“Last year, we learned to keep-flip-change …”

She sees my face wince, but proceeds anyways.

“The way that goes is you keep the original fraction, change the division to multiplication, and then flip the second fraction.”

“OK, but why does it work?”

Shrug.

“Who else learned it this way?”

About three-fourths of the room raises their hand. (The other fourth still seemed to struggle with dividing fractions. More later.)

“OK, out of those of you who learned it this way, why does this work?”

I get a few “because the teacher said so”‘s, “because it just works,” “because I did well on the test” and just kept shaking my head gently while listening. Sure, the easy thing to do at that moment would have been to punch down (metaphorically, people) and tell them that their last teacher was wrong for unveiling a trick without giving them the magic behind it. The harder conversation was sitting right there, and I had it.

“What does it mean to divide?”