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This three-part
example illustrates the use of iteration, recursion, and algebra to model
and analyze the changing amount of medicine in an athlete's body. This
example is adapted from High
School Mathematics at Work, a
publication from the National Research Council (1998, p. 80). These activities
allow high school students to study modeling in greater depth, as described
in the Algebra Standard.
In the first part, Modeling the Situation, an interactive environment
is used to become familiar with the parameters involved and the range
of results that can be obtained. In the second part, Long-Term Effect,
the interactive environment is used to investigate how changing parameter
values affects the stabilization level of medicine in the body. In this
part, Graphing the Situation, an interactive graphical analysis provides
a visual interpretation of the results. Through multiple representations
of a common concept, better insight into, and a deeper understanding of,
the problem situation can be achieved.
So far you have investigated this situation primarily numerically, by looking for patterns in the values of A(n) generated by the interactive figure. To get a better understanding of the patterns and why they are occurring, it is helpful to do a graphical analysis.
[How to Use the Interactive Figure]
In this example, students have used multiple representations to analyze a real-world situation. They have used equations, tables, and graphs. By analyzing the problem using all three representations and seeing the connections among the representations, students develop a richer understanding of the problem, its solution, and the important mathematics involved.
This example also illustrates the use and power of recursion. A recursive point of view is used to generate the equations and tables. This approach makes this problem accessible to more students. The equation NEXT = 0.4 NOW + 440 (start at 440), described in the Algebra Standard, is easy for students to generate and understand. This leads naturally to the more formal equation A(n+1) = 0.4 A(n) + 440, A(0) = 440. Using these equations, spreadsheet tables and graphs can be generated and analyzed.
The nonrecursive equation (called the explicit formula or closed-form equation) for this situation is A(n) = –293.3333(0.4)n + 733.3333. This is a more difficult equation for students to generate and work with. In fact, students are able to use this approach only when and if they get to more advanced high school mathematics. In contrast, the recursive approach can be undertaken early in the high school years. At an appropriate time, the closed-form equation should also be brought into the analysis of problems like this.
Even though students may not be able to write an explicit formula for A(n), they should realize that A is a function of n. The graph displays the fact that A(n) has a horizontal asymptote at 733.33. If the initial dose is greater than 733.33, then A(n) decreases to the asymptote, and if the initial dose is less than 733.33, then the function increases to the asymptote.
Finally, problems like this are important to study and teach for several reasons. They provide a rich environment in which to use important processes of mathematics. This example helps students develop skill in problem solving, mathematical modeling, communication, reasoning, finding connections, and using multiple representations. Such problems also provide experience with important mathematics content. The basic equation used in this example, expressed in several different formats, is equivalent to A(n) = r • A(n – 1) + b. If r = 1, then this equation represents arithmetic sequences and linear change. If b = 0, then this equation represents geometric sequences and exponential change. In applications, equations like this can be used to model and analyze many situations that involve sequential change, like the growth of money in an investment program, year-to-year population growth, or daily change in the chlorine concentration in a swimming pool.
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Take
Time to Reflect
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National Research Council.
High School Mathematics at Work: Essays and Examples
for the Education of All Students. Washington, D.C.: National Academy
Press, 1998.
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