# Engaging students: Exponential Growth and Decay

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Angelica Albarracin. Her topic, from Precalculus: exponential growth and decay.

How could you as a teacher create an activity or project that involves your topic?

During my freshman year of high school, my school offered AP Human Geography. One of the most important figures you learn about in this class is Thomas Malthus, who was an English economist and demographer during the late 1700s and early 1800s. Malthus was most known for his theory that population growth would outpace the world’s food supply. His argument was that since population grows at an exponential rate, and food supply at the time was increasing at a linear rate, then the world would run out of food in a short amount of time. Of course, today we know that Malthus’s theory was incorrect because it did not account for the profound effect that the industrial revolution would have on agriculture. However, if this theory were to be explained to a group of people who may not know what the difference between a linear and exponential function is, the usage of a graph as a visual aid would be extremely helpful.

Given this premise, students may be asked to create a graph with given coordinates to compare the difference between a linear and exponential graph, allowing students to see for themselves why this theory may have been extremely alarming to people during this time. After this, the students may be presented with several different scenarios such as “Graph a constant population of 1 billion vs. a rapidly declining food supply due to locust swarms” or “Graph a sudden population boom 5 years prior to a boom in food supply that increases at twice the rate of the population”. Students could be asked questions such as “Will the population have enough food to survive?” or “How many years will it take for there to be enough food to feed the entire population?”. I think this would be an extremely engaging activity for students as the premise behind it is an interesting piece of mathematical history and students’ imaginations can be engaged during the different scenarios.

How can this this topic be used in your students’ future courses in mathematics or science?

Exponential growth functions are commonly used to model the population growth of a species in Environmental Science. An important concept in Environmental Science is carrying capacity, which is the largest population a habit can support without degradation. Due to the carrying capacity, we typically see S-curves in the population models in Environmental Science as opposed to the normal J-curves. When students are familiar with the rapid rate in which exponential functions can grow, it provides intuitive reasoning for why carrying capacity exists in nature as habits very clearly have a finite amount of resources that cannot possibly support an infinitely growing population.

The concept of radioactive decay and half-lives is also very important in Chemistry. A half-life is a measure of the amount of time it takes for half of a radioactive isotope to decay.  While not all isotopes are radioactive, the ones that are decay at an exponential rate. Having knowledge of an isotopes half-life enables scientists to handle such material safely. Typically, scientists wait to handle such radioactive material until it has decayed below detection limits, which occurs around 10 half-lives. Beyond this, doctors must also use their knowledge of half-lives when using radioactive isotopes to help treat patients. For a radioactive isotope to be useful in this manner, its radioactivity must be active enough to treat the condition, but not too long as to harm healthy cells.

How has this topic appeared in the news?

Historically, exponential growth and decay graphs have been used to model the spread of epidemics/pandemics. Recently, with the advent of the Covid-19 epidemic, we are constantly seeing such graphs all over the news and agency websites such as the CDC. In the graph depicted below, we can see exponential growth in the number of cases around March, a small decline, and then another bout of exponential growth around June. Of course, in the real world, very few data follow an exact mathematical form so using the phrase “exponential growth” is an approximation. However, this exponential trend demonstrates just how contagious this virus is as we can see how thousands of people can be affected in a short amount of time.

During the Australian bushfires that occurred during January 2020, many articles began to attribute this disaster with climate change due to human activity. Though the causes of wildfires are highly variable and difficult to track, many scientists felt that Australia’s record warmth and dryness during the previous year, at the very least, allowed the fires to spread much quicker.    In the graph below, we can see a slight trend between the climate change seen in Australia (as recorded by the Australian Bureau of Meteorology (BOM)) versus the average climate change seen around the world by 41 models. A line of best fit has been drawn through the graph of 41 climate models, though hard to see, allows us to see more clearly that this data set increases at an exponential rate. While it is still difficult to determine whether this climate change can be directly attributed to the wildfires, we can still see our risk for such disasters increase as time goes on.

References:

https://www.britannica.com/biography/Thomas-Malthus