As with a bunch of these WitCHes, one dearly wishes that it were a PoSWW but it probably isn’t. In any case, there is more stupidity in this one than (hopefully) first meets the eye. It is a spin-off of a recent PoSWW, and is again a question from WA: 10(b) from the 2024 Mathematics Specialist “calculator assumed” exam, together with the marking key.
And, before we get going with the WitCH, a quick question. I understand that most Australian mathematicians take the University of Melbourne approach of not giving a stuff about school mathematics. Nonetheless, one has to ask: Is there even a single decent WA mathematician paying any attention whatsoever to what kids are being fed in their backyard?
Here is the exam question:
And, here is the provided solution, with the marking guide:
Nitpick: the lines should not extend below
in the diagram as they are meant to be rays. (Which means the marking guide is wrong in the third point…)
Comment: This question would not be out of place on a VCE exam.
Fair comment re the rays. But not the biggest issue here. I also find the use of “i.e.” instead of implies in the marking scheme kind of curious. But I guess you can’t make mathematical implications from a nonsense statement.
The writing of
instead of 
is also sloppy
Thanks, Simon. I’m fine with the i.e., since the two statements are essentially identical (to the extent that nonsense statements can be anything).
I agree on the italicised Arg: Not by a mile the worst issue here, but pretty klutzy.
Thanks, RF. The double-directed lines is pretty idiotic but not nearly as bad as the complex inequalities.
Huh. Sorry, RF, I just read your parenthetical comment, and of course you are correct. The double-direction lines are wrong, but the marking guide confirms the cluelessness. Plus, the terms “boundary” and “boundaries” are wrong.
Inequalities with complex numbers is a special kind of nonsense… I wonder if they noticed when they got to writing the marking scheme but just ploughed ahead anyway
Thanks, Simon. I see no reason to suspect that WA ever recognised their nonsense.
You guys are missing something.
Huh… I thought the
and ‘cancelling’ 
in an inequality was so bad that nothing could top it in such a short question.
You may well be onto something there… dividing both sides of an inequality by
is something I totally overlooked (so did the exam setting panel, it would seem).
So, for the more knowledgeable out there, can this actually be done and, if so, what precautions have to be taken? (I’m assuming since they are different dimensions, one should never divide by a purely imaginary number and expect the inequality to remain valid).
Thanks, RF.
One simply cannot define inequality of complex numbers: ordering is fundamentally a “one dimensional” kind of thing. Of course we know what the WA clowns were *trying* to do: they wanted to order just the complex numbers on the imaginary axis, which one can do. But, the way to indicate that, the way to compare ai and bi, is to write a ≤ b or Im(a) ≤ Im(b). That is explicitly ordering the real numbers, the imaginary parts of the purely imaginary numbers.
Now, presumably the WA clowns will not get it, and will say “all the students knew what we meant”. I don’t have the time but there are various reasons why that attitude does not cut it. Suffice to say, it is madness for a state educational authority to simply wing it on such things (although they were probably too stupid to realise they were even winging it).
Has this nonsense occurred in Victoria? There’s been no shortage of complex idiocy, of course, but I haven’t seen this one. Nelson, maybe. They may be stupid enough to have done it.
VCAA avoid complex inequalities, instead resorting to Re(z)>0 etc. (see NHT exam 2019 question 1f)
Cambridge and Jacaranda follow a similar pattern but, as you guessed, Nelson offends in the way this WACE question does. (Vic Maths units 3 & 4 Specialist Mathematics, page 159, example 8 for anyone interesting in looking it up)
Thank you, Wilba, and FFS. Do I win a prize?
I looked through Nelson, and, yes, Example 8 is the exact same inequality idiocy (although Nelson did include the Arg hole). I couldn’t see any subsequent exercise with the same nonsense, so I’ll probably leave off doing a PoSWW. (There was a later question, a reproduced VCE exam question, with “z ≤ 2”, but that turned out to be just a glitched version of |z|< 2.
