OK, reason number 872 why I love my summer job:
There's a kid in my class who's been working on learning about hyperbolas for the past two days. He keeps saying "I don't get it" but can't explain what it is he doesn't get. Today, I was going over some problems with him, and I realized that he didn't understand what an asymptote is. I explained to him, "It's a line that your graph keeps getting closer and closer to, but never touches." (Not quite a perfect definition, but good enough for his level.)
He said, "But doesn't it have to touch? If it keeps getting closer, it'll have to touch it eventually." I tell him, "OK, try this. Look at that wall over there." (About four feet away.) "Now, take a step that covers half the distance between you and the wall." He does. "Now, take another step that covers half of the distance that's now between you and the wall." He does that. "Now, keep doing that." He does, and it's like a lightbulb lights up in his head. I say, "You see? No matter how many steps you take, there's still the other half of the distance left."
He keeps trying it, taking tinier and tinier steps, until the intervals get so tiny that the toes of his sneakers hit the wall. He turns around and smiles and say, "See? I got to the wall." I roll my eyes and say, "That's because we're in physical space, and you can't really see intervals that small. In mathematical space, it'll work." He says OK, and picks up his book to start heading back to his seat.
On the way back, he has to step up onto a riser and walk across it to the next step. I notice him doing the step over half, step over half, repeat, thing on the riser. I let him do it for about 15 seconds, then, laughing, tell him to get back to his seat. He turns back to me and says, "I can't. The step's my asymptote."