Every year in some class or another, I've "taught" or "re-taught" or "reviewed" the simplifying of radicals. And finally, FINALLY, I've come up with a way that has about 90% success rate with my classes, and I'm tickled pink.
In the past I've taught "trees" or breaking down into prime factors and gathering "pairs" and all sorts of various ways, but invariably some kids (more than I'm comfortable with) are confused and it doesn't stick and wa wa wa, "we don't get it". Whew! This year, my key thing was, (amongst other things), first having them come up with the first 6 perfect squares (we discuss how to get them and list, 0, 1, 4, ...) after they try on their own .... then I have them box or highlight, 4, 9, 16, 25 ... then say that any time they want to simplify radicals, see if 4 is a factor then if 9, and so on and then (say) stop at 4 and DON'T take it down to 2 x 2 ..... and bla bla bla .... and it seems to stick. Practically ALL of the class got it and kept getting it for days on end and even when I "spiraled" it. Go me! Go them :)
So, all in all, a pretty good day.