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.