This is our post for discussion of the 2022 NHT Mathematical Methods exams, which have now been posted, here and here. (See also our corresponding 2022 NHT Specialist exams post.) We’ve had a quick browse, and found plenty of irritants but no glaringly large errors. We don’t plan to look further or to post on the exams, except to follow up on whatever people find worthy of comment.

### UPDATE (06/09/22)

The Methods (and Further) exam reports are now available, here and here.

Paper 1 – Question 8c ii.

The use of the word HENCE bothers me greatly here. If the word wasn’t there, I think it would be fair for a student to realise the integral in part d must be equal to 1 and use this to work out the integral in c ii.

However, my understanding of VCAA-speak is that HENCE in c ii means the result from c i MUST be used.

Which is essentially guiding a student through integration by parts (except they can’t say as much because it is not in the curriculum…)

I think the “hence” is ok, since (d) is implicitly using the result of (c). But the structure and the wording of the questions are pretty bad.

I don’t think there is anything wrong with the question itself, but doesn’t part (d) kind of give away the answer to (c ii)?

Yes. That’s structurally bad.

I agree with Marty about 8cii, it’s the same wording they always use for integration by recognition questions. Perhaps they could have set up 8c to find that coefficient in 8d to avoid the spoiler?

I have no issue with Q8c – it is pretty standard.

8d though… as a question it is fine. It just doesn’t sit right with me as the question to immediately follow 8c.

Sure, by that part of the exam students are probably just trying to finish and may not have used reading time well and so may not realise the answer is effectively given away… meh.

Some quick thoughts on Exam 1 and with sentiments similar to Marty’s:

Vanilla.

Question 3: The value z = 2 is explicitly given, with an expectation that this value is used. But the 2022 NHT Specialist Exam 1 Question 8 requires that students have memorised z = 1.96. Clearly VCAA cannot make up its mind and establish a consistent expectation across all of its exams.

Question 6: What a stupid ‘real-life’ context. Another ‘real-life’ snow job. This really irked me.

Question 7 (b): An odd/disconnected question to follow from part (a). (Then again, the NSW exams often do this. So I’m not criticising, I just thought it was odd).

Question 8: I agree with RF.

And I generally hate all 1 mark “Show that …” questions. For the reason that it’s impossible to know what trivial steps VCAA will demand in order to get the 1 mark.

Given the relative irrelevance of the NHT exams, I think the writers get a pass mark for this one. There are no glaringly large errors. But …. the continual shifting of the goal posts for the statistics content of VCE mathematics (particularly in Specialist Maths) – given that we have this gratuitous content forced upon us – continues to greatly irritate me.

Thanks, John. Q6 is hilarious.

Minor comment on Exam 1, 3c says ‘no solution’ rather than ‘no real solution’.

Thanks, Alex. I don’t mind that, at least in Methods, and would prefer VCAA go the other way. The continual “” and the like, as if could be anything else, is needless, cluttering and distracting.

Yeah, it’s not the end of the world but if there’s any chance that VCAA would mark a student down for writing ‘no solution’ instead of ‘no real solution’ for a quadratic equation for instance, they should at least be consistent on their end.

That’s one of the reasons I get students looking at exam questions as early as I reasonably can so they get used to seeing the set notation everywhere and knowing they can generally gloss over it. I get VCAA’s point that it defines all the other pronumerals and tells you it should be a value rather than a variable and sometimes helps check your answer is reasonable (for write your answer in the form a/b where a,b in Z types of questions) but especially the ‘in R’ ones don’t help a lot otherwise.

I’m not trying to defend the “k is Real” inclusion, but given how often recently we have had “a,b are integers” or similar, perhaps it is there to reassure students that non integer k-values are acceptable…?

I can’t think of a reason for it otherwise.

Ritual. That is the only “reason”.

I never said it was a good reason…

And I am guessing when it comes to VCAA and “reason”.