“Arg hole”. That should be the title of a blog post…
Yeah, I shoulda saved it. Running out of titles.
gse … certainly applies to “English” too::
please do excuse a math-illiterate Hums person!
Oh, definitely the complex inequalities is not topped here, Simon: that was the main thing. But I was actually directed to look at this question after posting the 2025 WA thing, as a second example of the Arg hole. Then, when I looked, I noticed this further, and greater, madness.
OK. I was waiting for someone more knowledgeable to start this discussion, but I have an issue with the word “locus” here.
A region is not automatically a locus of points and since there is no general point moving around in space, I cannot see a locus being created by a rule.
It feels like the question writer (or maybe editor) was trying to be more… complex? (bad pun) than was needed.
That said, I’m not a mathematician and so I’m not 100% confident on the actual definition of a locus so please correct me if this word has actually been used correctly here.
I appreciate your bravery RF.
I think “locus” is fine here. A locus is a geometric-picture thing, the collection of points satisfying some conditions, some equations or inequalities. It needn’t require a sense of motion, or the points being created in some manner.
What is *not* OK, is going the other way: to give the kids a picture but with no clear specification of what is being pictured, and then ask the kids for equations/whatever for “the locus”. That’s what WACE did in 10(a) of the PoSWW (and 9(a) is similarly and more obviously nuts).
I was taught that “locus” means “path”.
The Oxford Languages has:
“Mathematics
a curve or other figure formed by all the points satisfying a particular equation of the relation between coordinates, or by a point, line, or surface moving according to mathematically defined conditions.”
So I guess it’s the first part here.
Thanks, Tom. So, maybe both. My 70s texts use it in the equations sense. Hunting around a little, and not checking, it seems that the idea of “locus” was a Fermat thing, who used it in the equations sense. But I guess these equations are commonly parametrised, so it’s not a big slide to think from equation-locus to path-locus.
I’m surprised no-one, including the marking scheme, has mentioned that there should be an open circle at z = 1 – i.
Thank you, John. Ditto for 10(a) on the original PoSWW.
Hey Ray, do we include the point where you originate?
“No, it’s (too) complex.”
Is there a case for inclusion, similar to 0-vector and its perpendicularity?
Anyway, 2023 complex question (exam 2 vcaa,q2 part d.) Did not pay for closed circle?
Thanks, Banacek. On trying to define Arg(0), no, there is no natural way to do it. There is no one natural limiting value, since the argument depends upon the angle of approach. (Note that setting Arg(0) = 0, for example, would not save the question above, since the point 1 – i would still be excluded.) It’s the same block to productively defining
(of which someone should inform Smiling Eddie and the Desmos nitwits). The only thing one can do is define (little a) arg(0) = R, which has some naturalness, although I’m not sure I’ve ever seen it.
The 2023 VCAA question you’ve noted is undoubtedly stuffed, and is included in the error list, but it is a little more complicated in its stuffedness. As I wrote on the error page,
Part (ii) asks for the polar equation of “the ray” drawn in (i). This equation, however, will necessarily exclude the ray’s starting point, which would legitimately and naturally be included in the answer to (i). (17/04/24) The exam report draws the ray with the starting point excluded, and is silent on whether students were permitted to include that point. This is cowardly and dishonest: VCAA stuffed up the question and they should indicate how they dealt with the stuff up. It is a minor stuff up and it is difficult to imagine that students were docked a point, but VCAA have an ethical and professional obligation to clarify.
So, in answer to your question as to whether the closed circle (correct for (i) and incorrect for (ii)) was permitted: God knows.
Of course other things were happening in 2023, and also, with Honest Steve in charge, I wasn’t of a mind back then to pursue VCAA on such matters, rather than simply ridiculing them. But, now, Boss Smith has a chance to prove his worth (or lack of it), and I intend to provide him the opportunity. I will be approaching VCAA very soon – like tomorrow – with a number of 2025 examples of this level of grading-report nonsense, and much worse.
The marking scheme was less than ‘holesome’.
