Secret 2021 Methods Business: Exam 1 Discussion

Methods Exam 1 was today. We’ve been handballed a copy of the exam but haven’t looked at it yet, and we’ve not yet heard any reactions. People are free to give their thoughts in the comments below, and we’ll update this post as the dust settles.

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UPDATE (09/11/21)

Alright, it’s time for us to stop wagging school and get down to work. Obviously Methods 2 and Specialist 2 were the, um, more interesting exams, but we’ll begin with the relatively straight-forward ones.

There’s not much to say about Methods 1. It was too easy,* except for the final question Q9, which was nasty and, in parts, stupid. Other than that, there were no real issues that we could see. Below are our question by question thoughts.

*) In general “too easy” is significantly worse than “too hard”, although this is probably not the day to be trying to convince Specialist students of that.

Q1 Easy differentiations.

Q2 Way too easy antidifferentiation.

Q3 Easy trig question.

Q4 Easy enough graphing question, although there’s enough minus signs to probably catch people.

Q5 Easy and meaningless question on polynomial functions and transformations. The x-intercepts of g and h are most easily found by thinking of transformations, but presumably most students will do it the long way.

Q6 A nice binomial distribution question.

Q7 A very easy probability question, with pdf f(x) = k/x2 on [1,2]. The clarification that “k is a positive real number” is needless and clumsy. There’s also some standard and pointless pedantry, declaring the pdf to be “0 elsewhere”,  rather than just sticking to the natural domain.

Q8 An OK antidifferentiation question. Part (b) is most easily done with the second derivative test, which is not part of Methods. (We assume SDT is nonetheless permissible in Methods, although we have heard from an unreliable source that this is not the case.)

Q9 Not an intrinsically awful question, but an extrinsically awful question. As commenters have noted, it was nasty and idiotic to have all the difficulty in the exam concentrated in one final 20% question. And, as commenters have noted, there are parts of the question for which it is likely that many students will know exactly what is going on but will be tricked and slimed out of proper reward.

Part (a) in effect requires students to “show” that the line \boldsymbol{y = -\frac{x}{\sqrt3} + \frac2{\sqrt3}} is tangent to the circle \boldsymbol{x^2 + y^2 = 1}. As commenter John Friend noted, the question is most naturally and most easily done by plugging the line into the circle, and showing there is just the one solution. BUT, as John also noted, the VCAA assessors have traditionally and idiotically interpreted “show” to mean something like “derive”, precluding natural and completely valid “we know where we want to get to” proofs. It is difficult to believe that VCAA would be so stupid to enact such a policy on this question, but VCAA assessors can be pretty damn stupid.

The wording of Part (b)(i) is atrocious, but in effect asks students for which vertical dilations the line from (a) will still interesect the circle. The easy answer is -1 ≤ q ≤ 1, where q is the stretch factor, and that answer is wrong; as commenter Damien noted, the trap is that q = 0 is not permitted. Presumably, then, students who answer -1 ≤ q ≤ 1 will score 0/1 on this question. But note that students don’t have to score 0/1 for this obvious answer; since q = 0 has been excluded in the preamble to the question, this exclusion can then be considered to apply to any answer as well. So, if the Methods assessors decided, for once, to not be assholes, …

This is just a 1-mark question, but it is the kind of completely avoidable crap that makes VCAA so loved by so many. OK, sure, even though q = 0 makes perfect sense as a transformation, projections are not part of Methods and so you exclude it. But why not just exclude it by restricting q to be positive? You’d lose absolutely nothing of value in the question, and you’d stop kids stressing whether VCAA is, once again, going to be dicks.

Part (b)(ii) requires students to find for which vertical dilations the line from (a) will intersect the circle twice at points with positive coordinates. The question is ok, but it is over-egged, and it is way too much work for 1 mark.

Part (c) involves the determination and maximisation of the area of an associated triangle. The question is ok, although the not-obviously restricted domain will probably trick many students. In any case, we doubt students will do well on (c), since many will have been flustered by the earlier parts of the question.

64 Replies to “Secret 2021 Methods Business: Exam 1 Discussion”

  1. Well, since giving thoughts is free, here are some quick ones ….

    Questions 1 – 8: Benign.
    But the eyes of some students will \displaystyle glaze over at Question 6(b).
    The more convenient choices in Q8(b) were to test the sign of \displaystyle odd values of x on either side of 3.

