##
* *Reflection: Real World Applications
Can You Buy the Xbox One? - Section 1: Start Up

**Connection to Economics**:

Part of the conversation here will be around the concept of **hyperinflation**. Students will be curious how so much money could be worth so little. I suggest giving very brief explanation and demonstration of how money could reach such a point. In Zimbabwe in 2006, 1.25 US Dollars = 1 Zimbabwe Dollar. However, after a series of changes in currency, the Zimbabwe dollar lost value exponentially.

Yugoslavia in the 1990's is another great example:

In the 1990's the Yugoslavian currency was the dinar. However, Yugoslavia underwent hyperinflation and kept switching dinar systems. Each switch had a catastrophic effect on the Yugoslavian capital. Anything they had saved or owned became worthless. Here is what happened:

In 1990, you would pay 10,000 old dinars to purchase a dinar from 1990.

In 1992, you would have to pay 10 dinars from 1990 for a new dinar from 1992.

In 1993, you would have to pay 10^6 dinars from 1992 to buy a new dinar from 1993.

In 1994, you would have to pay 10^9 dinars from 1993 to buy a new dinar from 1994.

Again in 1994, you would have to pay 1.3 x 10^7 dinars from early 1994 to buy a new dinar from late 1994.

Altogether: 10^4 x 10^1 x 10^6 x 10^9 x 1.3 x 10^7 = 1.3 x 10^27

This means that 1.3 x 10^27 pre 1990 dinar = 1 dinar in 1994

So you would need to own 1,300,000,000,000,000,000,000,000,000 dinars in 1990 to be worth a single dinar in 1994. However, this number is enormous. The total world GDP is about 70,200,000,000,000. Just putting these numbers next to each other tells us that this hyperinflation would crush the world economy. If the world experiences this type of hyperinflation, the currency we had before the hyperinflation would be worth less than a dollar!

I ask them, "If the world GDP experienced this type of hyperinflation, what would the world GDP be worth after the hyperinflation took effect?"

To solve this, students might set up a ratio:

1.3 x 10^27 old dollars : 1 new dollar

7.02 x 10^14 old dollars : x new dollars

x = .00000000000054 dollars

To bring back a conversation about place value, I would ask "how many dollars is this?" Or "how many cents is this?"

*Connection to Economics*

*Real World Applications: Connection to Economics*

# Can You Buy the Xbox One?

Lesson 21 of 21

## Objective: SWBAT to understand the magnitude of loss involved in hyperinflation.

#### Start Up

*20 min*

When student enter the room, I display the XBOX One Image with the question, "Did I give you enough money to buy the Xbox One?" At this point students are wondering, "what money?" I then hand each group an envelope with a wad of cash.

The denominations I give out are Zimbabwe currency notes:

- Zimbabwe 1 Billion
- Zimbabwe 5 Billion
- Zimbabwe 10 Billion
- Zimbabwe 100 billion
- Zimbabwe 50 trillion
- Zimbabwe 100 trillion

I mix up the amounts in each envelope, but they should have amounts that are either above, below or exactly 343 or 344 trillion (see below for calculated cost of Xbox One in $Zimbabwe). The idea is to give them amounts that either *are * or *are not* enough to make the purchase. I like to give exactly $Z 343 trillion, which is about $Z 312 million short of the required amount.

I ask students, "what do you need to know in order to figure out if you have enough?" Here students will recognize that the price of the Xbox One is given in US dollars. So knowing the ratio of Zimbabwe dollars to US dollars is the key. I show them this graphic and then let them count the cash in their envelopes. The graphic shows that in the year 2008, **688 Billion Zimbabwe dollars = 1 US Dollar**.

This makes the problem essentially about multiplication with scientific notation:

**499 US Dollars (cost of an XBox One) x 688 Billion $Z per $US**

I encourage students to use scientific notation, rounding, and estimation.

**Scientific Notation:**

(4.99 x 10^2) x (6.88 x 10^11) = 3.43312 x 10^14 = 343,312,000,000,000

This is rounded to about 343 trillion.

**With Estimation** (I love to encourage students to estimate, which gets them into the habit of developing instinct with numbers):

(5 x 10^2) x (6.8 x 10^11) = 340,000,000,000,000

This is 340 trillion.

Estimation is important for many reasons. I like to ask the class questions like:

- Why is (5 x 10^2) x (6.8 x 10^11) manageable mentally?
- What about 5 x 6.8 is easy?

The idea is that you can almost see that 5 x 6 = 30 and 5 x .8 = 4 (since 5 x 8 = 40) and thus 5 x 6.8 = 34. Then you have 34 x 10^13 = 3.4 x 10^14. I think it is amazing that we can deal with such large amounts in our head.

