Secret Methods Business: Exam 1 Discussion

OK, we should have thought of this earlier. This post is for teachers and students (and fellow travellers) to discuss Methods Exam 1, which was held a few days ago. (There are also posts for Methods Exam 2, Specialist Exam 1 and Specialist Exam 2. We had thought of also putting up posts for Further, but decided to stick to mathematics.) We’ll also update with our brief thoughts in the near future.

Our apologies to those without access to the exam, and unfortunately VCAA is only scheduled to post the 2020 VCE exams sometime in 2023. The VCAA also has a habit of being dickish about copyright (and in general), so we won’t post the exam or reddit-ish links here. If, however, a particular question or two prompts sufficient discussion, we’ll post those questions. And, we might allow (undisplayed) links to the exams stay in the comments.

UPDATE (21/11/20) The link to the parent complaining about the Methods Exam 1 on 3AW is here. If you see any other media commentary, please note that in a comment (or email me), and we’ll add a link.

UPDATE (23/11/20) OK, we’ve now gone through the first Methods exam quickly but pretty thoroughly, have had thoughts forwarded by commenters Red Five and John Friend, and have pondered the discussion below. Question by question, we didn’t find the exam too bad, although we didn’t look to judge length and coverage of the curriculum. There was a little Magritteishness but we didn’t spot any blatant errors, and the questions in general seemed reasonable enough (given the curriculum, and see here). Here are our brief thoughts on each question, with no warranty for fairness or accuracy. Again, apologies to those without access to the exam.

Q1. Standard and simple differentiation.

Q2. A “production goal” having the probability of requiring an oil change be m/(m+n) … This real-world scenarioising is, of course, idiotic. The intrinsic probability questions being asked are pretty trivial, indeed so trivial and same-ish that we imagine many students will be tricked. It’s not helped by a weird use of “State” in part (a), and a really weird and gratuitous use of “given” in part (b), for a not-conditional probability question.

Q3. An OK question on the function tan(ax+b). Stating “the graph is continuous” is tone-deaf and, given they’ve drawn the damn thing, a little weird. The information a > 0 and 0 < b < 1 should have been provided when defining the function, not as part of the eventual question. Could someone please send the VCAA guys a copy of Strunk and White, or Fowler, or Gowers, or Dr. Seuss?

Q4. A straight-forward log question.

Q5. For us, the stand-out stupidity. See here.

Q6. An OK graphing-integration question, incorporating VCAA’s \boldsymbol{f = f^{-1}} fetish. Interestingly, solving the proper equation in (b) is, for a change, straight-forward (although presumably the VCAA will still permit students to cheat, and solve \boldsymbol{f = x} instead). As discussed in the comments, the algebra in part (c) is a little heavier than usual, and perhaps unexpected, although hardly ridiculous. The requirement to express the final answer in the form \boldsymbol{\frac{a - b\sqrt{b}}6}, however, is utterly ridiculous.

Q7. This strikes us as a pretty simple tangents-slopes question, although maybe the style of the question will throw students off. Part (c) is in effect asking, in a convoluted manner, the closest point from the x-axis to a no-intercepts parabola. Framed this way, the question is easy. The convolution, however, combined with the no-intercepts property having only appeared implicitly in a pretty crappy diagram, will probably screw up plenty of students.

Q8. A second integration question featuring VCAA’s \boldsymbol{f = f^{-1}} fetish. Did we really need two? The implicit hint in part (c) and the diagram are probably enough to excuse the Magritteness of part (d), but it’s a close call. Much less excusable is part (b):

“Find the area of the region that is bounded by f, the line x = a and the horizontal axis for x in [a,b], where b is the x-intercept of f.” 

Forget Dr. Seuss. Someone get them some Ladybird books.


56 Replies to “Secret Methods Business: Exam 1 Discussion”

  1. Thanks for kicking off the annual VCAA Mathematics exam discussions, Marty.

    I’d like to acknowledge your “We had thought of also putting up posts for Further, but decided to stick to mathematics” comment first and foremost, as that made me chuckle audibly loud.

    I’m sure you’re aware of all the media coverage surrounding the two Maths Methods exams, so I await your views on it. Personally I think those exams were cruel to the students who have had a tough enough year as-is, and any and all hard work that they’ve put in during the year and during preparation was effectively wasted. Those exams gave no opportunity for students to show the results of their hard work.

