Busy week, huh?

### UPDATE (04/11/23)

Thank you all for your comments. Mostly the exam was OK (at least as much as it could be given the idiotic mark distribution), but was hampered by bad wording. Again a few comments, mostly amplifying what has been noted below, but a couple other thoughts as well.

**Q1. **The minor problem is bad wording: “Show that the rule for the function …” should be replaced by “Show that the function …”.

The major problem is that, in the VCAA world, this appears to be a very bad question. The simplest (and perfectly valid) approach to the question is to start with x + 2 – 4/(x – 1) and establish the* equality *by writing the expression over a common denominator. But will VCAA accept that? And, even if VCAA will, I’m betting plenty of kids didn’t do the problem this obvious way, exactly because they fear that VCAA won’t accept it.

**Q3. **The wording for (b) is just nuts. If infinite limits are in the curriculum (are they?), then talk properly in the language of limits and/or use limit symbolism.

**Q4. ** Asking for an answer in the form -π√a/b with a and b positive integers is sloppy. Yes, it’s standard in VCE, but it’s sloppy.

**Q6.** There’s lots of words in this question. Like a real lot of words. I fell asleep reading this question and then woke up, looked at the question and went back to sleep. I can’t even count the words. But it’s a lot. There’s a lot of sentences, and a lot of words in each sentence. I think there’s an average of ten words a sentence, with a standard deviation of who knows and who cares.

Plus, as commenters have noted, but have not sufficiently emphasised, part (b) is screwed: there are infinitely many answers for a and b, with two being very natural and non-CAS findable. Even if there were only one findable answer, it doesn’t matter: you don’t ask for “the value” when multiple values answer the description.

Plus, the proper word is “equals”, not “equivalent”. If a teacher, or education authority, is using the word “equivalent” then they almost certainly using the word incorrectly.

**Q7. **The phrase “open hollow surface of revolution” will become legend. And, π(a√b/c – d) is multidimensionally sloppy.

**Q8. **Clumsy wording throughout.

**Q9. **Show that AB = … and AD = … . Why is this hard?

Q6b. Suppose that the intended answers were . Given the symmetry of the normal distribution, should be marked as correct as well, or does the question’s use of the word matter in this case?

I would presume that both answers would be marked correct, however, I doubt that many students went for the second one. This question question 7 further exemplify how VCAA does not care about questions with many solutions.

I think it’s not worth noting that apart from the the obvious and the less obvious answers, there’s an infinite number of other answers (but you’d have to be out of your gourd to worry about them in a CAS-free exam (or even a CAS-active exam)).

The question reads: “Find the values of a and b”.

The question is wrong. It is not wrong in the newspaper sense, but it is grossly wrong in the “Why the hell doesn’t VCAA employ some mathematicians?” sense.

Seriously, this is bad.

Is

“Find a pair of values (a, b).”

better?

Yes. Better to rejig the question to make a and b unique, but your rewording is correct, and good enough.

Seriously, this is basic logic, basic maths, basic grammar. You don’t ask for “the” thing when a number of things answer the description.

The thing about VCE teachers is that, rightly, they are playing the game, doing their best to get their students the highest marks they can. So, if a question is “benignly wrong”, in the sense that the taught mechanical methods will skate over the error, then teachers don’t stress it: no harm, no foul. But it is foul.

I understand teachers not caring about this kind of thing (sort of), but VCAA damn well should. This simply should not happen.

For the kiddies at home who are still interested, see attached.

2023 Specialist Mathematics Question 6 part (b) – Further calculations

Question 7, for the surface area of revolution, is it fair to say that the surface area should include the ‘inside’ and ‘outside’ of the surface (i.e. multiply the surface area formula by 2), because it specifies the surface as “open [and] hollow”? I did this on the exam, hoping I didn’t completely overthink it.

Oh, geez. I have thoughts, but I’ll let others comment first.

I also think that because the “formula” for surface area is given on the formula sheet, they will expect the result given by the use of this formula and not the doubling.

Then again… a nice examiner may realise that the question is ambiguous and pay both answers…

No niceness required. Given the wording, it’s impossible to sanely argue against giving full marks for finding the correct area of one side and then doubling it.

Dark and malevolent thoughts, I’m sure.

No. “That’s what happens when they try to listen to me” thoughts.

Yes, right… I do not think this is what the examiner(s) intended.

I think their use of and was to remove some of the ambiguity around similar questions where teachers and students have asked if the circular disk “ends” need to be added.

Of course, if they had simply added the disks in this case there would be no ambiguity!

“Find the area of the surface generated by rotating …. about the …-axis over the interval ….”

‘Nuff said.

Adding the disks every time, just to agree with their screwy wording on the support materials, is absurd. At some point, they have to bite the bullet and establish proper terminology.

