# Exponential Growth and Decay

**Geologic context:**radioactive decay, population growth, changes in atmospheric CO_{2}# Teaching Exponential Growth and Decay

^{by Jennifer M. Wenner, Geology Department, University of Wisconsin-Oshkosh}

*Jump down to:*Teaching strategies | Materials & Exercises | Student Resources

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) = N_{0}*e*^{kt}* (exponential growth)*

or #### N(t) = N_{0}*e*^{-kt}* (exponential decay)*

#### where:

**N**is the initial quantity_{0}is time*t***N(t)**is the quantity after time*t*is a constant (analogous to the decay constant) and*k*is the exponential function (*e*^{x}**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 CO
_{2}

### 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

### 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, CO_{2} 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

- Demonstration of Exponential Decay Using Coins
- M&M's Model for Exponential Decay
- 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 CO**– excellent examples of exponential growth._{2}Increase

## Student resources

** Geomaths** has a page explaining the math behind radioactive decay with a link to a very nice MathHelp tutorial on exponential functions.

**Mathworld**has some very good resources explaining:

- the constant e (more info) ,
- exponential decay (more info) ,
- exponential growth (more info) , and
- graphical and mathematical representations of the exponential function (more info)