WitCH 68: Implicit Offense

This is an old one, from the 2013 Specialist Exam 1. The examination report gives the answer as c = -3/4.

UPDATE (02/11/21)

The question is weird, talking about a point(s) on a curve that has yet to be defined. It’s just not a natural question. The major problem, as commenters have noted, is that it is simply ridiculous to talk of “a gradient” at two separate points which have different gradients. One can argue whether the question officially makes borderline mathematical sense. One cannot argue whether or not the question is ridiculous.

56 Replies to “WitCH 68: Implicit Offense”

  1. When a connected component of the solution set to the given equation contains points of the form (1,y), it either contains one point with an undefined gradient or it contains two points (of the form (1,\pm y_0)). There is no “gradient where x=1“. There are two candidate numbers (and two candidate implicit functions). Due to the reflective symmetry across the x-axis they will be \pm m(c).

    The answer to this question is: such a c does not exist.

    We can of course assume that whoever wrote the question doesn’t understand what they are talking about, and find the value of c such that m(c) = 2. But that doesn’t fix the question, it doesn’t answer it as stated. The solution is wrong, and furthermore the whole question is representative of how the entire curriculum is completely inadequate around the study of solutions to f(x,y) = c. What a sad situation.

    1. What’s the problem? There’s no rule requiring y to be a function of x, it’s a perfectly well defined curve and has a well defined gradient at each (x,y) on the curve except for y = 0 where it is vertical.

      1. ET, to steal from Gertrude Stein, there is no there there.

        The expression “where x = 1” is just really weird, since for the supposed correct value of c there are two such points, one with gradient 2 and the other with gradient -2. As I suggested below, there is a reading of the question which makes the report’s answer valid. But it’s a strained reading, which doesn’t save the question from being pretty crazy.

      2. No, it isn’t a perfectly well-defined curve. It is not even a curve. It is a collection of curves. The question as written has an answer: no such c exists. The question asks for a gradient when x=1. The solution set, as I explained, does *not* have a gradient when x=1. There is a set of two possible numbers that could be gradients. The question does not ask for one member of the set of gradients when x=1. It does nothing to eliminate the second possibility.

        That means the exam was marked incorrectly.

        Now I don’t know about you, but I’d say that a question with the wrong answer is a serious problem in any exam.

        1. Glen, the VCAA mathematics unit doesn’t give a flying Philadelphia fuck that this or any other question is wrong. It has no conscience, no morals, no ethics, no remorse. But it has plenty of arrogance.

          Look at it’s response to Specialist Maths 2019 Exam 2 Section A Q12 even in the AMENDED Examination Report (the original Report was silent on this error. The Report got amended following strong criticism of this question but still couldn’t admit error).

          Look at the Methods 2016 Exam 2 AMENDED Report for Q3 part (h). I’ve attached the original report. In neither Report is there acknowledgment of and remorse for this inexcusable error. Heads should have rolled. The way I hear it, it took over a year for this Report to get amended – weak as the amendment is – because it was continually stone-walled until someone higher up with some integrity was forced to step in.

          The error discussed in this blog is small fry compared to the above two errors. Yes, I agree. Whichever way you look at it – a defective question or a question with the wrong answer – it’s a serious problem in any exam. But VCAA mathematics does not care, because it’s not held accountable. Year after year, its lackeys ‘present’ at Meet the Con Artists and refuse to speak of these errors. And attendees to this circus fully sanction this. Year after year ‘Exam Solutions’ are sold that deliberately and knowingly whitewash these errors. And teachers keep buying.

          Until the VCAA Mathematics unit and its stooges are held genuinely accountable by teachers, it will continue not to care.

          2016mmexam2report-original

          1. Thanks John. I agree with you, of course. Here are two things I think of while reading that: (1) I care, and so long as I care, I will continue to denounce it. (2) VCAA *should* care.

            Thanks for the links, they are outrageous.

