Shamelessly stolen from a comment on Dan’s blog:
For things like exponent rules, students are expected to act like a computer programmer and be meticulous in how they work through the problems. Because exponents are just symbols to my students, sometimes they make up their own rules (as any teacher probably knows).
If you know they’re going to make their own rules, then make that your lesson. Start with this as your warm up:
You may happily debate me about whether that second one will help or hurt – just seeing a fraction can make some kids freeze up. You may need to sneak them later in as a notational convenience instead. But what’s not up for debate is that finding patterns is like candy for kids – they are far less reluctant to do this than other types of math, especially if it’s just “guess the next number”. The barrier to entry on that is pretty low. Even if most of them don’t get it, they’ll be willing to guess, and enough kids will understand so that the others will want to know why (this is gold). Let the kids explain their rules.
Notice that we haven’t actually addressed exponents at all yet. we’ve just let them make rules for patterns. This is where you do the lazy math teacher thing – you just rewrite then numbers in exponent form under the patterns, and ask the kids to make the same predictions you did earlier.
After that, it’s a matter of formalizing the rules. Have the kids write up explanations, and then start throwing those explanations under the document camera, and let them debate. Let them decide the wording, let them argue about meaning. Let them do it with their neighbors, let them do it with the class. Let them make the rules.
Same idea for the other exponent rules. Throw this up on the screen:
Give them 3 minutes, and then start stealing student papers and throwing them under the camera. Get good answers, get bad answers. Let them debate whether the answer is fake or real. Use their answers to let them define the rules, both for what works, and what doesn’t work. This is probably more free wheeling and unstructured than the first, so don’t forget to constantly solicit why’s and whynot’s, and poll for who agrees and disagrees, and have them share with each other whenever you don’t get enough responses to the above.
If you’re careful, you’ll never explain a single rule to them.
They’ll make em up all on their own.
Worth noting: This is not a huge multimedia overhaul. I’m sure there are great real world problems you can involve in this. This is not THE optimal exponent lesson. This is just one option to get rid of the “explain then practice” model of teaching. If you get them thinking, it doesn’t have to be fancy.