    Question 9:
    (a) Unfair. Very unfair. Through its Examination Reports, VCAA have told students for many years to “show” something by finding it. There are numerous examples from Examination Reports where students were penalised for substituting the given result to get a desirable conclusion. And yet here we have a question where the only straight-forward way of answering it (it’s worth 2 marks, not 4 marks …) is to substitute the given equation into \displaystyle x^2 + y^2 = 1 and then find there’s only 1 intersection point (at x = \frac{1}{2}).

    Otherwise (for 2 marks) students might to do the following (boneheaded calculation):

    \displaystyle m = \frac{dy}{dx} = -\frac{x}{\sqrt{1 - x^2}} …. 1 mark

    Let x = b at P. Then:

    \displaystyle m = -\frac{b}{\sqrt{1 - b^2}} …. (1)

    \displaystyle m = -\frac{\sqrt{1 - b^2}{(2 - b)}} …. (2) (gradient joining the points A and P)

    Solve equations (1) and (2) simultaneously:

    \displaystyle -{\frac{b}{\sqrt{1 - b^2}} = -\frac{\sqrt{1 - b^2}}{2 - b} …. 1 mark

    \displaystyle \Rightarrow b(2 - b) = 1 - b^2

    \displaystyle \Rightarrow b = \frac{1}{2} …. 1 mark

    \displaystyle b = \frac{1}{2} \Rightarrow m = -\frac{1}{\sqrt{3}} …. 1 mark

    \displaystyle y - y_{1} = m(x - x_{1})

    \displaystyle \Rightarrow y - 0 = -\frac{1}{\sqrt{3}} (x - 2) …. 1 mark

    \displaystyle \Rightarrow y = -\frac{x}{\sqrt{3}} + \frac{2}{\sqrt{3}}.

    And there aren’t even enough writing lines for “Showing” in this way!!
    This change-of-heart by VCAA is very unfair and dishonest in the face of past advice given by VCAA through its Examination reports. Then again, I suppose students could simply use the right-triangle AOP to get the angle at A and then take the tan of the angle to get the gradient … (That would have saved on latex code!!) OK, I take it all back! Nice “Show” question, VCAA. Good “Show” question, VCAA. Now sit!

    (b)(i) Nice understanding of the effect of a vertical dilation on a line is required. But 1 mark …? Sheesh. It’s easily worth 2 marks:

    \displaystyle y = -\frac{q}{\sqrt{3}} x + \frac{2q}{\sqrt{3}} …. 1/2 mark

    Required inequalities for the gradient:

    \displaystyle \frac{1}{\sqrt{3}} \leq -\frac{q}{\sqrt{3}} \leq -\frac{1}{\sqrt3}} …. 1 mark

    \displaystyle -1 \leq q \leq 1 1/2 mark

    (b)(ii) Seriously …? Only 1 mark?

    Required inequalities for the gradient (the left inequality is worth 1 mark all by itself):

    \displaystyle -\frac{1}{2} \leq -\frac{q}{\sqrt{3}} < -\frac{1}{\sqrt{3}} 1 mark

    (gradient OP)(-1/Sqrt[3]) = -1
    => gradient OP = Sqrt[3]
    => tan(theta) = Sqrt[3]
    => theta = Pi/3
    => Max area = sin(Pi/3) = Sqrt[3]/2.

    And I wonder how many students who get (c)(i) will say 1 for c(ii) …?

    1. Yep, I said 1 for 9c(ii), I’m so annoyed at myself…

      For 9a, I said \cos\angle POA = \frac{1}{2} (using the right-angled triangle) \implies \angle POA = \frac{\pi}{3} to find the coordinates of P, and then used that to calculate the line. Is this right?

      1. Yep. But note that this blog is not typically to tell students whether their exam answers are right or wrong. So this comment is a one-off freebie.

        A free plug to the free solutions at itute: They’ll probably be published in the next 24 hours. Students can do post mortems until their heart is content using them.

        1. I have no problem with students asking such questions, although I probably won’t work so hard myself to answer them. Not at least today. If other commenters want to answer or ignore, or something in between, it’s up to them.

      2. Ah same. Annoying, considering the model does work for pi/2>theta>0 but the domain is restricted. Better make sure to read things more carefully tomorrow.

        Hopefully marks are awarded for finding the correct derivative in the next part either way.

      1. I just send Marty a message about this rather than read what was already here (woops!)

        Am I missing something, why can’t q be zero? q can be zero. I think it’s great if q is zero! Maybe I’m wrong. I’d love to be convinced that I am.

      2. Here’s my thinking on the matter:

        I thought question 9bi from today’s methods exam might be WITCH worthy. I feel that there is no need to make q non-zero, as this transformation would map the linear line onto the x-axis (which satisfies the condition of the question). I.e. -1<= q <= 1 is a good way to think about the question/transformation, but students will be punished for not remembering VCAAs weird q =/= 0 restriction.

