0 Fraction Exponents. Easy.

Have you ever found yourself teaching a certain thing a certain way for years, and then one day you think about changing your explanation just a teeny tiny bit? And the new way makes infinite more sense to students, and the thing that used to be impossibly hard is now easy? And then you wonder what took you so long to find that more easy/obvious way of explaining something?

That happened to me today with fractions as exponents.

I won't bother to mention how I used to teach it. It was bad. Very bad.

Today, I started by showing them this, and they all shouted several people went "x squared!".

Then I asked them HOW they knew it was x squared. Somebody pointed out that three x-squareds multiply to equal x^6. Someone else said that you divide 6 by 3.

I made a big deal about writing x^(6/3) before writing the answer as x squared. And we did several more of these including some square roots, so they could see that you divide by two.

Then I showed them this, and gave some time to think about it and write down what they thought the answer should be.

Almost every single student wrote down x^(2/3)! And there were angels singing.

Then we worked on going backwards, which was no biggie at all.  Given x^(1/2), students could easily rewrite as square root of x and so on.

And then I didn't know what to do, because it used to take me a full class period to teach that. And some students would still be sitting there going, "Huh?". But today they just got it in, like, five minutes.

I realize that there is nothing earth-shattering about this method. The thing is, I've taught it this way before. Only I didn't lead with this part. I ended with this part. All I did today was change the order.

Oh, I love these moments of finding the tiniest little change that makes a huge difference.