## Trout Pond Population

This investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, tables, and technological tools are used to investigate the effect of varying parameters on the long-term population.

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.

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.

### Reference

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.

**Extensions**

- How do you think the population will change over time if the parameters are changed? That is, what will happen if a change is made to the initial number of fish, the rate at which the population decreases, or the number that is restocked each year?
- Move on to the next lesson,
*Numerical Analysis*.

**Question for Students**

What will happen to the population of trout in the pond? Will it increase, decrease, or level off?

**Teacher Reflection**

- Did you use the activity sheet to guide students' investigation? If so, did it provide more guidance than was necessary? If not, would using the sheet help the next time you have students explore this situation?
- How did you organize this open-ended exploration to avoid classroom management problems?

### Numerical Analysis

### Graphical Analysis

### Symbolic Analysis

### Learning Objectives

Students will:

- Use iteration, recursion and algebra to model and analyze a changing fish population.
- Use graphs, equations, tables, and technology to investigate the effect of varying parameters on the long-term population.

### NCTM Standards and Expectations

- Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.

- Use symbolic algebra to represent and explain mathematical relationships.

- Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations.

- Draw reasonable conclusions about a situation being modeled.