How I see exponent rules (and log rules)

Exponent rules can be difficult to remember, and memory has never been one of my strong suits. When I was in high school myself learning exponent rules, I would get mixed up just trying to remember them individually, and had to come up with a different way of thinking to condense it into one idea.  What I came up with deals with the levels of complexity of operations:

So, you've got your simple functions on the bottom, multiplication and division are a little more complex, and then exponents and roots are more complex. The actual chart above I created after the fact when trying to explain the idea to students later in life.  So, basically when it came to exponent rules all I had to remember was to "go down a level" of complexity.

So multiplication becomes addition, division becomes subtraction, an exponent to an exponent becomes multiplication and a root with an exponent becomes division. The chart also helps for remembering when to distribute. Operations distribute on to the tier below them. (exponents distribute over multiplication and division for example)  

My results with trying to get students to see the same thing I do has been mixed.  I usually end up doing an exploratory learning exercise with exponents , then going through the rules individually and only quickly going through this chart idea on the side.  While it doesn't really connect with every student, when a particular student gets the rules mixed up it can really help because it at least gives them a plan rather than just relying on straight memorization.

Later on, when I learned about logs it turned out that log rules (surprising to me at the time, not so much anymore) followed along the same lines, with exponents becoming multiplication, multiplication become addition, etc. 

Anyone else ever think of it this way?  Have some other strategy for helping students get exponent rules straight?  Let me know!

Student centered learning using WolframAlpha

Recently I had a really successful lesson on exponent rules.  Every year I go through exponent rules with my algebra 1 class, and though it's likely they've seen much of them before, their retention is such that it feels like they are seeing everything for the first time.  This year I tried to change it up a bit in order to do two things that I've been trying to get right this year:  Increasing my student centered learning and using wolfram alpha as an effective learning tool in class.





At the beginning of class, students were given what was last year's quiz I gave after teaching and practicing some exponent rules.  The instructions:

1)       Answer every question with your best guess.  We haven't learned this stuff yet, so I don't expect you to get many of the questions right, but I do want you to try to make some sort of guess for each.

2)      Use your laptop to go to wolfram alpha to check each of your answers.  Write the correct answer separate from your original answer and try to figure out what each exponent rule is.

3)     For each answer you got wrong originally, make up 3 similar problems that use the same rule(s), answer them and check them using Wolfram Alpha.

4)    Once you are confident that you could get every problem right without help, I will give you your quiz.

The result?  Students did better on the quiz than in previous years where I taught the material.  Granted, there are some other variables involved in that, but it was clear from moving around and talking to them that figuring out the rules themselves gave them a better understanding of what was going on. The whole thing took two 45 minute periods, part of which was getting them used to typing the expressions and a discussion about why they shouldn't use wolfram alpha to do all of their homework for them.

You may be saying to yourself "Hey wait a minute, you could've just given them an answer sheet and done the same thing."  Well yes, and no.  Answer sheets would take away the flexibility of being able to check answers to the problems that they make up.  The making up step also helps them to understand what they are really looking for and is a good strategy they can use to help study for tests.  Also, part of the point(which I discussed with them) was to get them used to using Wolfram alpha to check their work any time, like for homework or studying for instant feedback.

Now as pleased as I am with how this went I'm certainly not ready to give up explaining things at the board ever.  Exponent rules lend themselves well to this sort of activity as they are reasonable to figure out with the answers and I'm definitely going to look for more topics that would work well with this kind of activity in the future.  Can you think of any?