  2. (Please help with the LaTeX if my formatting is wrong…)

    I was not teaching VCE at the time and so have not actually done this question until today, so please forgive any glaring errors…

    I approached the question in two ways:

    y^{2}+\frac{3e^{x-1}}{x-2}=c
    So 2y \frac{dy}{dx}+\frac{3(x-2)e^{x-1}-3e^{x-1}}{(x-2)^{2}}=0
    Then we are told when x=1, \frac{dy}{dx}=2
    So we can write 4y-6=0 and therefore y=\frac{3}{2}
    Then \frac{9}{4}-3=c
    So c=\frac{-3}{4}

    It is my second approach that I’m not so sure about (even though there is clearly something wrong with my first approach given this is a WiTCH):

    When x=1, y^{2}+\frac{3e^{0}}{-1}=c
    Which means y^{2}=c+3
    And therefore y=\pm \sqrt{c+3}

    The presence of the \pm symbol now has me worried that there may be two answers.

    Or, have I missed the point entirely (again)?

    1. Hi RF. I think drawing the graph of \displaystyle y = \pm \sqrt{c - \frac{3e^{x-1}}{x-2}} for a value of c, such as \displaystyle c = - \frac{3}{4}, will ameliorate your worries.

      By the way 1. Re: Latex. You can make your fractions bigger by using the command \displaystyle at the start of the latex environment. Compare c = \frac{3}{4} with \displaystyle c = \frac{3}{4}.

      By the way 2. VCAA continues it’s love affair with superfluous brackets that – to me, anyway – just make the equation aesthetically uglier than it needs to be.

      1. Thanks JF. I really don’t use LaTeX much as very few (I suspect myself and one other, on a fixed-term contract) of my colleagues are able to edit documents if I do it this way… nonetheless, your assistance is appreciated.

        Brackets… yes… they do have a purpose but does writing e^{(x-1)} instead of e^{x-1} serve that purpose? I think not.

        1. Hi, “PN”. I renamed you on your three comments, to avoid confusion with another guy who signs themselves PN and has posted a lot here. I’ll change your name again on these comments to whatever you want (within reason).

          Also, and it may sound strange on this blog, coming from me, but I’ll ask you to tone down your language. (I’ll be asking, and re-asking, others as well.) I’m obviously no prude, and sometimes you just gotta scream. But in general I’ll be looking for people to use bad language sparingly, and only for purpose.

    2. RF, your first way is what was expected, and your second way is correct: for c = -3/4 there are two gradients, at different points where x = 1.

      What your first approach shows is that IF there is a point with x = 1 and dy/dx = 2 THEN c = -3/4. But, the IF-THEN is not reversible. It doesn’t rule out another point with c = -/34 and x =1, and with possibly (and actually) a different gradient.

  3. I think

    “… where x = 1 and y >0.”

    would fix the question.

    (I often wonder whether people writing questions like this actually look at a graph of the relation they’re giving).

    1. Well, you could specify that we take a logical and with the condition

          \[y<0\]

      . There are always ways to fix questions.

      The question as asked has an answer though: no such c exists.

  4. The more I ponder this question, the more worried I am that it was not noticed by the vetting panel (I assume there was one) that any curve that starts with y^{2} will, most likely, not be a function (in the VCAA sense of the word) and therefore the notion of a unique gradient is… misplaced?

    I’m also guessing that no-one on the vetting panel tried to sketch the curve to see what it looked like, despite VCAA’s infatuation with technology for this purpose.

    1. RF, I’m wondering if perhaps the writers and vetters were aware of the double-answer thing but thought it was ok. The question requires the curve has *a* gradient of 2 where x = 1. That doesn’t preclude it also having other gradients where x = 1.

      I’m not for a minute defending the question, with or without that crazy defense. The question is also plain weird anyway. But I agree, it is difficult to see how everyone could have missed this.

      1. I think the answer to the final part is that they weren’t looking.

        A more important question to me is why the question was A) written and B) allowed into the exam.

        My guess is that someone with a reputation for being a decent question setter wrote the question and then the vetting panel, knowing who wrote the question, just checked the given answer and moved on.

      2. The question is a bit weird, but I’m not as bothered by it as some of the other commenters on this blog.

        The family of curves is pretty cool and has one connected component for c<0, two for 0<c<3e^2 and three for c>3e^2. It’s fun to explore these curves and then try to derive/prove some of their properties – maybe a SAC question?

