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.

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.