Exponent Mindfulness

In preparation for our exponential/logarithmic function unit, I decided to try something I called  Exponent Mindfulness. Mindfulness because that's an initiative the team I work with has been working on, and exponents because being able to not only evaluate expressions involving them, but recognising numbers as powers, is the key to this whole unit. After all, working exponents out backwards is what logs are all about.

First we spent a week on Exponent Boot Camp, in which I review WHAT the exponent properties are, and also WHY they are, including negative exponents, rational exponents, and rational bases with positive and negative exponents. That covered the first part - evaluating expressions with all kinds of bases and exponents.

On this day, however, it was all about going the other way, developing those exponent lenses. After warming up with a few evaluation examples, I put 16 on the board and asked how can we write this number as a power? The answers I got were as expected:

Then I asked - that's it right? No other possibilities? Waited and asked about the possibility of a negative exponent. Here's where things got interesting. Even though everyone was fine moments ago with how to figure out a fraction to a negative exponent, the idea that you can get 16 from a negative exponent suddenly seemed to be mind bending. (It's always more fun to give your students the answer and ask them to come up with the question.)

Anyway, so I showed them this:

We spent a few minutes working out each of these, just to re-convince everyone that these were in fact all equal to 16.

Back to 16, and I asked again, "That's it now, right? No other possibilities?"

My students know me well enough to know that answer to that. So we moved on to rational exponents:

I got a few really great answers added here, like 65536^(1/4) and 1048576^(1/5).

Me: So that's it right?

Students: Nope, that's never it is it?

So now that we all knew that there were in fact infinitely many ways to express a number as a power,  I asked everyone to write a power on the board that equals 81, specifying that you can't use one that's already there. That went really well. Definitely you should do this again next year Future Audrey.