read it again. Label. And again." they did great. Then I'd walk away and come back and find an answer without a sentence. "What does that mean? What does x represent? Read the problem. Read it again. Write down exactly what you just said." Basically my job in this class is to refocus their attention on the task and refuse to do things for them while convincing them I believe they can do it. For other 9th grade classes this might be a breeze. For plenty of middle school classes too I'm sure. But if you have kids who need spaced repetition this isn't too painful of a way to mix in some fractions and percents while doing Algebra. It reinforced rules of solving equations that some of my students are still shaky on: 5x means five x's and we divide by five in 5x=80 because we need to split that 80 up among the 5 boxes. Oh! I'm so glad that finally clicked for you kid, missing half of the introduction to solving equations wasn't helpful but we got you there eventually!

This is what happens when I see a kid working quietly and independently and don't go check on him:

Didn't anyone teach you that math makes sense? There are -2 stickers? Where? There aren't even any gift bags in the equation. I'm sorry that you learned that math is a magical land where you put numbers and operations together to get a mystery number, I'll work on undoing that belief.
(I cut this problem from the sheet above because the bar model isn't super helpful nor is there a particularly interesting equation to write, so I cut it for some extra space to write on the other problems. Forgot to fix the numbering, sorry!)

Interestingly though, this student did expect the problem he wrote to make sense. It looks like he solved it by saying 40 (collars) - 20 (left) = 20 (who knows what unit) but then realized this didn't make sense, so he fixed it. This is the only problem he wrote a sentence for, and the only one where his equation matches the context. I also love all the student problems that don't have questions!

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