How would you talk about this problem?

I’m not even sure the title is asking the right question, but hopefully it is close enough to the right question!

Work has been really busy for me lately and both of the kids have also been a bit sick.  Because of these two little problems I’ve been doing a lot of review work with the boys rather than trying to cover too much new material.   We are still plodding along, though, and I started Chapter 20 in Art of Problem Solving’s Algebra book with my older son earlier this week.  The first section was a review of radicals.

Maybe not the most interesting topic in general, but there were a few neat sample problems.  For instance, we had a really nice discussion about the problem:  “Explain why \sqrt{x} > \sqrt[3]{x} when x > 1.”  Watching my son struggle with how to explain this fact was fascinating.

We’ve spent the last couple of days on the problems at the end of this section, and with only one problem left to do today, I’d actually planned to jump to the next section on absolute value.    For no particular reason, though, I changed my mind and got a great surprise.

Here’s the problem:

20.1.6.  Solve the equation \sqrt{x} + \sqrt{x + 4} = 2 \sqrt{4x - 5}

This turned out to be one of the most interesting 45 minute discussions that we’ve had on a math problem in a long time.  I’m so happy that I didn’t skip over this one.

As I’ve mentioned several times already on the blog I’ve never taught any elementary math before and I’m just learning as I go.  Sometimes I feel as though I’m covering the material better with my younger son for the laughable reason that I’ve already been through the topics once, but with my older son teaching just about everything is new to me.   That was certainly the case here.

With that background, I often struggle trying to understand how my son sees a problem that requires essentially no thought from me to solve.  This one was a real eye opener.  Among the interesting points of discussion:

(1) How do you deal with all of these square roots?
(2) Once you figure out that you should square them, now what – there are still square roots?

(3) Ah ha – rearrange and square again to get a quadratic equation.

(4) Wait – how come one of the solutions of that quadratic equation doesn’t solve the original problem?

(5) Really, seriously, wait, how could that be, didn’t we always do the same thing to both sides of the equation?

Anyway, though it was hardly perfect from start to finish, the discussion we had was super fun.  I’m going to take another full morning to talk through this problem one more time tomorrow because I think there are so many lessons hiding in it.  But, if you’ve read this far, what I’m really interested to hear is what I wrote in the title – how would you talk about this problem?

** update on 4/24/2014 **

Here’s how we talked it through this morning:

** update 2 on 4/25/2014

Talking through the algebraic solution part 1:

Talking about why only one of the two “solutions” we found in part 1 actually solve the original equation