| Problem | #1 | #2 | #3 | #4 | #5 | #6 | Total |
|---|---|---|---|---|---|---|---|
| Max grade | 5 | 5 | 5 | 5 | 5 | 5 | 30 |
| Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| Mean grade | 4.75 | 2.32 | 4.61 | 3.25 | 4.34 | 3.87 | 23.14 |
| Median grade | 5 | 3 | 5 | 5 | 5 | 5 | 25 |
79 students took this quiz. It was the first "controlled" (limited
time and help) test situation for these sections. Numerical grades
will be retained for use in computing the final letter grade in the
course. Each of the 6 problems was worth 5 points. The correct answer
earned 1 point, and 4 points could be earned by the requested
"supporting evidence". Each problem was worth 5 points.
The exam mostly tested algebraic skills which are essential for this
course and in many other situations which use mathematics. Most of the
problems were copied from textbook homework problems.
Students whose scores were less than 20 should be very concerned about their likely success in this course. Such students should ideally perceive their results as an alarm. Students should be spending at least 8 hours a week outside of class working on course material, and students with low scores should work on every suggested problem in the course syllabus. The course is relentlessly cumulative. "Catching up later" is practically impossible and students who think this are deceiving themselves.
Problem 1 (5 points)
Factor the top and cancel. Bad insertion or deletion of a minus sign
in the algebra (after correct factoring!) and the answer earned a
grade of 2 out 5: 0 for the answer and 2 out 4 for the supporting
evidence.
Problem 2 (5 points)
I gave 3 points for substituting and getting the answer 0 if "plugging
in" was presented clearly. I gave 5 points if in addition the word
"continuous" was used correctly. I wanted some explicit reason here
corresponding to the work needed to earn credit in other parts. If you
got 3 out of 5 (which 50 out of 79 students earned in this problem!)
maybe you could comfort yourself with the idea that, well, 28 out of
30 is a "good" grade.
Problem 3 (5 points)
2 points for factoring correctly but mistakes after that earned
nothing more.
Problem 4 (5 points)
I looked for multiplication by "the conjugate". A sign error in the
manipulation and answer was penalized 2 points (1 for the answer and 1
for an algebra error). Bad algebra with a correct answer will earn 1
point alone for the answer. I wanted correct supporting
evidence!
Problem 5 (5 points)
I looked for the answers alone. The first answer (correctly given!)
earned 1 point, and each of the next two earned 2 points each.
Problem 6 (5 points)
I looked for supporting algebraic evidence. Again, a sign error in the
manipulation and the answer was penalized 2 points (1 for the answer
and 1 for the algebra error).
| Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | #9 | Total |
|---|---|---|---|---|---|---|---|---|---|---|
| Max grade | 12 | 6 | 14 | 12 | 12 | 8 | 12 | 14 | 12 | 99 | Min grade | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 18 | Mean grade | 9.96 | 3.95 | 8.78 | 7.09 | 8.73 | 1.73 | 7.5 | 8.88 | 5.38 | 62 | Median grade | 11 | 5 | 9.5 | 6.5 | 10 | 1.5 | 10 | 9 | 5 | 65 |
Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this course.
Here are approximate letter grade assignments for this exam:Students with grades of 50 or less These students should carefully consider dropping this course. Students with very low grades in the first exam almost always cannot catch up with the material in the course. The first segment of the course spent substantial time reviewing ideas assumed to be known already. Almost all students who got D's or F's on this exam showed unsatisfactory understanding of background material and there will be no course time devoted to further review. Working on other courses where there is more chance of success may be an overall more productive strategy. Please discuss this with your academic advisors, and consider this recommendation seriously yourself.
| Letter equivalent | A | B+ | B | C+ | C | D | F |
|---|---|---|---|---|---|---|---|
| Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |
Problem 1 (12 points)
Each part is worth 4 points. Full credit is earned by the answer
alone. Minor errors (such as labeling 5·3 as 12, for example,
or a sign error) will be penalized 1 point. Errors in the Product or
Quotient or Chain Rule lose 2 points.
