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Double, Double, Toil and Trouble

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Warming Up Graphing the Results Real World J Curves Crashes Follow-up Activities Resources

My thanks to William Shakespeare for the title (Macbeth, Act 4, Scene 1), and my apologies to those of you hoping for an introduction to the play.  Instead, the following is a way to introduce exponential growth, line graphs and environmental crises.  It may not be Shakespeare, but I hope you'll find it valuable and intriguing anyway.  

Depending on your students' abilities, this activity could easily take more than one class period.  Feel free to pick and choose the parts that will work best with your class.

  1. Warming Up
  1. Explain that 
  1. you are about to ask a question.
  2. once you've finished posing it, you want each person to quietly think through an answer.
  3. after 2 or 3 minutes for thought you'll allow time for discussion.
  1. Ask students to decide which they would choose, a guaranteed salary of $10,000 per week (feel free to lower this number if you think it too large for your students) for one year; or a salary beginning at one penny per week that doubles each week for a year?
  2. After 2 or 3 minutes of silent thought, poll the class to find out how many would take the $10,000 per week (or the number you chose), and how many the beginning salary of one cent doubling each week.
  3. Let those on each side explain their answers.
  4. Assuming that an individual would keep the job for one year, attempt to reach a consensus on how much one would have earned at the end of the year under each scenario.
  5. If they are unable to arrive at the answers, explain that at $10,000 per week a person would earn $520,000 in a year.  Beginning at one penny, then doubling the salary each week, a person would earn $45,035,996,273,705 in one year.  That is $45 trillion, 35 billion, 996 million, 273 thousand and 705 dollars; or approximately 87 million times as much as $520,000.
  6. Click here to see a table that illustrates the process and results.  If you want, duplicate, distribute and discuss it with your students. If they are unfamiliar with data tables, show them how to use the column and row headers to find and make sense of information.  If they have calculators with an exponent key (XY) and they already know or are ready to learn about exponents, you can take this opportunity to show them how to use it to calculate the salary the person would earn at the end of each week.  They'll need to begin with X=2 and Y=the week you want minus 1, then divide the result by 100.  This results in the salary for week Y+1.  (For example, X=2 and Y=10 divided by 100 results in 10.24 - the result for week 11.)  Alternately, if you know how to manually calculate exponents and your students are ready to learn that process, you could show them how to do that.  You can also show them how to use a calculator's constant function to calculate weekly earnings.  Begin with .01 then multiply with a constant 2.  (Different calculators do this differently I believe.)  Each time you press the = key, you'll get a result for the next week.  The result of the first push is week 2; the second, week 3; etc.  If you want to use this activity to help your students as they learn to design and build computer spreadsheets, here is the link to an Excel workbook I've built to solve this and related problems
  7. Discuss the results with your class.
  8. Points you'll want to ensure emerge from your class discussion
  1. The two salary scenarios illustrate straight line and exponential growth.
  2. Straight line growth is steady and usually sustainable.
  3. Exponential growth begins very slowly, but eventually takes off and can not be sustained.
  4. For example, it would be impossible for anyone to earn or accumulate $45 trillion.  The CIA estimates that the value of everything produced in the world in 2001 was approximately $49 trillion, for 2011 its estimate is $79 trillion.  For one person to make anything close to that would mean that everybody else in the world would have to give up everything.  With no food, housing, clothing, etc., there would be mass death and nobody left to pay the salary of the "lucky" recipient.  Clearly human civilization would collapse, or stop the accumulation, long before it occurred.  
  5. Unchecked, exponential growth leads to the collapse of the system in which it occurs.
  6. Before a collapse, things can look pretty good for those on the skyrocketing curve.  Think how happy a person earning that exponentially growing salary would be in the middle of the year.
  7. To head off such a collapse, exponential growth must be recognized and stopped early.

  1. Graphing the Results
  1. Give each student a sheet of graph paper.
  2. Show them how to identify the X and Y axes.
  3. Label the Y (vertical) axis as Total Earnings $, and enter 0 to 600,000 at intervals of 100,000.
  4. Label the X (horizontal) axis as Week #, and enter 1 to 52 at intervals of 10.
  5. Title the graph, Weekly Earnings at $10,000 per Week.
  6. Show them how to plot points where each identified week and its total earnings meet.
  7. Have them plot the points.
  8. Have them connect the points with a line.
  9. If they did everything correctly, they should end up with a straight line.
  10. Click here for an image of a graph like the one they should see.
  11. Repeat steps a-g with the data from the total earnings when the salary doubles column.  At step c, enter 0 to 50,000,000,000,000 (50 trillion) at intervals of 5 trillion (5,000,000,000,000). At step e, title the graph Weekly Earnings: Exponential.  When they begin to enter points, they'll probably find it impossible to do so before week 37 or so.
  12. If all of those zeros are a bit numbing, take this opportunity to show your students how to use scientific notation.  If you do, allow them to label the Y axis using scientific notation.
  13. This time they will need to connect the points with a curved line.
  14. Here are two images of a graph like the one they should see: using scientific notation or not using it.
  15. Lead a class discussion where you compare the two graphs.
  16. Points you'll want to ensure emerge from your class discussion
  1. When plotted, straight line growth produces a straight line on the graph.
  2. When plotted, exponential growth produces a J curve, a line that barely moves above the X axis until it takes off; at which point it moves almost straight up.
  3. One of the best ways to spot exponential growth is to plot data and look for J curves.

