Completing the Square

It's amazing what a difference a change in teaching technique can do.

Taught my 20-1 honors class completing the square today.  You know, the thing that you do to get from standard form of a quadratic to vertex form.  Yeah, I can feel you cringing in your seats right now.  See, when I was taught completing the square, I had no idea why you  had to do any of the steps.  It made absolutely no sense, but being the good math student that I was, I went along with it and eventually got good at completing the square.

I had no idea why I what the process meant.  I just got very good at following steps.  What an awful thing to impress on kids.  "Don't question this, just know that it gets you the answer.  You don't have to  understand, just do it."

Ugh.

And the sad thing is, that's exactly how I taught it last year.  These are the steps, do it 100 times until you get good at it.  Got it?  Good.  Let's move on.

Today I changed things, I decided to teach completing the square using algebra tiles.  Students could see that you were literally COMPLETING a SQUARE.  This was something that I could never get students to  understand, why you have to divide the middle term and square it to get the final term.  My students now get it, they understand that they have to have enough 1s to make a square, and then to subtract whatever they added to keep it balanced.

They get it.

They don't just get how to do it, they get why.  And that, ladies and gentlemen, is math.

Granted, I'm still teaching them this very arbitrary thing of finding a vertex form of a very specific shape (parabola)...but one step at a time.  I consider this a mild success.