Exception In c), an omitted multiplier of 3 (so (sec(3x))2·(cos(2x))2 beginning
the quoted answer instead of (sec(3x))23·(cos(2x))2), although a
Chain Rule error, will only lose 1 point.
Problem 2 (6 points)
2 point for the correct formula for the first derivative of sin(g(x)),
3 points for the correct formula for the second derivative of
sin(g(x)), and 1 point for the correct answer, which must include
evaluations of the trig functions. Again, 1 point off for a minor
error and 2 points for an error in the Product or Chain Rule.
Problem 3 (14 points)
a) (10 points)
2 points for substitution of x+h into f (or something equivalent). If
somehow the student writes 5-x+h instead of 5-(x+h) the student loses
these 2 points, but they can be the only 2 points lost of the 10
if everything else is correctly done.
2 points for correctly instantiating [f(x+h)-f(x)]/h.
3 points for multiplying top and bottom by the conjugate and combining
the fractions correctly.
2 points for cancelling the h's.
1 point for the answer, which must
be the correct answer for the student's work.
2 points off if there is no mention of limh-->0 in an
otherwise correct solution.
The answer alone or the answer
obtained algorithmically earns no credit.
b) (4 points) The value of the derivative at 4 earns 1 point. A
correct tangent line equation earns 1 point. As for the graph: a
correct curve (the top half of a parabola opening to the left!) earns
1 point and a correct tangent line earns 1 point. The line must be
tangent to the curve, or 1 point will be lost.
Problem 4 (12 points)
The correct derivative earns 4 points. Deduct points for
differentiation errors here, but try to "read with" the student's
result, if it is not trivial. For example, a student who begins by
asserting that the derivative is 6x would (and should!) earn no points
for the whole problem. Simplification to yield a correct equation with
no quotient (so knowing that TOP=0 is enough) which finds where
f´(x)=0 earns 4 points (2 points for f´(x)=0 and 2 more
points for declaring that TOP=0). Deduct points for algebra errors
here. The two coordinates correctly written earn 4 points. Some of
that is deducted if incorrect "simplifications" are made.
Problem 5 (12 points)
3 points for s in terms of x (only 1 of 3 if there's some error in
Pythagoras); 3 points for the area of the square; 3 points for the
area of the triangle; 1 point for a correct formula for f(x); 2 points
for a correct domain (with or without either or both endpoints).
Only 3 points are earned for a formula in terms of s and x.
Problem 6 (8 points)
The word continuous in relevant context earns 2 points. The
phrase Intermediate Value Theorem in relevant context earns 2
points. This means that just scrawling (scrawl: "write in a
hurried untidy way") the word "continuous" or the phrase "Intermediate
Value Theorem" will not necessarily earn credit. 1 point is earned for
a function value which is positive, and 1 point is earned for a
function value which is negative. Some mention of the "size" of values
of cosine should be made for 1 point. Finally, an explicit correct
answer supported by the student's work earns 1 point.
Comment The problem statement contains this sentence: "Give
careful evidence supporting your assertion, including specific
results from Math 151." This instruction was interpreted strictly.
Problem 7 (12 points)
a) (7 points) Restricting attention to 0 and 2 is worth 1 point. 2
points for getting a constraint at 0. 2 points for a similar
constraint at 2. 2 points for solving and reporting correct values of
A and B.
b) (5 points) The graph should be continuous (1 point). There should
be a straight line segment from (2,0) to (2,1/2) (2 points). There
should be decreasing behavior as one moves to the left on the curve,
but not lower than 1 (1 point). There should be decreasing behavior as
one moves to the right on the curve, but not lower than 0 (1 point).