    What’s interesting is that this is (AFAIK) the universal view across VIC, normally one person says the year xxxx’s exam is “hard”, another says it’s “easy”, and everything in between.
    However, this year, the universal saying has been “fucked”.

    I’ll leave it at that, and invite others to share their views too.

    1. Thanks, Steve. I actually haven’t seen (or looked for) media coverage of the exams. I saw one link to a 3AW interview with a disgruntled parent but, as far as I could tell, it was a content-free whine. That doesn’t mean there wasn’t valid cause for complaint, but there was no way to tell from the interview.

      Of course I’m more than ready to entertain complaints about Methods exams, and I’ll follow up a link to any substantive discussion you or anyone throws at me. As it stands, I’ve only gone through Methods 1 which, question by question, seemed ok to me.

      1. Re: The 3AW talkback from disgruntled parent (

        Spare me. I have no love for VCAA. However, I listened to that caller using the above link and it’s the typical mewling that you hear every year. But apparently a pandemic year warrants giving it oxygen (there were more people today mewling on 774 ABC Melbourne to Virgina Trioli about it). Where was the mewling in the media in 2016 (for example) about the glaring mathematical errors?

        The exam this year was too long and parts of it were undoubtedly challenging. However, the caller should get her facts straight:

        1. There were 8 questions, not 10.

        2. The format was NOT unfamiliar. It was the same format as every other year (and included the ‘generic’ Q1). As well, VCAA had previously provided the cover page to the exam as well as the adjusted Formula Sheet.

        3. Many of the questions were of a Yr 11 (Methods Units 1&2) standard (Q2, Q3, Q4, Q6a,b, Q7 (but c was tricky)) and could/should have been answered well by a reasonable student. Furthermore, there were plenty of routine Yr 12 questions (Q1, Q4 Q8). A reasonable student had access to at least 20 – 30 of the 40 marks.

        The 2020 exam was completely fair – there were no mathematical errors and there was no inappropriate content. *Every* student sat the same exam and *every* student gets considered for disadvantage as a result of the pandemic. Many students suffer socio-economic disadvantage every year but this year, more than any other year, the school a student attended was going to be a significant factor in his/her performance. I found the claim that a Methods teacher at the school could not do one of the questions very interesting. I can both believe and not believe this claim in equal measure.

        It’s been a strange year, made even stranger by my defence of VCAA. Parents who want to mewl and bleet about how unfair the exam was, that the questions were completely different to what their son/daughter expected etc. etc. should vent their spleen at their child’s school, not VCAA. Maybe some teachers told their students it would be an easy exam (or maybe some students just hoped it would be) because of the pandemic. VCAA have no case to answer this year.

        1. I have to say that it was an unfamiliar format, to an extent.

          Previous years only had one or two 7-9 mark questions, whereas this exam had three 8 mark questions. These questions were generally at the level you would expect them to be (difficult but not so difficult as to be unreasonable), but the fact that there were three of them likely caught many students off-guard.

          As for this: “I found the claim that a Methods teacher at the school could not do one of the questions very interesting.”

          Some of my friends who go to other schools have teachers for Methods that can’t solve some exam or trial exam questions, so I don’t find this very surprising at all. The majority of my friends taking Methods have private tutors, from which I gather are much more helpful than their teachers.

          1. Hi 2020 Methods Student.

            Re: this exam had three 8 mark questions [and] likely caught many students off-guard.

            The Cover page was available at the end of Term 3 and showed 8 questions. 40 marks and 8 questions – three 8 mark questions is not a big surprise.

            I spent a few minutes discussing the cover page and its possible implications with my students – I would have hoped most teachers did the same but maybe this didn’t happen (maybe there were students who didn’t even know about the cover page). Three 8 mark questions was always a likely scenario.

          2. Thanks, MS. I find it difficult to judge the level or difficulty of such exams, and it doesn’t really concern me. I gather that many students found the exam too long although, as JF suggests, perhaps the format should not have been a surprise.

            For me, the fundamental idiocy is having a 1-hour exam, a byproduct of VCAA’s tech-free tokenism, which is in turn a consequence of VCAA’s tech-fellating.

        2. Thanks, JF. I also didn’t see the exam as that hard, although I’m not the one to judge. I think you have to be a little careful with the “every student sat the same exam” argument, however; the more an exam is unreasonably hard or weird (or wrong), the more random effects will enter into each student’s grade. I’m not suggesting that is an issue with this exam, but I’m generally wary of the “same boat” argument.