        As Glen said, the curve either has zero points with x=1 (for c<-3), one point (1,0) with undefined gradient for c=3, and two points (1, \pm\sqrt{c+3}) with gradient \pm\frac{3}{\sqrt{c+3}} for c>-3. Clearly, to get a gradient of 2, you need c>-3 and y>0. Once we are there, then the gradient is monotonically increasing wrt to c and there is a unique solution that the question was after and others have calculated above. For fun, here is a place to play with this graph and its tangents: https://www.desmos.com/calculator/ejsc7f8hik

        As Marty pointed out, the question only required students to find c that gives *a* gradient of 2 when x=1… but in an exam, it is not that nice to have an awkward question like this. As JF said, just stipulating that y>0 would have made everything much nicer and probably should have been suggested by the vetting panel! But then again, most students did ok in this question, so either they didn’t notice any issues or it didn’t bother them (probably the former).

          1. I’m serious – maybe I’m just undereducated, but this question doesn’t bother me so much.
            Your “I’m not for a minute defending the question” defense convinced me!

            1. It shouldn’t have. My defense may be enough to defend the question against execution, but that’s about it. The question is, at best, nuts.

      3. Marty, that’s garbage. I completely reject that defense.

        When we teach students about gradients of functions, what do we tell them? We have a curve given by the graph of a function y = y(x) and define y'(x). When it exists, it is unique. This is the gradient.

        To say that oh no, a solution set to a given implicit equation may now have several gradients at a given x coordinate, is wrong. To even speak of its gradient at a given x coordinate is wrong, without specifying which curve we are talking about.

        Yes, there are ways to fix the question, but as stated, I still stand by my “no such c exists” position.

        1. How dare you talk to the owner of this blog in such a manner! Aren’t you aware that I am King here!?

          More seriously, of course I’m not in a mood to fight hard to defend this question, if only because then I’ll probably have to haul Simon back to Planet Earth. But I think you have to distinguish between (a) something that makes no sense and (b) something that (maybe just) makes logical sense, even if no sober mathematician would ever write it. I think there is a case for (b), and I certainly think there is a strong chance that the argument for (b) is what was in the writers’/vetters’ minds. I don’t see how they all could have been unaware of the double-value.

          1. I’m a little surprised, but let’s keep on with it.

            This is the essential point on which we disagree:

            “something that (maybe just) makes logical sense, even if no sober mathematician would ever write it”

            Do you truly think that such things belong in mathematics exams? Grammatical errors, weird modelling things, poor physics… these are excusable (sometimes) in mathematics exams. Not in exams in their discipline (I’d argue).

            But this is a mathematical mistake. It is not a matter of opinion. This is an exam question assessing particular parts of a taught curriculum. This is not a discussion question up for debate or investigation that students should write an essay about.

            The gradient is not defined (in high school) for anything except for a graph. Furthermore, the gradient is always a unique value when it exists. The solution is not a graph at x=1, if it exists there at all. The setters of the question do not care about this, very likely, and I think they should. The correct answer to this question is that no such c exists, because “a gradient” (in the sense that has been taught) is not defined at x=1.

            Exam questions should not require leaps of logic beyond what has been taught in order to make an answer possible. Surely you don’t believe otherwise?

            1. Glen, this is in the What is this Crap Here series. What do you think are the odds that I think the above nonsense should be in an exam? I am simply curious, as a fifth order issue, as to *why* it was in an exam. To my mind there is a reading, albeit a strained and idiotic reading, that gives a (strained and idiotic) sense to the question.

              1. Yes, thus my surprise…. but then why call it a “defense”? Are you just trying to give a broken argument for the question being in the exam, knowing that it is broken?

                It is not unheard of that you have in mind some things as being problematic in questions, and other things that you do not consider to be problematic. So, although I can safely assume that you don’t want the question as a whole in any exam, I don’t know for sure that you consider the crime I outlined as reason enough to be condemned. I do consider it to be reason enough.

                The way that I read your first (or second) comment, it sounded as though you thought otherwise.

