This is my 500th post. I started writing something off the cuff about where I was and how far I’ve come as a teacher since I started blogging (which is when I started teaching). But then I realized that in the past two years (I’ve been teaching for four), I’ve stagnated in my evolution, and I got all depressed and wrote something that would probably resonate some of you, but also that would elicit pity comments. And I those depress me more.
So instead (DEFLECTION ALERT!), I thought I’d post something I came up with last year to deal with the Fundamental Theorem of Calculus, Part I.
Although it is easy enough to tell kids “this is what is says, this is how you apply it” (and they can do it), what I have always had a problem with is explaining: this is what the FTC PT I means, and this is why it works.  The reason? It’s freakin’ scary! There are lots of variables being thrown around… I thought I posted it last year after writing it, but apparently I did not.
I came up with a guided worksheet that breaks down the ideas into individual pieces, and helps students work through it:
Excuse the fact that it keeps on referring to FTC Part II… I always conflate which is which.
Regardless, I was pleased at how much better my kids last year understood the theorem. They understood the idea of the dummy variable. And that the integral was simply giving an accumulated area from some starting value to some indefinite, variable value in the future.
I’m hopefully going to start this tomorrow, so keep your fingers crossed.
 The cop out way is to only explain: “it’s the derivative of an integral, and since they undo each other, you’re left with the original function.” I feel doing so elides the mechanics of what’s going on. It’s a surface-y (and useful to some degree) way to think about it, but it lacks depth.