In our textbook slope fields come during the differential equations unit, which for the last 8 years made sense to me. But every year there were groans from students and comments about how “pointless” they are.

Well, here’s why students think they’re pointless.

They already know antiderivatives. So they can take many differential equations and find the family of functions whose derivative is given. Many of the basic slope field problems can be somewhat easily antiderived, especially once students know about separation of variables.

“But, wait. Not all differential equations can be antiderived!”

This is true. And I point this out to students. They don’t really seem to care when they have to plot 25 line segments on a sheet of paper. For one problem. Especially once I show them how easily a computer can plot slope fields. I also think it’s in part because they don’t connect to very much. We just do the section and move on.

So I’ve been thinking about this for a few weeks, on and off, trying to figure out how to motivate the lesson. Here’s what I’ve come up with.

When I teach a derivative rule (the power rule for example), the next day I teach the antiderivative. It’s fairly easy for students to follow and then remember, because they just learned the derivative the previous day. Antiderivatives are just the process of undoing a derivative. This is as opposed to how my textbook does it, which is save all antiderivatives for the integration chapter. Anyway, instead of doing that, I think I’m going to show slope fields in between. In other words, I’ll teach a derivative, then use a related differential equation to get students to think about what a “slope equation” means and plot it, and THEN teach the antiderivative rule.

It would go something like this:

- Teach the power rule.
- The next day give students something like dy/dx=x – 1 and ask them to think of it as a “slope” equation. Inputs are whatever you want, outputs are slopes. Ask, “What will this look like?” Help students work out a sketch of the slope field.
- Then teach the antiderivative rule.

There will certainly be some students that figure out the antiderivative rule prior to or during work on the slope field(s). But at least this provides more of a motivated use for a slope field:

“What does the function whose derivative is ______________ look like? Well, we have this slope equation and it gives us the slopes of tangent lines at whatever point we want. Let’s use that to build a graph that gives us an idea of what that function (or family of functions) looks like.”

You could use this throughout the units on derivatives, including things like implicit differentiation. For instance, “We know how to derive x^2+y^2=9. But what if you’re given something like dy/dx=2x/y? What would that family of curves look like?” And then you can use a slope field.

It’s possible that this isn’t a great idea. Maybe it’s better to just leave slope fields as their own thing, drag the students through them, and move on. But I do think it would help students understand that a differential equation is a “slope equation”, and that it’s useful. I also think it might help students better understand what a “solution” to a differential equation is and how we might visualize it. Last I can imagine students would come to the “differential equations” unit already fairly comfortable with a lot of the concepts in it.

Any thoughts on this? Drop them in the comments!

Update: The first activity inspired by this post is located here.

Another update: The second activity, involving trig functions, is here.