Sunday, May 5, 2013

Teaching Algebra without algebra.

I used to be terrible at teaching how to solve systems with elimination. I blame my textbook. Pearson starts their chapter on elimination like this:

"The Addition and Subtraction Properties of Equality can be extended to state, If a = b and c = d, then a + c = b + d. If a = b and c = d, then a - c = b - d.
You can use the Addition and Subtraction Properties of Equality to solve a system by the elimination method. You can add or subtract equations to eliminate variables."

BORRRRRINNNNNNG!!! And my typical student would have no clue what they're talking about. The book then shows how to solve the system 5x - 6y = -32 and 3x + 6y = 48. Of course, I'm the idiot because for so many years I've taught it the same way.

The other day, I was thinking about how horrible this lesson is and how it doesn't make much sense to the kids. I wondered if there was a way to solve a system of equations without using algebra and equations in the traditional sense. So I presented this problem:

I went to Dunkin' Donuts to get some of my colleagues some doughnuts and muffins. I bought five doughnuts and two muffins for $4.95. While this made some of my colleagues really happy, I noticed that a couple of them gave me dirty looks in the hallway as they didn't get anything. So the next day I had to buy seven doughnuts and two muffins for $6.25. Later, one of the teachers insists that he pay me for the muffin that I got him. I realize that I had no idea how much a muffin cost. Can you figure it out?

Some kids really struggled with this at first. They insisted that there wasn't enough information to figure it out and they were waiting for me to give a clue. After about thirty seconds, I noticed that some students were frantically writing. A little later a student raised her hand to tell me that she had the answer. Some of the other students began to grumble and assumed that she was wrong. I walked over and gave her the approving nod.

A little while later, after many appear to be making no progress. I showed how only knowing the cost on the first day didn't help. I showed that there were several possibilities for the price of each item such as a penny for each doughnut and $2.45 for each muffin. Or the two muffins were free and the doughnuts were $0.99 each. But none of these values would explain a cost of $6.25 on the second day. (Some were beginning to suspect that this had something to do with systems of equations.)

I asked them to think about how the situation changed from one day to the next. It was at that point that the light-bulbs began to turn on. Students started to see that the price went up by $1.30 and I only added two more doughnuts.

They could then find the price of one doughnut.

Once they had the price of one doughnut, they could go back to the first day and find the cost of the muffins.

We reviewed the same problem as a system of equations. It wasn't too difficult to see all of the similarities between the two representations.

The best student asked what would happen if I didn't buy the same number of muffins on each day. I wondered that too and asked what would happen if four muffins were bought on the second day. Of course I pretended to not know the answer and let the students come up with ideas. After a few minutes of discussion, we were about to give up when one student asked what would happen if we doubled one of the orders. I couldn't help but smile.

Note: Yeah, I misspelled doughnut. What of it? Many of my students thought I spelled it wrong in class because, according to them, doughnut is really spelled D-O-N-U-T. Thank you Dunkin' Donuts for helping kids with their spelling!


  1. This is great! Thanks for sharing. Oh - and you absolutely spelled doughnuts correctly :)

  2. It's blasphemous that you talk about doughnuts without mentioning Eddie Izzard. You wrote on CMC notepad paper, and seeing that makes me miss you even more. Great lesson, context always helps! Thanks, Nathan.

    1. Fawn, you're a doughnut. I was hoping you or Andrew would recognize the CMC paper. Thanks for your awesomeness!

  3. I have been thinking a lot about systems since I saw Christopher Danielson's Oreo presentation last week. One question that came up along the way is, how many calories are in a unit of stuf? My mind rightly went to systems of equations, as it did for most of the audience. But then he and others subtracted the equations, which I always thought is a procedurally clunky way to do it. Seems to me that when people subtract and there are negatives involved, the ground is fertile for errors. My algebraic preference has always been to obtain opposite coefficients and add the two together. Of course, we usually don't assign any meaning to the system, so that made a lot of sense.

    But this had meaning, so when Christopher put up a picture an audience member drew (looked something like this), subtraction made a whole hell of a lot more sense. Balancing procedural efficiency with contextual importance is a challenge for many, myself very much included.

    Bottom line here Nathan is that you hit this one out of the park and made me hungry for donuts (lexical efficiency).

    1. It's weird that you mentioned these two methods for systems: subtracting vs finding the opposite and adding. I had the same conversation with a colleague this morning. We both used subtraction, but another colleague of ours' used your method. I completely understand why you would want to do it that way. Personally, I'm thinking, I just spent this whole year doing subtraction of negatives with the students. I'm not running from it now. And besides, I really want them cluing in on the idea that they are eliminating terms by either adding or subtracting them. Subtraction also makes a lot more sense when all of the numbers are positive...something that will usually happen with a systems word problem.