The next question is dependent upon the estimation and shifts the focus from multiplication to division: **Is this estimation acceptable?**

It is interesting because students will first notice that our estimation is over 3 trillion Zimbabwe dollars. But does this matter? To best answer that question, we can convert amounts to US Dollars.

Since **688 billion Zimbabwe Dollars = 1 US Dollars** and **3 trillion $Z dollars = x $US** we can solve with division:

(3 x 10^12)/(6.88 x 10^11 ) = 4.36 $US

Now we can better answer the question, "Is our estimation acceptable?"

*expand content*

#### Investigation

*20 min*

In this part of class, students go "shopping" for other items. The question is: **How much would that cost in Zimbabwe Dollars circa 2008?**

I print out a variety of fun objects to "purchase" and let students pick blindly from a shopping bag. Their job is to evaluate the cost of the item they pick. The two sources I used to find these items are listed at the bottom of this section. The photos are listed there as well with prices in the name of the file.

For example, students might pick the Aston Martin with a price tag of $1,850,000.

Once students have their items, they begin their analysis. It starts by asking questions that mirror the process in the Start Up and transforms to more questions around division.

**I post this question on the board along with several followup questions**:

- How much would your item cost in Zimbabwe Dollars?

**Followup questions**:

- Find a way to estimate this value.
- Compare the estimate to the exact value.
- Is the estimate acceptable? How do you know?
- How many Zimbabwe dollars does the class have in total?
- How many US Dollars is this worth?
- If I had 10^6 Zimbabwe dollars, how many US dollars would I be able to buy?
- If I had 10^9 Zimbabwe dollars, how many US dollars would I be able to buy?
- If I had 10^12 Zimbabwe dollars, how many US dollars would I be able to buy?
- If I had 10^80 Zimbabwe dollars, how many US dollars would I be able to buy?
- If I had 10^100 Zimbabwe dollars, how many US dollars would I be able to buy?

**Web resources:**

A list of "most expensive" from the web site Born Rich.

24 ridiculously expensive everyday items from the web site Cool Material.

*expand content*

#### Summary

*20 min*

Here the focus of activity is for students to share the cost of each item in both US and Zimbabwe dollars and then discuss the algorithms s/he used to calculate the cost. I am particularly interested in the students' ability to break down and estimate numbers. I want to know if students can find reasonable ways to estimate, and, if they can determine if their estimate is reasonable. This opens up a debate for the class.

For example, if a student has an item that cost 1,250 US Dollars, they would find the exact amount in Zimbabwe dollars by multiplying 1,250 x 6.88 Billion. However, they could estimate and solve mentally doing 2 x 6.8 Billion. The question becomes, "Would this estimate be *above* or *below* the actual amount?" We would also discuss why this might be a better estimate than "1 x 688 billion," which is typical in rounding (students see 1.25 and think "I should round down.") We could get into more detail by discussing if 2 x 6.8 billion is a better estimate than 1.3 x 6.8 billion. Here, we would talk about the impact of place value on the estimates we make.

*expand content*

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- UNIT 1: Starting Right
- UNIT 2: Scale of the Universe: Making Sense of Numbers
- UNIT 3: Scale of the Universe: Fluency and Applications
- UNIT 4: Chrome in the Classroom
- UNIT 5: Lines, Angles, and Algebraic Reasoning
- UNIT 6: Math Exploratorium
- UNIT 7: A Year in Review
- UNIT 8: Linear Regression
- UNIT 9: Sets, Subsets and the Universe
- UNIT 10: Probability
- UNIT 11: Law and Order: Special Exponents Unit
- UNIT 12: Gimme the Base: More with Exponents
- UNIT 13: Statistical Spirals
- UNIT 14: Algebra Spirals

- LESSON 1: The Sand Reckoner
- LESSON 2: How High Can You Count?
- LESSON 3: The Mechanics of Scientific Notation
- LESSON 4: Deconstructing Standard Form
- LESSON 5: Fluency with Large Numbers
- LESSON 6: Ongoing Assessment during the Scientific Notation Unit
- LESSON 7: One Quadrillion Yen
- LESSON 8: Fabulous Fab
- LESSON 9: Sand, Stars and Water Drops
- LESSON 10: Strings for Small Numbers
- LESSON 11: Fluency with Small Numbers
- LESSON 12: Homework and Scientific Notation
- LESSON 13: Million and Billions and the Business of Microsoft
- LESSON 14: Don't Eat the Hot Sauce
- LESSON 15: The Powers of 10
- LESSON 16: Astronomical Units
- LESSON 17: The Most Annoying Sound on Earth
- LESSON 18: Products and Quotients in Scientific Notation
- LESSON 19: Study Time (Game)
- LESSON 20: 1.21 Gigawatts
- LESSON 21: Can You Buy the Xbox One?