An interactive environment is used to become familiar with the
parameters involved and the range of results that can be obtained.
The interactive figure calculates
the amount of medicine in a person's body immediately after taking a dose. In
this scenario, the individual takes an initial dose of medicine followed by
recurring doses, taken faithfully at fixed intervals of time. The interactive
figure allows the initial dose to be different from the recurring doses.
The simulation requires three
the amount of medicine in the body immediately after taking the nth
recurring dose of medicine. So A(0) = initial dose, A(1) = amount
of medicine in body after first recurring dose, A(2) = amount
of medicine in body after second recurring dose, and so on.
This interactive figure calculates
the amount of drug in the system just after taking a dose of medicine.
If the amount of medicine in the body just before taking the nth
dose is desired, subtract the amount of the recurring dose (or initial dose
for n = 0) from the value calculated for A(n).
The interactive figure in this example illustrates
calculation features that can be implemented in spreadsheets or graphing
calculators. Spreadsheets or calculators with iterative capabilities
can be very useful for investigating and understanding change—whether it
is due to growth or to decay. In computer and calculator spreadsheet
programs, students have a powerful tool that permits them to calculate
the results of multiple dynamic events quickly and accurately. The ease
of calculation frees students to focus on the effect of changing one or
more of the problem parameters. In this example, an athlete takes a
constant dose of medicine at regular intervals. Using a calculator or a
spreadsheet, students can determine the effect when changes are made in
the initial dose, the recurring dose, or the percent of medicine
eliminated from the body.Obtaining explicit formulas that
capture such effects is often quite difficult and in some cases,
impossible. In order to have had the experience that will lead them to
an appropriate closed-form equation with which to model such situations,
students generally must be at a fairly high level of mathematics. A
recursive approach, especially when supported by a calculator or an
electronic spreadsheet, gives students access to interesting problems
such as this earlier in their schooling. It also informally introduces
them to an important mathematical concept—limit. In this initial
phase of the investigation, students should recognize that the level of
medicine in the body initially rises rapidly but with time increases
less rapidly. Although one might question whether the accuracy of the
recorded answer affects this observation, it appears that the level
eventually stabilizes, so that after about seventeen periods, the value
seems no longer to change. In other words, the athlete's body is
eliminating the same amount of medication as she is taking. This
observation can be mathematically verified by showing that (733 1/3)0.6 =
440. As students play with the various parameters in the
problem, they might make a number of observations. For example, the
initial dose has no long-term effect; the level of medication still
stabilizes at the same value, regardless of the value of the initial
dose. Changing the recurring dose does change the level at which it
stabilizes. This line of discussion is extended in the next part of the
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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