WitCH 90: Tiles and Tribulations

This is late, and it isn’t very Christmasy. But it came up as part of another project, and it’s been bugging us, so Bah Humbug.

The following is a question from 2022 Mathematical Methods Exam 1. It’s been discussed some on this post, including a concerning rumour (edited 24/12/22), and we also hammered some of the wording on this post. It is clearly deserving of its own WitCH.

25 Replies to “WitCH 90: Tiles and Tribulations”

1. marty says:

Indeed. I humbly apologise for not WitCHing this crap immediately.

1. Red Five says:

You have many fish to fry Marty, so to quote Charli Hebdo, “All is forgiven.”

There is also an awful lot of crap this year to unpick. Just wait until the examiners reports come out…

1. marty says:

Maybe I should stop frying fish.

1. Red Five says:

You catch it, you cook it, you eat it.

Or as the dolphin said: “So long, and thanks for all the fish.”

2. John Friend says:

1. and 2. below are bad, but what I’m explicitly damning as the baddest of the bad is the following rumour (if true):

“One more bit of insanity. For question 3 (finding k such that the pair of linear equations have infinitely many simultaneous solutions), writing the equation k^2 + 8k = -15 is not sufficient for the 2nd method mark, students needed to write k^2 + 8k + 15 = 0.”

1. Red Five says:

Different question (or did I miss something?).

Agree though – that would be the middle finger to rule them all (if true – which I doubt we could ever prove; ironic really since VCAA now wants students to prove things…)

1. Glen says:

Probably this is already noted earlier (sorry I’ve not been up to speed with the blog) but the question doesn’t say that tiles need to be placed without rotation, does it? Or that they need to line up perfectly? Maybe I missed it. (This affects the value of a part and also the last part, because if you allow rotation then a “continuous pattern” can’t be guaranteed for all ways in which tiles “can be placed”.)

3. Red Five says:

(apologies to anyone and everyone who may have pointed this out earlier).

Condition 2 states that the tiles must form a continuous pattern. However, the example given the lack of a domain for any of the “rules” you could argue that the pattern will never be continuous unless only the trigonometric option is chosen.

1. marty says:

That’s too nitpicky. But, if the grading for this question was as is feared, then it would be sauce for goose.

4. Glen says:

Ahhh the tile question. It is all so easy and weird, I worry for the marking.

For example, part c… do students need to consider the cases:

AA
AB
BA
BB

explicitly and say that in each case the boundary points are equal? Or do they then need to also argue that a row will consist of a union of such pairs? Do they need to explicitly state that f and g are continuous?

Or can they simply note the boundary values (assuming the a is as specified in an earlier part…. another issue) and then say, clearly it’s all fine.

I’d be stressing in the extreme doing this question.

1. Red Five says:

I think all four permutations are covered by the statement, but I never know for sure unfortunately.

As for rotations… I think this was raised on the exam discussion post.

5. marty says:

Could the marking scheme not have then been later altered?

1. marty says:

Don’t be ridiculous. Of course it would be relevant.

6. marty says:

OK, I’ve deleted a number of comments, which then also resulted in a number of replies being deleted.

Forget the rumours about the grading, no matter how solid. The question is sufficiently appalling as it is. Issues with the grading can wait for the examination report. Issues with the absurd waiting time for the examination reports will be part of near year’s campaigning (and is already underway).

7. marty says:

Just to be clear, commenters are permitted to hammer “VCAA”. Hard. But I don’t want any comments, even oblique comments, aiming at individuals. Even if these comments are objectively fair and objectively reasonable.

I might let some such comments through, and might make them myself at times. The more a person is in charge and/or is a public representative of VCAA or ACARA or whatever, the more the person is fair game. But the default position is that comments containing implicit or explicit personal attacks will be deleted.

