Negative Exponents by Patterns

Right in between our unit on linear functions and our unit on exponential functions, I did a little "review" of zero and negative exponents.  I tried to teach it strictly by patterns.  Last year I taught the rules and then explained via the patterns why the rules were there.

I start with what students know.  I ask students to evaluate 3^1, 3^2, 3^3, and 3^4.  They can do this and they can even show why it works.  We recorded that part in this table (sorry all I have pictured is the final, but imagine half of it is blank :)

We notice that the the pattern from 3 to 9 to 27 to 81 is times 3 (duh!), but still important to state.  Then I ask students to think about the pattern backwards... Divided by 3!  So we continue that pattern to see 3 divided by 3 is 1 :)  And we have to continue the pattern on the top showing that 3^0 is 1. So then we talk about how multiplying 0 threes (or 0 anythings is 1).  It's still hard for them, but they can see.  I try to draw the connection to 0 in addition  (the identity) and 1 in multiplication (the identity).  But they are freshmen, so that is advanced.  If a few students make that connection from our quick discussion I'm happy.

Then we continue dividing.  1 divided by 3 is 1/3 (yes I have to force them to use fractions--I just say that they will recognize the pattern better if they use fractions).  Then by 1/9 there is a bunch of "oh"s and almost every students (or at least every students who is still paying attention) can say that the next in the pattern is 1/27.

We use the word reciprocal (instead of opposite as they tend to) and come to the rule that to evaluate negative exponents, we evaluate the positive first.  At this point I gave them several problems to practice in their notebooks.

That about sums it up.  If you have other questions about this, let me know :)  Or other ideas to make it better I'd be happy to hear it!!!!

-Kathryn