          1. The exam was long and challenging in parts but it was not unfairly hard (as I’ve remarked above). It makes me wonder what’s happening at the Methods 1&2 level (but I don’t wonder too much because I know exactly what’s going on at those levels). So much of the VCAA Methods Exams every year can be done solely from Methods 1&2.

            I say to Methods Units 1&2 students that they are actually doing the first year of a two year course in Methods Units 3&4.

            This year has been a shit year. It’s the sort of year that can magnify the good but certainly magnifies the bad. I think that major contributors to the issues many parents and students have been complaining about (unfamiliar format, questions on stuff not taught etc.) is the result of:

            1) the magnification of poor teaching.

            2) the magnification of socio-economic imbalance.

            3) the magnification of the inherent difficulties of a student who would have been mathematically borderline at the best of times.

            The “Every student will be considered for disadvantage” crap that the Merino Sheep (MS) irresponsibly bleated, ambushing VCAA, VTAC and schools alike, will address 2) but it’s impotent with 1) and problematic with 3). By the way, I’m calling out what the MS said as crap because there is and has always been a robust process for considering disadvantage:


            *THAT* is the process that should have been followed and would readily have dealt with 2) and 3). As it is, I have no doubt some schools will abuse the open slather on disadvantage that the MS encouraged and this will be to the detriment of schools that have tried to do it fairly, as well as to the detriment of students who might have performed better than otherwise expected as a result of drawing on inner qualities that often do not get valued as much as they should.

            As for 1), the MS knows what we all know but would rather bleat rhetoric than effective policy. It is 1) that these mewling parents should be targeting. But I guess it’s easier to blame the faceless VCAA and its exams than face the reality of 1) and 3). And just to take a free kick at the Vampire Institute of sucking money from Teachers, I don’t see that teaching standards have improved one little bit after 12 or more years of those feckless bloodsuckers: 1) is alive and well and kicking. But I see a lot of extra bullshit that teachers have to put up with (which arguably hinders good teaching by wasting valuable time).

            Maybe the mental health of students and teachers would be significantly improved if the current VCE system and tertiary entry and all the gravy-train add-ons went the way of the dodo. If 2020 drives significant change in that direction then some good will have come from it.

            1. Thanks, JF. Your suggestion that the plague has magnified existing weaknesses and inequalities rings true. I also have no doubt that this year will be the start of substantial changes to schooling and assessment; I very much doubt that these changes will be for the better.

              1. Indeed. Expect to hear more about the *blended classroom*. (How else is the Govt going to pay for tutors in every school?)

                I think the only reason there’s a perception that the exam was too hard, unfair etc. this year is because the media gave some mewling parents a platform to bawl from this year. Every year students, parents and some teachers bleat about how the exam was too hard, unfair etc. This year the bleating is getting oxygen (presumably because it fits in with the whole 2020 narrative) and so there’s an exaggerated sense of how hard the exam was. The factual basis of some of what I’ve heard is so non-existent as to be almost Trumpesque.

                I also think there’s been an exaggerated sense of entitlement (I’ve had a shit year so I deserved an easy exam, it’s been a shit year so my son/daughter deserved an easy exam). Seriously, how could someone who’s done the past VCAA exams and seen the cover page not be prepared for it?? It’s much ado about bugger all but highlights a deeper problem.

                (Boo hoo the race was up hill, against the wind and it was raining. So unfair, I didn’t expect it. I wasn’t prepared for it.
                Well guess what – where you finish in the race is still relative to everyone else, regardless of uphill, wind, rain etc.)

  2. IMHO – the exam was weird. More so than I’ve come to expect. And it felt long when I was working through the paper later that day. Not as many “free” marks and a lot of places where a student who was reading too quickly could make a mistake and be heavily penalized; none of which I think helps give a fair assessment of ability.

    That said, Paper 1 exams recently have felt a bit on the easy side, so maybe this was the pendulum swinging back the other way? At least until we saw Paper 2…

      1. Will update next week (have deliberately left my notes at work to ensure the lawns get mowed this weekend), there were specific questions I thought were unusually difficult (for early in the paper) and the exam, on the whole, had a weird feeling about it that for the moment I cannot articulate.

