This is a short post to archive some thinking I did on the subway home from work today. I had a Geometry class today and it was clear to me that their understanding of radicals was … not so good. And I don’t think it is their fault. I remember teaching Algebra II years ago and tried building up some conceptual understanding so puppies don’t have to die… and it was tough and I didn’t really succeed:

(Poster made by the infinitely awesome Bowman Dickson.)

I also remember having this exact same conversation with my co-teacher last year. We considered the following “thought exercise.”

How would you explain to a student in Algebra I why ?

I would like to add the corollary “thought exercise”:

How would you explain to a student in Algebra I why ?

And so on the subway home, I thought about this, and had the same insight I had last year.

We define (at least at the Algebra I level) to mean “the number you multiply by itself that yields 15.”

I want to highlight the *concept* more than the notation, so let’s call that number .

So for us is “the number you multiply by itself that yields 15.”

Now let’s similarly call “the number you multiply by itself that yields 5.”

And let’s call “the number you multiply by itself that yields 3.”

We know from this . Why? Because that’s the definition of “square” for us.

But we also know and for the same reason.

Thus we know .

Here’s the magic.

Let’s rearrange:

.

Study this a minute. It takes a second (or it might for students) to see that .

Now remember I used symbols because I wanted to focus on the *meaning* of these objects, not the notation.Let’s convert this back to our “fancy math notation.”

So that gets at our first “thought exercise.”

I wonder if trying the same with the second thought exercise might work? The tricky part is that we’re trying to *show a negative statement*. I know… I know… most of you probably say “hey, just show the kids .” But that doesn’t stick for my kids!

So let’s try it: for us is “the number you multiply by itself that yields 15.”

Now let’s similarly call “the number you multiply by itself that yields 10.”

And let’s call “the number you multiply by itself that yields 5.”

So:

.

Then challenge students do something similar to show that . They hopefully will start failing in their endeavor!

I predict they will start with: . Yay. That’s true… So from that true statement, they are going to try to show that .

But they can’t really go anywhere from here. They’re stuck. I still predict some weaker students may say: “But clearly we can just say . It’s like you have “half” of each side of the equation!” But it is at this point you can ask students to do two things:

1) Ask ’em to show the algebraic steps that allow them to make that statement. There won’t be valid steps. And in this process, you can see what *other* horrible algebraic misconceptions your students have (if any).

2) Or say: okay, let’s see if you’re right. If , then I know . And as soon as you start distributing those binomials, they’ll see they don’t get (our original statement).

Okay I just needed to get some of my initial thoughts out. Maybe more to come as I continue thinking about this…