Criterion C: Mathematical Process
Type II, Mathematical Modeling:
Developing a Model
Achievement level
0 The student does not define variables, parameters or constraints of the task.
1 The student defines some variables, parameters or constraints of the task.
2
The student defines variables, parameters and constraints of the task
and attempts to create a mathematical model.
3 The student correctly analyses variables, parameters and constraints of
the task to enable the formulation of a mathematical model that is
relevant to the task and consistent with the level of the course.
4 The student considers how well the model fits the data.
5 The student applies the model to other situations.
C0 Type II The student does not define variables, parameters or constraints of the task.
C1 Type II The student defines some variables, parameters or constraints of the task.
It is important to define your variables (t
represents time in hours since the first dose was given, etc.) and
state what parameters and constraints are given in or implied by the
model. Constraints are often things like domain and range restrictions (t
> 0, for instance) and parameters would include facts like,
in the drugs task, the initial dose was 10 micrograms. (The term
"parameter" is most often used to represent the coefficients to be
found in a function; in f(x) = ax2 + bx + c, the parameters are a, b, and c.) It
is possible to earn C0 by neither stating nor implying anywhere what
the variables, etc., are. Be sure to avoid this trap. And while implied
definitions might avoid C1, they're not as good as explicitly stated
ones are.
C2 Type II
The student defines variables, parameters and constraints of the task
and attempts to create a mathematical model.
This
student completely ignored the requirement of the model that it have a
rate of change proportional to the amount of the drug remaining. As you
will learn later (if you haven't already), this means that the function
must be exponential in nature. He did create a model, but it's not
relevant to the task or consistent with the level of the program, since
it ignores, or perhaps completely misinterprets, vital mathematical
information. This could be ignorance of the meaning of the rate of
change requirement or carelessness; there's no way to tell from the
work. This also happens to come from a student who is mathematically
very capable... and who did a lousy job on the assignment anyway.
C3 Type II The student correctly analyses variables, parameters and constraints of
the task to enable the formulation of a mathematical model that is
relevant to the task and consistent with the level of the course.
The
model that this student found was appropriate, and his use of
technology that followed was quite good. However, he misread the fit of
the data. His statement about fluctuations in the amount of the drug is
just wrong; the drug showed a steady decrease. While it did not fit the
data perfectly, he seems to have expected it to be dead on all the
time. That's just not the nature of modeling a real-world process.
C4 Type II The student considers how well the model fits the data.
Here,
the part of the work shown is not where the student discussed goodness
of fit. She did do that, and it was fine. This piece was selected to
show why that work did not reach level 5. You can see that there's a
problem with the direction in which she claims to shift the graph,
although the formulas she gives do in fact cause a horizontal shift to
the right. What she didn't check was whether the rate of change
requirement was met with her new function. The original model did meet
this requirement, but the new one does not, so her attempt to apply it
to the new situation is incorrect. (Also, her attempt at piecewise
notation is not quite right. That's assessed in criterion A.)
C5 Type II The student applies the model to other situations.
This
is a piece of work which shows how you can score well on one criterion
and really poorly on others. The student earned A1, B1, and D3.
However, his goodness of fit analysis and his application of the model
to the situation with additional dosages were just fine. The table you
see here is how he got the values in the graph, and he used them to
find functions with a calculator for each part. His process is
fundamentally fine. But the formulas that he comes up with should be a
single function, written piecewise, and he doesn't do a very good job
of indicating what he's up to. The reason that it's so hard to read is
that he wrote in pencil; I had to edit it with graphics software to
make it show up this well.
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