            2. Glen, I think we all agree that the question as worded should not have appeared on the exam.

              I suggested a simple amendment that would have fixed it. A simple amendment that the vettor should have advised (and maybe did but the writer is an imbecile …)

              Marty is moon-lighting as a psychiatrist (he’s a specialist in abnormal psychology), trying to make sense of \displaystyle why VCAA thought it was OK to include it as written. He’s trying to get inside VCAA’s head, a simple thing to do since there’s so much room. Alternatively, a difficult thing to do since it’s a pinhead. Hmmm … getting inside an empty pinhead. There’s a Zen paradox somewhere in all this.

              1. Alright, what I’m railing against here is the idea that we should allow “readings” of questions that literally change what is written. I haven’t seen anyone else here say that there is no c that satisfies the question as written. So while we may all agree that the question is crap, and shouldn’t be in any exam as written, I don’t see that we are really in agreement.

                Which is not a problem of course..!

                1. Glen, there are two worlds:

                  1. The real world.
                  2. VCAA world.

                  In the real world, the question has no answer. There’s no value of \displaystyle c. Undisputed.

                  In VCAA world, the question \displaystyle does have an answer. Just like the Specialist 2019 Exam 2 MCQ 12 has an answer.

                  We live in the real world, but it’s VCAA world we must prepare students for. If I taught a student with a parent who was a QC, I’d encourage that student to live in the real world; knowing that the student would be pay to see their exams and that QC parent would be ready, willing and able to hold VCAA legally accountable when it insisted that “No value of c exists” was not the correct answer. However, until that happy day comes, or when teachers unite and rally and cry “No more!!”, we must tolerate VCAA world. And that includes understanding VCAA world.

                  VCAA gets away with its reprehensible, abhorrent, repugnant, contemptible behaviour because its not held accountable for its behaviour. What I’d love to see is the AEU Victorian Branch launch a class action against VCAA and its lunatic world.

    2. RF, I think ‘troubled’ is a more fitting word than “worried”.

      Mor generally, we should all be very troubled at the consistent mathematical incompetence demonstrated by VCAA writers, vetters, auditors etc. As well as VCAA’s consistent pig-headed failure to express remorse over its numerous errors and blunders.

      I’m not surprised that this error was not noticed by the vetting panel. As we’ve seen many times, VCAA vetting panels are as useful as inflatable dartboards. Why expect them to sketch a graph to see what’s going on? VCAA can’t even check that a set of equations it gives has solutions (eg. 2019 Exam 2 Section A Q12 – discussed at https://mathematicalcrap.com/2020/02/14/witch-35-overly-resolute/).

      1. John, I guess you’re right but I’m still astonished. I’m also astonished no one has ever pointed out this error to me.

        I am currently taking a very quick look through the older exams, to see if there are obvious errors to add to the lists. It took me approximately 1.5 seconds after reading the question above to go “Huh?” I assume it was the same for you and Glen, and plenty of others who read this blog. So how is this nonsense not known nonsense? Or is it known nonsense and I just don’t have any friends?

        1. There might be a few factors at play here, but I would suggest the following as possible (not defending any individual or organisation, just offering suggestions based on experience):

          1. Some teachers probably do spot these errors but they are the only teacher in their school who understands, so they don’t say anything to anyone (because the teacher exam feedback no longer feels like a useful endeavour and it takes a lot of time they would prefer to spend doing other things).

          2. When teachers do find errors, their first question is not “how the expletive did this happen?” (think the CASIO v TI issue a few years back) but “how can we prepare our students for the next time this happens?”

          3. A teacher with a podium/soapbox from which to complain about said errors is a rare species. Refer to point (1).

          4. Speaking only for myself, I’m rarely confident that I’m right and VCAA is wrong when ever I see something like this. I’m not a mathematician.

          5. The “errors” that get reported more widely tend to be from the FM exams (think the “famous” 50c coin problem…

        2. That’s right, Marty. Occam’s Razor. You’ve got no friends. However, some alternative explanations might be:

          1) 2013 was 8 years ago. I (and others?) wasn’t quite as cynical then as I am now. As you say,

          “the question requires the curve has *a* gradient of 2 where x = 1. That doesn’t preclude it also having other gradients where x = 1.”

          So I (and others) probably saw the error at the time, rationalised it in good faith, calculated the obvious/expected answer in good faith and then moved on. Out of sight, out of mind.