          1. OK, so I found nothing *wrong* with the paper and IF you ignore the rubbish “context” for some of the questions, I thought some of the questions were pretty good.

            HOWEVER, I did feel that the difficulty gradient was a bit steeper this year – by about Q3 I was starting to think of those students who obtain a study score of 22 – 28 and wondered how many of them were feeling under pressure a lot earlier in the paper than in other years. There were less “easy marks” I felt, although there were possibly also less “really difficult marks”.

            Did I think the paper was fair? That is a very difficult concept to define. There was nothing on the paper outside the curriculum, a few skills were tested a lot more than others, nothing new here. For a student who was going to get a study score of 35+ it was probably a very fair exam and will give a fair ranking for those students. For the lower end… I’m not so sure.

            While I am all for testing algebra skills, this paper I don’t think will spread the lower end as well as some years. Is that a problem? Possibly not, I have no idea how the scaling is going to work this year.

  3. Not a 3/4 Methods teacher, but my reaction was:

    1) Very few easy marks
    2) About 5-10 minutes too long, because of all the bits and pieces 1 mark / 2 marks questions and also because some questions unnecessarily and unreasonably prescribed answers in a certain form.
    3) BUT nothing tricky, because everything was either a standard / straightforward question, or had appeared in exams from prior years.

    Maybe slightly harder than average, but not unreasonably difficult.

    1. Thanks, SRK. As I replied to Methods Student, the 1-hour format is insane. In particular, a little extra fiddling over a little extra itsy-bitsy marking can very easily screw up a sprint exam.

  4. Hi guys,
    I have to admit, I think the exam was different this year.

    The forcing of particular solutions has always been prevalent in Specialist Maths, however, it’s much less likely in Methods and usually it’s a basic logs or indices question.

    I have taught Unit 1-4 Methods and Specialist for 15 years in schools (mainly public) and it’s really obvious that the Methods papers were written by Specialist & Further teachers. Since when have we referred to the gradient as the slope in the methods exam? Let’s confuse the shit out of students with terminology we have never used in the past Methods exams.

    In regards to teaching standards… Oh my… I left teaching two years ago as I was starting to get stressed out by incompetent colleagues… I was constantly picking up the pieces for their inability to teach and stressed out of my brain, never getting to go to the toilet during the day as I had lines of students from other classes seeking help in their free periods.

    I have an ex-student who struggled at Maths Methods who is now a Leading Teacher at a school as the head of maths and there is no way they could get 70% on Exam 1 in the time alloted.

    Students are being taught the wrong thing (if they are being taught anything… Students now complain at my office that teachers write down the notes from the textbook on the board then sit down… No explanations and rarely help…) And VCAA changes its mind certain areas of the course by the year in the paper.

    Recently they have told us to teach students that endpoints are not differentiable, then they get students to find the equation of the tangent at an end point in exam 1 in 1997.. As this is now possible (yes, we know it’s possible if the function is continuous on a larger domain etc…), But it confused one of the most brilliant students I have ever taught (he dropped to a 46 in methods in year 11… Then in year 12 was awarded the premiers prize in spesh, after he dropped one mark in exam 2 due to that stupid fucking water fountain question… How far from the top of the fountain…. Blah blah) … They regularly throw stuff into the exam that’s not in the study design and I am finding myself teaching students many concepts “just in case” they look at gradients of inverse functions or intersections of inverse functions… Another concept that is lost on them…

    Anyway… Long story short… The exam wasn’t hard… In fact, I thought the concepts where easy, but to examine the concepts they did with the solutions they wanted is enough to intimidate the best students. Make one slight numerical mistake and you’re stuffed…

    They’ve never the given a rational function to integrate with a fractional substitution… Why would you do that?

    Many of the questions were worth more marks than the paper suggested… Hence, why it was considered long! I would love to see the marking scheme… It will be such a pile of shit… It won’t reflect a consistent marking scheme… It can’t…

    Anyway… I think I went well and truly off on a tangent… Students in most public schools are screwed now, as there are not many capable teachers in the system and the ability to mentor new teachers is impossible will the workload imposed on the more competent teachers.

    And why would you teach if you can tutor students in groups and earn $300-600k per year? And work less… Have less stress?

    1. OK, serious question…

      I’ve now heard from multiple Methods teachers that they think this years paper was written by a Specialist teacher.

