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04/30/2013 06:08 pm ET | Updated Jun 30, 2013

What Do I Have to Do to Get an A in This Class? (A Mathematical Perspective)

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It's that time of the year again. Final exams in colleges and universities are getting close, and one of the most popular questions students are asking their instructors is, "What do I have to do to get an A in this class?" (In other cases it may be, "What do I have to do to get a C in this class?", but the concern, like the mathematics involved, is similar.)

The question is right up there with "Will this be on the final?". There are no prizes for guessing what information is being fished for there, and what study habits will result from a "no" answer. That's why the responses of many in the teaching profession range from "maybe" to "of course." (There's also, "What a great idea!", an answer which is guaranteed to have the rest of the class wishing the question had not been asked in the first place -- it should certainly stifle further curiosity along those lines.)

A third perennial favourite, which seems to hold a strange fascination for students, is, "How many questions will be on the final?". It's a lot less clear, for mathematics courses anyway, what behaviour will result from any specific answer to that. Will study patterns really vary depending on whether 10, 15 or 20 questions are promised? If so, in what ways? (Has any research been done on this?)

As for what is required to get a particular grade in the class, the answer should be determinable by the well-delineated grading policy in the course syllabus. Students don't always read that document carefully, of course, as some of them unwittingly reveal two months into the semester by asking, "Where is your office?" or "What are your office hours?". (Or, "How many tests do we have?".)

The "What do I have to do to get an A in this class?" chestnut tends to be asked late in the semester, when opportunities to earn most of the credit for the course (homework, quizzes and tests) are in the past. It seems like a realistic question for a student who has an 85.75 percent average going in the final, but is it? Can such a student come out with 93 percent overall if the final exam is worth 30 percent of the course credit?

It's less realistic for students who may not have devoted sufficient attention to the class up till now, or those who did try hard but for one reason or another didn't do as well as they'd hoped to. It's out of the question for students whose average is so low that they'd be lucky to come out with a passing grade overall. Yet, it seems that every semester it's asked by at least one student who assures the instructor that her performance is about to switch from D or C quality to A level, starting tomorrow. (It's never today, for some reason.)

For many students, starting from where they are, the best they can do is aim for the highest possible overall mark for the class, and that translates into getting 100 percent (or as close to it as possible) on the final. The reality is that sometimes an A is still beyond reach, for good mathematical reasons, unless the instructor is open to the idea of abandoning the stated grading policies and awarding an overall A to any student who literally scores 100 percent on the final.

The overall score for a course is typically computed from various component scores, including tests, quizzes and homework. They tend to be weighted differently, and the individual components may also have different maximum scores. Perhaps there are three tests with max scores of 50, eight quizzes each graded out of 100, and 12 homework assignments with max scores of 10. Not to mention a final exam graded out of 200. Let's assume that the overall score (out of 100 percent) is based on the following breakdown:

Tests 15% each
Quiz average 15%
HW average 20%
Final 20%
Total 100%

It's surprisingly tricky to compute with this information, which may explain why students (even mathematics students) sometimes have difficulty working out their own (real or projected) averages.

At first sight it might seem that (T1+T2+T3) x 0.15 + Q x 0.15 + HW x 0.20 + F x 0.20 is the appropriate weighted average to work out, but that ignores two points: (1) The total of 100 in the table is correct but requires careful incorporation of the three Test scores, and (2) Not all components have a maximum score of 100. For a student who got everything correct it leads to a puzzling: (150) x 0.15 + 100 x 0.15 + 10 x 0.20 + 200 x 0.20 = 79.5. Something is wrong.

(For the record, while all three test scores can be used like this, if we modify the formula, the Quiz and HW averages need to worked out in advance, adding up individual scores and dividing by 8 and 12 respectively. Appropriate adjustments can be made if some Quiz or HW scores are dropped.)

To take account of the max score issue, each component needs to be scaled as well as weighted.
One way to effect this scaling is to adjust each component so that a perfect score comes out to be 100, by inserting an appropriate factor before its weight. We also need to take account of the fact that we are using all three test scores. The following formula works here:

(T1+T2+T3)x(100/50) x 0.15 + Q x (100/100)x 0.15 + HW x(100/10) x 0.20 + F x (100/200) x 0.20.

It's quicker to work with this in the simplified (but more mysterious) form

(T1+T2+T3)x 0.30 + Q x 0.15 + HW x 2 + F x 0.10.

A student with perfect scores gets (150)x 0.30 + 100 x 0.15 + 10 x 2 + 200 x 0.10 = 100 as expected, and a student with test scores of 34, 40 and 42, a Quiz average of 84.2, a HW average of 8.8 and a final exam score of 187 gets (34+40+42)x 0.30 + 84.2 x 0.15 + 8.8 x 2 + 187 x 0.10 = 83.73.

There are online grade calculators that deal with weighted components and others that deal with max points but not so many that handle both at the same time.

In trying to answer the "What do I have to do...?" type question, it's not so hard if we know the overall average to date. Here's a simplistic example. Suppose a student has attained an average of 75 percent on the parts of the course leading up to the final exam, where those components account for 70 percent of the overall course grade. The final exam is worth 30 percent, and a little algebra, as implemented by the grade calculator linked from the bottom of this Calculating a Weighted Average page, can be used to determine that in order to get an overall average of 93 percent, the student would need to get 135 percent on the final. There's the rub: for this student it's just not going to happen.

It's more attainable for a student going into the final with an average of 90 percent in the same class: the same grade calculator shows that an overall average of 93 percent can be pulled off by scoring a perfect 100 percent on the final. (This is also verified by just computing 90 x 0.70 + 100 x 0.30 = 63 + 30 = 93.)

A moment's reflection suggests that going into the final with less than a 90 percent average, for a course like this in which the final exam is weighted 30 percent, puts a 93 percent course average beyond reach. (All the moreso if the final is worth less than 30 percent.) However, if you have a student in such a course with an 85.75 percent average going into the final, who's convinced that when he puts his mind to it he never gives less than 110 percent, tell him to go for it. He can still come out with 93 percent overall if he gets 110 percent on the final exam. For students restricted to a maximum of 100 percent of the final, it's too much to hope for.

Some goals are simply out of reach, as we've just seen. Here's another example, in a different context. Recall our final challenge from Mean Questions With Harmonious Answers: Chris decides to do a 180-kilometer bike ride and sets himself the goal of averaging 30 kilometers per hour overall. He slacks off for the first half and only goes 15 kilometers per hour on average. How fast does he have to go in the second half to hit his target average? Unlike in the case of test scores, here there is no (theoretical) limit on now well (i.e., fast) he can do in "the final" (i.e., in the second half). Yet, the surprising answer there is that he can never catch up. He won't average 30 kph (kilometers per hour) overall if he he goes 45 kph in the second half---one of the points of that posting was to illustrate the counter-intuitiveness of averaging in some contexts---nor will he pull it off if he straps the bike to the top of a high performance car and completes the second half of the race at an average speed of 145 kph. Or even if he hitches a ride on the space shuttle going at its top speed of 28,292 kph. Slacking off earlier can never be made up for.

As for the real answer to "What do I have to do to get an A in this class?", it's not rocket science. The computation of overall grades from component scores doesn't involve the harmonic averages which arose in Mean Questions With Harmonious Answers, but the truth is still obvious enough. "Study hard and ace all of the homeworks, quizzes and tests along the way" should be one's mantra from the beginning.

It's not always what students want to hear as they approach final exams, especially bearing in mind that the desired grade may no longer be within reach, but the truth hurts sometimes. We all need to remember that there are unavoidable consequences of decisions made in more carefree times.

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