Modeling Exponential (or Geometric) Growth

(Download the MS Word version of this document.)

Purpose

To use dice to model exponential growth; to examine our values concerning population growth.

Materials

10 to 20 dice
graph paper
notebook paper
pen
pencil

Procedure

In this experiment you will use dice to model population growth. Each die represents a person. Each throw of the dice represents a year. A three or a six represents the birth of a child, so each time one of them comes up, you must add a roll of the die. If a one comes up, a death has occurred, so remove it from the population.

Questions

Answer all questions in complete sentences.

1. What is the ratio of births to deaths in this model population?

A. UNRESTRICTED EXPONENTIAL GROWTH (UEG):

Put all dice into the cup. Shake the cup and dump the contents (carefully) onto the table. You lose a point for every die that hits the floor! Remove and count all the "ones" that appear. A "one" represents a death. Record the number of deaths on a chart. Count up all the "threes" and "sixes" that appear. Since they correspond to births, you must add a roll of the dice for each of them. Fill in the required information in the chart. Repeat the procedure for UEG until the total population exceeds 500 people. You must complete an entire year before stopping.

2. How many "years" did it take you to reach a population of 100?

3. After you reached a population of 100, how many more "years" did it take to reach a population of 200? How many more years to reach 300? 400? 500?

4. Use your data table to plot a graph of unrestricted exponential growth of a population. Use "Years" as the x-axis and "Total Population" as the y-axis. Be sure to give your graph an appropriate title and to label the axes of the graph.

5. Using this experiment, define exponential (or geometric) growth.

6. In what way do exponential (or geometric) growth rates differ from arithmetic growth rates?

B. WORLD POPULATION TRENDS

The following is a listing of estimates of world population from 1650 to 1988. Figures such as these are compiled by the United Nations and are published in most almanacs. (I have rounded the figures to make them easier to graph.)

1650 = 0.5 billion	1930 = 2.1 billion	1970 = 3.6 billion
1750 = 0.7 billion	1940 = 2.3 billion	1980 = 4.4 billion
1850 = 1.1 billion	1950 = 2.5 billion	1985 = 4.8 billion
1900 = 1.6 billion	1960 = 3.0 billion	1988 = 5.0 billion

7. Using the above information, plot a graph of world population versus time from 1650 to 1990. You will need to estimate what you believe the population was in 1990. (Hint: Use your graph to make this estimation by extending your graph line. Be sure to use a dashed line to show that you are not certain of your data.)

C. DOUBLING TIME OF A POPULATION is the number of years it would take to double a population if its current growth rate were to remain constant. To determine doubling time, divide 70 by the populations' growth rate. For example: for the world, the current growth rate is 1.8% per year. 70 divided by 1.8 equals about 39; if the world's population continues to grow at its current rate, there will be twice as many people on Earth in just 39 years. U.S. population in 1992 is about 251 million. With many "baby boomers" now having children, even though they are having small families (average 1.9 children), the U.S. has about 1.75 million more births than deaths a year. Immigration also contributes significantly to continuing U.S. growth. Altogether, our population is growing by approximately 2.4 million per year.

8. Calculate the doubling rate for the population of the U.S. (Show your work.)

9. Is the U.S. overpopulated? Explain your answer using several examples.

10. Use your graph to predict world population for the year 2000. (Hint: Use dotted lines to extend your graph into the future.)

[printer-friendly]