      What does that mean? I teach all subjects, so am I a Specialist teacher?

      Do exams written by Methods teachers who do not also teach Specialist have a different feel/look? How?

      Genuinely curious.

      1. It’s a strange one, RF, for sure. Maybe it’s a reference to the number of questions requiring an answer in a particular form …? Maybe it has become a meme …?

        I think it’s a ridiculous, meaningless and unfair statement, a strange attempt to somehow justify the baseless claim that the exam was hard and unfair.

        I think only someone who’s made this proclamation can explain its rationale.

  5. Hi W.

    Re: “just in case they look at gradients of inverse functions”

    I assume you mean questions like the ones on VCAA 2019 Examination 2 Section A Q15 and VCAA 2018 NHT Examination 2 Section A Q15 where you need to know that if \displaystyle g(x) = f^{-1}(x) then \displaystyle g'(x) = \frac{1}{f'(f^{-1}(x))}.

    Yep, you won’t see that in the Study Design but it’s become a very popular type of question. The Study Design is a useless piece of shit. It lacks clarity, specificity and examples. And then there are its really bad points …

    Re: “or intersections of inverse functions”.

    VCAA have had an unhealthy love-affair with this for many years now. It’s inexplicable. And in the tech-free exam it always requires solving f(x) = x without any justification. Ditto all the idiot writers who include it on their trial exams.

    Re: “Many of the questions were worth more marks than the paper suggested… Hence, why it was considered long! I would love to see the marking scheme… It will be such a pile of shit… It won’t reflect a consistent marking scheme…”

    I agree with all of this. The VCAA marking scheme would be an interesting read. But of course it’s secret (unlike in NSW where the marking scheme gets published – within weeks of the exam!) and there’s usually only one reason why stuff like that is kept secret (you’ve already touched on it). Of course, if you’re lucky, a VCAA marking scheme might blow into your face on a windy day while you’re out exercising.

    Re: “They’ve never the given [sic] a rational function to integrate with a fractional substitution… Why would you do that?”

    What do you mean? There’s no question like that on the Methods Exam 1 that I can see. Furthermore, integrating using substitution is not on the Methods course. But that doesn’t stop some idiot writers from including it on their trial exams ( see Q3).

    Re: “teachers write down the notes from the textbook on the board.”

    Yes, I’ve seen plenty of that. It makes me really angry. Many of those teachers would have loved remote teaching this year. It would be a contributing factor for all those parents mewling on talk-back radio about how hard and unfair the exam was.

    1. Hi John,

      Just wondering as a student: is your only objection to the g'(x)=\cdots result that it’s not listed on the study design? The first time I stumbled on those types of questions, I derived the result by noting that g(f(x)) = x, so g'(f(x)) \times f'(x)=1 by differentiating both sides, before substituting x\to f^{-1}(x). Assuming this is correct, I thought this was a really nice/creative* application of the chain rule, and it seems like a shame something like “applications of the chain rule” isn’t codified into the methods SD (as it is for specialist.)

      *with respect to the rest of the course, at least.

      1. Hi, Student. I’m sorry about the delay in your commenting appearing. I just found it in the trash. Not sure how it got there.

      2. Hi Student.

        What you’ve done *is* correct *and* nice. It’s how I derive this formula for my classes. But I don’t think students should have to derive this result by themselves, especially for a multiple choice question. And if it was required for a short answer question I’d expect some simple scaffolding. So yes, my objection is that it’s not listed in the Study Design. Such applications of the chain rule should be explicitly listed.

        Related to the chain rule: The best Methods students might see a link between the chain rule and integrating f'(x)/f(x). Nevertheless, such integrals are not listed in the Study Design and students should not expect to meet them in a tech-free exam. Students should not have to derive the required result for a short answer question – I’d expect suitable scaffolding if such an integral appeared in a question, in which case the question becomes a simple ‘integration by recognition’ question.

        I think you’ve raised a broader issue: What should be explicitly listed in the Study Design? Another example that comes to mind is the whole shonky business of solving f(x) = f^{-1}(x) in Exam 1 by solving f(x) = x. VCAA have an unhealthy love affair with this type of question. Given the extensive examination precedent, it should be explicitly stated in the Study Design that solution of the former is expected in a tech-free context by solution of the latter *when f(x) is a strictly increasing function*.

