Secret 2023 Methods Business: Exam 1 Discussion

The usual. Go for it.

UPDATE (04/11/23)

Thanks to everyone for your comments. I’ve gone quickly through the exam, without computing everything (or much of anything). I’ll make a few general comments and then pick on a few questions; mostly I’m just repeating commenters’ observations, but there are a couple things to say, or at least to say louder.

Obviously the exam was pretty fraction heavy, which is intrinsically ok. Except, the exam wasn’t also heavy in more Year 12-ish ways. There don’t appear to be any “errors” in the newspaper sense of the word, but there are definitely a few things there to irritate (at least) mathematicians.

Q2.  It’s a good thing they emphasised to solve for \boldsymbol{x\in R}. I was just about to solve for 4. (Alternative joke: I was just about to solve for x a quarternion.)

Q4. I used trapezia/rectangles of constant height 1, the first based on [1, 3/2] and the second based on [2, 5/2]. Tell me I’m wrong.

Q5. This question seems good in principle but poor in execution. As Bugle noted in a comment, part (b) can naturally be tackled with trig symmetries. However, the range of k to be considered seemed fiddly enough that it was probably simpler, or at least no harder, to solve the problem by autopilot computing. It’s not good to invite cleverness that turns out to not be beneficial.

Q7. (24/11/23) Someone emailed me about (d), and they’re right: the wording really, really sucks. It’s not an error, since the question specifies “regions” (plural), but it’s way too easy to overlook that, and to hunt for a single region bounded by the three functions.

Q9. Walking tracks. Those two stupid curves are walking tracks.

VCAA’s real-world flavour text is always silly and pointless, but Q9 is even sillier than usual. A track is a track: who cares whether, with some particular orientation, the thing has a “turning point”? Beyond that: the point P is undefined; the question should have referred to “stationary point” rather than “turning point”, unless they really mean the latter (in which case 2 marks is insufficient); the two stupid tracks have nothing and essentially nothing to do with the stupid park; the use of “verify” cannot mean much different in this question, but appears to have confused people.

 

In sum, there don’t appear to any hanging offences, but Q4 and Q9 are each worth a slap. All four of the above questions could have and would have been significantly improved by proper mathematiciany scrutiny.

36 Replies to “Secret 2023 Methods Business: Exam 1 Discussion”

  1. Felt way too easy conceptually, I think a lot of people will get tripped up by the seemingly massive fractions, but they had nice simplifications.

  2. Q8c. I’m fine with questions that test students’ ability to manipulate fractions, but this felt a bit much and is not really testing a higher-order skill by this point.

    Q7d was a nice surprise – good if you saw the shortcut (I expect a lot of students will realise there are two equal areas…)

    Q5b was my favourite. Nice question.

    1. 8c and 7d I agree with you

      5b, I’m conflicted. During the exam, I hated it for making me do those integrals just to get a basic trig equation, but I realised after that there was a neat way to do it geometrically by considering areas under curves. I doubt that anyone did this during the exam as opposed to the obvious algebra route, though.

  3. Seems like they recognised last year’s exam being way too easy and as expected opted to rectify that by fiddling with the numbers lest they put in the effort to create conceptually harder questions that are also not incorrectly worded

  4. 9b. “Verify that f(x) and g(x) both have a turning point at P”.

    Will students be expected to verify that the stationary point at x = 2 is also a turning point – ie. that f'(x) and g'(x) change sign?

    1. Probably not given that the question is only worth 2 marks. In previous methods exams you’re often allowed to assume that when f'(x) = 0 it is a turning point unless otherwise stated in the question.

      1. “In previous methods exams you’re often allowed to assume that when f'(x) = 0 it is a turning point unless otherwise stated in the question” – evidence for this? Not saying you’re wrong, but I’m not aware of this practice.

        I agree that, in this case, it won’t be required, because otherwise 2 marks is too stingy. But the question should have used “stationary point” rather than “turning point”.

        I wonder if non-calculus approaches would be accepted, say (1) for f(x), its graph is a vertical translation and reflection about the x-axis of y=x(x-2)^2 which has a turning point at x=2, as can be observed by the double root at x=2, (2) for g(x), this is a quadratic function, and its turning point occurs at x = - \frac{12}{2\times -3}=2.

  5. Like others have said, quite easily conceptually, however also differs a bit from previous methods exams.
    Q1 It says to simplify which is not normally the case.
    Q4 Asks to estimate the area using trapeziums under the graph which has not previously been tested.
    Q5b A pretty nice question
    Q6b) “Use z = 2” Something that marty might take issue with, however, z is used on the confidence interval formula sheet.
    Q8 Not a big issue but upper case T and lower case t are used to basically represent the same measure.
    Q9 Verify question, doesn’t normally occur on methods exams. part (b) Not sure whether you would have to show that the stationary point is a turning point and not a point of inflection but given only 2 marks, probably not.

