SWBAT to model with an exponential function.

Exponential growth is amazing to talk about, but even more wonderful to see.

10 minutes

I start off this lesson with the story of "Chess and the Monsters," a goofy spin off of the classic chess problem and have kids read the two main roles in the story: the "evil king" and the hero named "Chris."

The powerpoint tells the story and has the lines as well (there are only a few lines for students to read and they tend to make the class laugh): chess warrior

The basic idea of the story is that an evil king stole all the money in the land. However, the hero Chris sets up a scheme to win it all back. He has his scary monster friend "attack" the town and the evil king panics. The evil king offers a reward (1 billion pennies) to stop the monster and Chris saves the day. However, Chris doesn't want the 1 billion pennies, instead he asks for 1 penny and then twice the amount tomorrow and again after that and so on for two months. The king agrees and keeps track of the payments on his chess board (64 squares = two months).

The finishing question is, "did the king make a wise choice?"

25 minutes

This lesson is meant as an introduction into the world of exponents, so we let students solve this puzzle without much initial discussion. We want to see how they interpret the problem and give them as many resources as we can to approach with their intuition. Procedurally, students might recognize the exponential growth, but the comprehending the vast numbers created by the process of doubling is something that is hard to grasp.

I make laminated chess boards for the students and let them use it as a type of whiteboard. Other students can use paper copies of boards and write on the paper. I also include a small amount of pennies at each table.

They are simply asked, "did the King make a wise choice?" Some students will quickly realize that the amount of money growing on the board is vastly greater than 1 billion pennies and others will enjoy finding the detailed amounts on each square.

Students use calculators to keep track of their work and model the penny process with the coins I give them (at least on the first few squares).

I place other questions on the board and ask students to go a bit further when they feel comfortable saying the the king did *not* make a wise choice.

**Follow Up Questions:**

1) Is there a function for the number of pennies on any square?

2) Is there a function for the sum of pennies up to the nth square?

3) How many pennies will be on the 34th square?

4) How many pennies will be on the 35th square?

5) How many pennies will be on the 64th square?

Questions 3 and 4 will introduce **scientific notation** to the students on the calculator.

20 minutes

We start the summary with a general discussion of why students think that the king did or did not make a wise choice and then support that discussion with the follow up questions we gave during the investigation:

**Follow Up Questions:**

1) Is there a function for the number of pennies on any square?

2) Is there a function for the sum of pennies up to the nth square?

3) How many pennies will be on the 34th square?

4) How many pennies will be on the 35th square?

5) How many pennies will be on the 64th square?

I start by talking about 3 and 4, since that allows us to discuss the larger numbers on the other squares. I mention that this "form" of a number is called scientific notation, but encourage them to think of it intuitively, where 2^34 = 1.7 x 10^10 means that we multiply 1.7 by 10 ten times. Then I use a calculator that shows the full number (something like wolfram alpha or excel)

Question 1 asks students to find the function 2^(x-1) and I have students share how they reached that equation. If they struggle with this, I show a table and discuss how we could "see" the function by listing out the first couple of values. There is no absolute process for finding any function (that I know of), but here we can talk about some basic approaches in pattern hunting. "I look at the input and output values, the connections between the step we are on and the value of coins."

Question 2 is *much* more complicated and is something I only discuss if some students were able to make progress with it. One intuitive way to see that the function is (2^x) - 1 is to look at some up the smaller sum sequences.

I might show them a couple of sequences like this:

Sum of squares 1,2 and 3 = 1 + 2 + 4 = 7

Value of pennies on square 4 = 8

"I notice that all the pennies put together on the first 3 squares is one less than the number of pennies on the 4th square."

**Students are amazed by this process but also feel overwhelmed with the idea that they should be able to find such functions. **I try to explain that many of the problems are only solved by pattern hunting, by playing with numbers until you see something. "Its like writing a song. You might have a song in your head from the beginning, but sometimes the song just comes as you play. You just keep playing with the instrument until something works. Sometimes you need to *play* with math as well."

I like to finish with number 2^64 or 18 quintillion and *show them how large this really is.*

http://www.kokogiak.com/megapenny/

I like to show them all the slides, from 1 penny to 1 Quintillion

As I go along, I like to compare this amount of pennies to other large numbers, like the current world GDP is about $72 trillion or $71,830,000,000,000. I ask questions like, "is it better to 18 quintillion pennies or 71 trillion dollars?"

The slide show is a nice way to end the lesson and it always wonderful to see their reaction to the volume of a quintillion pennies.

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