It seems like it takes the edge off when the variable isn't there. But today one of our warmup problems was: 5x + 1 = 2x + 7.
I have been amazed at how many of my students have been willing to attack equation solving by using a guess and check table. I've never taught it that way, but some kids have just taken to it. After today, I may start to encourage it. One kid noticed that when you let x=1, the right side is greater than the left side. But if you let x=10, the left side is greater. When the balance of power shifts, you know that the answer is between your last two guesses. Of course, typical guess and check strategy. But the thing I like about it when dealing with these linear equations is that they are beginning to think in terms of linear systems and how the point of intersection acts as a dividing point between which equation has greater value. They're teaching me something.
But Brandon took the cake. He says, "Mr. Cox, you can tell the left side is going to be 6 because 5+1=6 and the right side is going to be 9 because 2+7=9."
"What does x have to be for that to be true?"
"X=1. But as we make changes to x, the other one is growing faster."
"How fast is it growing?"
"The left side is growing by 5 and the right side is growing by 2. So eventually, we know that the left side is going to be greater than the right side."
"Yeah. So when are the 1 and the 7 important?"
"Only at the beginning."
It took all the self control I could muster to keep from talking about initial condition or rate of change at this point. I'm glad I didn't because I think I would have ruined an authentic learning moment for this kid. The thing I wanted to encourage the most in him was the fact that he looked for patterns and then asked questions to help make sense of those patterns.
One warmup which I expected to spend 5 minutes on turns into 20 minutes of slope, y-intercept, linear systems and problem solving strategies all because a few students took an approach I've never taught.
Another example of the kids re-writing the lesson plan.
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2 comments:
I love that you let the students be the mathematicians. It is hard not to step in when they're making those discoveries and sum it up for them. However, I totally agree with you that it takes away from the "authentic learning" in which they are engaged. I do think it is better for them to let those ideas ferment for a while.
Did you have this student share his observations with the class or was it a side conversation?
I think I actually used the word "frement" when I talked about this with a colleague yesterday. I think that the one contribution that I can give these kids is to let them do it. I am not a good lesson planner so I have to let them do it. The one thing I can bring every day is the "question." I know I am good at that.
This conversation was definitely with the whole class. As a matter of fact, when a student has something to share with me, I get as far away from them as I can so even the side conversation becomes a class discussion.
I have to give you a bunch of credit for suggesting the analytical approach to geometry with my 8th graders because it has caused me to start using a more problem based approach to the algebra with my 7th graders. I don't know if this conversation would have ever happened had it not been for my PLN.
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