Inside Collection: Advanced Algebra II: Teacher's Guide
Summary: A teacher's guide to extending the concept of exponents to students.
This is one of my favorite class discussions. You’re going to talk almost the entire class, and just give out an assignment toward the end.
So far, we have only talked about exponents in the context of positive integers. The base can be anything: for instance, we can find
So, let’s plop ourselves down in an imaginary point in history where exponents are only defined for positive integers. We are the king’s mathematicians. The king has just walked in and demanded that we come up with some sort of definition for what
So we start with a brainstorming exercise. The object is to come up with as many possible things as you can think of, for
By the way, somewhere in class there is a smartaleck who knows the right answer and therefore won’t plan. “It’s
OK, so after a few minutes, you start collecting ideas from the groups.
The answer is—the definition should be as consistent as possible with the one we already have. Of course it won’t mean the same thing. But it should behave mathematically consistently with the rules we have: for instance, it should still obey our three laws of exponents. That sort of consistency is going to be the guideline that we use to choose a definition for the king.
And hey, what exactly do negative numbers mean anyway? One way to look at them is, they are what happens when you take the positive numbers and keep going down. That is, if you go from 5 to 4 to 3 and just keep going, you eventually get to 0 and then negative numbers. This alone is a very powerful way of looking at negative numbers. You can use this to see, for instance, why positivetimesnegativeequalsnegative and why negativetimesnegativeequalspositive.


OK, you may or may not want to get into this, but I think it’s cool, and it does help pave the way for where we’re going. (It also helps reinforce the idea that even the rules you learned in second grade have reasons.) The numbers I’ve written on the left there show what happens to “multiplyby5” as you count down. Clearly, looking at the positive numbers, the answers are going down by 5 every time. So if that trend continues as we dip into negative numbers, then we will get –5 and then –10 on the bottom: negative times positive equals negative.
On the right, we see what happens to “multiplyby5” as you count down. Since we already know (just proved) that positivetimesnegativeequalsnegative, we know that
As I say, you may want to skip that. What is essential is to get across the point that we need a new definition that will cover negative exponents, and that we are going to get there by looking for consistency with the positive ones. Then they are ready for the inclass assignment: it shouldn’t take long. Do make sure to give them 10 minutes for it, though—you want them to finish it in class, and have time to ask you questions, so you know they are ready for the homework.
“Homework: Extending the Idea of Exponents”
"This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"