Our second (and last for now) NHT WitCH is due to the ever-vigilant John the Merciless (who shall, to begin, hold his fire …). It comes from the 2019 Exam 1 of Specialist Mathematics (calculator-free):
The examination “report” gives the answers as: (a) (51,65); (b) 0.02, 0.03 accepted.
Our second WitCH of the day also comes from the 2017 VCE Specialist Mathematics Exam 2. (Clearly an impressive exam, and we haven’t even gotten to the bit about using inverse trig functions to design a brooch.) It is courtesy of the mysterious SRK, who raised it in the discussion of an earlier WitCH.
Question 5 of Section B of the (CAS) exam concerns a boat and a jet ski. Though SRK was concerned with one particular aspect, the entire question is worth pondering:
The Examiner’s Report indicates an average student score of 1.4 on part a, and comments,
Students plotted the initial positions correctly but significant numbers of students did not label the direction of motion or clearly identify the jet ski and the boat. Both requirements were explicitly stated in the question.
For part i, the Report indicates an average score of 1.3, and comments,
Most students found correct expressions for velocity vectors. The most common error was to equate these velocity vectors rather than equating speeds.
For part ii, the Report gives the intended answer as (3,3). The Report indicates that slightly under half of students were awarded the mark, and comments,
Some answers were not given in coordinate form.
For part i, the Report suggests the answer (with the displayed answer adorned by a weird, extra root sign). The report indicates that a little over half of the students were awarded the mark, and comments,
A variety of correct forms was given by students; many of these were likely produced by CAS technology, including expressions involving double angles. Students should take care when transcribing expressions from technology output as errors frequently occur, particularly regarding the number and placement of brackets. Some incorrect answers retained vectors in the expression.
For Part ii, the Report indicates the intended answer of 0.33, and that 15% of students were awarded the mark for this question. The Report comments,
Many students found this question difficult. Incorrect answers involving other locally minimum values were frequent.
The Report indicates an average score of 1.3 on part d, and comments;
Most students correctly equated the vector components and solved for t . Many went on to give decimal approximations rather than supplying the exact forms. Students are reminded of the instruction saying that an exact answer is required unless otherwise specified.
We’re back, at least sort of. Apologies for the long silence; we were off visiting The Capitalist Centre of the Universe. And yes, China was great fun, thanks. Things are still tight, but there will soon be plenty of time for writing, once we’re free of those little monsters we have to teach. (Hi, Guys!) In the meantime, we’ll try to catch up on the numerous posts and updates that are most demanding of attention.
We’ll begin with a couple new WitChes. This first one, courtesy of John the Merciless, is a multiple choice question from the 2017 VCE Specialist Mathematics Exam 2:
The Examiners’ Report indicates that 6% of students gave the intended answer of E, and a little under half the students answered C. The Report also comments that
I guess if you’re gonna suggest a painful, ass-backwards method to solve a problem, you may as well fake the solution:
Checking directly that P(1 – i√2) = 0 involves expanding a cubic, and more, which the text does in one single magic line.
The painful multiplication of the products for part b is much more naturally and easily done as a difference of two squares: (z – 1 – i√2)(z – 1 + i√2) = (z – 1)2 + 2, etc.
After all that the third factor, z – 1, is determined “by inspection”? Inspection of what?
AS RF notes, it is much easier to spot that z = 1 solves the cubic. Then some easy factoring (without long division …) gives P = (z – 1)(z2 – 2z + 3). Completing the square then leads to the linear factors, answering both parts of the question in the reverse, and natural, order.
Alternatively, as John notes, the difference of two squares calculation shows that if z – 1 + i√2 is a factor of P then so is the quadratic z^2 – 2z + 3. That this is so can then be checked (without long division …), giving P = (z – 1)(z^2 – 2z + 3), and so on, as before.
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.
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.
It’s a long, long time since we’ve had a WitCH. They have been not-so-slowly accumulating, however. And now, since we’re temporarily free of the Evil Mathologer, it is the WitCHing hour.
Due mostly to the hard work of Damo, all of the outstanding WitCHes have been resolved, with the exception of WitCH 8. That one will take time: it’s a jungle of half-maths. Our new WitCHes are not so tricky, although there is perhaps more to be said than indicated at first glance.
The Examiners’ Report gives the answer as . The Report also indicates that the average score on this question was 1.3/5, with 98% of students scoring 3 or lower, and over a third of students scoring 0.
To be honest, we’re not sure the exercise below is a PoSWW. It may simply be a minor error, the likes of which are inevitable in any text, and of which it is uninteresting and unfair to nitpick. But, for the life of us, we have no idea what the authors might have intended to ask. Make of it what you will:
UPDATE: For those hoping that context will help make sense of the exercise, the section of the text is an introduction to factoring over complex numbers. And, the text’s answer to the above exercise is A = 2, B = 5, C = -1, D = 2.
Question 3(f) on Section B of Exam 2 was a clumsy and eccentrically worded question that covered material outside the curriculum. Unsurprisingly the Report made no mention of these issues. But, what about a blatant error by the Examiners? Would they remain silent in the face of such an error? Again?
Question 6 on Exam 1 (not online) required students to find the “change in momentum” of an accelerating particle. Unfortunately, the students were required to express this change in kg m s-2. The Exam had included the wrong units, just a careless typo, but a blatant error. The Report addressed this blatant error with the following:
Students who interpreted this question as asking for the average rate of change of momentum to be dimensionally consistent with the units and did this correctly were awarded marks accordingly.
That’s it. Not an honest word of having stuffed up. Not a hint of regret or apology. Just some weasely no-harm-no-foul bullshit.