          How times change. The well of good faith is completely dry these days. These days I look at every VCAA exam knowing that there will be errors in it. And my loathing, disgust and contempt is such that I want to nail them to the wall for these errors. All of which is a damning indictment on VCAA and its lack of mathematical leadership and integrity.

          2) I think that the serious mistakes made by VCAA and its arrogance and total lack of remorse over the last decade mean that errors such as this one are viewed with much greater seriousness and contempt in hindsight.

          3) It’s too easy for a teacher to develop the bad habit of subconsciously seeing what VCAA probably meant, and subconsciously calculate that ‘intended’ answer. Particularly when there’s a subconscious bias that there must be an answer … (I think 2019 Exam 2 Section A Q12 is a good recent example of that).

          4) Following from 1), until the last few years there’s been no forum to air these errors. This is one of the many valuable services your blogs provide. The MAV \displaystyle should provide such a forum (particularly via it’s ‘Exam Solutions’) but experience has clearly demonstrated that it will NOT (and actively avoids discussion of exam errors in its ‘Exam Solutions’ even when the correct answer only makes sense when the error is acknowledged – eg. pointing out the error in 2018 Exam 1 Q6(b) was verboten). Understandable given the incestuous relationship it has with VCAA. And of course we all know that VCAA ignores exam feedback (even though it pretends to seek it) and does not provide any receipt of feedback given.

          As an aside, it’s possible that your blogs will one day be part of the evidence submitted to prove that VCAA and it’s goons is unfit to be a regulatory educational body. A Royal Commission on Victorian education is long overdue.

          1. I should have added 4) (I thought I did but it disappeared once the edit window expired):

            4) Following on from 1), until recently there’s been no forum to air VCAA’s errors. This is one of the many valuable services Marty’s blogs provide. The MAV should provide such a forum (particularly in its ‘Exam Solutions’), but it has clearly demonstrated that it will NOT (even in its ‘Exam Solutions’ when the correct solution only makes sense when the error in the question is pointed out – eg. pointing out the error in 2018 Specialist Exam 1 Q6(b) (https://mathematicalcrap.com/2019/02/24/the-vcaa-dies-another-death/) was explicitly verboten. And the MAV ‘Exam Solutions’ explicitly weaselled out of pointing out the error in 2016 Methods Exam 2 Section B Q3(h) (https://mathematicalcrap.com/2017/05/08/the-median-is-the-message/)). This attitude is explainable via the very ‘special’ relationship it has with VCAA (which makes inbred hillbillies look genetically diverse by comparison).

        3. Thanks, RF and John. Of course I’m aware that there has been no general forum for discussing and compiling such errors. And I know plenty of teachers will take note of errors and the like for themselves and maybe a colleague or two, without the errors receiving broader attention. And, yes, it’s a pretty old question. But still, on occasion people will tell me about this or that question from the past. That I haven’t had the question above flagged to me suggests that it hasn’t received the whacking that it might.

          I guess it’s something like what you guys suggest. Without access to a subculture suggesting that VCAA is generally nuts, and thus any given question may well be specifically nuts, teachers just make strained sense of things as best they can.

          1. It’s not just maths in VCAA. Teachers from lots of areas have similar problems with ongoing issues in their exams, rarely any contrition in the examiner’s report, and no forum to really talk about it. I can think of recent conversations with Economics, Computing and English teachers about this issue.

              1. This question in particular was no big deal – it was clunky but interpretable even for students under exam conditions. But other questions in VCE exams are incomprehensible, wrong, or even more insidious, ambiguous and then marked according to only one valid interpretation.

                I think the VCAA process (at least, what I’ve seen) is flawed – and that leads to things getting onto exams that shouldn’t. There is a good Craig Barton podcast with the chief examiner for AQA – their exam writing process sounds amazing cf to VCAA. Although, they have a much larger cohort than us – so maybe their resources are proportionate http://www.mrbartonmaths.com/blog/chief-examiner-trevor-senior-how-gcse-maths-exams-are-written/

                Marty: Have you applied to be on the panel for methods or specialist? You definitely have the knowledge and drive to do a good job – I don’t see how you wouldn’t get a spot!

