How Big is a Trillion?

Learning goals for this example include:

• Understanding large quantities,
• Comparing quantities,
• Unit conversions,
• Identifying assumptions, and
• Verifying information.

This example requires very little in the way of prerequisites and can be used early on in a quantitative reasoning course.

Part I: Buying Power of $1.2 Trillion

David Leonhardt's January 17, 2007 New York Times article What .2 Trillion Can Buy is one source that can be used to begin discussions of large numbers. This article describes how one might make sense of $1.2 trillion. Certainly, understanding such a quantity depends heavily on what one understands beforehand, namely the cost of other items. After reading the article, students can answer the following study questions:

1. Of the writer's list of programs that could be funded with the $1.2 trillion, which two are the best for helping you understand this large amount of money? Why are the two you selected most helpful?
2. How many $150,000 homes would the $1.2 trillion buy? How can we make sense of this number of $150,000 homes? Determine an appropriate measure (e.g., population of a particular state or city) that would help someone make sense of the number of $150,000 homes that $1.2 trillion could buy.
3. Compare the writer's estimated cost of the war in Iraq to the Pentagon's estimate of the cost of the war using a ratio and a percent.
4. Compare the writer's estimated cost of the war to Lawrence Lindsey's estimate using a ratio and a percent.
5. Compare the estimated cost of the war per day to the annual budget of the National Cancer Institute using a ratio and a percent.
6. Choose some measure of the size of $1.2 trillion (other than homes) that helps you understand its magnitude and express the $1.2 trillion in your measure. The answer to #2 is an example of this. Explain why this measure is meaningful to you.

Part II: Visualizing a Trillion

There have been numerous graphics appearing in the nation's papers which try to give readers a sense of the relative sizes of millions, billions, and trillions. Some of these graphics can be found online as part of blogs and other commentaries. Sources and suggested study questions follow:

Graphic 1: Start at this blog and follow the link to the pdf file of the graphic.

1. Verify the calculations that accompany the three graphics labeled "Time".
2. What assumptions have been made to arrive at the claim that "A trillion dollars could give every high school student in the united states a free college education"?
3. Which of the facts on this page startled you the most? Why?
4. Ask a friend to estimate an answer to the following question: If you had gone into business on the day Jesus was born and made a million dollars every day, how long would it take to make a trillion dollars? Compare responses you receive with the quantity in the graphic.

Graphic 2: What does a trillion dollars look like? - Pagetutor.com offers a graphic to compare the orders of magnitude for $100, $10,000, $1 million, and $1 billion

1. Verify that the graphics at this site are reasonable. Explain your calculations and any assumptions that you had to make.

Graphic 3: Visualizing One Trillion Dollars - Appearing at Mint.com, this graphic seems to come up the most often under internet searches:

Ten colorful posters convey the buying power of one trillion dollars.

1. Which of the posters expressed the value of $1 trillion in terms that were meaningful to you?
2. Pick one of the posters and verify the information it contains. Did you have to make any assumptions? Were the assumptions reasonable? Did you have to research any facts or figures?

Sample solutions for Part I: Buying Power of $1.2 Trillion

1. Of the writer's list of programs that could be funded with the $1.2 trillion, which two are the best for helping you understand this large amount of money? Why are the two you selected most helpful?
The list of programs is:
• Doubling of cancer research funding.
• Treatment for every American whose diabetes or heart disease is now going unmanaged.
• Global immunization campaign to save millions of childrens' lives.
• Universal preschool for every 3- and 4-year-old child across the country.
• The city of New Orleans could also receive a huge increase in reconstruction funds.
• The recommendations of the 9/11 Commission that have not been put in place -- better baggage and cargo screening, stronger measures against nuclear proliferation -- could be enacted.
• Financing for the war in Afghanistan could be increased to beat back the Taliban's recent gains.
The choice of the best two depends on the experience and understanding of the responding student. For example, saving millions of childrens' lives and universal pre-school in the US are more familiar to many students than is the genocide in Darfur.

2. How many $150,000 homes would the $1.2 trillion buy? How can we make sense of this number of $150,000 homes? Determine an appropriate measure (e.g., population of a particular state or city) that would help someone make sense of the number of $150,000 homes that $1.2 trillion could buy.
$1.2 trillion will buy eight million $150,000 houses. This is approximately one house for every person in the state of Virginia or nearly three houses for every person in Arkansas.

3. Compare the writer's estimated cost of the war in Iraq to the Pentagon's estimate of the cost of the war using a ratio and a percent.
The Pentagon's estimate is given as $50 billion. The writer's estimate of $1.2 trillion is 24 times $50billion. The Pentagon's estimate is approximately 4% of the writer's estimate.

4. Compare the writer's estimated cost of the war to Lawrence Lindsey's estimate using a ratio and a percent.
Lindsey's estimate is given as$200 billion. The writer's estimate of $1.2 trillion is 6 times $200 billion. Lindsey's estimate is 16.67% of the writer's estimate.

5. Compare the estimated cost of the war per day to the annual budget of the National Cancer Institute using a ratio and a percent.
The war cost is estimated at $200 billion per year or about $550 million per day. This is about 9% or 1/11th of the annual budget of the National Cancer Institute. So, eleven days of the war costs as much as the National Cancer Institute's annual budget.

6. Choose some measure of the size of $1.2 trillion (other than homes) that helps you understand its magnitude and express the $1.2 trillion in your measure. The answer to #2 is an example of this. Explain why this measure is meaningful to you.
Answers will vary.

Part I
A variety of formal and informal assessment strategies are possible. One possibility:

• Individual: For homework, each student is required to read the article and write up a short response to Part I, item 1.
• Group: Small groups work through items 2-5 during class. Participation and/or attendance credit is given.
• Class: Small groups report out on their responses to item 2. Instructor picks one of items 3,4, or 5 to discuss. During discussion, instructor needs to make sure that students are comfortable using both ratios and percents to compare quantities. Also, make sure students use the correct language when stating comparisons.
• Individual: Students should prepare individual responses to one of items 3,4, or 5 (instructor assigned) as well as item 6 to be collected the following class day.
  

Part II
As with Part I, a variety of assessments are possible. One approach would be to use two of the graphics (e.g. Graphic 1 and Graphic 3) to form the basis for some class discussions and group work (informal assessment only). The remaining graphic can be used for formal assessment either as a turn-in homework assignment or, perhaps, as an in-class individual assessment test item.

1. Leonhardt, David "What $1.2 Trillion Can Buy", New York Times, January 17, 2007. This article may be directly viewable from your browser. If not, visit nytimes.com and search the archives for "what $1.2 trillion can buy david leonhardt".
2. McClatchy-Tribune, Visualizing a trillion, February 9, 2009 as seen on Cleaveland.com.
3. Pagetutor.com, What does one TRILLION dollars look like?, as seen on August 24, 2009.
4. Mint.com, Visualizing One Trillion Dollars, as seen on August 24, 2009.