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tive impact on our climate.

(Martin, 2011, p.11).

During the 1970’ s “ energy crisis ” the Western world, and in particular the United States, experi enced considerable gas shortages and elevated prices. We see this today as a result of global insta bility, in particular, the Russia - Ukraine war. A thoughtful and comprehensive approach to identi fying problems associated with “ sustainable growth ” using non - renewable resources has been available since the mid - 1970 ’ s, if not earlier. Over the two decades between 1976 and 1996, the late Albert A. Bartlett, Ph. D, former Professor of physics at the University of Colorado in Boulder, Colorado, wrote a series of eleven papers titled, The Exponential Function - Part I through The Ex ponential Function - Part XI, published in the Phys ics Teacher Journal . His most significant papers are collectively called Environmental Mathematics that reside in the volume, The Essential Exponen tial . Dr. Bartlett is known for his pithy quotes associat ed with exponential functions and nature. They include:

Anyone who believes in indefinite growth in anything physical, on a physically finite planet, is either mad or an economist, Ken neth E. Boulding, economist This article builds on the sequence of topics that begins with Conceptual Understanding of the Ex ponential Function that includes class activities and a simple doubling mathematical model, fol lowed by the section, Mathematical Functions, that model consumable resources, before concluding with closing remarks. Conceptual Understanding of the Exponential Function In what follows, are several well - known, yet unre alistic, examples of exponential growth. In particu lar, we focus our attention on the “ explosive ” growth of the function, because this part is less un derstood or known by many people. The following two examples are exercises teachers can use in the classroom to develop conceptual understanding about the explosive growth innate to the exponen tial function. Paper - folding Activity Take a sheet of paper 1/100 th of an inch thick, in which a pad of 100 sheets is about 1 inch thick, and fold it thirty times. Quickly students realize that this is not possible. Suppose you could do this. We can use mathematics to find out how thick the paper would be after the 30 folds. We start with a single fold, which is twice as thick as the single paper. After two folds 2 2 it is four times as thick. Then, after three folds 2 3 it is 8 times as thick, 2 4 then 16 times as thick. The pattern continues, so after 30 folds its thickness in inches is

The greatest shortcoming of the human race is our inability to understand the exponential function. We must realize that growth is but an adolescent phase of life which stops when physical maturity is reached. If growth continues in the period of ma turity, it is called obesity or cancer. Prescribing [sustained] growth as the cure for the energy crisis has all the logic of prescribing increasing quanti ties of food as a remedy for obesity. Sustainable growth [with finite re sources] is an oxymoron.

Other notable quotes from scientists and econo mists addressing growth and nature include: Facts do not cease to exist because they are ignored, Aldous Huxley Humans fooling other humans is quite com mon. When humans are dealing with Nature, however, honesty is imperative; for Nature cannot be fooled ,

Note:

so the thickness in inches is

Let us translate the 10 7 inches into a larger unit. Using miles, as our larger unit, we find

Virginia Mathematics Teacher vol. 48, no. 1

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