Grades in Math 251:1 & 5 in spring 2011


The first exam

The future, student performance, exam construction, and course grades

I will repeat some of these rather risky comments in class before I return the graded exams (at the end of each class meeting on Monday, March 7 [after that the exams can be picked up in my office]). The remarks are risky because, even though what I write is true, some of my comments are not particularly nice.

You are time-travelers. You will almost surely be working in the year 2050. Things will change dramatically. To have any chance of significant success in life, you will need to learn a great deal and you will probably be required to reinvent yourself many times. You should start now! Develop good habits.

Math 250 is a straightforward course. One aim of the exam was to minimize arithmetic computations. Machines will likely do such tasks and therefore you need to be able to "think". There were some computational questions but they were minimal. Most of the exam asked you about the logic of the subject and, a few times, to explain your conclusions. Currently machines can't do such things well. There are many, many students worldwide who want do these things very well, and you will be competing with those students for jobs. If you want meaningful success, especially in this competition, you must practice!

About a third of the students are absent on any day of the course. And about a third of the students who attend spend much of their class time dreaming, texting, and not concentrating: yes, it is that obvious. Attending a math lecture is fairly hard work, even for a relatively straightforward topic. If you want to learn, you must participate actively, ask questions when you don't understand, read the textbook before and after lectures, and do homework. You can consult me and work with other students. I have made these recommendations several times.

Grading in this course will be simple – this is a low-level math course with no way to earn any "extra" credit. Magic opportunities to improve grades will not appear near the end of the semester. A colleague of mine remarked that I don't give grades but I do attempt to report student performance accurately. You give yourself grades. Students who are listed as (possibly!) graduating seniors should note this especially.

The last day to withdraw from a course with a W is March 21.

Exam results

Problem#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 Total
Max grade 16 10 8 10 8 4 12 10 8 6 8 99
Min grade 0 0 0 0 0 0 0 0 0 0 0 20
Mean grade 7 7.05 7.17 4.12 2.53 3.04 6.93 3.2 4.73 2.19 4.76 52.73
Median grade 7 8 8 4 2 4 6 2 5 2 8 52

The versions of the exam were very similar. Numerical grades will be retained for use in computing the final letter grade in the course.
Here are approximate letter grade assignments for this exam:

Letter
equivalent
AB+BC+ CDF
Range[85,100][80,84][70,79] [65,69][55,64][50,54][0,49]

Discussion of the grading

An answer sheet with full answers to version A (with the yellow cover sheet) is available. The two versions of the exam were very similar. Here is a more compact version of this exam. 75 students took the exam.

Grading guidelines
The grader will read only what is written and not attempt to guess or read the mind of the student.
The student should solve the problem given and should not invent another problem and request credit for working on that problem.

Problem 1 (16 points)
2 points for specifying the Animal.
a) (2 points) For the answers.
b) (4 points) For the answers.
c) (4 points) For the span.
d) (4 points) For the span.

Problem 2 (10 points)
2 points for specifying the Animal.
2 points for the answer (linearly dependent). 2 points for a non-trivial linear combination. 4 points for supporting work.

Problem 3 (8 points)
2 points for the answer (the rank and nullity of the student's result!), and 6 points for supporting work (row operations).

Problem 4 (8 points)
a) 2 points for a correct answer, and 3 points for supporting evidence.
b) 5 points for an explanation.

Problem 5 (8 points)
a) (4 points) 1 point for taking the transpose of B–1 and 3 points for the correct multiplication. –2 for an error in the order.
b) (4 points) 2 points for multiplying the given column vector by B–1, and 2 points for multiplying the result by A–1. –2 for an error in the order.

Problem 6 (4 points)
1 point for a 5x4 matrix, M. The rank of M is worth 1 point, the nullity of M is worth 1 point, and the nullity of MT is worth 1 point.

Problem 7 (8 points)
a) (9 points) Writing [W|I4] is worth 2 points. Row operations are worth 5 points. The correct answer is worth 2 points.
b) (3 points) Writing W multiplied by W–1 and getting I4.
Ethical comment A number of students had an incorrect candidate for W–1 yet reported that the product of that matrix with W is I4. This certainly can't be a correct or accurate computation. I note that individuals who are credentialed as professional engineers are supposed to have ethical standards: would you want to work with or for someone who reported a false product of two matrices because it was convenient? ... or drive over a bridge designed by such a person or use a heart stent built by ... ETC.

Problem 8 (10 points)
2 points for specifying the Animal.
2 points for clearly indicating which w's will imply that the span is all of R4. 4 points for describing why the "good" w's do indeed give all of R4 and 2 points for indicating why the "bad" w does not give all of R4.