    1. Approximation using the trapezium rule is a new addition to the course this year and was expected to be on the exam this year. My students were pleased to see this question.

    2. Upper case T denotes the random variable. Lower case t denotes values of the random variable T. They are conceptually different things.

  6. Some thoughts on an error-free (not withstanding some clumsy wording) and unremarkable exam:

    Question 3: I think some students will fall for the simple trap of giving \displaystyle x \leq 2 rather than \displaystyle 1 < x \leq 2. The graph in part (a) should help most students avoid this.

    Question 4 part (b): Students who first find the general solution might get confused with two 'parameters' or might use k as the parameter in their general solution and screw up.

    Question 6: I think some students might confuse the preamble to part (b) as a separate question and think VCAA screwed up by not providing writing lines. This could have been worded and structured much better. Having said this, I think it’s a good thing that students are told what critical value of z to use. Past exams (mainly Specialist) have often left students wondering whether 2 or 1.96 should be used. I have consistently argued that critical values of z should be included on the Formula Sheet.

    Question 7: It would have been helpful if parts (b) and (c) were on page 9 and the set of axes was larger. This would make it easier for students to see the required areas for part (d). The VCAA has a habit of wanting students to find areas that are microscopic when you try to shade them on their diagrams.

    Question 8: It was pleasing to note that the given function was indeed a probability density function.
    Part (a): The perennial mystery is how much calculation must be shown to get the 1 mark.
    Part (c): I thought the required arithmetic was somewhat gratuitous.

    Question 9:
    Part (a): The perennial mystery is how much calculation must be shown to get the 1 mark. For example, will the VCAA really require
    \displaystyle f(0) = 12 = a - 0(0 - 2)^2 = a - 0 = a
    as part of getting 1 mark for showing the value of a AND the value of b? I suppose we'll know in a few years time when the Exam Report is published.

    Part (b): How much calculation is required to verify? Must the nature of the stationary point at x = 2 be tested or can students state "By inspection of the graph …"
    I think some students will forget to state the coordinates of P in the excitement of getting the correct x-coordinate. Will such a slip-up get penalised 1 mark. It seems that some 1 marks require much more work than others.

  7. I wonder what the original form of 9c was. Weird to spend time setting up these two very odd walking tracks and then create a theme-park for which one of those tracks is just a nuisance visual.

    The amount of fraction work required seems a bit high for such a short exam.

  8. This was hinted at in an earlier comment, but I have a big issue with the word verify in Q9.

    Part (a) would have been a perfectly good “show that” and I cannot see how “verify” makes it any better/easier but I can see the choice of words making it MORE confusing for students.

    As to how, exactly, you are meant to verify that both graphs have a turning point at P… YUCK!

    If a student were to actually show that P is a turning point on both graphs, will they lose a mark because they were asked to verify? We will likely never know.

      1. It is from my (albeit limited) understanding that:
        “verify” is a “use the values supplied in the question to show that they work”.
        e.g. Verify that P(2,4) is a point on y=x^2, or something the rather.
        and “show that” instead requires you to derive the values supplied in the question through logical and sequenced working out.
        It is heinously inconsistent, from what I’ve seen. 9a) looks fine to be a ‘verify’ qn in my opinion, but b) is much more unnatural and ambiguous to what is presumably required (given it is 2 marks).

          1. I would say so, but in this case, it is so crazily confusing.

            Re: b) I believe the main problem with this question is that it has stupidly asked students to “Verify that f(x) and g(x) both have a turning point at P”. No values have been given about the co-ords of P, so even if you were to verify that P(2,12) is a T.P. at f(x) and g(x), you would first need to show that P=(2,12). The question is much more natural and far less confusing as “f(x) and g(x) both have a turning point at P. Find the co-ordinates of P.” There is no need for snuggling in any of this “show that” or “verify” nonsense to b).

          2. I have an issue with the use of the word verify on two, related grounds.

            1. In the past, the examiners reports have gone to great length to say that “show that” requires a specific process to be followed. I am not confident that “verify” means the same thing in the eyes of the marking scheme and I am concerned that students who completed this question as though it were a “show that” may not receive full marks for a reason known only to VCAA.

            2. Similar to KB’s comment above, in Q9b, I see no data to actually verify. There is no value to substitute and check. To show that P is a turning point on both y=f(x) and y=g(x) means, from my (perhaps wrong) understanding, showing all of the following:
            f(p)=g(p)
            f'(p)=g'(p)=0
            – That both y=f(x) and y=g(x) actually change gradient at x=p

            Seems a lot for the mark allocation.