I wonder how they will go with “logic and proof” in 2023…

For the scared Methods teacher, RF’s comment and what follows relates to the new Study Design for Specialist Maths – They’ll play it safe. I give you the Friendly guarantee that there will be a proof by induction question on one of the exams (probably Exam 1) followed by a “Hence find/prove/show …”

I give you the Friendly guarantee that the logic questions will be multiple choice in Exam 2 and that there will be a truth table question in Section B.

The questions will be tried and tested – copied from interstate exams to avoid criticism and to be conservative.

What I’m interested to see is if VCAA finally has the guts to use the word “Prove” instead of the ridiculous slippery and slimy “Show” …

More leeks to follow …

I like leeks.

Leek fried with bacons will be tasty.

Regarding MM next year, how about assessing Newton’s method with a question in exam 1? 3-4 marks?

How about central difference approximations? And points of inflection with second derivatives?

Noticeably MM2 NHT, ERQ5:

Finally a kinematics question appears in MM. Well…since 2016 absolute value was removed, this time it has sneakily slid into the question for distance travelled. One may argue that a graphical integration approach with technology could have been adopted but…

ERQ4: I am on the same page as Alex.

Will be curious to see what sorts of “correct explanations” would be accepted for the confidence interval question. I guess we can’t be “95% certain/confident” that…

Ritual abuse of notation uses.

Exam 2

MC8 “The range of the function with rule … is contained within the interval”. I’m not one to normally complain about excessive CAS use, but that is egregious. Not even asking the range but an interval that contains it. May as well have put the graph on the page.

Minor point, MC13 uses a centred dot for a product function, didn’t think that was assumed knowledge (favours TI users over Casio)

I can’t wait to be underwhelmed by whatever sample answer VCAA gives to question 4giii (interpreting the confidence interval).

Thanks, Alex. MCQ8 is a bad, and badly worded, question.

Just ran through Exam 1, some thoughts:

Q3 Is it fair to assume people know what z signifies? (Though if not, I can’t think of a better way to phrase the question)

Q5c) I don’t like but understand conditions like ‘… in the form… where m,n in Z’, but wtf is the point of saying ‘ln(m/n), where m,n in R’ ?? The entirety of question 5 felt a bit weird; you can presumably test more than just algebraic manipulation.

Q7a) ii. “Find the value of b… where b>0” is technically redundant given the specified domain. I very much appreciate 7ai.

Q8) Is there any actual point to making the domain (-infinity, 0]? It slightly complicates the ‘general solutions’ you need to then find (e.g. only positive integers, etc.), but I’m disappointed if that’s the only reason for doing so.

Hi Kell.

All fair and reasonable observations.

Q3: Yes, most people would make the reasonable assumption that is the critical of z in this case, where . One can argue that it's reasonable to understand this notation but at the end of the day it's sloppy notation by VCAA and assumptions need to made. VCAA are quick to expect teachers and students to make assumptions about what it means and intends, but not so quick to extend similar generosity towards students (and very churlish when this is pointed out)

You say that you can't think of a better way to phrase the question. I'm sure you could, starting with a simple "where is the critical value of z for the test. Of course, you wouldn't even need to say this if the critical values were included on the formula sheet, but that would be expecting too much common sense from VCAA.

Q5(c): VCAA like to make sure there are an infinite number of possible forms of correct answer, to maximise a students opportunity for guessing it. Only kidding. I understand why VCAA ask for answers to be expressed in a particular form (I do it myself on assessments so that I'm not spending 5 minutes figuring out if some weird answer is numerically correct), but there are ways of doing it that make the form unique. VCAA does not understand this.

Q7(a)(ii): Yes, I liked part (i) too. But then VCAA reminds us all why VCAA is also redundant ….

Q8: I assume the restriction is ultimately to accommodate the support of the pdf in part (d).

Thanks, Kell. JF has pretty much said it, but I’ll also quickly reply.

Q3. Yes the undefined is absurd.