Problem 9 (8 points)
a) (4 points) 1 point for the answer, and 3 points for justification (an explanation).
b) (4 points) 1 point for the answer, and 3 points for justification (an example).

Problem 10 (6 points)
a) (2 points) The definition.
b) (4 points) 2 points for a valid example, and 2 points for evidence supporting the assertion.

Problem 11 (8 points)
The explanation.


The second exam

Exam results

Problem#1 #2 #3 #4 #5 #6 #7 #8 Total
Max grade 15 10 15 6 12 12 12 18 100
Min grade 2 0 0 0 0 0 0 0 19
Mean grade 10.95 5.64 11.95 2.71 6.5 8.24 7.20 12.77 65.91
Median grade 11 6 14 2 6 8.5 8.5 14.5 70.5

Numerical grades will be retained for use in computing the final letter grade in the course.
Here are approximate letter grade assignments for this exam:

Letter
equivalent
AB+BC+ CDF
Range[85,100][80,84][70,79] [65,69][55,64][50,54][0,49]

Discussion of exam construction

The two versions of the exam had the same problems, so problem 1 in B is problem 3 in A, 2 in B is 4 in A, 3 in B is 1 in A, 4 in B is 2 in A, 5 in B is 5 in A, 6 in B is 8 in A, 7 in B is 7 in A, and 8 in B is 6 in A. What follows refers to problems in the order of the A version, as do the numbers in the table above.

Problem 1 was given for essentially the third time to students in these sections. Problem 4 was given previously on a quiz. Problem 5 was given in review material, as were parts a) and b) of problem 6 while part c) was given previously on the first exam. Problem 8 was given previously on a quiz. That's 63 points out of 100 on this exam!

The other problems generally resembled examples discussed in class and the textbook and/or given for homework.

Discussion of the grading

An answer sheet with full answers to version A (with the yellow cover sheet) is available. Here is a more compact version of this exam. 66 students took the exam.

Grading guidelines
The grader will read only what is written and not attempt to guess or read the mind of the student.
The student should solve the problem given and not invent another problem and request credit for working on that problem.

Problem 1 (15 points)
5 points for each part: 1 point for the dimension, and 4 points for the basis requested (2 for each vector).
In part c), 1 point will be deducted for reporting row vectors, since "vector" in this course is supposed to refer to column vectors.

Problem 2 (10 points)
5 points for the definition, which should be given in complete English sentences. 5 points for showing exactly how the definition fails for the collection of vectors described.

Problem 3 (15 points)
a) The characteristic polynomial earns 2 points, and each eigenvalue earns 1 point.
b) Each eigenvector earns 2 points.
c) 1 point for each of D and P.
d) 3 points for P–1.
e) 1 point for PD and 1 point for successfully computing PDP–1 and getting A.

Problem 4 (6 points)
2 points for the answer and 4 points for an explanation.

Problem 5 (12 points)
6 points for each part. 1 of those is for the answer, and 5 for the process.

Problem 6 (12 points)
In each part, 1 point for a correct answer and 3 points for some adequate supporting explanation.

Problem 7 (12 points)
Indicating an intention to compute det(A–μB) with correct entries in the 3x3 matrix earns 3 points. Computing the determinant correctly earns 5 points. Getting the roots and stating a correct answer to the problem is worth the last 4 points.

Problem 8 (18 points)
In each part, 1 point for the answer (YES/NO), 1 point for the eigenvalue, and 4 points for some adequate supporting explanation.


The final exam

The exam seems to have been more difficult than intended. In particular, questions 6 and 17 did not get good scores and probably were new to people. These questions do resemble some real uses of linear algebra. In recognition of the difficulty, I've decided to rescale the totals proportionately, so that 180 would become 200 (that is, I modified the total by multiplying that score by 2/1.8). Using those numbers, statistical results for the total were: high=172.22, low=55.56, mean=128.93, and median=134.44. The raw score results (not rescaled) were high=155, low=50, mean=116.03, and median=121.

The rescaled numerical grades will be used in computing the final letter grade in the course.
Here are approximate letter grade assignments for this exam:

Letter
equivalent
AB+BC+ CDF
Range[170,200][160,169][140,159] [130,139][110,129][100,109][0,99]

Discussion of the grading

No answer sheet is available. Here is a more compact version of this exam. Two minor changes were made in what's posted (an "a)" was inserted in question 16, and 7b) had the word "real" added). 61 students took the exam.
I graded the exam rapidly, then waited a day and regraded it. I believe most if not all grading errors were detected and fixed.

Grading guidelines
The grader will read only what is written and not attempt to guess what the student's solution is or read the mind of the student.
The student should solve the problem given and should not invent another problem and request credit for working on that problem.