        1. Hi MyCool.

          I agree with \displaystyle q = 0. But you have to remember that Pedant is VCAA’s middle name. VCAA likes to do things like this, to trip people up to try to make themselves look clever. (It tripped me up, no hard thing to do). Wait for the sanctimonious comment in the Examination Report (some time in 2025).

          VCAA is the Frank Burns of Victorian Education – Pedantic, incompetent, sanctimonious, arrogant, “infallible” …

          1. Yep, it tripped me up too…maybe that’s why I’m bitter. I think I quite like the essence of the question, but I feel bad for the kids who have a good grasp on transformations, but won’t get to separate themselves from the pack here because of that silly trap. There aren’t that many opportunities to shine on this paper, hopefully there’s some good opportunities today.

            1. Good luck today, MC, and everyone. I’ll be interested to hear how you go, and what people think. (I’ll put up a companion Methods 2 post soon after the exam finishes.)

    2. Also, do you think they will want a piecewise function for 9ci to properly account for always using the furthest left intercept for p’ as theta goes past pi/3
      Edit: realised restricting to 0<theta<=pi/3 covers all required q values so no piecewise needed

    3. Whoops, I timed out …

      (b) (ii) continued:

      1 (gradient OP)(-1/Sqrt[3]) = -1
      => gradient OP = Sqrt[3]
      => tan(theta) = Sqrt[3]
      => theta = Pi/3
      => Max area = sin(Pi/3) = Sqrt[3]/2.

      I wonder how many students who get (c)(i) will say 1 for c(ii) …?

    4. I agree about 6b – it was quite a setup to parse… but once a probability tree diagram was drawn it was straight forward. [I’m out of practice on those sorts of questions – and missed the solution the first time through]

      Q9a) I think you’ve gone overboard. Just use the triangle to find the angle and/or slope of the tangent line and go from there. The only fact you need is that the tangent is perpendicular to the radius – and that is given in the diagram for you.
      Q9bi) 1 mark is a bit stingy – but I think they just expected a quick drawing and some thought – no working, so hard to allocate another mark fairly – or maybe not.
      Q9bii) Maybe one mark for setup up and one for answer? But again, it would be a mess to mark…
      Q9c) I’m not sure why ci was worth two marks and bi / bii only one.
      Looking at the teacher facebook group, quite a few teachers said 1 for part ii 🤷🏽‍♀️

      1. Oops – I must have misread or missed your edit. You gave the triangle method for 9a!

        Also, you must be using quite a bit of Mathematica recently – the Sqrt[3] notation!

    1. send me an email to remind me and I’ll send you scans of the papers.

      Methods today and tomorrow, Specialist Friday and Monday.

      1. RF,

        How about sending scans to Marti to upload for non Method Bloggers before itute gets its act together?

        Steve R

        1. Sorry I don’t want this kind of discussion on the blog. If someone hints a reddit link or whatnot, that’s fine. But I’m enough of a target for VCAA without becoming a file-sharing site.

        2. Speaking for Marty, unfortunately he can’t do this because of copyright. He can post links but not the exams themselves (not even old pre-VCAA exams). A bit petty of VCAA, I think, given that it would be for educational purposes.

          1. Guys,

            Sorry for potentially causing trouble … I can wait

            I hope todays exams were fair and well vetted even if they were uninteresting

            Steve R

            1. Hi Steve.

              You haven’t missed much. The allocated marks in Q9 were not what I’d have used, and there were some odd restrictions for no good reason (that I could see). But in fairness to VCAA, the exam was dull, uninteresting and benign. And I mean that in a nice way. ‘Safe’ would be a fair assessment. (But others might see things more noteworthy of discussion than I did)

              And yes, I am seeing a doctor tomorrow about the concussion I suffered today just before the exam became available.

        1. Email addresses are not shared at these blogs. Some of us know email addresses through other channels.

          A tip: No-one should be disclosing an email address in a public domain, even if its a one-use-only address.

          I’m certain bootleg copies of the exam will appear in the coming days on the internet. And most teachers who visit here should be able to borrow a copy from their school’s VCE Coordinator. (And some non-teachers will know teachers …)

  2. Was definitely one of the easiest Methods exams for a while; most likely a reaction to the response last year. Much less computation required. The last question did require some thinking but that’s about it.

  3. I thought that it was a good exam overall to be honest, question 9 was challenging but nice, although I would have preferred more generous mark allocations there.

    Thank goodness VCAA killed the petition (https://www.change.org/studentvoicevic) to change the exam formats, imagine if it had to be rewritten. The whole thing would have been a complete trainwreck.

