Illuminations: Trout Pond

Trout Pond

Symbolic 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 and analyze a changing fish population
  • Use equations to investigate the effect of varying parameters on the long-term population

Materials
 
Computer and Internet connection

Instructional Plan
In the previous parts of this unit, students carried out graphical and numerical analyses of the trout population problem. They also worked with some equations. To further their understanding, students will find additional equations or formulas in this lesson.

 

Reasoning About the Recursive Formula

The equation A(n) = 0.8 × A(n - 1) + 1000 models the trout population problem recursively. (This type of equation is recursive because it involves using the previous population to determine the present population.) Allow students to answer the following questions in regards to this equation:

  • How can this equation be used to determine the long-term population? Use this hint if you need it.

  • Explain why solving x = 0.8x + 1000 will give the long-term population. How is this seen in the cobweb graph that was generated in the Graphical Analysis lesson? Use this hint if you need it.

 

Finding an Explicit Formula

Try to find another equation that represents this situation - an equation that has this form:

A(n) = "some expression involving n but not A(n-1)"

Use this sequence of hints if you need it.

Such an equation is sometimes called an "explicit formula," because it gives the population explicitly in terms of n. Also, it is sometimes called a "function formula," because it gives the population as a function of n.

Questions for Students
 
Use the explicit formula to find the population after 10 years. Explain how this method is different from using the recursive formula to find the population after 10 years.
[The explicit formula is A(n)] = -2000 × 0.8n + 5000. For n = 10, the value of this formula is -2000 × 0.810 + 5000 ≈ 4785.]

Use the explicit formula to find the long-term population. Explain your method and compare to the result you got in previous lessons.

[As n gets very large, the value of 0.8n will get very close to 0. Therefore, the value of the formula will get closer and closer to 5000. This result is similar to the result obtained through numeric and graphical methods.]

Consider the general form of the recursive formula in this situation:

A(n) = r × A(n - 1) + b; 0 < r < 1.

What is the explicit formula for A(n) in this general case?
[The explicit form above can be used to find a general solution.]

Reason about the general explicit formula above to find a general formula for the long-term population in situations like this. Use this general formula to find the long-term population in the original trout population problem. Use this hint if you need it.

As n gets very large, the value of rn will get closer and closer to zero. Consequently, the value of the explicit formula will approach b / (1 - r).

Assessment Options
 
  1. Have students complete a journal entry summarizing how tables, graphs, and equations can be used to analyze the changing trout population.

Extensions
 
  1. Allow students to perform a web search and find other examples of patterns that can be represented with recursive formulas. Some possibilities include the Fibonacci sequence and the Hailstone problem. Even factorials (e.g., 5! = 5 × 4 × 3 × 2 × 1) can be written recursively: n! = n × (n - 1)!.

NCTM Standards and Expectations
 
Algebra 9-12
  1. Use symbolic algebra to represent and explain mathematical relationships.
  2. Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.

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

 Activities
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