Problem 1 (12 points)
a) 4 points for the definition. Only 2 points are earned if the phrase "linear combination" is used with no explanation. These 2 points can be recouped if the student defines "linear combination" correctly in part a) of problem 15.
b) 8 points. 2 points of these are awarded for explaining what the matrix manipulations have to do with the requested verification.

Problem 2 (18 points)
9 points for each of a) and b): 2 points for the dimension, 2 points for one vector in a basis, and 5 points for the other (orthogonality earns 3 of those points).

Problem 3 (16 points)
a) 4 points: 2 for the characteristic polynomial, and each eigenvalue earns 1 point.
b) 5 points: each eigenvector earns 2 points, and checking orthogonality successfully earns 1 point.
c) 4 points: 1 point for D and 1 for a possible P, and 1 point each for correct normalization.
d) 1 point for P–1.
e) 2 points: 1 for PD and 1 for multiplying the result by P–1.

Problem 4 (6 points)
6 points for an explanation using results and language of the course, and related to the specific situation of this problem. A simple quote of a textbook or lecture result will earn only 2 points.

Problem 5 (8 points)
1 point for the value of the determinant, and 7 points for the process (good luck if you expand along a row or column!).

Problem 6 (18 points)
Obtaining the (general) characteristic polynomial is worth 2 points. Getting eigenvalues using the quadratic formula earns 4 points. There are three cases. Analysis of each case is worth 4 points: real distinct roots, complex roots with non-zero imaginary part (square root of a negative number), and double root (multiplicity 2).

Problem 7 (10 points)
a) 5 points: 1 point for the answer, and 4 points for justification (an explanation).
b) 5 points: 1 point for the answer, and 4 points for justification (an example).

Problem 8 (10 points)
a) 5 points: 1 point for taking the transpose of B–1, 1 point for reversing the order, and 3 points for the correct multiplication. If the multiplication is correct but the order is wrong, 3 points are earned.
b) 5 points: 1 point for reversing the order, 2 points for multiplying the given column vector by B–1, and 2 points for multiplying the result by A–1. If the multiplications are correct but the order is wrong, 3 points are earned.

Problem 9 (12 points)
a) 8 points: 6 points for v1 (an answer needs to be "instantiated", with all numbers, but an explicit answer does not need to be evaluated) and 2 points for v2.
b) 2 points for the answer, which need not be simplified.
c) 2 points for the answer.

Problem 10 (10 points)
Rewriting things as dot products earns 3 points, and then the balance is obtained by completing the explanation. 1 point off if no · is used (the "dot" in dot product), and 2 points off if || ||'s are misused.

Problem 11 (12 points)
a) 7 points. Note that the answer can be checked! 2 points for the answer alone with no method indicated.
b) 2 points for the answer (this can use the student's A–1).
c) 3 points: 1 point for the answer, and 2 for some explanation.

Problem 12 (12 points)
In each part, 6 points: 1 point for the answer (YES/NO) and 1 point for the associated eigenvalue. 4 points for some explanation.

Problem 13 (10 points)
a) 6 points: 2 points for Av and 4 points for ||Av|| (of those, 2 points are earned for asserting or using v1·v2=0).
b) 4 points for the answer.

Problem 14 (10 points)
2 points for the rank, 2 points for the nullity, and 6 points for the process.

Problem 15 (12 points)
a) 4 points for the definition.
b) 8 points: 2 points for the answer, and 6 points for the process.

Problem 16 (12 points)
a) 7 points: AB computed is worth 5 points, and an appropriate comment about BA is worth 2 points.
b) 5 points: knowing that det=0 is connected to non-invertibility is worth 2 points, and then using this along with knowledge about how det works with matrix product is worth the other 3 points. The remark that a product of an invertible matrix with a non-invertible matrix is not invertible earns 2 points. No points are earned if some formula such as (AB)–1=B–1A–1 was quoted.

Problem 17 (12 points)
a) 5 points: computation of AP and PA is worth 3 points. The conclusion correctly derived is worth 2 points.
b) 5 points: computation of AQ and QA is worth 3 points. The conclusion correctly derived is worth 2 points.
c) 2 points for remarks supporting the conclusion.


Course grades

I added the two in-class exam scores and the rescaled final exam scores. To that I added 50/90 multiplied by the total quiz score for each student, and then I added 50/212 multiplied by the total homework score for each student. The total possible score which could be obtained was 500 points. I then put these total scores in bins which corresponded to the bins used for each of the exams (so the boundaries would be {250|275|325|350|400|425} for F/D/C/C+/B/B+/A).

Rutgers regulations require that the exams be kept for a year. Students may look at their exams and check the grading.


Maintained by greenfie@math.rutgers.edu and last modified 5/11/2011.