  1. Real World J Curves
  1. Explain that exponential growth (resulting in J curves when plotted) exists in the real world.  It is most often seen in the natural environment and in stock markets.
  2. Perhaps the most famous natural example is human population growth.  The U.S. Census Bureau provides data tables of world population estimates from 1950 (with projections to 2050).
  3. The pressure placed on the environment by the exponentially growing number of humans has created any number of environmental J curves.  Jay Hanson explores many of them on his Brainfood web site.
  4. Albert Bartlett, a retired professor of Physics from CU Boulder, has presented the talk captured in this video (Arithmetic, Population and Energy) more than 1,600 times since 1969. His message about population growth and resource consumption is compelling. Links to papers that Bartlett has authored for general audiences may be found at http://www.hubbertpeak.com/bartlett/
  5. World Population Balance has posted an easy to understand demonstration of exponential growth. You may step through it at http://www.worldpopulationbalance.org/bacteria_exponential_growth_curve
  6. Throughout 1999 and into early 2000, the NASDAQ Composite (made up primarily of technology and "Internet" companies) grew at exponential rates.  It crashed in the spring of 2000.  The Dow Jones Industrial Average grew exponentially during the 1920s, the 1950s and 60s, and the 1990s.  Here is a graph of the Dow Jones Industrial Average Yearend Closings from its founding in 1896 through the depths of the Great Depression of the 1930s.  With it you can see the J Curve form in the 1920s, and the subsequent crash of 1929-32.
  7. Lee Anderson, late professor of Political Science at Northwestern University, pointed out other J curves in travel speed, human life expectancy, the explosive power of weaponry, and the number of books published per year.  Ray Kurzweil pointed out another one in computing power.

  1. Points to make about crashes
  1. Unchecked, exponential growth leads to the collapse of the system in which it occurs.
  2. It is difficult to accurately predict when growth will stop and a crash will occur.  Hence the debate throughout the last third of the 20th century over whether humans had pushed the world to the brink of environmental catastrophe.  Despite many thoughtful people predicting imminent calamity, we have yet (at least as of this writing near the end of 2000) to see such dire warnings fulfilled.  But, that does not mean they won't be.  It only means that the exact timing of crashes is hard to predict, and possibly that human ingenuity can forestall them. Think about this as you examine the Brainfood site, which can otherwise be very scary.
  3. If growth can be slowed before a crash takes place, it seems possible to avoid the crash.  This is what U.S. Federal Reserve Board chair Alan Greenspan, with his much publicized attempts to negotiate "soft landings", has attempted to do with economic growth throughout the 1990s.

  1. Follow-up activities
  1. Here are assignments I've given to help students begin thinking about human population growth.
  1. What drives human population growth?
  2. Human population on the 50th Earth Day
  3. You may also use the calendar activity described in Putting Time in Perspective with the following milestones in human population growth to illustrate population growth's J Curve properties.
Scale  120,000  years = 365  days
         
Event # of years ago

Would appear on 

 at
Modern humans appear 120,000 January  1  12:00:00 AM
population reaches 1 million (10,000 BCE) 12,000 November  25  12:00:00 PM
population reaches 10 million (6,000 BCE) 8,000 December  7  04:00:00 PM
population reaches 100 million (500 BCE) 2,500 December  24  09:30:00 AM
population reaches 250 million (950) 1,058 December  28  06:45:57 PM
population reaches 500 million (1600) 408 December  30  06:12:57 PM
population reaches 750 million (1720) 288 December  31  02:58:33 AM
population reaches 1 billion (1804) 204 December  31  09:06:28 AM
population reaches 2 billion (1927) 81 December  31  06:05:13 PM
population reaches 3 billion (1961) 47 December  31  08:34:08 PM
population reaches 4 billion (1974) 34 December  31  09:31:04 PM
population reaches 5 billion (1987) 21 December  31  10:28:01 PM
population reaches 6 billion (1999) 9 December  31  11:20:34 PM
population reaches 6.7 billion (July 2008) 0 December  31  11:58:14 PM
       
