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.

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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.

]]>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.

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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…

Thanks!

]]>I’ve been working on a conference presentation called “Lies My Education Teacher Told Me” and if you don’t mind I might yoink this one.

]]>I didn’t think anyone still read this.

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