>The idea is that if we are to introduce such a quantity, and then assume it abides by the same sort of operations as integers, then we must conclude that 1/8 x 1/8 is a quantity quite a bit smaller than 1/8. It's a more formal way of looking at it, but many students, in my experience, seem to like it.
The "multiplication makes things bigger" meme is something I never experienced, but its mentioned often. If people really experience this a puzzle, confusion, stumbling block, it would seem to be because they missed something along the road. My instincts would be to go back to where they were supposed to learn that "half of something" and "three of something" are conceptually similar, and are going to be captured in arithmetic as (fractional) multiplication, along with the rules for doing so numerically.
Your phraseolgy is more like a kind of proof, or derivation from some other known (algebraic) relationship, and I wonder if kids who missed the "step" alluded to above are really getting it, even if they happily assent to your demonstration. In other words, I don't know how you can miss the notion that "half of" and "three of" both map to multiplication, but be happy with the longer explanation that involves reasoning from notions of inverse operations.