Radical Expressions to Rational Exponents

My first unit during student teaching has been on radical expressions. I am following the textbook closely, but am trying to challenge the students with conceptual questions in each section, not just teach them how to do the procedures.

I found a reference to a Park Math problem (Book 5, p 66; and I will reference the blogger who mentioned it when I find him/her: Michael Pershan) that helps students think about the relationship between radical expressions and expressions with rational exponents. I tweaked it and wrote up the following two questions that I posed to my Algebra II classes: 

If x ∙ x = x^N, what is N?
If x^a ∙ x^a = x, what is a?

I let them sit on it for a couple of minutes, and I asked them not to shout out the answer. Within about three minutes in one of my classes a student had it. Still the other students sat thinking. I gave them another minute, and then I asked the student to tell the rest of the class what he got and how. He said a = .5 because .5 + .5 = 1. The thing that surprised me about it all is that the student who first got it normally struggles in my class. He sits in the back by himself and pays attention, but it takes him a little while to catch on. I know he can get it because when I work with him individually he'll get it after I explain it. I love that he the one to figure this problem out. It gave him a change to shine.

Funny enough, the same thing happened in my other Algebra II class. A girl who struggles with math even more than the student mentioned above figured out that a = 1/2. Her explanation: "It's common sense."

This little event reminded me that one of my goals for teaching math is to show students that there is more than one kind of mathematical thinking. There is doing all the procedures - and those kids know if they're good at this because they've been doing it their whole math career - and then there are a whole slew of other skills, like reasoning, estimating, thinking abstractly, puzzling, finding resources, visualizing, solving a simpler problem to help you figure out a harder one, etc. I believe that every person has a mathematical sense, it's just not utilized or made known to them. Most students think math is a set of rules because that's how it's often presented, and that's how I've been presenting it, too. How can I make math seem coherent, logical, and useful to students? That's a challenge I expect will take me a career to figure out.