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One of the things I love about Desmos is that it allows students and teachers to keep the focus where it should be. Working on a linear approximation problem in Calculus? It’s easy to get caught up in algebraic and numerical details and lose sight of the big (somewhat amazing) picture: We can approximate a crazy curve with a simple line! (Provided we stay in the neighborhood, of course.)

And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what’s happening visually in linear approximation?

Here’s a problem from last year’s AP Calculus review workbook:

There’s a lot of great work on the page.

• Take the derivative? Check.
• Find the slope of the tangent line? Check?
• Find the equation of the tangent line? Check.

But that’s where things fall apart.

Now, imagine you’re a calculus student. You’ve been hammering away at this thing for several minutes. Maybe you don’t even remember what the problem’s asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. “Great news, everyone! That’s on the list! Well done, folks. On to the next problem!”

Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.

Here’s what I built with a pair of students last year (with a link to the live graph here):

Students can’t use Desmos on the AP exam (for now, anyway), so I’m not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that “other” graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what’s happening in this problem so they’ll be more prepared for something similar in the future.

Here’s a short list of what this Desmos graph did for us in this scenario:

1. We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we’re still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
2. We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! (“Alfred! Get down here and pick up your comic books!”) So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don’t name it, it’s difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we’re going to take over the world, with math as our trusty sidekick.)
3. We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There’s a function. We called that f. There’s a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it’s easier to avoid simply evaluating f(4.2) or l(4.2), because we know we’re dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
4. We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
5. We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.

Do you need Desmos to teach this stuff? Maybe not. But given the option, I’ll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples.

So this year? We went straight to Desmos and put the emphasis right where it belongs.

P.S. I love GIFs.