When I lecture I like to improvise. I prepare notes, of course, the more detailed the more precise I need to be, but my performing instincts are most satisfied when I just go in front of the class with some key points to hit and maybe a few key lines worked out ahead of time. But I did recently make an iconic mistake, repeating the mathematics instructor’s equivalent of the lawyer asking in court a question without already knowing what the answer will be. Improvisation has to be carefully prepared.
The lecture was going over areas for basic geometric figures — rectangles, parallelograms, triangles, circles, and so on — and we’d got to the funkiest-looking formula of the shapes in the book. What’s the area of a trapezoid? The trapezoid is a four-sided figure with two parallel lines. If the length of one of those parallel lines is b1 — and since we’re choosing how to describe these things, it can be, by definition — and the length of the other of those parallel lines is b2 — once again, we can — and the distance between the two parallel lines is a, then we have a simple formula for the area of this trapezoid. It’s a * (b1 + b2)/2.
(The letters might seem a bit arbitrary. b for the lengths of bases is probably almost mnemonic; at least, it’s got the use of b-for-a-base-length in common with the area formula for triangles. a seems to be a harder match. I like thinking of it as the “altitude” of the higher base above the lower; altitude is used to describe how far other polygons reach, too.)
And I mentioned to my students that they could just memorize this, but if they liked, they could work it out from knowing the area formula for triangles. My students seemed, rather credibly, skeptical. So I thought I’d give it a try, and I did, and got stuck. Happily I’m not too uncomfortable with dead air, standing in front of a group with nothing to say, but even better someone asked a marginally related question and I was able to go to that, and leave the unfinished demonstration behind, with nobody asking aloud how we were to get from where we left off to where we wanted to go.
Afterwards I did realize the way out of where I got stuck, and after one false start — at home — figured out the derivation. This I wrote up and sent them as a shared file on the class web space, hopefully saving my honor without taking up more class time.