  9. I found this exam quite frustrating to say the least. As others have pointed out, the arithmetic manipulation was time consuming and I also felt that the exam wasn’t testing much other than if I could process long fractions and sums. It lacked the rich problem solving and critical thinking aspects of many other exams in the past so I was disappointed that I didn’t get to engage with these types of questions, instead having to brute force my way through. I think this exam disadvantaged students who had done their study which is a bit annoying and confusing for me. Just hope methods 2 is a bit less dull and bland in flavour.

        1. Thanks Marty – I do try.

          Sometimes.

          Seriously though, I do complete this survey every year because:

          1. I live in hope that it will be read and understood.
          2. If I don’t then I’m saying that I’m fine with the exam(s)
          3. It is 40 minutes of my life that is probably going to be wasted at Dan Murphy’s anyway
          4. If enough teachers give feedback, VCAA *might* do something.

      1. Everybody’s a comedian.

        (One tip about leaving feedback – you don’t get sent a confirmation copy of your feedback. So type it up in a word file, date it and then copy and paste into the on-line feedback form).

  10. What a disappointing exam. I felt so bad for my students who walked out of the exam today feeling deflated. Our top-ranked students were not given the opportunity to show off their mathematical knowledge and were instead bogged down in calculating large fractions and using index laws from the yr10 curriculum. I still don’t understand the point of the trapezium question, VCAA only made it more challenging by having fractions, yet the mathematical concept is incredibly basic. Year 7 students can find the area of a trapezium, so why does it need to be on the yr12 exam? I suppose the only reason is that the exam had to address those meaningless changes to the study design.

    1. Re: “why does [the trapezium rule] need to be on the yr12 exam?”

      I agree. It really frustrates me that the left/right-rectangles were removed and replaced with the trapezium rule. The rectangle rule is the simplest sort of Riemann sum and made it easy to explicitly calculate area from limits. I liked to show this. And I really liked doing this in Specialist Maths Units 1&2 as part of the Sequences and Series topic. The trapezium rule is another bit of the Study Design that feels like a shag on a rock.

  11. Thank you to everyone for your comments. I’ve only briefly looked at the exam, and at this stage don’t have much to add. I take it that the exam is error-free, at least in the VCAA sense of the term.

    The one thing that stood out for me was in Q9(b), that the point P is completely undefined. Sure, the meaning is clear enough (even if there’s some serious dispute over what the question requires). But it’s the kind of sloppy wording that mathematicians, if not others, find really irritating.

    1. For what it’s worth, it bugged me too. But by the time I got to commenting on Q9 I was running out of steam and I was too irritated by part (a). It wouldn’t have hurt to have no point labelled P and to word part (b) as:

      “Show that f(x) and g(x) have a common turning point and give its coordinates.”

      Then later on it can be named P if required (Spoiler alert: It’s not required. It’s obviously named P for Pointless).

  12. I think it’s worth making the obvious and perhaps predictable observation that there was no pseudo-shit on Exam 1. It is with sincere regret that I 100% doubt the same observation will be made for Exam 2.

    1. BiB, don’t game the discussion (and watch your language).

      Methods is an atrocious subject, and Exam 2 tends to have the worst questions of all of the exams. But it doesn’t tend to have “errors” in the VCAA sense of the word. At the moment, the focus is specifically on Exam 1 and generally on errors. Don’t pretend otherwise.

  13. Thank you all for your comments. I’ve now gone through the exam *very* quickly, and have updated the post with a few thoughts of my own.

  14. Re: Update comments.

    I agree – no newspaper worthy content in this exam. Apart from some pointless arithmetical demands, it was mostly harmless. Question 8 part (c) is a good teaching example for exam technique – cut something loose (and return later) if the calculation obviously takes more time than getting the 1 mark can justify. There are many students with OCD that can’t do this but nevertheless must learn (or be trained) to do it.

    Question 2 – I think \displaystyle x \in R is fair enough. I initially raised an eyebrow but then decided that, even though Maths Methods deals exclusively with reals, you still probably want to be clear and explicitly exclude non-real solutions (which do exist). It certainly doesn’t impact the clarity of the question (but it makes you wonder how there can be attention to detail on things like this, but lack of attention to detail on things like defining the point P in Question 9).

    Question 4 – I won’t say you’re wrong. But I’m trying to imagine what your trapezia look like. Probably my bad for having zero ability to visualise.

    Question 5 – I agree that “It’s not good to invite cleverness that turns out to not be beneficial.”

    Question 9 – (*Warning – The following contains metaphoric language). Yes, a slap (maybe several, and with a wet fish) is deserved.

    1. Q2 is hardly of the greatest concern, but you’re wrong.

      “Solve the equation …” is all you need, and all you should have, unless there’s reasonable doubt about what you’re solving for or over what range. Here, there is zero doubt of either.

      Teachers – all teachers – use too many words.

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