I like to begin class with a little overview of the new topic. We're going to begin our study of exponential equations and functions by first digging a little deeper into the idea of exponentiation itself.
As I hand out Exponents 1, I'll explain that today this is one of those lessons in which a calculator will actually make things harder. The point today is to focus on patterns, not simply on "answers". While a calculator makes the answer easy to obtain, it blinds us to the patterns that might have helped us to understand that answer. (MP 5 & 8)
They'll begin by simply writing out the first few powers of 3, then they're asked to look for and list any patterns they've found. Finally, students will be asked to evaluate some other powers of 3, including zero, negative powers, and rational powers. The important point here is that they're to try making use of the patterns they've found in order to do this.
It's worth noting that much of the content of this lesson really belongs to Algebra 1 under the CCSS. See this video to hear why I think it's important to revisit it.
I like to let my students work individually on the worksheet at first. They shouldn't need more than a few minutes to calculate the powers of 3, and then they should begin the search for patterns. It's important that everyone have a chance to observe some of these on their own since it can help build confidence in less proficient students. Once they've all begun to notice patterns, I'll allow them to work in small groups for the remainder of the period.
Among the things I expect the students to notice are the properties of exponents that they learned about in Algebra 1. These include the multiplication and division of powers, as well as powers of powers. I also expect students to notice some patterns specific to this base. For instance, the ones digit of the powers of 3 follows a simple pattern of 3, 9, 7, 1. It's important to acknowledge all of these patterns and encourage your students to look for others you may not have thought of. Some of them may prove useful in the future, some may not, but the search is worthwhile. (MP 8)
At this point, I like to have a brief discussion in which the different groups will each share out one or two patterns that they noticed. This is the time to find some joy in the beauty of mathematics and marvel at each new revelation.
I'll be sure to bring up the properties of exponents if none of the students do: "Did any of you notice that 3^4 is the square of 3^2? Why should that be? Is there a similar relationship between 3^8 and 3^4, or between 3^6 and 3^2?" These are the patterns that will be important for the next few problems.
Please see this video for some more thoughts on the patterns.
For the remainder of the class period, students will continue to work in small groups to complete the remaining problems. It's important to remember that the goal of these problems is not so much the "answer" but the way we get there.
Encourage your students to find simple ways to use the powers of 3 that they already know, along with the patterns and properties we've discussed, to evaluate the rest of the powers. (MP 8) For example, 3^27 can be found by cubing 3^9 or by multiplying 3^10 by 3^10 by 3^7.
If some students have trouble with 3^0 or with negative powers, I will do the following. First, all ask them how they got from one power to the next in the list of powers at the top of the sheet. They'll answer that they simply multiplied the previous power by 3. Ok, I'll say, so we can move forward from one power to the next by multiplying by 3. How can we move backwards? By dividing by 3, of course. At this point, I'll write 3^0, 3^-1, and 3^-2 at the top of their worksheet above 3^1. This is almost always a clear enough "hint" to help them make sense of negative exponents.
Rational exponents are much harder to make sense of, but I'm confident that some of my students can come up with a meaning on their own. For the rest, I will point their attention to the relationship between 3^3 and 3^6 or between 3^2 and 3^4. In each case, the exponent has been doubled; what has happened to the power? It's been squared. Ok, then what should we expect if the exponent is halved? This should be the same as taking the square root of the power! Exactly. This is about as helpful as I will be today.
Tonight, the homework is to continue working on these problems. It might be a good idea to ask the class to check their answers with a calculator tonight. This will ensure that everyone has the same "data" to work from during tomorrow's discussion of rational exponents.