        The current Study Design (and it’s previous iterations) is a useless piece of shit. It lacks clarity, specificity and examples. Teachers are constantly having to guess aspects of the curriculum by what appears on the Exams and Examination Reports*. And things become exponentially confusing when VCAA justifies the inclusion of stuff NOT on the course with mealy-mouthed “there’s a natural connection to the Study Design” statements (see here: The problem is that the Study Design gets written by … *ahem* I need to bite my tongue here and be circumspect in what I say … people who in most cases don’t have the necessary skill-set to produce a decent document.

        *Even worse, some (inexperienced) teachers look to trial exams for guidance. This creates real problems when idiot writers include stuff on allegedly reputable trial exams that is not on the Study Design. See Q3 here:

      3. Hi, Student. That’s a nice argument, although it is worth noting that there is a gap in it. Your argument assumes that the inverse function g is differentiable, but this assumption is unnecessary: the differentiability of g follows from f having a non-zero derivative. The proper proof, which hovers around but doesn’t use the chain rule, requires some subtle but not-too-hard fiddling. In the high school context, however, the argument you’ve given is fine. (Being a mathematician of little brain, I find the Leibniz form, dx/dy x dx/dy = 1, much easier to think about.)

        Of course the main point, as JF points out, is that none of this inverse pondering should be your job. Either inverse derivatives are in the curriculum or they aren’t, and it’d be kinda nice if VCAA made up their damn mind.

    2. “teachers write down the notes from the textbook on the board”. I had a teacher when I was a graduate student in the US who came to class with the text book under his arm; he did not refer to it during the class; and he wrote constantly. I took copious notes…until I realised that in a class he had written 5 pages of the text verbatim on the board without referring to the text. After that I did not take any notes. Instead I would go to class with an envelope and pencil and *listen*, occasionally noting something I thought was important. I knew that I could go home and read the details. A memorable year, and an unforgettable teacher. PS It was an excellent text – a classic.

      1. Not sure what your point is in this context?

        The teachers copies the notes onto the board and they don’t explain them, or talk about them for that matter. Their idea of teaching is writing the notes from the textbook verbatim to the board…

  6. Sorry, I meant integrating a rational function with terminals that were rational… Typing on an iPad not my computer as I CBF getting my PC, so I was trying to get it across without typing too much… Being lazy…

    Hmmmm… Marty, I was taught by the best… Monash Uni 1997-2000… I used to have a crazy Einstein looking guy who didn’t wear any shoes… (LOL I also hate shoes… Taking them off is frowned upon in a school… But my students got used to it!) I played you in squash too about 20 years ago…

    I also remember another lecturer… latanzi? Possibly… Funny fucker… Very dry sense of humour…we got along well!!!

    I was not a great fit for most schools as principals don’t like to have anyone questioning them!!! With one exception… Gerry Schiller… Great man and an exceptional leader…. I left Glen Waverley when he was about to retire 🙂

    1. Hi W.
      Re: Sorry, I meant integrating a rational function with terminals that were rational.

      I still don’t understand. Can you clarify? The only rational functions that can be integrated within the scope of the current Methods course are functions that algebraically reduce to (sum of powers of x) + (constant)/(Linear). Can you give an example or a question from a VCAA Methods exam that clarifies your comments.

      (Maybe you’re referring to some methods *trial exams* where the idiot writer gives an integrand of the form f'(x)/f(x) where f(x) is not a linear function: see Q3)).

      Re: Glen Waverley.
      So you would have taught with Mr Marshall (now retired). An outstanding maths teacher and person.

      1. JF, I think Worm means integrating a power function with a rational-but-not-integer exponent, with rational-but-not-integer terminals. (Question 7c from this exam).

        1. Yes. I can’t believe I just wrote the same thing twice… and explain it incorrectly.

          Non polynomial.. index rational… not a rational function…

          I’m tired. This last 6 months has killed me. Bloody online learning…

        2. Hi SRK. I think you mean Q6(c). If that *is* the salient question, overlooking yet another stupid intersection of f and finv requirement, I don’t see what the problem is with calculating the integrals. Surely we should/must have an expectation that a semi-reasonable Maths Methods student can calculate \displaystyle \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{2}} \sqrt{x} - 2x^2 dx and \displaystyle \int^{1}_{\frac{1}{2}} 2x^2 - \frac{1}{\sqrt{2}} \sqrt{x} dx …? (4 marks)

          Furthermore, my unapologetic expectation is that a semi-decent Methods *Unit 2* student should be able to find the value of the derivative of \displaystyle \frac{1}{\sqrt{2}} \sqrt{x} - 2x^2 at \displaystyle x = \frac{1}{2}.

