Geologic context: radioactive decay, population growth, changes in atmospheric CO2
by Jennifer M. Wenner, Geology Department, University of Wisconsin-Oshkosh
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Exponential growth and decay are rates; that is, they represent the change in some quantity through time. Exponential growth is any increase in a quantity (N) -- exponential decay is any decrease in N -- through time according to the equations:
N(t) = N0ekt(exponential growth)
or
N(t) = N0e-kt (exponential decay)
where:
- N0 is the initial quantity
- t is time
- N(t) is the quantity after time t
- k is a constant (analogous to the decay constant) and
- ex is the exponential function (e is the base of the natural logarithm)
Teaching Strategies: Ideas from Math Education
Put quantitative concepts in context
There are a number of geologic contexts in which to introduce the concept of exponential growth and decay. Some of these include:
- Radioactive decay
- Population growth
- Increases in atmospheric CO2
Use multiple representations
Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. Here are some ways that exponential growth and decay can be represented.
- Tabular representation:
Students can be given a table of data illustrating how N changes through time. (This example (Excel 12kB Jun20 04) contains the results for two made-up exponential equations - shown at the top of each column - illustrating the effect.) Have students examine at the data and explain what is happening to the numbers. You may wish to have them describe what this would look like on a graph.
- Graphical representation:
Show (or have the students construct) graphs (Excel 18kB Jun20 04) that illustrate the change in N through time. Explain why the graphs are curved and why the lines never seems to intersect the x-axis.
- Symbolic representation:
Use pictures or animations that illustrate growth or decay. There are a number of excellent animations available on the web showing the quantitative aspects of radioactive decay, including this one: Decay animation (more info)
- Algebraic/numerical representation:
After showing the plot of exponential decay, depending on the level of your students, you can show them the equation for this plot. Jump to equation above
Use technology appropriately
Students have any number of technological tools that they can use to better understand quantitative concepts -- from the calculators in their backpacks to the computers in their dorm rooms. Exponential growth and decay can make use of these tools to help the students understand this often difficult concept.
- Graphing calculators
- Computers
- Demonstration of Exponential Decay Using Coins
- Participants of the Quantitative Skills Workshop in 2002 developed a template for teaching mathematical functions. Included in this activity page are exercises for teaching Population Growth and Atmospheric CO2 Increase -- excellent examples of exponential growth.
- the constant e (more info) ,
- exponential decay (more info) ,
- exponential growth (more info) , and
- graphical and mathematical representations of the exponential function (more info)
Graphing calculators are an easy way for all students to enter data and to see what a curve of that data looks like. All graphing calculators are slightly different and students may need help with their particular model. There are some helpful hints for some calculators at Prentice-Hall's Calculator help website (more info) .
Exponential growth and decay provide an excellent opening for an introduction to the use of spreadsheet programs. Students are likely to encounter spreadsheet programs in many of their classes and they are excellent tools for visualizing the shape of an equation.
Work in groups to do multiple day, in-depth problems
Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.
Exponential growth and decay is a concept that comes up over and over in introductory geoscience: Radioactive decay, population growth, CO2 increase, etc. When each new topic is introduced, make sure to point out that they have seen this type of function before and should recognize it.
Teaching Materials and Exercises
Student resources
Mathworld has some very good resources explaining:
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