Well, the 1/8 thing was just something I was throwing out there as perhaps an alternative. Generally, if n is an integer, 1/8 is the symbol representing the quantity which, when multiplied by 8, yields 1. 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. If one looks at the properties of 1/8 as something to be explored rather than being set in stone, students can examine 1/8 x 1/8 as an exploration, and don't have to think of the "answer" as something to be learned by wrote. There's a rationale behind it.