Title
Between Burning Eyes and Slimy Bottoms (Jo Ann Ellerbrock)
Goals
(1) Students will explore using recursion to model real world situations.
(2) Students will explore writing a closed form for mixed recurrence relations
(3) Students will communicate their findings.
Abstract
This activity, set in the context of maintaining an appropriate amount of chlorine in a pool, focuses on how differing amounts produce different effects. Students are asked to find the amount of chlorine needed on a daily basis to maintain the optimum concentration in a pool. This activity provides a basis for introducing using recurrence equations to model more complex drug dosage problems. Students are then asked to write a "closed form" equation as well.
Problem Statement
Have you ever emerged from your community swimming pool with your eyes burning? Have you ever seen a pool that was green with slime and mold? Where does it come from? This activity will allow students to explore the complexities of maintaining just the right amount of chorine in a pool.
Instructor Suggestions
(1) This problem provides an excellent opportunity for students to demonstrate the "four faces of a function."
(2) A mixed recurrence relation is one in which both multiplication by a constant and addition of a constant is required. In general, the closed form solution for the mixed recurrence relation
(3) Students working in small groups can use the guess and check method to determine the appropriate interval.
(4) Discuss with students whether their model is appropriate for continuous or discrete data and what this means in terms of the amount of chlorine that should be added.
(5) For more advanced students, this problem lends itself well to discussing a limiting value.
Materials
"Between Burning Eyes and Slimy Bottoms" Activity Sheet, Pencil, Paper, Graphing Calculators or Computer with Spreadsheet software.
Time
Students should be able to complete this activity at home after working on it in class for one period.
Mathematics Concepts
Discrete Mathematics Concepts
Recursion, Mixed Recursion, Closed Form
Related Mathematics Concepts
Arithmetic Sequences and Series, Geometric Sequences and Series, Percents, Inequalities.
NCTM Standards Addressed
Problem Solving, Reasoning, Algebra, Functions, Discrete Mathematics
Colorado Model Content Standards Addressed
Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)
Curriculum Integration
This activity would be appropriate for an algebra two or pre-calculus class at the end of a unit on arithmetic and algebraic sequences.
Further Investigation
See the Fall, 1992, Consortium for a more complicated model on drug dosage.
Variations/Comments
This activity was developed from information found in the Modeler's Corner of the Consortium. It is based on a talk given at the 1991 NCTM Convention by Joan Reinthaler.
References/Resources
Crisler, N., Fisher, P., & Froelich, G. (1994). Discrete mathematics through applications. New York: W.H. Freeman and Company
Kenney, M.J., & Hirsch, C.R. (Eds.). (1991). Discrete mathematics across the curriculum, K-12. Reston, VA: National Council of Teachers of Mathematics.
Cozzens, M.B. (1991). Consortium, Fall, 1991, p. (6)