      1. Actually, the Victorian SRC managed to secure a proper meeting with VCAA and DET in which they successfully lobbied for the recommendation 2, the staggered return to school. Over 5000 signed the petition, it is quite clear that it was and had to be taken seriously.

    1. The “student voice”. Spare me.
      Proposals (1) and (2) were lobbied by several professional organisations (including schools). I don’t think the ‘VicSRC’ gets sole credit for (2) being successful, but I acknowledge that its input wouldn’t have hurt.
      As for proposal (3) …. Spare me. This is the hair-brained sort of thing that triggers the “Spare me” eye-roll.

      5,247/(50,000 plus*) is approx 10% of all Yr 12 students sitting exams, at best.
      Yr 11’s sitting a Unit 3&4 subject are not included in the 50,00 plus (but are possible included in the 5,247).
      * https://vcaa.vic.edu.au/news-and-events/latest-news/Pages/VCEExams.aspx

      Who vouches for the authenticity of the signatures? Did Yr 11’s sign? How do we know mum and dad didn’t sign too?
      I’m not familiar with this stuff, is 10% (at best) a good result? Something to be taken seriously? Seriously enough for proper meetings with VCAA and DET?

      Where’s the VicSRC petition for better quality VCE Mathematics exams?? Would a signature proportion of 10% (at best) secure a proper meeting with VCAA and DET?? I would have thought better quality exams was an important issue for student (and teacher) well-being. Maybe we should all align with the VicSRC and get some real action started.

      1. Personally, I feel that the number is an incredibly good result, even considering that many non-students would have signed. Even apart from that, the Victorian SRC is surprisingly influential in lobbying, they are involved with the senior secondary reform and the merging of VCE and VCAL, which they had previously advocated for. This influence is what made the petition so dangerous.

        Getting the Victorian SRC onto VCAA’s incompetence would be great, only problem is that they are 80% funded by the Victorian government and usually tend to avoid rocking the boat too harshly. Doubt Marty’s rhetoric would appeal to them.

        1. Aha. A conflict of interest, due to funding. Well then, it looks like the VicSRC does/(can do) nothing more than lobby for vanilla ice-cream. Rather than being a true agent for meaningful, positive change.

          Any change that doesn’t require rocking the boat is either superficial or trivial.

          Rhetoric? Marty’s blogs don’t contain rhetoric. Their content is based on cold, hard, incontrovertible facts. These facts are in almost every VCAA exam.

          Facts that are no doubt very inconvenient for some people and organisations.

      2. “Where’s the VicSRC petition for better quality VCE Mathematics exams??”

        Unfortunately it appears most teachers are unaware of the pitiful quality of VCE Mathematics exams, let alone students. In fact, as a student that is doing methods and spesh this year at a reasonably decent private school, the SACs we have done make the VCE exams look like flawless gifts from the heavens. It’s probably better at the top schools in the state (e.g. Scotch, Haileybury, Melbourne Grammar etc.) but it’d probably be much worse at a typical underfunded public school.

        1. “Unfortunately it appears most teachers are unaware of the pitiful quality of VCE Mathematics exams,”

          Yes, and that’s nearly the whole problem. Add apathy and the problem is complete. As well as the fact that some of those teachers end up writing and vetting VCAA exams.

    2. Huh! why doesn’t anybody ever tell me anything? I didn’t know anything about this.

      Of course VCAA has been a Morrisonian screw-up, but I agree with John: (3) was a pretty stupid idea.

  4. From a methods student..

    Q1 – Just standard differentiation stuff, comparatively easy to other years’ Q1

    Q2 – Simple antidifferentiation, again pretty easy but I feel it clashes with later question

    Q3 – Pet peeve, let ‘find the range and period of g(x)’ be one question like it usually is in exam 2. Interesting that they made us find general solutions instead of a restricted domain, imagine this would trip people up.

    Q4 – Fine again, just graphing. I imagine b will confuse some people as y2 but they will only solve y>=3 and leave it as x>=1… Like myself…

    Q5 – a is easy, surprised they didn’t use a matrix for b

    Q6 – a&c are fine, b is interesting and I think it would trip some people, despite it not being too hard, by merit of g not fitting directly into the conditional probability formula.

    Q7 – Easy integration, definitely worth only one and two marks.

    Q8 – A was probably the worst question in my opinion. Not difficult but the fractions demand more time than they’re worth. What’s more is this question tests the same concept as Q2, except for the antidifferentiation of the square root. B is nice though, maybe not worth 2 marks though. In my opinion, replace Q2.

