In this unit, students will be investigating the numeric, graphical, and symbolic representations of a recursive function. Specifically, students will explore a scenario in which a trout pond loses a portion of its population to natural causes, but the pond is restocked with fish each year.
To begin the lesson, have students think about the following situation:
Each spring, a trout pond is restocked with fish. That is, the population decreases each year due to natural causes, but at the end of each year, more fish are added. Here’s what you need to know.
- There are currently 3000 trout in the pond.
- Due to fishing, natural death, and other causes, the population decreases by 20% each year, regardless of restocking.
- At the end of each year, 1000 trout are added to the pond.
Allow students time to think about this situation. It might be advantageous for students to work in pairs to discuss their findings. When students have had ample opportunity to investigate the situation, ask the following questions:
- Do you think the population will grow without bound, level off, oscillate, or die out? Explain why you think your conjecture about long-term population is reasonable.
- Let the word NEXT represent the population next year, and NOW represent the population this year. Write an equation using NEXT and NOW that represents the
assumptions given above.
To allow students to focus solely on the mathematics of this activity, you may wish to have them use the Trout Pond Exploration activity sheet to structure their investigation.
With a chart to tally data and a list of guiding questions, this sheet will help students focus on the concept of recursion rather than on the skill of organizing data. However, if you wish to have students devise their own methods for organizing information, this activity sheet may provide too much guidance.
