Illuminations: Trout Pond

Trout Pond

Numerical Analysis

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.

Learning Objectives
 

Students will:

  • Use iteration, recursion and algebra to model the Trout Pond exploration
  • Use graphs, equations, and tables to investigate the effect of varying parameters on the long-term population

Materials
 
Computer with Internet connection
Trout Pond Spreadsheet (Excel) or Illuminations Spreadsheet Application

Instructional Plan
In the Understanding the Problem lesson, students investigated what happens to the population in a trout pond under particular circumstances. In particular, they considered 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.
  • 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.

At the end of the previous lesson, students were asked to derive a NOW/NEXT equation. The trout pond exploration can be modeled using the following relationship:

NEXT = 0.8 × NOW + 1000
(start at NOW = 3000)

Have students answer the following questions:

  1. Explain what the words NOW and NEXT represent in this equation.
    [NOW represents the number of trout currently in the pond. NEXT represents the number of trout that will be in the pond next year.]
  2. Where does the factor of 0.8 come from?
    [The population decreases by 20% each year. Therefore, 100% - 20% = 80% of the trout remain at the end of each year, and 80% = 0.8.]
  3. Explain why the equation represents this situation.
    [The number of trout in the pond NEXT year is equal to the number NOW in the pond times the growth factor (0.8) plus the restocking amount (1000).]

Encourage students to provide a more formal representation of this relationship. That is, instead of using NOW and NEXT, have them use functional notation. Assign a variable to represent the number of the year, and create a recursive equation to describe the relationship.

A(n + 1) = 0.8 × A(n) + 1000
A(0) = 3000

If students are not able to generate this equation on their own, then present it to the class, but be sure to have them consider how the recursive equation relates to the NOW/NEXT equation above. In particular, have them answer the following questions:

  1. Explain what n, A(n), and A(n + 1) represent in this equation.
    [The variable n represents the number of the year.
    A(n) is the number of fish currently in the pond, and
    A(n + 1) is the number of fish that will be in the pond next year.]
  2. Use A(n) and A(n - 1) to write another equation that also represents this situation. This equation should begin
    A(n) = ...
    Explain why this new equation and the A(n + 1) = ... equation above are both accurate representations of this situation.
    [The recursive equation
    A(n) = 0.8 × A(n - 1) + 1000, with A(0) = 3000, is also an accurate representation of the situation. The relationship between A(n) and A(n + 1) is equivalent to the relationship between A(n - 1) and A(n), becuase both pairs represent expressions for consecutive years.]

To investigate this situation numerically, students should create a spreadsheet. Students can use the Illuminations Spreadsheet Application, or they can use the Trout Pond Spreadsheet (Excel).

Extend the spreadsheet so that the population is calculated for 50 years. (Refer to the instructions below when using the Illuminations spreadsheet application.)

Instructions
  • In the spreadsheet, click on the cell for n = 25.
  • Hold down the Shift key and use the down arrow to move to the cell where n = 50.
  • Click on the Edit button, and choose Fill Down.
  • Use the same process to extend A(n) to 50 years.

Based on their work with the spreadsheet, students should analyze the results and answer the following questions:

  1. The general equation for this spreadsheet is
    A(n) = r × A(n - 1) + b
    For this particular situation, what is the value of r? What is the value of b?
    [r = 0.8, b = 1000]
  2. Was your conjecture about the long-term population correct?
    [Answers will depend on the original conjecture made by students.]
  3. When does the trout population reach 5000? Does the mathematical model ever actually yield 5000? Explain.
    [During the first 50 years, the trout pond gets very close to but never actually reaches 5000. (You might find it interesting to restrict the numbers in the Excel spreadsheet to integers. Due to rounding, a value of 5000 will appear in year 38.) If the spreadsheet were extended further, a value of 5000 would appear in the Excel spreadsheet at year 69, due to rounding; however, a value of 5000 will not appear in the Illuminations spreadsheet application.]
  4. Both in mathematical terms and in terms of what you'd expect to happen with a trout pond, explain why this long-term population is reasonable.
    [When 20% of the trout are lost, the population is decreased to 4000. When the pond is restocked with 1000 fish, it returns to 5000. It therefore makes sense that the population would remain constant.]
  5. Does the trout population change faster around year 5 or around year 25? How can you tell?
    [It changes faster initially, because the restock amount greatly exceeds the number of fish that is lost. Later, the restock amount is nearly equal to the number that is lost.]

After investigating the original situation with tables and spreadsheets, students should answer the question, "What if?" That is, they should attempt to find out what happens when the assumptions change.

There are three key factors in this problem:

  • the initial population, A(0);
  • the annual restocking amount, b; and,
  • the annual population growth factor, r.

Ask students to think about the following three questions. Before they investigate the situation, have them make conjectures about what they think will happen.

  1. If the initial population doubles, what will happen to the long-term population?

  2. If the annual restocking amount doubles, what will happen to the long-term population?

  3. If the growth factor rate doubles, what will happen to the long-term population?

Students can use the calculator below to test their conjectures. To use the calculator, simply enter values into the boxes, then press Calculate to see the results. (Note that the growth factor is found by subtracting the annual population decrease rate from 1. For instance, if the decrease is 20% annually, then Growth Factor = 1 - 0.2 =0.8.)

If they prefer, students can also use the Illuminations Spreadsheet Application instead of this calculator.

Enter Values
Initial Population:
Restock Amount:
Growth Factor:

Were there any surprises? Did students notice any patterns?

Students should systematically investigate the effect that other changes in these three factors will have on the long-term population. Require students to keep track of the results of their investigations, and have them describe any patterns that they see.

Questions for Students
 
What effect do changing the initial population, the growth factor, and the restock amount have on the long-term population of the pond?
[The initial population affects the number of trout in the pond for the first few years, but it does not have a great effect on the long-term population. The growth factor and restock amount are far more important to what will happen in the future. The population will reach a constant value when the annual decrease rate is equal the restock amount.]

Assessment Options
 
  1. Suppose that the annual decrease rate is 20% and the initial population is 3000, as in the original situation. But now suppose that you want the long-term population to be 7500. What restocking amount should be used?

Teacher Reflection
 
  • How did technology assist with this lesson? Were students able to discover more because of technology?
  • How did the use of technology change the way students thought about the mathematics of the situation?
  • When using technology in the classroom, what procedures need to be implemented to ensure that students stay on-task?

NCTM Standards and Expectations
 
Algebra 9-12
  1. Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
  2. Understand and perform transformations such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on more-complicated symbolic expressions.
  3. Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations.

References
 
  • 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.
  
1 period   

NCTM Resources

Navigating through Algebra in 9‑12

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