      2. Ian is a legend! Great fun… great sense of humour! Taught maths extremely well for a Chemistry Teacher! 🙂 He was exceptional with the middle school kids as well… I always struggled with this ..

    2. “crazy Einstein looking guy who didn’t wear any shoes”. Nope, doesn’t ring a bell.

      You’re referring to John Lattanzio. He was a good guy, most of the time, and anybody who has a Loudon Wainwright III song written about him about him can’t be all bad. But he and his semi-mathematical colleagues made the place very painful to be.

  7. I don’t think the calculating of the integral is a problem… but they’ve never done anything like that before…

    Also, subbing in 1/2 to x^(3/2) is something they’ve never done before (that I can recall) in an exam…

    This threw a lot of students off, especially seeing as you have the rationalise and simplify the equation and put it in a specific form. There is so much room for error and is it really testing the skill they want too?

    I know the question is easy for us, however, Methods is *only* the middle maths in VCE and I think there needed to be some questions that wanted to see the skill of antidiff, without the surds etc…

    Personally, I think the Methods exam required more algebraic manipulation than the spesh exam… and it shouldn’t…

    1. There are many examples from VCAA Exam 1 (and commercial trial exams and textbook questions) where integration involving surds is required. 2017 Exam 1 Q9 immediately comes to mind. A reasonable student who has done a reasonable amount of work would not be put off by the integration required in this year’s exam.

        1. Worm, the 2017 question is discussed here; to my eye, the manipulation in the 2017 question was similar to but simpler than in the 2019 question. (The 2017 question was a monumental screw-up for other reasons.)

          On the algebraic fiddliness of the 2019 exam, I agree with you, and with John, and with myself. I think you are right, that the exam feels more algebraically fiddly than usual. Plus, the requirement for answers to be in particular forms has blossomed from an occasional nervous tick into a full-blown psychological disorder. It’s not just that these forms can cause more work; the request for these forms is insane. But John is also correct, that it is absolutely ridiculous if the manipulation required in 6(c) is now considered too difficult.

          I also don’t know how you get from there to here. Even assuming VCAA now has clear and reasonable expectations of VCE students – they don’t, but let’s pretend they do – how do they now implement those expectations? How does VCAA reestablish the standards that they’ve been trashing for the last thirty years?

        2. The integral is easy. And yes, I think the substitution is easy, and the simplification to the required form is easy:

          \displaystyle \left[ \frac{1}{\sqrt{2}} \times \frac{2}{3} x^{3/2} - \frac{2}{3} x^3 \right]_{0}^{\frac{1}{2}} + \left[ \frac{2}{3} x^3  - \frac{1}{\sqrt{2}} \times \frac{2}{3} x^{3/2}\right]^{1}_{\frac{1}{2}}

          \displaystyle = \left[ \frac{1}{\sqrt{2}} \times \frac{2}{3} x \sqrt{x} - \frac{2}{3} x^3 \right]_{0}^{\frac{1}{2}} + \left[ \frac{2}{3} x^3  - \frac{1}{\sqrt{2}} \times \frac{2}{3} x \sqrt{x} \right]^{1}_{\frac{1}{2}}

          \displaystyle = \frac{1}{\sqrt{2}} \times \frac{2}{3} \times \frac{1}{2} \times \frac{1}{\sqrt{2}} - \frac{2}{3} \times \frac{1}{8} + \frac{2}{3} - \frac{1}{\sqrt{2}} \times \frac{2}{3}  - \left( \frac{2}{3} \times \frac{1}{8}  - \frac{1}{\sqrt{2}} \times \frac{2}{3} \times \frac{1}{2} \frac{1}{\sqrt{2}}\right)

          \displaystyle =  \frac{1}{6} - \frac{1}{12} + \frac{2}{3} - \frac{\sqrt{2}}{3} - \left( \frac{1}{12} - \frac{1}{6}\right)

          \displaystyle = \frac{1}{12} + \frac{2}{3} - \frac{\sqrt{2}}{3} - \frac{1}{12} + \frac{1}{6}

          \displaystyle = \frac{4}{6} - \frac{2\sqrt{2}}{6} + \frac{1}{6}

          \displaystyle = \frac{5 - 2 \sqrt{2}}{6}

          Sure, a bit tedious but not difficult, particularly if each step is written out. And the terminals have been found in an earlier part so it’s essentially 4 marks (= 6 minutes) for the above. Hardly the sort of stuff that should throw off a semi-decent Methods student. The exam is not for trained monkeys and parrots. If a student can only do things that s/he has rote learnt and is ‘thrown off’ by anything a little bit different (and I’d hardly consider the above calculation to be unusual or different), then that student should not be doing Methods Units 3&4.