I thought it was OK – a bit light on integration techniques? Only substitution and by parts. The stats and probability question had way too much to read to do something pretty simple – You have to carefully read half a paragraph several times to make sure you haven’t missed something and then add three numbers. I don’t have the attention span for those questions!

Amber, I totally agree with your comments about Question 6. My eyes were glazing over. It would have benefitted from having a good Editor. I hope most students had the presence of mind to spend a minute or so summarising that slab of text:

etc.

Otherwise it’s too easy to misread a datum and get the wrong answer as a consequence.

Agree. Also, parts a and b could have been interchanged and the question may have made more sense.

Odd that Methods had an explicit limit to an infinity question, but Specialist asks for “the value that the velocity of the particle approaches as x becomes very large.”

Yes, I raised my eyebrows at that too.

(Perhaps “Find the limiting speed of the particle.” is better)

Note the wording of the 2023 NHT Specialist Mathematics Exam 1 Question 7 part (b):

“Write down the limiting (terminal) velocity of the particle in .”

No good reason why “limiting velocity” could not be used in the November exam.

I don’t like reading essays to answer a 4 mark stats question.

Induction with differentiation was more interesting than I thought VCAA would go with.

Overall it felt easier than necessary (other than “open hollow”)

Re: Induction. Yes, but perhaps more obvious in hindsight. I was expecting a divisibility proof by induction.

Yes, the wording of Question 7 left me with a feeling. I’m expecting to see Marty turn the dial to 11 once he catches his breath.

Well, I would. But I somehow feel this is my fault.

If VCAA has over corrected in an attempt to “Marty-proof” their exam then:

1. That is on them.

2. Thanks Marty!

The more I think about the “open surface” question, the more I feel that doubling the final answer is very much a valid answer, since the question goes to too much length to specify the resulting shape is .

The law of unintended consequences strikes again!

You better keep it at 10 then, Marty.

When I first saw the wording my first thought was that VCAA might have listened. If only they’d asked as well. Thanks, RF. ‘Over-corrected’ is exactly the phrase my first thought was groping for.

The worst part is that if VCAA see this they’ll probably think “damned if we do and damned if we don’t”. That is NOT the case! Here is my plea: Please listen ask.

But VCAA were pretty stuck. They had written in the support materials that a curve sweeps out a “solid region” or whatever. So, what are they then to do? If, in the exam, they had written “to form a surface of revolution”, which is the standard terminology, plenty of kids would have added in the disk.

Short of purchasing a time machine, I don’t see VCAA had (or now has) a good way out.

There’s always a way out. In this case it’s easy:

1) The VCAA gets someone attentive and competent to make every necessary change to its resource and reference material so that the standard is acceptable.

2) The VCAA says the following in the first VCAA Bulletin of 2024:

“Mathematics update:

The VCAA has analysed student examination responses related to questions on the new Study Design content. These responses have been used to amend the resource and reference materials to ensure that this material is more reflective of the types of response required.”

It’s spin doctoring 101. My invoice is in the mail.

Maybe VCAA have a way out now. They didn’t have a way out yesterday.

A routine exam in my opinion with maybe one differentiating mark, giving the domain of the parameter for the general solution to the very last question such that t>=0 (not t element of R) which some may have not considered

I think the substitution in Q7 would not have been immediately obvious to all students and therefore a nice question, even if the number work at the end ruined it a bit.

Regarding Q10d, I suspect quite a few students will have written is a positive integer, omitting the possibility that , again, mostly due to time pressure.

Which is why these exams are so hateful. You can do all the work, slightly misremember some who-cares constraint, and get proportionally hammered on a 2-mark question.

Here is a question for those familiar with VCAA’s …

Suppose a student had written for Q10d that

Would this be marked as correct?

Now that’s a very good question, RF. When does VCAA like to say and when does it like to say it’s not? Because VCAA likes it both ways.

Unfortunately, BOTH are mentioned at different points.

I think there was a blog post dedicated to this very matter late last year or earlier this year.

However… since VCAA has now written this question, I might send them an email to innocently inquire (after the report comes out of course).

The audit report, that is!

UPDATE – If VCAA were to choose to be consistent, would have to be marked as wrong.

I refer to the examiners report for 2021 Paper 1, Q7.

What is the definition of a open surface?

This is a genuine question that I’m trying to find the answer to, and I’ve only been able to find the definition of a closed surface on Wikipedia.

Tell me when you find out.

My intuition and extrapolation from the definition of an open set makes me think that the surface in question 7 is not an open surface because it has a “boundary” that you can’t put “balls” (or whatever the analogy would be for surfaces) on that are subsets of the surface.

