Little Things

I used to have a value in my teaching that students should figure out as much of the math as possible on their own. “Never say anything a kid can say” or something like that.

That’s not a value of mine anymore. Instead, my value is that students should spend as much time as possible doing math. Doing math might look like figuring out something new, but it might also look like applying what they already know in new ways to practice, consolidate, and extend their understanding. I’m unafraid to tell students things if it means more time for doing math.

The issue with telling students things is it works best in small chunks. The longer I’m talking and the more new ideas I’m introducing, the harder it is for students to follow along.

In class yesterday I was working on conic sections in Precalc, and I wanted to introduce graphing circles. First I wanted students to get their heads around simple graphs of circles centered at the origin, like

Then I wanted them to translate those circles around the plane, graphing equations like

Students have seen similar transformations before, but likely in a different form, working with equations written as a function of x. The way x is transformed might be familiar, but doing the same thing for y is new. I could do some explaining to fill in the gap, but that can lead to a mix of blank stares and questions that lead us down confusing rabbit holes. This is tricky; I’m building off of students’ prior knowledge, but it’s a totally different structure than what they’re used to, so those connections are unlikely to be clear. And there’s nothing worse than telling students they should know something that feels confusing and counterintuitive to them. How can I bridge that gap?

I started class by giving students a set of parabolas to graph in groups:

There were a few false starts, but the ideas came back quickly. After graphing a few of these, we were able to discuss and summarize the core property of these transformations: subtracting something from one variable translates the function in the positive direction, and adding translates the function in the negative direction. Counterintuitive, but consistent with what students have seen before — and suddenly this rule holds for translations in both the x direction and the y direction.

With this knowledge explicit, translating circles is a cinch. I can introduce the standard form of a circle and get students practicing quickly, and the transformations feel like something that builds off of what they already know, rather than a mysterious new idea. My favorite part is that the introduction to transformations with parabolas probably cut the time of my explanation by half, if not more. I got to start class with students doing math, and we had more time to build off of that knowledge and solve harder problems with circles at the end of class as well.

2 thoughts on “Little Things

  1. GW

    I completely agree that extending an idea of translations to the y-coordinates is not necessarily a trivial concept just because students recognize the pattern for the x-coordinates. One way I have addressed this when teaching circles to my students is to start with the distance formula, since the definition of a circle is the set of all points equidistant to a center (h, k). Let (x, y) be any point on the circle. Then the distance from (h, k) to (x, y) is the radius r. When you substitute (x, y), (h, k) and r into the distance formula, then square both sides to eliminate the radical, we have derived the general equation for a circle. The issue of subtraction from both variables feels natural because a distance in the horizontal direction is calculated identically to a distance in the vertical direction.


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