Q5. Parts (a) and (b) are ok, except that they don’t lead anywhere, but (c) is ridiculous. Ignoring the infinity of correct answers, what is the point?

Q7(a). The wording of both (i) and (ii) are needlessly clunky.

Q8. Yes, a poorly composed question. Needlessly cute.

Marty, I left the door open for you with Q3 …. I was waiting for you to say:

A better way to phrase the question would be to have no question.

I can’t even gather sufficient energy for that. I just ignore stats garbage.

Stats Garbage ?

Hi Marty,

I’ve been comparing MM2022 NHT Exam 2 Q15 with MM2019 Exam 1 Q6 with in the hope of trying to better understand what it is all about!

The latter question has a well defined large population where each member either has an attribute or not; in this case pegs are faulty or not faulty; so that the idea of a population proportion p is clearly established. In part(b) we go onto determine a probability relating to the sampling distribution of P hat = X/12 , the proportion in the presumed simple random sample size 12 which are faulty; we do this by relating it to the underlying binomial distribution of X which is Bi (12,1/6).

Now in the 2022 Exam Q, we are simply given the probability distribution of some rv X which is NOT binomial since E(X) = 8/3 and n = 4 implies p = 2/3 but V(X) is not 8/9. We are then asked to compute a similar probability concerning P hat. But just what exactly is P hat? And what does P hat = X/4 actually mean? Am I missing something?

Hi, Rob. Ignoring that it’s stats garbage, I think the 2022 question is ok. Doesn’t simply mean the sample proportion, so in this case is just X/4? So, asking for ≤ 1/3 is just a too-cute way of asking for X ≤ 4/3.

Thanks Marty.

As I understand Binomial rv’s, when sampling from a Bi(4,p) distribution with Pr(success)= p, a single value of the variable X is obtained namely, 0,1,2,3,or 4. P hat = X/4 is then clearly defined to mean the proportion of successful outcomes in the sample.

HOWEVER, when taking a samples size 4 in a non binomial environment such as the one in MCQ15, we simply obtain 4 values of X such as { 1,2,4,4} or {0,2,3,4} etc

Hi Rob.

Read the attachment. Construct the probability distribution table for . We see that the required calculation is equivalent to finding . This is what VCAA .

But what VCAA intends versus what it says are often two different things and we are constantly required to guess the intent.

You have a very valid point:

There is NO defined large population where each member either has an attribute or not. And so the idea of a population proportion p is NOT clearly established. We have to assume VCAA’s intent, yet again. In fact, it looks to me like the given distribution for X is pre-supposing a sample of size n = 4 is taken from a population.

It’s not “too cute”, it’s simply a stupid, poorly worded question.

Sampling distribution template

Hi Rob. I think you may have a point. Let me think.

Hi, Rob. I think you’re right. I think the question is screwed.

The question begins with a probability distribution X, which can take the values 0, 1, 2, 3 and 4. But it’s just a probability distribution with five possible outcomes. So, imagine the same probability distribution, call it Y, except that Y can take the values A, B, C, D, and E. Then we take a sample n = 4. What then is the “sample proportion”, let’s call it , for Y? It would appear that makes no sense.

The point is, in order to create for X with its intended meaning, we have to think of X as the distribution for counting the number of things with a certain attribute. But, there is no reason given to interpret the possible outcomes of X in that manner. There is no indicated underlying attribute.

Further, even if X were the distribution of the number from a population with a given attribute, how can that automatically indicate anything about the random sampling from that population?

Exam 2 thoughts:

MCQ:

Very confused by the options given for Q8. You already know A and B can’t be correct (otherwise E and D would also be respectively), and the function’s clearly unbounded from below leaving only D.

I liked Q6 and Q18 – a bit of geometric insight immediately solves the former, and I appreciate the connection between trigonometric definitions and the latter.