That, or the definition is related to the definition of an open manifold here (https://en.wikipedia.org/wiki/Closed_manifold#Open_manifolds), but I don’t have the mathematical knowledge to understand much on that page.

Or I could just be overthinking it and an open surface is just one that isn’t closed.

Forget it, Joe. VCAA’s use of “open” has absolutely nothing to do with the mathematical definition of an open set (in a topology).

VCAA’s use of “open” is “Please, please, whatever you do, don’t add in any disks”.

Answer: Either grass, Rebound Ace or clay.

I really wonder why they didn’t stay with the wording used in Exam 1 Sample Question 8 (or even Q9).

Edit: RF nailed it, I think. VCAA listened and over-corrected.

Took me a minute to work out the reference there BiB…

Nice one.

Q10c – I did this at first as an integral, and found it easy enough. Some students told me that they (correctly) recognised the path as part of the circumference of a circle and solved for that way.

I hope both methods are paid, because I really think this second method is quite clever under the time pressure.

Not to mention totally valid. (Which may or may not mean anything)

There is no sane argument for not giving full credit to both methods.

However, your hopes are redundant because it’s only worth 1 mark and it’s not a “Show” question. I’ll wager VCAA won’t even care about the method, only a correct answer.

Which should mean that a student who used the fraction of a circle method and still got Pi/8 is awarded the mark… right?

We assume (hopefully correctly) that when a question is 1 mark, it is an answer mark.

I’m never totally sure on these matters.

I have seen multiple marking schemes, and outside of show thats 1 mark questions are always answer mark only. It should be basically safe to treat them like so.

Very smart method, but have to careful the parametric equations here are sin(2t) and cos(2t), not sin(t), therefore the answer is pi/8, not pi/4. My top student got pi/4 by using this method without noticing sin(2t), cos it is one mark question, she will get 0! Very sad!

But so… all good.

(Sorry, bad joke)

Ouch!

I repeat my comment from above: having an exam bloated with 1-mark and 2-mark questions over-rewards pedantic care and way under-rewards deeper thought. These exams are hateful.

Interestingly, no pseudocode or algorithms and well as differential equations. I can smell an ugly logistic equation in exam 2 if the pattern of MM is to be repeated. In general, a routine paper gives rewards to students making the fewest number of mistakes .

It would be a new level of insanity to examine pseudocode in Exam 1. How it was examined in the Maths Methods Exam 2 was one of the very few sensible ways of doing so in a mathematics subject.

(For the unfortunate readers who don’t have a copy of Methods Exam 2:

Pseudocode was examined in a single multiple choice question (1 mark out of 80):

Pseudocode for Newton’s Method was given. Students were essentially required to extract the equation being solved, the initial guess, and the number of iterations used. They could then use this data to run Newtons Method on their CAS and get the answer).

If this is more or less how pseudocode will be examined, most of my ill-will towards it being added to Methods and Specialist will disappear. It would be a great thing if the VCAA agreed and stated this.

I am expecting/hoping for a similar question in Specialist but in the context of Euler’s Method.

Exam 2 lends itself to a Logistic equation question. I still don’t understand why it’s treated in the Study Design like a separate type of DE. It’s just another example of an autonomous DE, just like Newton’s Law of Cooling, radioactive decay etc.

I agree that a routine exam is an opportunity for students to do a Bradbury. I dislike this.

Thank you all for your comments. I’ve update the post with some thoughts.

Not that it matters, but I find it amusing that the instructions still say “Take the acceleration due to gravity to have magnitude g ms^-2, where g = 9.8”. Isn’t this irrelevant in the new study design?

Still has relevance in kinematics, though I don’t know why they can’t just specify this in the question, when they already fill half a page about Josie’s trip to work for some probability muck. I can imagine an Exam 2 modelling question involving a skydiver, a plane, and a squishy end.

Re: Update comments.

Q1 – VCAA have said in their 2021 Exam 1 Exam Report that

“A reminder that ‘show that’ questions require a reasoned argument. The answer is given and students are required to provide a detailed progression to the answer. Practise setting out working in a sequenced and legible manner. Ensure that each step is a clear and logical development and follows from correct mathematical reasoning”

So any student that does the simplest thing that you mention will be penalised. Gonzo I know. But that’s the rule.

I agree that it is possible that VCAA will act in this manner. But for VCAA to apply a “start to end” mandate to this question would be an utter perversion of mathematics, logic and language. It would be way, way worse than their standard “show that” perversion.

Typically, VCAA uses “show that” to mean that a “reasoned argument” is required, as your quote indicates. More accurately, VCAA uses it to mean a derivation is required. This distinguishes, for example, “showing” x = 3 is a root of p(x) by factoring from “showing” it by plugging in.

This a thoroughly idiotic use of language, going against common usage and common mathematical usage. But you can shake your head, translate the phrase and you know what to do.

