This mini-project shall help students to understand that exponentially growing quantities will grow beyond any practical limit, even if the rate of growth does not look particularly alarming. This is demonstrated using the current rate of growth of the earth’s population. After introducing the notion of doubling time and a reflection about the properties of exponential growth, the question "When will mankind have populated the entire universe?" is posed, whereby it is understood that its growth rate remains constant. The surprising answer shall stimulate students to think about properties of dynamical systems of this type, as well as about the evolution of real systems that show exponential (or approximately exponential) growth. The computations necessary shall be performed by the students themselves.
The material provided consists of a lesson plan, 7 worksheets for students and a summary of the solutions the students should achieve.
The mini-project is appropriate for a little more than one hour of work in small groups (preferably teams of 3 students). The worksheets shall be handed out to the groups one after another.
Prerequisites are either some familiarity with the notion of logarithm (and the ability to solve an equation of the form with respect to ) or the use of a scientific calculator allowing to compute arbitrary powers .
The worksheets are designed so that students can record their results on them. If they are re-collected in the end, the teachers get an idea of which pieces of content were understood how well.
The file ExponentialGrowth.zip contains
- a lesson plan
- 7 worksheets for students
- and the solutions of the problems as the students should achieve them.
As background information on exponential functions and the logarithm, a web page such as http://www.themathpage.com/aprecalc/logarithmic-exponential-functions.htm can be recommended to the students.