Short answer:

The ‘landscape designing’ context for Q2 is hilarious and utterly pointless. I appreciate Q2e) though, requiring students to think about the image of the curve rather than blindly manipulate to ‘find’ the rule of the transformed function.

Q4a) i. I’m not sure how you can properly ‘show’ the result of the integral in just three lines. I’m also completely lost on what g) iii. means by ‘interpret’ – if anyone here knows, I’d genuinely appreciate clarification.

Q5 is awful, especially c) ii. and d). Pretty much the entire question is calculator spam

Thanks again, Kell. Quick thoughts in reply.

MCQ8. Yes, a ridiculous question.

MCQ6. I’m not sure what you mean. How are you looking to solve it geometrically?

MCQ18. I wasn’t crazy about this question. The intrinsic problem is nice, but the wording is too cluttered.

Q2. Yes, the “modelling” is insane. I fail to see the attraction in Part (e).

Q4(a) Yes, an interesting use of “show that” no a 1 mark question, given VCAA’s unceasing sermonising on the meaning of the expression. Here, I assume it means writing nothing more than writing the correct integral and then claiming CAS gives the indicated value.

Q5. I hadn’t looked carefully but, yes, it is awful. Part (a) is meaningless, enough I think to add it to the error list. Yes, (c)(ii) is awful: “maximum positive rate of change” is terrible wording, and do they really mean that’s what they’re looking for?

Yes, Question 8 is dumb. I don’t know what exactly the writer is trying to test but whatever it was, it could be done in a better way. I assume the “contained” business is because the range is but is a very clunky irrational number and a decimal approximation to obviously doesn’t work if you want to write down the range.

I they’re trying to test whether a student understands the use of brackets and how to round numbers. (IF so, there are much better ways of testing this). The left endpoint of -4 in the incorrect options is stupid. I don’t see how the question and its wording can be rehabilitated. It’s just a stupid question.

Re: Section B Question 2. Yet again VCAA has to have some dumbass, lame-brained, pin-headed context.

Re: Question 4. Part (a) (i): Yes, another stupid 1 mark “Show that …” question. It is there so that students who couldn’t get the value or who would otherwise get the wrong value can still do part (a) (ii) and get 1 mark. The value is not needed after this, so a lot of stupidity for setting up 1 mark in part (ii). Marty’s assumption for what has to be ‘shown’ is undoubtedly correct. And yet, part (b) (ii) requires the value of the median in part (i) to get at least 1 of 2 marks … If you’re going to have your stupid 1 mark “Show that …” question, part (b) is where it should be.

At least the pdf integrates to 1, unlike some past VCAA so-called pdf’s.

Re: Question 5. Yes, a horror show in many ways.

The Methods (but not Specialist) exam reports are now up. I’ve added links, above.

I’m not sure if I’m misinterpreting what they’ve said on exam 1 question 7b “Setting 𝑦 = 𝑓(𝑥) and then later writing 𝑦 = the inverse function is a contradiction of ideas and should be avoided” but I think it conflicts with what they said on the 2012 exam 1 for question 3 where they let y=h(x) and later after rearranging to x=2y^3+1 they state y is the inverse of h(x).

I hate my job.

Thanks, Alex. It is, at best, anal-retentive nonsense, and I think it is probably worse than that.

The final answer should be in the requested form, f-1 (x) = … . But other than that, what is/are the examiners’ concern(s)? That students do might the algebra before switching x and y? That students might write something along the lines of “y = the inverse function”? That would be nuts. There is no “contradiction of ideas” whatsoever. And yes, it contradicts the 2012 report (which is also nuts). I’ll add to the Methods error list.

What this report says for Q7(b) is total crap. It says

“Let ” then follows with the “swap x and y for inverse” crap, and then solves for y which it then calls . This was written by a mathematics incompetent – they have done exactly what they pontificate in their first couple of lines to NOT do. This nitwit has totally contradicted themselves, let alone contradicted previous reports. Where does VCAA find these halfwits*? – is it a case of water always finds its own level?