(Having said that, I am not sure VCAA is consistent: I believe there are exam questions where you have to “show that” a given function is a solution to a given differential equation: there’s zero chance it means solve the DE, rather than plugging in.)

But in the question we’re discussing, there is zero logical difference between going “start to end” or “end to start”. Either way, we are applying a “reasoned argument” to establish the equality of two expressions.

If VCAA were to demand “start to end” here, it would be akin to demanding any induction proof must begin with LHS = …. . That would be insane: if the RHS gives you better stuff to manipulate then that’s where you start. Exact same thing here.

No argument from me.

Re: “I am not sure VCAA is consistent: I believe there are exam questions where you have to “show that” a given function is a solution to a given differential equation: there’s zero chance it means solve the DE rather than plugging in.”

I think they use the word “Verify” for this type of exam question. Which means it’s OK, I think, to plug stuff in.

And when it comes to “Prove the identity …” I think you can start from any side you want.

Verify, show, prove etc. VCAA has a ‘Glossary of command terms’ but these words don’t appear. It would be good if VCAA defined them somewhere – we might not like the definitions but at least there would be clarity and consistency.

There have been a few cases in past VCAA Specialist exams where it is asked “Show that the two particles collide at t=6” and it is not possible for students to equate each component and solve for a value of t (tech free), and in this case simply subbing in t=6 to the position function of each particle is VCAA’s intended method. From my experience doing past papers, this and the differential equation case mentioned above have been the only times VCAA ‘allow’ a student to use the information given to show the desired result, and yet did not indicate as such leaving many such as myself stuck trying to derive the result independent of the information given as this has been the requirement in every other case.

So VCAA can’t even follow its own rules. If that’s the solution VCAA intended, then the collision question should have said “Verify that …” not “Show that …”

It’s this lack of consistency by VCAA that frustrates the hell out of me. Surely VCAA has an exam writers guide that is used to maintain consistency – it looks to me like the writers just make it up as they go along, new writers come in and set their own rules. This is another reason why there should be a panel of attentive and competent university mathematicians that has oversight over every exam.

Thanks, SS. Interesting point.

SS, do you know the year of the exam that your “Show that …” question appeared?

Thanks, BiB. Just a quick follow up. I suggested in my comments that I thought there were past exam questions on DEs where “show that” amounted to plugging. But I checked and I think I was wrong, misremembering something else.

Clarification: Maths Methods Exam 1 2021 Report.

I’ve read the 2021 Exam 2 report and the 2022 reports: their comments on “show that” questions do not have the same sense of mandated direction. So, I’m more sceptical of VCAA being idiotic on the grading of the 2023 question. But my main point stands: VCAA has been idiotic enough on enough other questions that many students will shy away from using the best, obvious method.

I should have also mentioned the 2020 Methods Exam 1 Report:

“Show that …’ questions require a reasoned argument. Remember the answer is given and students are required to provide detailed progression to the answer given”

Re: “I’ve read the [Specialist] 2021 Exam 2 report and the 2022 reports: their comments on “show that” questions do not have the same sense of mandated direction.”

I agree. For ease of reference, the relevant comments are:

Taken from 2021 SM Exam 2 Report:

“There were three questions (Questions 3bi., 4a. and 4d.) for which students needed to show how a given result was reached. In these cases, steps that led to the given result needed to be clearly and logically set out to obtain full marks”

Taken from 2022 SM Exam 2 Report:

“There were four questions (Questions 2ai., 3aii., 5a. and 5c.) for which students needed to show that a given result was reached. In these cases, steps that led to the given result needed to be clearly and logically set out to attract full marks.”

“In a ‘show that’ question, students are required to clearly and logically show the steps that lead to the given result.”

“Some students did not clearly or logically show the steps required for a question with the ‘show that’ command term”

“Just as in other ‘show that’ questions, sufficient detail was required.”

However, we see a similar comment in the 2022 Maths Methods Exam 1 Report:

“This question was a ‘show that’ question and most students seemed to be alert to the need to show clear, logical steps in their solution process.”

What are we to think if we didn’t see or forgot the 2021 or 2020 comments!? Does this mean VCAA has back-flipped on its back-flip?

It’s as if VCAA suddenly realises that their past comments don’t accurately reflect their secret marking schemes and they decide to be clearer … My concern is that this is the case with the SM reports. And even if it’s not, this means there is an inconsistency between Methods and Specialist as to how these questions are marked. Or maybe there’s an inconsistency between Exam 1 and Exam 2 …

The way I see it, there is no clarity or consistency. We read Exam Reports that seem to suggest different things from year to year and from subject to subject. This is why I have consistently said that it would be good if VCAA defined them somewhere, with an example – the Study Design is the obvious place.