          And yes, the terminals on the 2017 question were easier (0 and 1). But maybe the integration and resulting fractions were a little harder …

          I’m happy for anyone to point out anything at all in the above calculation that is unreasonable and would throw off a decent student.

  8. “Re: “just in case they look at gradients of inverse functions”

    I assume you mean questions like the ones on VCAA 2019 Examination 2 Section A Q15 and VCAA 2018 NHT Examination 2 Section A Q15 where you need to know that if g(x) = f(x) then g'(x) = 1/{f'(f^{-1}(x))}.”

    I usually explain this to students as “x and y” have been swapped, so rise and run have been swapped and the gradient is therefore the reciprocal. Am I dumbing it down too much?, they seem to grasp it better.

    Thank god Chem was good today, its cheered up the troops.

  9. Marty, I’ll disagree with your comment that in Q3 “The requirement to express the final answer in the form \displaystyle {\frac{a - b\sqrt{b}}{6} … is utterly ridiculous”.

    Surely asking that \displaystyle \frac{5}{6} - \frac{\sqrt{2}}{3} be simplified in this way is not asking too much? (And it lets a student to check the ‘reasonableness’ of his/her answer: Can’t get the required form? Go back an check for a careless mistake).

    1. John, this one is not as bad as other “in the form of” questions, but it is bad. The form doesn’t serve any purpose, which means there is no point asking for the answer in that form. All I think it does is promote autopilot grading and, consequently, nitpicking. It’s not a great question to begin with, since the understanding is minimal and is pretty much just heavy computation, to no great end. You really think if a student gets through that slog and finishes with \frac56 -\frac{\sqrt2}{3} or \frac{5-\sqrt8}{6} or, a bunch of other perfectly reasonable answers, that they should be docked a mark?

      In all my ninety years of setting uni assignments and exams, I don’t remember ever demanding an answer be expressed in a particular form.

      1. OK, fair points. I agree that it’s probably done mainly to simplify the marking (by requiring a unique answer) and students certainly don’t deserve to lose a mark if they get to the finish line with a correct but unsimplified form.

        It’s easy for me to assume that VCAA puts laziness ahead of common sense and student welfare.

        But maybe someone who pays a lot of money to attend a

        ‘Meet the people who present the Examination Report 6 months before it’s published and don’t answer any questions outside the scope of that report’

        can ask what is the rationale behind ‘give your answer in the form’. (If you don’t ask, you don’t get [even though we all know you won’t get]).

          1. Yes, I was going to mention the non-uniqueness in general. But in this case the required form of answer actually is – surpise! – unique.

            In fairness to the required form in Q5(b), the required form of answer is unique in the sense that the of infinite number of correct forms, only one emerges ‘naturally’ from the calculation.

            Digression: The GAT. In previous years, some students have sought to slip the so-called GATchphrase into one of the writing tasks. This year’s leading contender, according to social media at least, is “GAT on the beers”, in honour of Premier Daniel Andrews’ stern warning not to visit mates’ houses for drinks.

            My point: I don’t think many students will be looking to slip an ‘unnatural correct’ form into this question.

  10. Marti and other frequent commenters,

    Are the 2020 papers available online anywhere so that the Magritte qualities of the comments can be fully appreciated

    Steve R

    1. Hi, Steve. They’re not officially online until sometime around the second coming of Christ. There are reddit-like links around. I won’t hunt them or post them, but if someone commented with such links, I probably wouldn’t delete the comment.

      1. Nothing wrong with seeing the publicly available solutions 😉

        As Marty implied, VCAA hate making the exams available in a timely way. It’s inexplicable, given that NSW publishes the HSC exams within a fortnight. But VCAA prefers laziness over common sense.

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