Quiz 8 (Take-home assignment): Writing an exam problem
Math 2300—008 / Due March 7, 2016
It takes a lot of thought to write an exam problem that tests the right concepts
while also being fair, clear, possible to grade, and an appropriate level of difficulty.
In this assignment you will write a good exam problem.
Format: You should separately address all three sections, although your problem
and solution with rubric can be on the same page. Your write-up can be in paragraph
form or bulleted points, as long as you communicate your ideas clearly. It can be
hand written, typed*, or some combination of the two. I will provide an example for
reference.
*If you wish to typeset math, you should use the program LaTeX. It requires a
small amount of programming so you can email me if you want a template.
I. (3 pts.) Topic
Describe the concept you wish to test and the specific points a student should
know.
You should address the following:
• Summarize the concept that is being tested.
• If appropriate, give some intuition for the concept.
• What are the big picture ideas and smaller details that a student should
know about this topic?
II. (3 pts.) Planning
Write a few sentences or bullet points with your thoughts on how you will test
the concept.
Things to consider:
• Is it clear what the problem is asking?
• Is it clear what a solution needs to receive full points?
• Is there more than one way to solve the problem correctly?
• Is it possible to guess the answer without understanding?
• How can you increase difficulty of the problem?
• Does the problem avoid coincidences wherever possible, so that you can
tell which numbers are coming from where?
III. (3 pts.) Problem, solution and grading rubric
Write up your problem with an ideal solution and describe how it should be
graded. You can do this by labelling certain parts of your solution with point
values or through a set of rules.
Things to consider:
• What mistakes do you expect to see? How will you grade them fairly?
• If someone makes an error early in the problem, can they still receive points
for work based on that error?
• Ideally, your rubric should be clear enough that two different people could
grade the same problem and get the same grade.
Presentation (1 pt.): Your write-up does not need to be typed and does not need to
use full sentences, but it should be organize and reflect that you put in some effort
and thought.
Topic. Your topic is assigned by last name.
• A - B: Integral test
• C - D: Limit comparison test
• G - J: Ratio test
• K - M: Alternating series test
• P - Z: Alternating series remainder test
Page 2
Sample Write-up.
Topic: The Direct Comparison Test.
(I) I want to write an exam problem that tests the Direct
Comparison
Test (DCT).
P
P
The DCT states that if you have two series
an and
bn , both with all
positive terms, then the following is true:
P
P
(i) If an ≥ bn for all n and
bn diverges, then so does
an .
P
P
(ii) If an ≤ bn for all n and
bn converges, then so does
an .
This allows us to use knowledge of the convergence or divergence of one series
to say something about the other. It makes sense because we know that if
something is at least as large as infinity then it is also infinite and vice versa.
To be able to use this test, students should be able to:
• Use intuition to guess whether the function converges or diverges, since
this will inform their choice of comparison function
• Understand which inequality is used to show convergence and vice versa
• Pick a good comparison function that satisfies the need inequality
• Confirm that both an and bn are positive
• Show that bn converges or diverges as necessary
• Draw the appropriate conclusion.
(II) I want to make sure that students use the DCT and not another test; to
do this, I could either state that you must use the DCT or write it so that
none of the other tests work out well. The sine function makes it hard to
calculate ratios, so I’ll use that in the numerator to eliminate the LCT and
ratio test. I also want to make sure that people don’t accidentally guess the
right comparison function, so I’ll make it need to be something other than
1 over a power of n. I can increase difficulty by having powers of n in the
numerator and the denominator and by having some of them be fractional,
such as inside square roots. Lastly, I want to make sure that people remember
to confirm the hypotheses, so I’ll remind them to do that in the problem
statement.
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(III) (10 pts.) Consider the series
∞
X
2n + 3n sin2 (n)
√
.
n5 + 1
n=1
Determine whether this series converges or diverges. State the names of any
test(s) that you use, and be sure to confirm all needed hypotheses.
Page 4