Patterns in the Digits of Powers

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Objective

SWBAT to use a pattern to understand certain characteristics of very large powers

Big Idea

We can use patterns to solve problems that computers can not deal with.

Start Up

15 minutes

This is a lesson about the power of patterns in mathematics. We start by giving a few warm up problems.

Can you solve this in your head? 

2^4

2^5

2^6

2^7

2^8

2^9

2^10

2^11

2^12

2^20

 

I give students 5 minutes to solve these. They can try and solve them all in their head or on paper. The beauty of these large numbers is that we are simply doubling, which is not as difficult as many other types of calculations, like tripling and quadrupling. 

If they are stuck, I nudge them by sharing my strategy: 

"I start with 2 and then double it to get 4 and then 8 and then 16. Since I doubled 2 four times and got 16, I know 2^4 = 16. If you doubled this, what would you get?"

 

I believe most students can reach 2^10, but might struggle with 2^11 and 2^12. A few students might reach 2^20 and get 1048576. When we share the answers here, I might not even reach 2^20. I would only discuss it if someone tackled it. Instead, I give them a hook by saying, "maybe there is an easy way to find out something about this number. Perhaps we struggle with the exact calculation, but there has to be something we can figure out."

Here I go back to the list of problems and add in 2^0, 2^1, 2^2 and 2^3. I ask them to hold off their calculations and look at the numbers. I give them a minute, "what do you notice about the numbers? What do you notice about the digits?" Students can make low inference observations, "I notice that the numbers are going up!" And this is a perfect starting place for a conversation. I might nudge them a bit by highlighting the last digits of each number

2^0 = 1

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

2^5 = 32

2^6 = 64

2^7 = 128

2^8 = 256

2^9 = 512

It is a great moment when students first see the patterns of 2,4, 6 and 8 repeating in the powers of 2. Except for that pesky 1, the pattern will always hold true. I talk for a moment about that pesky 1 and what we should do about it. The answer here is to ignore it. Many students want to know, "are we allowed to ignore it?" This is a great question and I love when I get it. It gives me a chance to shatter the idea that math is simply about the rules we are given. I use this as an opportunity to demonstrate that math is alive and a product of our perspective. Here we are looking for a pattern. The numbers behave the way they do because of some fundamental laws of a number system. But an infinite cacophony of patterns emerges from these simple rules. It is up to us to find and describe these patterns. I think of it like art. An artist might be given a certain medium and be limited in the colors they have, but it is the way that the see and use these colors that defines the work they produce. The way we see and use the patterns in math defines the work that we produce. 

Besides, 2^0 isn't special. I ask them, "Are there other numbers that also have 1 as a power?" Here the idea is that all numbers except 0^0 are universally defined as 1. So this property isn't unique.

2^0 = 3^0 = 4^0 = ....

Since all numbers (except 0) share this value, we tend to universally ignore it (this relates to the fact that we ignore 1 as a prime number). 

The exciting thing here is that students will naturally wonder, "do other bases have their own patterns?"

"Yes of course! That is what you are going to find today."

But first, I want them to revisit the original problem. "How can we use this pattern to figure something out for 2^20?" I let students take a few minutes to discuss. We review the idea that the pattern is in a "chunk of 4," so that the fifth number starts the pattern over (and the 9th, 17th and 21st). Students might recognize the modular logic and use division and remainders, but that is something we only discuss during the summary (I don't want to push too much in the intro unless they are ready for it). 

 

Investigation

25 minutes

The goal is to help students use the patterns they observe to find the last digits of large powers. I start them off with this guide: Patterns in the digits

The mild problems are meant to help students transition from counting the digit that appear to dividing. I try to encourage this by using multiples of 4. I circulate and help them see this algorithm. I let students use a calculator for this exercise, but it will really only help them if they understand the correct algorithm. The numbers are chosen deliberately to exceed the capacity of any calculator they can find. I might spend a moment or two reviewing how to enter exponents on the calculator as well. 

When they finish with the mild, medium and spicy problems, they grab a mystery card an analyze the base of the card. The goal is for them to analyze the pattern in the digits of the powers of that base and then create a few problems to share with the class. 

The "mystery cards" are usually just index cards with a colorful question mark on one side and a base on the other. I choose bases between 3 and 100, but usually only post about 16 cards (one for each partnership). 

Summary

20 minutes

This activity is meant to help students focus on the importance of patterns in mathematics. By using the patterns in these problems, my hope is that they will apply the process to other problems as well. 

In this summary, we review the worksheet and then share some student sample problems as exit tickets. 

In reviewing the worksheet, we ask students to share their algorithms for the mild problems. When we reach the medium problems, I ask them "I wonder if our calculators could do this?"

We enter 2^332 and notice that the calculator doesn't reveal the last digit. So we apply our algorithm and share the answer. 

Then we enter 2^333 and get an "overflow error." We discuss this and move to a more powerful calculator. "I wonder if Google could solve these?"

So I enter 2^332 and 2^333 and the calculator works fine. Except you can't see the last digit. I ask them if we can come back to this in a moment and then I enter 2^1023 into the search bar and everything seems great: 2^1023.png

However, when we enter 2^1024, we get an interesting message: 2^1024.png

Do you think that this is correct? Doe 2^1024 = Infinity? We talk about it for a moment and go to the most powerful online calculator that I know of: Wolfram Alpha. We enter in 2^332, 2^333, 2^1023 and then 2^1024. In each case we can see all of the digits and confirm that we were correct. However, the spicy example brings us to the brink. We enter 2^132853 and can see the full digit display: 2132853.pdf But what happens when we enter 213854? Again we lose visibility of the digits. As a check in, I ask them to take a minute and solve the problem for me. 

We finish class by reviewing some of the other patterns in the bases that they examined. From those we pick a mild, medium and spicy question for a 5 minute exit ticket. We collect these as they go see how the lesson went.