Off the track, but let me offer a comment on “calculator assumed”. Today I was with two other people for lunch. We enjoyed the meal and I went up to the counter to settle the bill. The young lady had a calculator and she worked out what we owed. It seemed a bit high to me; so she did the calculation again and got a totally different answer. A bit rattled, she did it again and got a third answer. And then a fourth answer. I went to the loo to give her a chance to work it out again. By the time I got back, a more senior staff member had taken over. She had got the same answer twice. This staff member explained to me that their calculators were a bit old and that was the problem.
I kept the receipt and have a good problem – from the real world – to share with my students tomorrow!
No I agree. The complex inequality is the worst aspect.
Hi Marty! Love your blog. Despite the all the issues inherent to teaching mathematics in Victoria, I still would like to. I am about to embark on a double degree in education and science, hoping to major in Mathematics and Chem. As I select subjects to study, naturally I’d choose the things I find interesting etc, but I’d also like to consider what would give me the best mathematical background to teach the material in VCE. Yes, I know the curriculum is stuffed but surely it is best to at least know what is truly correct and navigate the mess from there. Thanks for your time!
Dear amt, thank you for your very kind comment re the blog and best of luck with the mathematics and (ugh) chem and (double ugh) education. (The first ugh is just my personal ugh, that I really sucked at chem at HSC: no reflection intended on uni chem.)
I agree *in principle* with what you say: if you plan to teach then there’s a strong argument to choose subjects that will benefit your teaching. But, how that may play out can depend a lot on where you’re studying.
Other people can perhaps offer thoughts here. But if you also want to send me an email or a “contact Marty” through the blog, I’d be happy to have a chat offline.
Hi Marty, if you’re more comfortable with an email, very happy to do that… can’t find it though on the blog though. I am going to Monash (only place that offers this in VIC), which I remember was more your wheelhouse? Just posting this here in case others who’ve been have ideas 🙂
Dammit: replied to the wrong comment. Again …
Sure. Just click the “contact” thing near the top of the page and it’ll give you the email address (and a contact form).
Thanks to everyone who has offered suggestions to amt. I’ve also replied to offline. In brief, I suggested that amt ignore the maths: guys like Burkard are only going to explain the meaning of things, where what really counts is the pedagogy. So, just go for the pass on the maths subjects and work hard on the education stuff, particularly the Sullivan school of neo-CLT.
Hi AMT
I find that a lot of school mathematics makes more sense to the student if related to its use in Physics or Engineering. Less so for Chemistry.
Hmm, I am more Maths than I am Chem I think. If I seriously want to do Maths as my main thing, should I look to do further study in either of these areas instead? I do really like Chem and I’m good at it, but I’m willing to drop it if it’s sufficiently worthwhile
Hi AMT
Don’t drop Chem. Instead, keep both.
It’s definitely an asset in you if you could teach both disciplines.
Chemistry also has a great deal of good maths applications, such as using the rate of change to analyse reaction rates, reversible reactions, even involving solving quadratics in some sophisticated problems. Others include algebra…in a lot of problem solving…
Thanks for your thoughts Sentinel, really appreciate it.
In a 7-12 school, science covers physics, chemistry, biology, psychology, and geology, especially at the 7-10 level. Of course one cannot major in all these subjects. But it may be useful to have a smattering of these subjects to develop a broader degree. (BTW, I teach only mathematics, but I am not qualified to teach anything else.)
On average, each week, there are about 50 jobs for mathematics teachers advertised in government schools in Victoria. Many of these will be for teachers of mathematics and science; some will be mathematics and PE, others will be mathematics and something else. Many will be not in Melbourne; often these jobs outside Melbourne offer financial incentives for people who take them up.
It would also be useful to develop your skills in IT.
Keep an eye on the website School Jobs Vic for advertisements for jobs in government schools.
Good luck.
Thank you Terry for your kind words. I did actually consider IT as well, but I didn’t feel very confident given I didn’t do any of those subjects in high school (my high school was public so pretty limited options). I’ll keep an open mind if there’s an opportunity to study that though