    Q9 – Definitely the ‘hard’ question of the exam. A was nice, finding the angle then gradient requires knowledge of trig and perpendicular lines. Spesh people might fall into a trap, trying to use inappropriate strategies. Bi is nice too, though one mark is a bit cruel it can be solved just from looking at the graph. Bii should absolutely be two marks, as it requires a bit of working to find sqrt(3)/2. Ci, triangles are God. Restricting q is the hardest part of this question as many students may simply fail to consider it (For transparency – I did.). Cii is pretty fine, not having any local maximum sit in the domain is interesting but anyone who got this far would be able to notice the area is increasing for the domain and will solve for the correct value.

    Overall, pretty standard in my opinion.

    1. Hi Riley – you gave a really good breakdown of the questions!

      I especially like the observation that Q8a and Q2 test the same knowledge and Q2 could have been better used testing a different skill.

  5. I’m sorry for you guys that don’t have access to the exams. It’s idiotic, since there’s no reason for the paper to be public within 24 hours of the exam. But I’m not going to be the sharing guy.

    Sooner or later, here or elsewhere, someone will post a social media link of some type.

    1. Not sure if this was the invitation to share a link but here, if it’s inappropriate I apologise and feel free to delete it.

      It’s tied to a video, not ideal but it’s all I could find.

      ALSO some of the dude’s solutions are wrong in the video so beware.

      1. Yep, Riley, it was an invitation, and thanks very much. I also wouldn’t be too harsh on the guy: it was obviously designed to get rough solutions out as quickly as possible, and the video is labelled as such.

        1. Haha… Marty your interpretation was correct…

          28 mins to get it done and explain it a bit… Thought it was likely that I fucked stuff up! I hadn’t even glossed over it… Just jumped straight into it… I have created an updated video for future students.

          I have approval from VCAA to post the worked solutions… (After I received an angry phone call from he who shall not be named a few years ago 🙂 ) and I was trying to get them out for people to see the paper. I like to help students understand what VCAA wants… not what is always the correct thinking and/or correct answer!

          1. Thanks, Worm (and watch your language, please). I and plenty of others appreciate your hot takes. Feel free to link videos to the other exams, if you do them.

          2. I’m very curious about the VCAA approval – did it come from the Acting Mathematics Manager or from higher up? Email or phone?

    2. Apparently, and I’m only reporting what VCAA said, the exams don’t get published until after ‘Legal’ has signed off on them. Copyright issues etc. The sorts of things I would have thought had been sorted out \displaystyle before students sat the exam (except for English Exams: https://www.smh.com.au/education/sloppy-copy-in-exam-raises-ire-20111109-1n7eo.html#:~:text=By%20Jewel%20Topsfield&text=THE%20VCE%20exam%20body%20has,acknowledging%20she%20was%20the%20author)

      But apparently not and apparently it takes months.
      As Robert Ripley said about VCAA’s explanations, “Believe It or Not!”.

      Maybe VCAA should hire NESA’s legal team – the NSW HSC exams get published within days of students sitting them. And the \displaystyle Marking \displaystyle Scheme shortly after. Believe It or Not!

  6. Thanks very much to everyone who has commented. I’m not looking at the exam today, but will try to go through it reasonably thoroughly tomorrow.

    1. Actually. Neither of them… VCAA…

      Had nothing to do with the Head of Mathematics… that was who the original phone call came from to tell me to remove them unless I get approval.

      I emailed someone at VCAA who gives you the ok to use them. I couldn’t tell you her name?

      1. Worm,

        From one of those who don’t have direct access to the papers… many thanks for sharing your thoughts “legally”

        I liked your method of directly writing on the paper and thinking aloud on how best to use the Casio

        Steve R

      2. By Acting Mathematics Manager I meant the person currently acting as the Mathematics Manager at VCAA, not a school head of maths. I like to keep a score card of who does what at VCAA, but it’s no biggie.

        1. Yes that’s who I was referring to… I meant mathematics manager… head of maths at VCAA…

          I don’t work in a school.

          1. I see. How typical and unsurprising. No doubt some favour currying goon referred your Youtube to him. Good riddance to him, hopefully a new broom sweeps clean and can earn back the trust that has been totally lost over many years.

            @Marty: I’ve really bitten my tongue in this comment.

  7. Thanks again to everyone who has commented. It was very intreresting to follow, and it made my job much easier.

    I know everybody has moved on to pondering Specialist, or to getting drunk, or whatever, but I’ve posted my thoughts above. In the normal scheme of things, Q9 would definitely be a WitCH, but commenters have already hammered the question sufficiently hard. On to Specialist 1

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