The start up is only 5 minutes, because I use it to briefly explain how awesome this lesson is going to be. When my class enters the room, there are about 6 tedious looking multiplication problems on the board. They don't know it yet, but each factor is the power of the same base.
For example, I put up problems like 64 x 16, which is 2^6 x 2^4. I also put up Nastier_Looking_Problems:
125 x 5
81 x 27
216 x 36
343 x 49
215 x 64
I tell them that they are going to build a tool that can solve these multiplication problems. On their desks is an envelop and inside are the secret instructions to building this magical tool. I give them 35 minutes to construct their tool and prepare a presentation on how to use it.
Each "Magical Tool" envelope is different because each group is building a different slide rule. I give different bases to different groups and lay out some key instructions to help them understand how the tool works. I have a station in the room with scissors and tape, but all the other supplies they need is in their envelope.
Teacher's Note: To understand the mathematical background of this investigation, check out James Tanton's essay On Logarithms, which was one of my favorite reads this summer. I recommend reading it before exploring the lesson further.
The envelope contains a single page of instruction (there are 8 pages in the document for the 8 tables):
The envelope also contains a ruler template:
An important question that we will address in this lesson is: which problem from the board can you solve with your ruler.
On Logarithms. James Tanton. http://www.jamestanton.com/wp-content/uploads/2009/09/logarithm-essay_docfile.pdf (Accessed August 11, 2014).
During this time, I ask each group to submit their ruler to me and I pass it to another group. The group has 2 minutes to figure out what multiplication problem is being solved, writes it out on a sticky note and then passes it back to the original group for review. If there are any disagreements, we resolve them during the summary discussion.
During the summary, we discuss general findings with the magical ruler and I ask groups to describe what they were able to solve. Each group had a different power and as a class they all seemed to work. So I ask, "would this work for any base?" Here I want students to point out that it seems to work for all positive integer bases, but we haven't tested negative bases or non-integer bases. So we do so on the board and see that it does work for negative bases (I use -2) and it does work for fractions (I use 1/2). I show it all on the board by simply sketching the rulers and the numbers that would match. The big question here is, "what is going on?"
So we pick an example to focus on. I use the problems on the board as a basis for discussion. Why was the base 5 able to solve the 125 x 5 problem? Here the goal is to see that 125 x 5 = 3125 can also be written with base 5 as 5^3 x 5^2 = 5^5. After trying a few others and lining up the statements, I ask "what do you notice about the two exponents in the factors and exponent of the product?" This is where students observe the first law of exponents, that x^a x x^b = x^a + b
We make sense of this by using the associative property. For example:
5^3 x 5^2 = 5^5
(5x5x5)x(5x5) = (5x5x5x5x5)
So multiplying 125 by 25 is the same as multiplying 3 fives and 2 fives and thus is the same as multiplying a total of 5 fives. On the rulers, we would start at 25 or 5^2 (two ticks on the powers of 5) and then move up 3 ticks (or 3 powers of 5). The idea is that every tick on the ruler represents multiplying up by a power of 5.