The shape in question was , and when I claimed its area was finite, my students didn’t believe me. Actually, my students didn’t even know what I meant.
Back when my improv was as good as my planning, I used to love digressing into topics that I found especially interesting, and I indulged myself again this week by spending 10 minutes talking about the idea that a number that never stops getting bigger might have a limit. The idea I focused on this week to exemplify my point was the difference between 1 and 0.99999… “repeating.” I showed, through long subtraction, that there was no difference between the two numbers (the two representations of the same number?). I went to the “ends” of 1.0000… and 0.9999… and did the 1st-grade trick of borrowing from a more significant digit. When my students said, “but you’re still left with a 0.00..01!” I said, “well, we gave up too early!” and added a few digits. It was this last bit that really helped them understand, and I saw that wave of comprehension go across their faces (all of them, I think!).
Since my calculus class is not an AP class, I struggle with the decision to include limits at all. You don’t need them to explain the other concepts of calculus, and since I’m definitely not going to get all the way down to basic limit proofs it makes sense to me to skip them altogether. This breakthrough is a big one, though. If I can create a lesson that leads students to this thought process… I’m in!