The “keep change flip” soundbyte. I’m all for a good aoundbyte, and they’re how I remember things, but we always have to stop and retell the story of what that soundbyte means. Keep the first fraction, change the division symbol to multiplication, flip the second fraction so you’re multiplying by the reciprocal.

It’s hard to pull a one size fits all explanation. :) it might depend what age I’m talking to how I’d approach it, and which connections they already know. For my high schoolers, we spend a lot of time talking about “the story of x”, in which a fraction means “we are multiplying the numerator and dividing by the denominator” as we draw a little picture. After we interpret, we “come on back home” by undoing all the things that happened to the variable along the way. We talk about the fact that division is the inverse operation of multiplication. Instead of dividing by a/b, we can “undo the opposite” and multiply by b/a.