A student strained her knee in an intramural volleyball game, and her doctor has prescribed an anti-inflammatory drug to reduce the swelling. She is to take two 220-milligram tablets every 8 hours for 10 days. Her kidneys filter 60% of this drug from her body every 8 hours. How much of the drug is in her system after 24 hours?

This task can be approached in many different ways using numeric or algebraic methods, by hand, with calculators or with computers. Regardless of their approach, mathematically literate students will need to be familiar with the mathematical structure in this task. Whether the situation involves understanding effective drug dosages, population growth, bank accounts or loans, amortization, heating or cooling, filtering pollution from lakes and streams, models of learning and forgetting, or models of the economy, students should be familiar with some of the many situations which give rise to iterative processes—repeated processes in which future levels are determined by present levels. The importance of technology in dealing with tasks of this type is worth noting. Students should be able to think and work with iterative processes, and also should be sufficiently comfortable with technology to expect, as a matter of course, to consider long-term behavior and trends. All high school graduates need to be comfortable with both the structure of such tasks and the technological tools for their investigation.

Iterative models illustrate a mathematical tool that has become increasingly important in recent years: difference equations, the discrete-time analogs of differential equations. In the past, students did not study difference equations until well into their undergraduate or graduate courses. But with the aid of a spreadsheet or graphing calculator, students can handle such iterative problems with relative ease, provided that they can make the translation from a problem statement to an equation describing the iterative process, also called a recursion equation.

One approach is to create a table of values relating the number of 8-hour periods to the amount of the drug remaining in the student's system. To do this, some assumptions must be made about how quickly the drug gets into her system after it is taken. The simplest model assumes that the drug is active immediately after the medication is taken.

In this task, the information given is the rate at which the drug is eliminated by the kidney, whereas the focus of the tasks is on how much remains. The fundamental unit of time is an 8-hour period.

If the student takes 440 mg of the anti-inflammatory drug, after 8 hours her kidneys have removed 60% of the 440 mg, leaving 40% of the dose in her system. After she has taken her second 440-mg dose, the total amount of drug in her system in mg is then: (.4)(440) + 440 = 616. After 16 hours, she has 40% of the total for 8 hours, plus another 440. After 24 hours, she has 40% of the total for 16 hours, plus another 440. These calculations are summarized in Table 1.

One way to generalize this method is to use Table 1 to see patterns. First, let A_n denote the amount of drug in the system after dose n. Then the "drug remaining" column may be represented as in Table 2. The relationship in Table 2 between the "drug remaining"