Tuesday, May 19, 2015

The Importance of Understanding the Exponential Equation

Sometimes we, as teachers, teach concepts to our students that we understand are important and we understand would be quite beneficial for them to learn. We feel that, even if they do not use this particular skill in their future, they will at least understand the concept and have knowledge of how it applies to the world they live in. Essentially, this is what we strive for in developing life-long learners. But all too often, with a strict time schedule, we jump into skills and abilities without giving a non-generic real world connection, back ground or goal to the concepts and skills that are being developed.

Today, I would like to give an example of the importance of the exponential function and exponential growth. It is easy to explain half-life and depreciation through an exponential function. Albeit, far from the life experiences (and thus importance/interest) of our students. But why is exponential growth so important to us as individuals and as a species on this planet?

First off, as a teacher, listen to this short talk by Dr. Albert Bartlett from the University of Boulder Colorado. It will be your call whether or not to show this students and this will largely depend on the dynamics of your class. Dr. Bartlett is famous for stating that:

"The greatest shortcoming of the human race is our inability to understand the exponential function."



Now, for the classroom, here is a video from National Geographic that is an excellent visual representation of the current exponential growth that we as humans are in on Earth (Note: this is also good for Populations Dynamics in Grade 12):



The problem here is that exponential growth never ends and assumes (in the case of population) that resources are unlimited. This is not the case with our planet and we are currently beginning to face the problem of our inability to control (and understand) the exponential growth of humankind.

But here's the issue: this mis-understanding of limitless exponential growth does not only apply to population. Look at the growth of debt in the United States since 1940:



Finally, this series of graphs shows just how often this type of growth arises:


This is a dangerous trend which the vast majority of the population simply does not understand the basics behind and it arises in many important aspects of our species' existence on the planet. As we can see and much as Dr. Bartlett explains, it can be detrimental for us to not have any knowledge of exponential growth. One important area that exponential growth has arisen and that can be addressed by individuals on an individual basis is consumption.

Here's an example to run with students out of Washington that can easily be adapted for your area (for example, Lake St. Clair/Lake Erie or Lake Superior):


The activity is a very clear example of water consumption that runs through three scenarios. The resource limit is the entirety of Lake Washington (770 billion gallons of water). In the first scenario, a gardener takes 1 gallon of water from the lake to water her plants each day. The water source will last 2.1 billion years. In the second scenario, she gains one client each day that requires an additional gallon of water each day (consumption grows linearly) it takes 3,400 years to deplete the resource. Finally, in scenario three, the gardener is a true business mogul and doubles the amount of clients each day (consumption grows exponentially). This time, the entire lake is drained within 40 days.


This is an excellent activity that can be done with your class to show the severity of exponential growth. As this concepts arises in all grade 11 and 12 courses, having students understand why it is so important by giving it some real-world background can be very beneficial to students developing their mathematical understanding and ability. 

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