The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view. Instead of giving a starting value and asking for subsequent values, it gives an end value and asks about what happened in the past. The simple first question can generate a surprisingly lively discussion as students often think that the algae will grow linearly.

In this lesson, through various examples and activities, exponential growth and polynomial growth are compared to develop an insight about how quickly the number can grow or decay in exponentials. A basic knowledge of scientific notation, plotting graphs and finding intersection of two functions is assumed.

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year.

RUNNING THE ACTIVITY—PART 1

In the first stage of this Activity, ask everyone to crumple up a piece of (scrap) paper. Pick one person to come up to the front of the room and be the initial member of the population. Members of this population should follow these rules:

If you are standing at the front of the room, then throw your paper high in the air (at least several feet above your head) when the facilitator gives the “next generation” command.
If you catch your paper, then you survive and may “reproduce” by calling up another member of the audience to join the population.
If you don’t catch your paper, then you “die” and must sit down. Lead participants through several “generations.”
If the population crashes or becomes extinct (because all of the population members drop their pieces of paper), begin again, noting that sometimes populations will crash by chance or become extinct when their numbers are small. Record the size of the population over time. What is the maximum population?

Once the members of the audience are all standing at the front of the room, take a look at the graph of the population over time. Ask the participants to reflect

Did you ever count bacterial colonies growing in a petri dish? Or track the accumulation of money in your savings account? Or watch as mold spread over an old piece of bread? If so, you have observed population growth in action. As populations grow, the change in the number of individuals in those populations over time can be classified in different ways. Exponential growth and logistic growth are terms applied to specific patterns of population expansion. Both exponential and logistic growth are central to many processes and are the basis for many models. Perhaps because of their prevalence, or perhaps because of their “observable” quality, these types of growth are commonly described in biology, mathematics, and even economics courses. However, most textbooks present them in terms of somewhat sophisticated mathematical equations. Unfortunately, those equations do not provide a handy structure for thinking about growing systems from the “bottom up.”

In this Activity, you will get a chance to explore exponential and logistic growth models, relying on simple rules and observable aggregate behavior rather than on specialized mathematical techniques. This Activity approaches system modeling from the perspective of one individual or one creature—much like StarLogo does. The penny model presented in this Activity is an idea model. You might think about the advantages and limitations that this type of model provides for understanding exponential and logistic growth. You might also consider what modifications would be required to create a penny-based minimal model for a system.

ACTIVITY MODELING CONCEPTS

Discover how simple rules can define behavior that is typically described by complex mathematical equations.
Learn about the concepts of exponential and logistic growth.
Analyze the elements of a model that make it appropriate or inappropriate for thinking about specific systems.

Three Weeks in March is a data-driven approach to modeling the spread of coronavirus cases in the United States.In this activity, students will use a difference equation to model day-to-day changes in the known cases of coronavirus within U.S. borders, as reported by the Centers for Disease Control and Prevention, during the first three weeks of March, 2020. The solution to this difference equation is an exponential model. The activity can serve as an introduction to exponential models.The main goals of this activity are to:Learn the idea of difference equation;Model exponential growth using a difference equation;Simulate exponential growth and estimate a parameter value using software.These resources were created by Dr. Rob Kipka of Lake Superior State University.