An exciting MISTAKE

5 Oct

I usually start off the systems of linear equations section with a CBR activity where in their group they have to use two TI-graphing calculators and two CBRs to match 4 different graphs. It’s a great activity because it gives them exactly what the graph looks like in real time. This means that they can actually see the creation of the abstract graph while watching what is being represented.

This year though, I came upon a very interesting mistake that I too quickly put off at first. I kept thinking about it while they were doing their activity and in the end, I was convinced it was a brilliant mistake and one that must be addressed.

But first, here is the actual activity for those who might want it. I was actually debating whether I should even include this activity this year due to time constraints, but as soon as I heard the discussions that arose from just the first question, I knew it would be well worth it. Then of course, there was this “mistake” that made it THAT much better. =)

Picture 1
Picture 2
Picture 3

The Mistake
The actual mistake was made while one group was working on the second graph with the parallel lines. We had been using tape measures taped to the floor in order to create accurate graphs.

When I turned around, I saw that two people were walking away from their tape measures with the CBR pointed towards them. I was baffled. So I went over to ask what it was that they were doing and why they were doing it.

The response I got was “It looks like that, so we’re making it look like that.”

Ah~!! At first, I thought it was just a funny mistake. When I asked them to explain what the graphs actually meant and compared it with how they did the previous one, they all started to laugh too upon realizing what they did.

BUT, like I said before, the more I thought about it, the more I thought it was a brilliant mistake to address. So I took pictures of the tops of their heads at the end of the period to present the mistake to the rest of the class the next day.

So the next day, one of the things I did was to have the class actually think of what it was that the group was thinking. (In the Powerpoint, the multiple heads actually move one by one across.)
Picture 1

Picture 2

Then, in order to address the issue and have the class really understand why this would not give the right graph, I had them interpret 3 points on the graph and what they meant. Not only that, I drew on this slide, where you saw the x and y coordinate as actual DISTANCES on the graph.
Picture 3

I then had them re-think how this could have been done w/ the girl walking the same way, but simply moving the CBR, which would have looked something like this: (again, the powerpoint actually had them move instead of this mess of a screen capture.)
Picture 4

What I loved about this mistake:

  1. It made sense. When I had asked my students what they thought this group was doing, I almost immediately saw the wave of understanding pass through the class as they started to discuss it.
  2. The concept of seeing each point on an x-y plane having TWO values, TWO meanings per point is a tough concept for students to grasp. When they saw the tape measure as the y-axis, they ignored what the x-axis meant in also existing.
  3. Seeing the dissection of each of the coordinate point as a horizontal distance and a vertical distance is a concept that is seen all throughout math!! I suppose in any kind of graph reading, but I can think of unit circles, iterations, geometry, and even just slope! (which is probably another added reason to why they have such a hard time seeing it still.)

This is one of those things, like most things in this class as it turns out, that might not show up on a CST exam. I don’t even know how to test for understanding myself. I just found it fun and interesting to address. Of course, I believe it was crucial too. But *shrug*, what isnt?

(In case you want the activity, including the instructions on how to transfer the files, see widget to the right. The file is titled “CBR Systems” and the calculator instructions is called “TI Combining Data”.)

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