i don’t know if this question will apply to all classes, my students are definitely not honors. In Alg1, my students had already seen simplifying radicals in Math8. They were familiar with the factor tree method and circling the doubles.
The first question I asked them, before we actually tried the “procedure”, was:
Why is sqrt(15) simplified and sqrt(12) not?
Blank stares. Then someone said because there aren’t any double to circle. I asked why we care about doubles. Nothing. They had no clue except their teacher told them to circle doubles.
So we talked about perfect squares being hidden inside of that radicand and that’s why we can simplify it further. The “doubles” just find perfect squares.