Problem 8 (14 points)
a) (10 points) The graph should have three discontinuities (1 point
each), from left to right: one removable, one with a jump, and one
with asymptotic behavior. 3 points for correct sign behavior in each
interval (1 more point for f´(x)=0 for x>4). 2 points for
correct {in|de}creasing behavior. 1 point for the asymptotic behavior.
While I think there should be a vertical asymptote at only one side of
one discontinuity (as shown in my solution to this problem), I believe
that a reasonable person might disagree with this, so I will not
penalize answers which show another asymptote to the left of -1. I do
not think that the behavior near the middle discontinuity of
f´ is "asymptotic" -- that is, the left and right limits near the
value are finite and have different signs.
b) (2 point) The correct answers.
c) (2 points) the correct answers.
Problem 9 (10 points)
a) (4 points) A correct answer earns 1 point. 3 points for combining
the fractions, getting into simple fraction form, simplifying and
plugging in.
b) A correct answer earns 1 point. Some analysis (or even just
acknowledgment!) of absolute value earns 2 points, and then algebra
gets the third point.
c) (2 points) A correct answer earns 1 point. Some support (the word
"continuous"?) earns a second point.
| Problem | #1 | #2 | #3 | #4 | #5 | #6 | #7 | #8 | Total |
|---|---|---|---|---|---|---|---|---|---|
| Max grade | 12 | 12 | 12 | 20 | 16 | 10 | 10 | 8 | 93 | Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | Mean grade | 8.49 | 3.52 | 8.72 | 8.46 | 10.85 | 4.83 | 5.24 | 1.75 | 51.86 | Median grade | 10 | 1 | 10 | 9 | 13 | 4 | 6 | 1 | 50 |
Numerical grades will be retained for use in computing the final letter grade in the course. Students with grades of D or F on this exam should be very concerned about their likely success in this course.
General comments Attendance
Students who answered at least 15 of the 18 QotD given before the second exam scored markedly higher (both mean and median) on the second exam than students who answered fewer than 15 of the QotD.Comparison
The letter grades of 41 students went down from the first exam to the second exam (there really is much material in this course, even for students who may have studied calculus before). The letter grades of 21 students stayed the same (12 of those, however, were F/F combinations!). The letter grades of 9 students improved.
Here are approximate letter grade assignments for this exam (which are slightly different from the first exam):
| Letter equivalent | A | B+ | B | C+ | C | D | F |
|---|---|---|---|---|---|---|---|
| Range | [80,100] | [75,79] | [65,74] | [60,64] | [50,59] | [45,49] | [0,44] |
Problem 1 (12 points)
a) (1 point) Students should demonstrate knowledge that ln(1)=0 and
show that the equation is correct at (-1,0).
b) (9 points) Differentiating the equation earns 6 points: chain rule
for ln (3 points), product rule for xy (2 points), all else (1
point). Slope at (-1,0) earns 1 point, going through (-1,0) earns 1
point, and a correct equation for the line earns 1 point.
c) (2 points) The line should go through (-1,0) (1 point) and seem to
be tangent (1 point).
Problem 2 (12 points)
2 points for labeling the variables and getting a function for the
area; 3 points for differentiating the function correctly; 3 points
for getting the roots when the derivative equals 0 (1 of these for
setting the derivative equal to 0) and 1 point for rejecting one of
the roots; 2 points for explaining why the max has been obtained; 1
point for the correct answer.
Computation of dy/dx if no connection with the problem is given earns
0 points.
Comment Many students assumed that three-sided shapes were
triangles and somehow the appearance of several of these in the same
picture implied that the "triangles" were similar. First, if only two of the sides are line segments, the
result is NOT a triangle. Second, the
non-triangular shapes are NOT similar:
there is no reason to assume this is true.
Problem 3 (12 points)
a) (6 points) 1 point for an area formula, and 2 points for
differentiating it. 1 point each for the answers.
b) (6 points) 1 point for a diagonal formula, and 2 points for
differentiating it. 1 point each for the answers.