And VCAA has the gall to include Logic and Proof as an Area of Study in the new Study Design (do as we say, not as we do).

Furthermore, I loathe and detest the whole “swap x and y for inverse” crap. I think it’s mathematically lazy and sloppy and sets a very bad example for students. Typical VCAA. How hard is it to just start with:

Let where .

Then you solve for y.

Or more formally (and making more visible where it comes from), how about starting with

and solving for the inverse (which is exactly what my first suggestion says).

Seriously. How hard to set this as the standard?

* And I’m being generous with saying half.

Isn’t your first method still swapping x and y though? The argument VCAA has always made regarding this is that they don’t want students going between y=f(x) and x=f(y) without any reasoning.

I don’t mind your second approach but feel that writing the f^-1(x) repeatedly would be a pain (an issue I assume you’re trying to avoid with your first method).

You’ll get a kick out of this. The 2016 Exam 1 has an inverse question (q5) where they do what’ve you’ve written. Then exam 2 has an inverse question (q4bi) where they write the swap in. 2018 NHT exam1 q5 has x= f(f^-1(x)). 2018 exam 1 question 5 does the thing they tell students not to (go from y=f(x) to x=f(y) without saying anything). Consistency.

This is probably why nothing is written into the study design, they can’t make up their own minds about what they want.

No one should care about any of this. It does not matter. None of it.

I just want to know what to tell my students 🤣 They seem overly pedantic some years, then not at all others.

Just ask VCAA. If they don’t give you a straight answer, let me know.

Alex, you have to have a mind before you can make it up.

But it’s nothing to do about minds or the lack thereof. It would make errors all the more obvious. So it’s everything to do with a refusal to be held accountable.

Marty, I wish it didn’t matter. But it does matter. It matters a lot. VCAA’s pedantry has very unfair consequences. Teachers have to try and mitigate these consequences by VCAA-proofing their students. How to set out a solution for finding the inverse function is one of those things that students have to be VCAA-proofed against. This is made all the more harder by VCAA’s lack of consistency and its embarrassing history of mathematical ineptitude.

Plus, the contradictory nature of the solution given in the Report matters. It matters in all sorts of ways and is indicative of the sloppy mathematical thinking that pervades all of the Reports. And as I commented earlier, VCAA has the gall to include Logic and Proof as an Area of Study in the new Study Design. That’s like a vegan telling us how to cook meat on a BBQ.

John, I meant it didn’t matter mathematically. Of course it matters to VCE students. But Christ, there is no point guessing. Just ask VCAA.

Marty, in fairness to VCAA and the new Mathematics Manager, I think your suggestion is reasonable. From what I have seen and heard so far, the new Mathematics Manager is genuinely engaging with teachers and schools and is making an effort to *listen* and give straight answers to queries. From what I have seen, the new Mathematics Manager is personally responding to queries.

My advice to teachers would be the same as yours. I think there is cause for optimism that there will be more transparency and a greater willingness to *listen* and give straight answers. The new Mathematics Manager cannot change the past, but he can ensure more integrity and humility during his tenure and in doing so, undo much of the damage and mistrust that was done and created in the past.

Alex, we await the outcome of your VCAA enquiry.

McaNeill is clearly an improvement over DLL. But even if he weren’t (and it’s early days), the strategy should always be the same: ask him/them. If they give a reasonable reply, even if and especially if it contradicts previous VCAA nonsense, then all is sorted. If they give a bad or incoherent reply, then hammer again.

VCAA cannot possibly provide an answer here that will please me, since it will undoubtedly be too ritualistic. (All that is required here is, in either order, switching x and y, and doing the algebra. Everything else is fetish.) But they can, and can be forced to, provide an answer to VCE teachers that makes absolutely clear what is expected.

Yes, that would be good.

What would be really excellent is if VCAA’s statement said:

1) must be the starting point,

2) Students are allowed to define with another symbol to expedite solving for

3) Students perform appropriate algebra to solve for .

