SWBAT to model cost by analyzing the exponential growth of a digital video game economy

Video games are an interesting source contexts for mathematical analysis.

15 minutes

This lesson will hook all of your video-game-loving students right away. If nothing else, they will be shocked to walk into math class and find a lesson about a video game like Diablo 3, which has become wildly popular across the world. It has certainly broke many sales records:

http://en.wikipedia.org/wiki/Diablo_III#Sales

I start this lesson by introducing the game concept and some of the incredible stats around the game:

http://us.battle.net/d3/en/blog/9691895

A bunch of the stats are really interesting for the students. The game itself has a complex economy and they mention many fun facts based on the numbers and stats. There are many great topics to discuss.

In my class, I share some of my experiences with the game, which include playing far too much but also having the chance to connect with friends that live very far from me. "What better way to hang out with your buddies than to help each other rid the world of evil demons?"

I also show them my character:

I say, "Yeah, I am a wizard and I'm pretty awesome." Then I introduce the problem at hand.

"I want to make a Marquis Emerald, is it worth it?"

The discussion is about the factors that go into making *anything* worth it: **time and money**. From one perspective, students will argue that this whole game is a waste of time and certainly not worth it. But assuming I want to play the game, how do we figure it out?

I show them the Marquis Emerald as a starting point:

Here, it is helpful for students to notice that the Emerald is made from other materials. Once they recognize this, they need to know how these materials are acquired. The full emerald sequence is daunting:

The Marquis is the highest type of emerald from a list of 15 gems. Each gem can be combined to make the next. By using gems, items and gold that you collect or purchase online, you can eventually craft the Marquis Emerald.

With such a complex problem, I help students break it into manageable pieces. We discuss the parts of the problem, so that they can begin to break down. The worksheet and materials given in the next section are meant to help students process the mathematics and economics of the game.

25 minutes

To begin this exploration, I have my students explore the simple exponential relationship of **using 3 of one emerald type to make the next**. The goal is for them to realize that you need 3^14 chipped emeralds to eventually make 1 marquis, since there are 14 steps of 3 needed to reach the highest stone. Each denomination in between would therefore also be one less power of 3. This is the fundamental structure of the mathematical problem here (**MP7**). It is difficult to see this structure, because of the complex context, but the moment when a student sees how simple this Mild Problem is can be *really *exciting:

Once students process the initial idea of the number of emeralds involved, they can grab the cost table and begin to answer tougher questions:

This is where students can ask if it is cheaper to buy a lower end gem and mold it up the ladder to the marquis or if it is better to pay the hefty fines for the finer gems. I would ask students to start by comparing any pair (one lower like a chipped and one higher like a radiant star emerald). This is difficult because their are fees associated with each process. This would be a Medium Difficulty question. I am prepared to scaffold their work by saying, "How much would you have to spend at the auction house on chipped emeralds to be able to make the marquis?"

We reach for the Spicy Level of Problem when I say, "How much gold would you have to spend to make a marquis from chipped emeralds?" The assumption here might be that you need to purchase all the chipped emeralds. From here there are many extensions, but I write them on challenge cards and only offer them to students who are mastering the mild question (i.e., the Common Core specific) content. In this case, I am aiming for students to have proficiency in standards dealing with exponential functions (8.EE.A.1)

**Some possible extensions**:

- You can buy all of this with USD currency. Currently $2.13 will buy you 50,000,000 gold coins. So how much money would it cost to make the marquis (Students can consider various starting points).
- If you only played the game to collect money, gems, etc, how
*long*would this take?

**Some Useful Data**:

- My average gold per hour = 500,000
- My average flawless square emeralds per hour = 10
- My average tome of secrets per hour = 20
- My average pages of jewel crafting per hour = 5
- My average tomes of jewel crafting per hour = 5

**Here are some other resources:**

20 minutes

As we bring the lesson to a close, we answer the question ("Is it worth it?) and laugh at the absurd scenario (the number is quite high). I think it is also an appropriate moment for social commentary. This game is notorious for drawing players into endless game play with all of these incentives. Players advance characters to get meaningless upgrades in a never ending ladder that makes your character slightly better. This is dangerous, if people don't stop to think about it. I find it interesting that the laws of exponents and the 8th grade common core standards actually help us analyze a video game like Diablo and understand the impact it can have on our daily life.