Problem 4 (20 points)
a) (6 points) 1 point for each of the limits. 4 points for
justification (2 of these would be earned by just mentioning powers,
with no detailed explanation carried out).
b) (5 points) 2 point for the answers. 3 points for justification.
There must be use of the derivative.
c) (5 points) 1 point for the answer. 4 points for the justification,
which should mention both {in|de}creasing to infinity, and some idea
of continuity so that the line y=100 does intersect y=f(x). The
student should give some idea why there are at least two roots of the
equation given, and then some further idea why there are exactly two
roots.
Comment I think this was the most subtle part of the
exam. Although a number of students gave the correct answer, good
supporting reasoning was given by only a few people. What's needed is
a way of combining the limit statements and the derivative statements
and some reason why the curve actually intersects the line.
d) (4 points) 2 points for the labeled inflection points on the graph
(that's requested twice, once underlined), and 2 points for the
concavity intervals (1 point if one interval is given correctly, and
putting in/taking out endpoints doesn't matter here).
Problem 5 (16 points)
a) (9 points) 4 points for each vertical pair (x= and type) in the
table. Both entries must be correct to earn the point. 5 points for
convincing explanation (something resembling the First Derivative Test
earns 2 of these and remarks about the specific signs of this
derivative will earn the other 3).
b) (7 points) Each critical point correctly sketched earns 1
point. Going through (0,0) earns 1 point. A smooth curve defined on
[-1,4] which resembles a good answer will earn 2 points.
Problem 6 (10 points)
a) (5 points) Correct locations of A1 and A2
with labels each 1 point each. A1 should be to the left of
A and A2 should be to the left of A1, but to the
right of 0. The assisting tangent line segments in each case earn
another 1 point each. The limit earns 1 point.
b) (5 points) Correct locations of B1 and B2
with labels each 1 point each. B1 should be to the left of
B and B2 should be to the right of B1 but to the
left of -1. The assisting tangent line segments in each case earn
another 1 point each. The limit earns 1 point.
Problem 7 (10 points)
a) (4 points) A correct answer earns 1 point. 3 points for checking
L'H eligibility and differentiating (done twice). Be sure the second
derivative on the "top" function should be correct or 1 point is
lost.
b) A correct answer earns 1 point. 3 points for taking ln,
rearranging, checking for L'H eligibility, differentiating, and exp
back. Other arrangements (such as beginning by writing the function as
eln(function)) can also be correct.
c) (2 points) A correct answer earns 1 point. Some support (the word
"continuous"?) earns a second point.
Problem 8 (8 points)
The initial description of the model earns 3 points: a function and a
domain. "Distance=Rate·Time" is not a description of a
valid model. It is a recital of a formula which may or may not apply
to this problem. The formula is valid for motion in
a straight line with constant velocity and the problem statement does
NOT state these conditions and therefore
the formula cannot be used. The specific result needed is
the Mean Value Theorem, and, in context, this will earn 2 points. Use
of this result to get correct information earns the final 3 points, 1
of which is for a correct estimate. That 1 point can be earned with no
supporting evidence.
Comment One student estimated (version A)
that the distance between the towns ranged from 3600 miles (wider than
the continental U.S.) to 5400 miles (more than the distance from New
Jersey to Hawaii). Could this really be done by driving an hour and a
half at a speed between 40 and 60 mph? On the other hand, another
student asserted that the distance between the towns ranged from 6 to
9 miles, which also seems remarkable for a trip of an hour and a half
with speed at least 40 mph.
The final exam and course grades
Initial grading
Grading of the final exam was done on Tuesday, December 18, the day
following the exam. Each lecturer and recitation instructor graded the
same problems for all students. Certainly I should keep confidential
which problems I graded, so all I will say is that I don't want to
think more about trains leaving St. Louis, and I don't want to
consider, at least for a while, piecewise defined functions which are
to be both continuous and differentiable.