4) An exemplar is given.

It is the job of the Examination Reports to do this. But as we have seen, what’s said in an Examination Report gets forgotten over time and inconsistency reigns supreme. Therefore it is the sort of thing that MUST be embedded in the Study Design. In the VCAB era that preceded VCAA, it was. It wasn’t too hard to do then, it shouldn’t be too hard to do now!

I got a response first thing Monday morning, I’m waiting for a follow up just to clarify his response a little more.

“Hi Alex,

Thank you for your enquiry.

The two elements at play here involve the need for students to demonstrate the notion of an inverse, that being the swap of y and x, and also being mindful that function notation is important (should include a rule and a domain) and any new function developed cannot have the same name as a function that has already been declared. Students must avoid generating solutions that identify y as one function, then identify y as the inverse function, like:

y=x^2, swap y and x

x=y^2

so y=sqrt(x)

It must also be remembered that the external assessors report provides an indication only of the types of solutions which ought be generated, and do not, in and of themselves, constitute a marking scheme.

I hope that this information has gone some way towards clarifying your enquiry. Of course, if you have any other queries, please don’t hesitate to reach out.

Kind regards,”

Thanks, Alex. In general it’s better, at minimum, to paraphrase such private communications, although this one seems fine.

In brief:

*) The remark in the examination report is poorly worded and whiny, and what you’ve quoted here doesn’t help a lot. I think the practical issue of “what VCAA wants” probably just comes down to writing at some point, to make it clear enough what the y is then referring to.

**) Despite the report, and what you quote now, it is very standard to have y refer to two functions at the same time. If I find the area between and , I’m doing exactly that.

***) The report and quote have a point, albeit a minor and pedantic point. If you’re creating new functions out of old, then it is worthwhile labelling the different functions clearly in the context. But again, don’t make a fetish out of it. If I take the function and translate it up 1 unit, I have no problem referring to the new function as .

****) In summary, the VCAA is, as is typical, being pedants for no purpose other than to be pedants. It is an ignorant and crabbed attempt at mathematical rigour. But, the pedantry they are asking for here is (I’m guessing) pretty minor and not too unreasonable.

Fair, thanks for that.

*) That is my assumption too, I did ask about that in my reply. I’ll let you know what he says.

** & ***) I agree with using y for several equations. VCAA sometimes will use y=f(x) and y=g(x) and other times have y_1=f(x) and y_2=g(x) so it’s not something I’m going to stress over too much (for y).

Alex, it’s terrific to hear that you got such a detailed and timely response. There seems to be plenty of anecdotal evidence that the current Mathematics Manager is making the effort to give genuine and timely replies to all queries.

I really hope the current Maths Manager continues doing this, it’s going a long way to restoring good will between teachers and VCAA and can only lead, in a small but non-trivial way, to better outcomes for students.

(Under previous Manager-ment, you’d wait weeks to get a reply. When/if the reply finally came, it was often from an underling who had misunderstood the query (in some cases the query was not correctly communicated to the underling) and who said that they were told that the query didn’t make sense. It was a war of attrition to get answers/clarification to anything. I know of at least one query that took over 1 year to get a satisfactory response, and that was only because the Curriculum Manager ultimately intervened).

I agree with Marty that y = is often used by VCAA with several different functions. I also agree with Marty (and VCAA) that students should be giving a clear statement saying rather than y = …

As previously stated, I dislike the ‘swap y and x’ and teach a different way of setting out such solutions.

Unfortunately there are a several things that don’t matter mathematically but nevertheless matter … And some of these things are not known by the average teacher *. Some of these things are only told to Assessors (who are bound by confidentiality agreements to stay silent), and/or given in VCAA Marking Schemes that are kept secret.

* In fact, there are things that matter that the average teacher doesn’t even know that they don’t know, thanks to the VCAA shroud of secrecy.