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## Let them make their own rules

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).

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.

1. Jenni | April 15, 2014 at 8:26 am | Permalink

This is a great idea! I saw your comment on Dan’s website and wanted to see how you approached more “mundane” math topics like exponent rules. I like this a lot. I also see relevance in my geometry class when I teach inductive/deductive reasoning. There, students create and identify patterns to make a prediction. I might think to group those two topics together in the future, although I wouldn’t have thought of it in a traditional pedagogical sense.

Thanks!

2. Bryan Anderson | April 15, 2014 at 10:45 am | Permalink

Nice idea, although I would not go right to the “lazy teacher” way you mentioned. I might start the first sequence with 3 instead of 1.

3,9,27,81,…
After they found the pattern like you ask, I would then ask them to write the sequence in powers base three instead.
3^1, 3^2, 3^3, …

This makes the student generate the sequence to find the power rule.

Then after working with students on the last sequence:
32,16,8,4,… Which in power form would be 2^5, 2^4, 2^3, 2^2, …
Students should see that the power is decreasing, this would be a great link to negative powers and the Axiom of b^0 = 1. (I never liked axioms, let me discover a reasonable explanation why it is so)

A thought for the last slide, would an inherent power of 1 be better or would writing them with a larger power pull out more discussion on combining powers?
Such as:
7^2*7^2*7^2*7^2*7^2…

3. Mr. K | April 15, 2014 at 12:33 pm | Permalink

I might start the first sequence with 3 instead of 1

I might also, if I wanted to just use one sequence. I’m trying to establish patterns, though, and the big sneaky trick they think I’m pulling on them is that I switch to division in the last one. That’s the one I’d focus on to establish the pattern (I know I said patterns, and you’re right to point out where that’d go wrong. If I did try that, I’d realize my mistake in period one, and have a chance to fix it up later in the day), and the one I’d the write in exponential form (only for the part presented) and then have them extend the exponent pattern as well, to see that

Students should see that the power is decreasing

And

7^2*7^2*7^2*7^2*7^2*7^2

this is one of the patterns I’d expect to see very far down the line. I think the first thing I’d see is 7^12, and then half the kids would get stuck. The next one I might expect to see is 7×7^11, which is one of the things you need to work on as a special case using the traditional method of teaching this. Or, you might get 7×7×7×7×7×7×7×7×7×7×7^2. Either way, one kid will get it, and you can have the class argue for 5 minutes with each other over why it’s right. And then most of the kids should be able to come up with another, because they have a way of changing an expression to another equivalent one. That should get them off to the races, and provide you with a bunch of examples to have the discussions with, including hopefully eventually the one you listed.

4. Zach | April 15, 2014 at 5:32 pm | Permalink

Wow. Thank you for the response. I didn’t see the connection between exponent rules and geometric sequences. I could see this working really well. Thanks for the tip! :)

5. Bryan Anderson | April 16, 2014 at 6:55 am | Permalink

When I was referencing 7^2 × 7^2 × 7^2 × 7^2 × 7^2 × 7^2 it was more in mind of Jenni’s reference to the law exponents where (a^2)^3 = a^6. I was just wondering if students would be more apt to discover this rule themselves when the power is larger than 1. Otherwise I could assume that students would progress as you anticipated.
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I think the first thing I’d see is 7^12, and then half the kids would get stuck. The next one I might expect to see is 7×7^11, which is one of the things you need to work on as a special case using the traditional method of teaching this. Or, you might get 7×7×7×7×7×7×7×7×7×7×7^2. Either way, one kid will get it, and you can have the class argue for 5 minutes with each other over why it’s right. And then most of the kids should be able to come up with another, because they have a way of changing an expression to another equivalent one. That should get them off to the races, and provide you with a bunch of examples to have the discussions with, including hopefully eventually the one you listed.
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I am not sure students would make the jump from a^y * a^z to (a^y)^z. I see this first step as a very valid way to create a scaffolding of knowledge.

6. Mr. K | April 16, 2014 at 8:07 am | Permalink

I was just wondering if students would be more apt to discover this rule themselves

Remember that a key part of this is sharing student work. This lesson relies heavily on a document camera, or some other way of quickly sharing students work with the whole class. Someone somewhere is going to come up with 7^6×7^6 as one of the representations. That’s a natural starting point for (7^6)^2, and may require a bit of directed class discussion. However, once that’s in the open, the (intentional) choice of 7^12 allows for a number of other variations to be found before you start having them formalize the rules.