estimates from Wikipedia page on World population: http://en.wikipedia.org/wiki/World_population  
  1. Have your students research, or give them, a history of the Dow Jones Industrial Average yearend closing prices.  To avoid burdening them if you ask them to do the research, assign each student a different 3 or 4 year block.  Divide the Dow's age (104 as I type this) by the number of students in your class to determine the exact number of years to assign each student.  Collect and compile their research, then distribute it to all students before proceeding.
  1. Instruct them to plot the Dow's yearend closing prices on graph paper.  Discuss the result.  Is it straight line or exponential?  Is it possible to see the great crash of 1929-32 on this graph?  Are other crashes visible?  
  2. From the data, ask your students to identify and plot the Dow J curves that formed and crashed throughout the DJIA's history.  I see them from 1903-1907, 1914-32, 1932-1974, 1974-?.
  3. Should you want to eliminate the effects of inflation, use the Minneapolis FED's CPI Calculation Machine to generate a table in constant 1999 dollars; then graph its numbers.  If you do so, have your students compare and discuss the two graphs (Constant $ and then Current $).
  1. If you liked the warm-up, try it again using a thought experiment with paper folding.  Ask your students to determine how thick a piece of paper would be if one could fold it in half 52 times.  You might want them to begin by researching the thickness of a piece of standard notebook paper.  I assume it to be 1 millimeter or 1/254 or an inch.  Here is what I calculated.  Raju Varghese also looked at this problem.  Here are his calculations.

    Another perspective on this problem appears in Sara Steindorf's Christian Science Monitor piece, Speaking of Big Numbers (September 24, 2002).
  2. Other Classroomtools.com lesson ideas that allow students to work with numbers
  1. Interesting numbers
  2. Putting Time in Perspective
  3. Taxes Made EZ
  4. those containing data tables with which you might want your students to work

Warm-up activities

School Safety Facts

What is the most common crime committed in the U.S.?

When a person dies at the hands of a gunman, who most often pulls the trigger?

How much is an education worth?

How Much Money Do You Owe?

Main Events

Media use survey from Propaganda in the classroom

How much do they make?

The world in a room

Additional Resources

Books Population Issues Environment Excel workbook

Books

Grasping very large or very small numbers can be daunting.  These two books show how to do it in innovative ways.  They are definitely worth time any spent with them.  Both are currently out of print, but time seeking them from a library or used book store will be rewarded.

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Web sites dealing with Population issues

  1. The Why Files: The Year of Six Billion
  2. Population Reference Bureau
  3. Peterson's Population page
  4. PopNet (now a part of PRB's Country Reference and Statistics page)
  5. United Nations Population Information
  6. If statistics like those in this section are not dynamic enough for you, don't miss Hans Rosling's 2006 TED conference presentation, available for viewing via YouTube. Rosling is a  Professor at Sweden's Karolinska Institute. If you like his talk, you'll love the materials you'll find at his web site, Gapminder.org.
  7. 6 Billion Humans - This site was exceptional.  It allowed you (or your students) to work interactively with statistics about the human population of the planet.  It was presented by the Museum of Man in Paris.
  1. U.S. Census Bureau
  2. University of North Carolina
  3. The University Exchange Focus on Global Intelligence (world population)
  1. Zero Population Growth
  2. Population Action International
  3. Alan Guttmacher Institute
  4. Planned Parenthood
  5. National Geographic site on Overpopulation
  6. World Population Balance
  1. Overpopulation.com
  2. Population Research Institute
  3. Jeff Lindsay's Population page

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Web sites dealing with the Environment

  1. Internet resource list
  2. U.S. Environmental Protection Agency
  1. Earth Day Network
  2. Greenpeace USA
  3. Paul Ehrlich on Environmental Fables
  4. Woods Hole Research Center: The Warming of the Earth
  5. Worldwatch Institute
  6. True or False: Glaciers Have Been Growing, Not Shrinking (a Classroomtools.com fact check page)
  1. The National Anxiety Center
  2. Global Warming Information page

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Excel workbook

I've created an Excel 97/2000 workbook containing some of the data used in this activity.  Please download it if you want to create additional graphs.  You might also want to explore its construction.  Doing so might give you ideas for showing your students how to use a spreadsheet program to manipulate data when working with problems like those on this page.

To download the Excel file (double.xls), click on its link below.  A small window should open.  One of the options presented should be to save the file.  Click on that option.  You should then be prompted to specify where on your hard drive you want to save the file.  Put it in the folder where you save and open Excel spreadsheets.  Once the file has completed downloading, the small window will close on its own.

double.xls

Should you have problems with the file, or should you need a version in a different file format, please e-mail me.  I will help in any way I can.

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original web posting: Tuesday, November 14, 2000
last modified: Friday, May 10, 2013