                1. 1) “I think the VCAA process … is flawed”. Ya think!!?? And for those who were wondering: https://en.wikipedia.org/wiki/AQA

                  2) It’s not about resources. It’s about making sure the people doing the job are competent. It’s about accountability. As I’ve often said, my beef is not with the writers. It’s with the idiot vetters. And the idiots who demand ‘real-life’ contexts.

                  And I don’t have much sympathy for groups (and teachers) that \displaystyle could hold VCAA accountable but don’t (because of fear, apathy, or craven brown nosing).

                  3) Absolutely, Simon! Marty’s a sure thing. He can’t miss!

                  Unless, on the smallest off-chance of course, VCAA makes decisions with its heart and not its brain.

                  (And you were wondering whether what followed 3) was sarcastic …)

                2. Simon, you seem properly concerned with whether a question screws up the exam. I think you are under-concerned if a question is mathematically meaningless or mathematically perverse.

            1. History can be added to that list … (And I don’t just mean a one-off issue on the 2012 exam of a giant robot assisting socialist revolutionaries in 1917).

          2. Re: “And, yes, it’s a pretty old question. But still, on occasion people will tell me about this or that question from the past. That I haven’t had the question above flagged to me suggests that it hasn’t received the whacking that it might.”

            These days, I’ll usually recall a question from an ‘old’ exam that has an error in it only when a student draws my attention to it. (Hence our correspondence on horizontal asymptotes shown on graphs of functions defined on finite domains etc.)

            So to me, Simon’s suggestion that ” [students] didn’t notice any issues” in this question is very plausible. They don’t notice it, I don’t revisit it. Out of sight, out of mind.

              1. No, the suggestion is correct and that’s a bad thing!

                My comment notes that these questions are historical. Over time (8 years or more), error or no error, rage against the machine or no rage, they get forgotten. It’s only when a student comes across a historical question that causes them trouble that a re-collection of that question occurs. I’m suggesting this as a reason why errors like this one have not previously been raised.

                (Look at the forgotten historical antecedents to the infamous ‘intersection of a function and its inverse’ question …)

                Maybe there could be a contest – Find the best/worst ‘historical error’ in a VCAA maths exam (pre-2010, say …)

  5. Hi,

    Just to show IB is not immune from vetting mistakes …

    Check out this potential witch from the 2017 physics text

    Steve R

    1. Hi, Steve. I’m not sure why your link is not appearing. Maybe I put the “No Goddam Physics” filter back on. In any case, if you email me the link, I’ll hammer it into the comments somehow.

      (I have some repairs to do in the comments settings, but have had no time.)

    1. Sorry Steve, still no go. I’m looking forward to seeing it though, I haven’t found anything terrible in an IB exam before.

      1. Glen,

        Your right about IB exams in that they seem to vet them to a high standard.

        My example comes from the Pearson physics text though

        I’ll try to attach screenshot one more time

        Steve R

        1. Glen,

          Still no luck. If you have the Pearson 2nd edition text I was curious about the worked example on p346 on calculating power from a hydro dam using simplified version of Bernoulli’s equation for hydrostatic flow

          I have sent a low resolution screen shot to Marti

          Steve R

  6. OK, just quickly on Steve’s IB Physics question:

    0) I’ll look generally at the settings for commenters and attachments and the like, once I deal with the latest mountain.

    1) Steve, the screenshot is way too angled and low res. If you do a proper scan then I’ll seek to post it.

    2) One poor example in a physics IB text is hardly comparable to the continual, predictable wrongness and awfulness of VCE exam questions.

    Any sizeable institution/product will make/have errors. The Haese IB maths text, which is excellent, contains errors. The Oxford text seems not so good but is still miles better than any VCE maths texts. I do not know of any errors or weirdnesses in IB maths exams, but I have no doubt that they exist.

    The issue isn’t whether there are errors. The issue is whether there is a general culture of mediocrity and arrogant unaccountability. The VCE has this culture in spades. I have no evidence that there is any comparable cultural issue within IB.

    1. Marti,

      I’ll try and get a better resolution scan to you.

      I too found the Haesse books on Mathematics rather better than the Pearson Physics text
      in 2017 and 2018 .

      The question concerned seems to have a fundamental error in logic though rather than a typo

      Steve R

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