The following WitCH comes from (CAS permitted) 2018 Specialist Mathematics Exam 2:
The Examiners’ Report indicates that about half of the students gave the intended answer of D, with about a third giving the incorrect answer B. The Report notes:
Option B did not account for common factors and its last term is not irreducible, so should not have Dx in the numerator.
Update (11/08/19)
The worst kind of exam question is one that rewards mindless button-pushing and actively punishes intelligent consideration. The above question is of the worst kind. It is also pointless, nasty and self-trippingly overcute.
As John points out in the comments, the question can simply be done by pressing CAS buttons. But, alternatively, the question also just appears to require, and to invite, a simple understanding of partial fraction form. Which brings us to the nastiness: the expected partial fraction form is not a listed option.
So, what to make of it? Not surprisingly, many students opted for B, the superficially most plausible answer. A silly mistake, you silly, silly student! You shoulda just listened to your teacher and pushed the fucking buttons.
The trick, of course, is that the numerator factorises, cancelling with the denominator and leading to the intended answer, D. The problem with the trick is that it is antimathematical and wrong:
- As Damo notes, the original rational function is undefined at x = -1, which is lost in the intended answer.
- As Damo also points out, there is no transparent, non-computational way to check that the coefficients in answer D would, as demanded by the question, be non-zero.
- It is not standard or particularly natural to hunt for common factors before breaking into partial fractions. Any such factors will anyway become apparent in the partial fractions.
- To refer to the partial fraction form is actively misleading. Though partial fraction decomposition can be defined so as to be unique, in practice it is usually not helpful to do so, and the VCE Study Design never does so. In particular, if answer B had contained a final numerator of Dx + E then this answer would be valid and, in certain contexts, natural and useful.
- The examiners’ comment on answer B is partly wrong and partly incomprehensible. One can pedantically object to the reducible denominator but if that is the objection then why whine about the Dx in the numerator? And yes, answer B is missing the constant E, which in general is required, and happens to be required for the given rational function. For a specific rational function, however, one might have E = 0. Which brings us back to Damo’s point, that without actually computing the partial fractions there is no way of determining whether answer B is valid.
But of course all that is way, way too much to think about in a speed-test exam. Much better to just listen to your teacher and push the fucking buttons.
I’ve got three points that I would love to make, but I don’t want to spoil the fun for other crappers. So I’ll start off with just one.
The problem with option D is that they have cancelled out common factors, which is particularly problematic because the original expression was undefined when x=-1, whereas it is defined for option D.
That’s very funny. I didn’t even see that. Excellent point, Damo. (p.s. Perhaps “crapologist” is a more inviting term than “crapper”.)
Crapologist is certainly more inviting, not to mention more accurate. However, I would argue that the fact that ‘crapper’ feels a bit looser and dirtier is what gives it its appeal. I’m part of a community of crappers. Maybe it’s just me….
I’ll simply make the point that there is absolutely no mathematical understanding required in this question. None at all. Just press the buttons on the CAS calculator(or in Mathematica’s case, simply command Apart[…] like Moses). Furthermore, it encourages a student to have no mathematical understanding, to simply press some buttons and move on.
In fact, it actually penalises students who attempt to use some mathematical knowledge.
It’s a poor question that has no place on a CAS-Active Exam. What bugged me the most was the number of ‘copycat’ questions that appeared on commercial trial exams the following year ….
By the way, Damo. I love your comment. The so-called correct option is not even equal to the original function …. So none of the options are correct.
Thanks, John. Yes, however bad the question is (and it is), it is made much worse by being in a black box exam.
This terrifies me, John. The justification of CAS was that it would enable more ‘deep thinking’ by reducing the need for students to spend large amounts of time on calculations. This kind of questions exposes the lie of that justification. Students are rewarded for knowing how to push buttons, as opposed to rewarding mathematical knowledge. Unfortunately, I can’t see this situation improving – indeed, in light of the recent ‘review’, I am considering alternatives to teaching.
There is a problem with their note on option B. I’ll do my best to unpick it, although I’m not sure I’ve quite nailed it. It is not that the fraction should not have Dx in the numerator because the denominator is not irreducible, but rather: when one of the fractions has a quadratic denominator, then the numerator should be Dx+E and; because the denominator is not irreducible, we should be writing this fraction as the sum of two fractions, both with linear denominators (in this case).
As John said, I think that the big problem with this question is that the correct partial fraction is not there. I would suggest that it should be:
A/(2x+1) + B/(2x+1)^2 + C/(2x+1)^3 + D/(x+1) + E/(x-1)
Thanks, Damo, and I agree. Even on the Examiners’ own terms, their comment on Option B is way off key. People have pretty much nailed the awfulness of this question, and I’ll update when I next come up for air. But, I think there’s a little more awfulness to isolate.
That’s a correct option VCAA could have had which would have at least required more than simple button pushing. But that boat sailed the moment they specified that A, B, etc were non-zero real numbers.
Specifying that A, B etc were non-zero is an unnecessary addition to the problem – and potentially quite destructive, as students reading this problem may assume that this is generally true for partial fractions – but I don’t think it invalidates the above solution.
I get A= 34/9 B = -11/3 C = 1 D = 1/9 and E = -2
There might be a slip in your calculations, Damo (or mine) – specifically, I get A = 0 (I haven’t checked the other numerators of the fractions that would otherwise ‘cancel’):
Assuming (2x^2 + 3x + 1)/(2x+1)^3(x^2-1) = A/(2x + 1)^3 + B/(2x + 1)^2 + …. + E/(x + 1) I get
2x^2 + 3x + 1 = A(x^2 – 1) + B(x^2 – 1)(2x + 1) + C(x^2 – 1)(2x + 1)^2 + D(2x + 1)^3(x + 1) + E(2x + 1)^3(x – 1)
Substitute x = -1/2: 1/2 – 3/2 + 1 = -3A/4 => A = 0.
I think the other numerators corresponding to fractions of the fractions that would otherwise ‘cancel’ (C and E) will also be zero. So I think the non-zero statement is necessary to force the option that VCAA want.
Ha, yes. I was scratching my working on the back of an envelope, and at some point mistook the expression on the original numerator to be 2x^3+3x+1. Which it is not. As you were.
The amazing thing about these WitCHes is they are often worse than I thought when I posted. You guys commonly point out aspects that I hadn’t seen. And here you’re both correct: requiring the coefficients be non-zero absolutely kills any reasonable non-CAS approach.