I never anticipated the number of "aha" moments I would have by encouraging my students to talk or write about what they are thinking. The most recent was a simple warm-up on combining like terms. A student thought that 2x + 3x = 5x^2. Lately I've been asking students to give me "convincing arguments" about concepts. High school students like to argue, so I thought I'd go with it. So, my question to students was "Give me a convincing argument as to why 2x + 3x not equal to 5x^2?" That's a hard question to answer. Most students responded about the rule being that you shouldn't add the variables. They are "like terms" so x + x won't equal x^2. From there I was inspired by the "always, sometimes, never" questioning from the Mathematics Assessment Project. I asked "Is 2x + 3x = 5x^2 always, sometimes, or never true? Justify your reasoning and give counterexamples." Most of the students said it was never true, some said it was always true. I had to explain how to find counterexamples and we started substituting values for x. One student noted that x = 1 made the equation true. And then we puzzled about what else would make it true.

I grabbed Desmos, and showed students that 2x and 3x could be graphed using y = 2x and y = 3x. Two lines...so, what happens when you add two lines? You just get a different line...but it is still a line. The students predicted a shift, but instead the line just got steeper.

Then we graphed y = 5x^2 and saw the parabola. Quite clearly students saw that 2x + 3x cannot be 5x^2 because you can't add two lines and get a parabola. However, they also saw that 2x + 3x = 5x^2 was sometimes true...when the graphs intersect. We substituted x = 0 and x = 1 to show that was true.

I grabbed Desmos, and showed students that 2x and 3x could be graphed using y = 2x and y = 3x. Two lines...so, what happens when you add two lines? You just get a different line...but it is still a line. The students predicted a shift, but instead the line just got steeper.

Then we graphed y = 5x^2 and saw the parabola. Quite clearly students saw that 2x + 3x cannot be 5x^2 because you can't add two lines and get a parabola. However, they also saw that 2x + 3x = 5x^2 was sometimes true...when the graphs intersect. We substituted x = 0 and x = 1 to show that was true.

Students rarely think about an equation being false, let alone sometimes true. What a great way to show them visually what is going on. It wouldn't have happened unless I had been asking my students to give me convincing arguments, and using formative assessment to learn what they knew. I know that if students can make me curious, I will have more success making them curious.