An interactive graphical analysis provides a visual interpretation of the results.
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 points graphed are (n,
A(n)), with A(n) representing the amount of medicine
in the body immediately after taking the nth dose of medicine.
Placing the cursor over a
data point will display the value of A(n) for that point.
The horizontal scroll bar
at the bottom of the applet will allow you to observe the graph for larger values
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.
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
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
The non-recursive 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
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.
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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