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.
Trout Pond Exploration Activity Sheet
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.
Contemporary Mathematics in Context: A Unified Approach, from the Core-Plus Mathematics Project, Course 3, Unit 7. Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Harold L. Schoen, Gail Burrill, Eric W. Hart, and Ann E. Watkins, with Mary Jo Messenger and Beth Ritsema. Glencoe/McGraw-Hill, 1999.