SWBAT to use the associative and commutative property to compare and order powers

Instead of seeing exponents as simply "repeated multiplication," we want students to find elegant and simple ways of comparing exponents.

15 minutes

This lesson is based on James Tanton Essay 3 (2012) in which he writes about the AMC competition. His approach and conversation is the model for my lesson. You can find a lot of other great math resources and ideas on his website: http://www.jamestanton.com

I plan to start today's lesson by asking students to order several groups of numbers.

**Group 1:**

2^2, 2^4, 2^3

**Group 2:**

2,10,5

**Group 3:**

5^2, 2^2, 10^2

**Group 4:**

5^3, 2^4, 10^5

**Group 5:**

10^8, 5^12, 2^24

I give the students five minutes to work on this task. Afterward, we talk about their approaches to the first four groups. I plan to ask, "What is *easy* about the first four groups?" I will create and label an "Easy" table pm the board. Here are some possible student contributions:

- Group 1 is easy to compare because the bases are equivalent
- Group 2 and 3 are easy to compare because the exponents are the same
- Group 4 is easy to compare in standard form

However, nothing is easy about group 5. This brings us to our first major strategy, if something isn't easy, think **"how would I change this problem so that it was easy?"** My goal is to get students to play with Group 5 until they figure out a way to recreate some of the "easy" elements of Groups 1, 2, 3 and 4. Stanton calls this line of reasoning "wishful thinking". I love the phrase, so I like to use it with my students.

25 minutes

As Tanton notes in his essay, seeing exponents as simply repeated multiplication might slow students down a bit on this problem.

Instead of trying to do the repeated multiplication, we can write out the numbers in expanded form:

5^12 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

10^8 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

2^24 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

Here is a very abbreviated version of the possible flow of class discussion:

We could use brute force and calculate each, but when working on 2^24 its hard not to think that there must be a better way. So how? The key here is to group numbers in ways that you can compare them.

**Light Bulb 1**: Remember that if the exponents or bases were equal, then we could easily compare. Could we get the exponents equal in 10^8 and 2^24?

Sure, 2^24 is made from 24 twos or 12 pairs of twos or 8 triplets of twos. The 8 triplets is very useful, since 8 triplets is:

(2 x 2 x 2) x (2 x 2 x 2)x (2 x 2 x 2) x (2 x 2 x 2) x(2 x 2 x 2) x(2 x 2 x 2) x(2 x 2 x 2) x(2 x 2 x 2) =8^8

This means that 2^24 = 8^8 and students can quickly compare this to 10^8.

**Light Bulb 2**: Can we make the bases the same?

5^12 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

10^8 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10

Here we need to see that 10 = 5 x 2. Then 10 ^ 8 is rewritten:

10^8 = (5 x 2)(5 x 2)(5 x 2)(5 x 2)(5 x 2)(5 x 2)(5 x 2)(5 x 2)

Using the Commutative Property, we can reorder this:

10^8 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x **5 x 5 x 5 x 5 x 5 x 5 x 5 x 5**

and

5^12 = 5 x 5 x 5 x 5 x **5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 **

The bolded 5's are clearly equal, so we need to compare 2^8 and 5^4. However, 2^8 = 4^4 and 4^4 is easy to compare to 5^4, so 5^12 >10^8 and the answer is 2^24 <10^8 <5^12

**In the next section,** I will ask students to solve a few other ordering questions and then create one of their own to share with the class.

25 minutes

I ask the students to try two other problems and ask them to avoid using calculators. They need to play with the numbers strategically in order to compare them. The two main goals are two find a way to get equal exponents or bases.

Then I encourage them to create a problem of their own to share with the group. These problems are difficult to generate and I include several tips on the worksheet here:

However, I highly recommend for students to try creating a problem without these tips. The process of creating a problem will force them to ask questions and conjecture. If they are successful in creating a problem, then they will have really developed a strong understanding of the problem.

20 minutes

To bring the lesson to closure, I will start by asking several students to share observations and algorithms around the two examples from the Investigation.

After we explore the first two problems, I will have about three students share the problems that they wrote. This is always exciting, because I ask them to present the problems to the group. Then, I give the class up to five minutes to solve. Next, we compare our answer to the student sharing and discuss how it went. In this process, students are exposed to the problems from their classmates and given mini lessons during the review. It turns each student into a teacher and is one of my favorite aspects of teaching this lesson.