Checking and regrading
I spent the afternoon on the day after grading going through all of
"our" exams in detail. Since few students will see their final exams
(see more about this below) I felt a special responsibility to check
the details. I found no mistakes in addition. I did discover
many (well, maybe 6 or 7, so maybe not many!) discrepancies of
1 or 2 points in grading. That's to be expected in a big bunch of
exams. I found two further errors, one unjustifiably penalizing a
student 4 points, and another 11 point error. I'd feel very good about
my checking all of the exams, except that the 4 point error was in a
problem I graded, in my handwriting! What's this say about my own
accuracy and concentration?
The results, overall and locally
The exam was taken by 617 students, and the results, on a 200 point
scale, ranged from 0 (indeed) to 200 (yes). The overall median was
127. In our lecture, 65 students took the final exam. For this
population, the grades ranged from 26 to 198, the median was 140, and
the mean was 135.35. The letter distribution of grades on the final
exam was definitely skewed higher for the students in sections 7, 8,
and 9 than for the general population. The results showed that many
students prepared diligently for the final exam. There was only one
problem where "our" student solutions showed systematic weakness: the
optimization question. When/if I teach 151 again, I'll try to improve
my discussion of such problems (especially since the same weakness was
displayed in the second exam). Final exam grades were converted from
numbers to letters using the "bins" displayed below. The alert reader
will note that these assignments are quite similar (multiply by 2) to
those used earlier for the first and second exams.
| Letter equivalent | A | B+ | B | C+ | C | D | F |
|---|---|---|---|---|---|---|---|
| Range | [170,200] | [160,169] | [145,159] | [140,144] | [120,139] | [110,119] | [0,109] |
Use of these grades
The final exam letter grades are given to allow individual lecturers,
who are responsible for reporting course grades, to align the
performance of their specific groups of students with the overall
performance of students in the course. Random chance may give one
lecturer a group of students with better or worse performance than the
overall population, and, indeed, specific scheduling requirements may
force specific subpopulations with different math preparation to
enroll in some lectures. The common final exam and grading are part of
the faculty's effort to assign appropriate course grades, with equal
grades for equal achievement.
Course grading
The information I had included the following: grades for the three
exams (two during the semester and the final), attendance information
(for the lecture, from the QotD, and for the recitation, as reported
by the peer mentors), textbook homework scores (reported by the peer
mentors), workshop grades (reported by Mr. Yin, although I did some of
the grading also), and quiz grades (mostly reported by Mr. Yin,
although I had the results for the quiz on limits). This is a large
mess of numbers but my electronic friends handled the arithmetic with
little strain.
I computed a number for each student weighted as previously described. The ideal sum
added up to 593. The numbers obtained ranged from 164.106 to
555.554. I then summed the breakpoints for the three exams and
multiplied them by 1.4825 (that's 593/400 for those of you still
awake). Each student's total then corresponded to a tentative letter
grade. I examined each student's record to make sure that this process
had not distorted or misrepresented student achievement. I adjusted
some letter grades which were near boundaries. Almost all of these
adjustments were up. A few grades were pushed down, because
"students whose exam grades are all near bare passing or are failing
may fail the course in spite of numerical averages: students
must show that they can do adequate work connected with this
course independently and verifiably." I entered the course grades into
the Registrar's computer system on Thursday, December 20. I hope
students will be able to see them soon.
If you have questions ...
Rutgers requests that I retain the final exams. Students may ask to
look at their exams and check the exam grading. These students should
send me e-mail so that a mutually satisfactory meeting time can be
arranged. Students may also ask how their course grades were
determined using the process I described. Probably e-mail will be
sufficient to handle most such inquiries.
Thank youWarning Students who officially passed the course are allowed to go on to Math 152. Students who received consistently poor exam grades are likely to do badly in Math 152 without extraordinary effort. Math 152 is quite technical (maybe even dull!) and has much more computation than 151.
Maintained by greenfie@math.rutgers.edu and last modified 12/20/2007.