8. student says:

Somehow in part b), looking at my statement of marks, one examiner gave me 3/3 and the second examiner gave me 2/3. And I can bet I’m not the only one to lose half a mark on it… I remember I set up the integral and everything, purposefully avoided any shortcuts (eg. since the equation is a cubic, if the leftmost and rightmost point on the tile are halfway up the height of the tile, and have the same gradient sign, half the area would be shaded; I think this is a decent argument (correct me if I’m wrong) but I pictured VCAA not following/accepting it so I did the stupid integral) and I showed my calculations for the size of the square and everything. And it must have all been correct calculation for one assessor to give me full marks… I’d been concerned about ‘show that’ questions for weeks leading up to the exam, its never clear how much needs to be shown. I suppose nothing is trivial to VCAA (I’m inclined to say its because they’re stupid and bad at maths). Maybe they’re mad I didn’t quote the formula for the area of a square.

Of course the problem isn’t just a personal one: it’s that so many students are sitting there in the exam, and rather than being occupied by, y’know, year 12 maths, they’re stressing about whether they need to state how to find the area of a square. VCAA can’t discriminate between top students using hard questions (maybe they couldn’t answer such questions themselves) so instead they’re just really, really petty. (It was the only mark I lost on the exam).

1. marty says:

Thanks, student. I don’t understand your non-calculus cubic approach. Presumably the examination report will indicate what trivia VCAA decided to whine about, which resulted in your lost mark. Forgive my ignorance, but what happens when two graders disagree? They just average the two scores?

1. student says:

The exam has 40 marks, but you get a mark out of 80 for the exam which is the two scores added together (so yeah, they average the results, unless they differ significantly then I think they get a third examiner in, which is much more of an English thing than Maths). In hindsight my cubic approach was missing the necessary (and possibly sufficient?) condition that the cubic’s point of inflection was in the middle of the tile. Should have thought that through a bit more before posting, sorry about that. I hope the exam report will indicate what the problem was, but I’m not so optimistic…

1. marty says:

Ah, I see. A nice cubic argument.

9. student says:

Ok so the exam report did indicate where my half mark went. I don’t think I explicitly stated that 200cm^2 is half the area of the tile at the end. I think I did do it at the start though? But the “error” would be along those lines. >:( so petty imo

1. marty says:

What do you mean you “did do it at the start though”? You mean that, by memory, in some clear manner you explicitly declared at the start of your answer that you were looking for 200 cm2? Complete with units? And they still docked you?

1. student says:

By my (admittedly not very clear, and possibly wrong) memory, I think I started with 1/2 of 400 = 200, then did the integral to show that the area under the curve also equals 200. This does violate VCAA’s idea of what ‘show that’ means, since you have to go from LHS -> RHS and you can’t go LHS -> [something] then RHS -> [the same thing]. I guess this was the format I used, which is probably what VCAA docked me for.
I’m sure that the ‘LHS = [something] and RHS = [the same thing]’ format is a logical, if somewhat messier, way to show LHS=RHS. But VCAA doesn’t seem to care much about logic…

1. marty says:

Hi, student. It’s difficult to be sure. There was a suggestion that the assessors really nitpicked on this question, but whether your lost mark was from a lack of units, or from not saying at the death that “200 is half of 400”, or something other triviality, God knows.

I’m also a bit puzzled and concerned by your LHS –> RHS remark. There is no question that VCAA has no clue what “show that” means, but you’re suggesting something more. If you are supposed to “show that” LHS = RHS, and you do so by computing LHS = THING and RHS = THING, then this should be a valid “show that”, even by VCAA’s idiotic definition. If you, or anyone, has evidence VCAA thinks otherwise, I’m very keen to see it.

1. student says:

Thanks… Unfortunately, I’m pretty confident that VCAA doesn’t accept ‘LHS=THING and RHS=THING implies LHS=RHS’. For evidence:

All of my Specialist teachers stressed that ‘show LHS=RHS’ can only be done by starting at the LHS and progressing to get the RHS. I don’t have the paper anymore but I remember losing marks on a year 11 test for a ‘LHS=THING and RHS=THING’ argument. (I don’t remember the question, just the frustration). Teacher’s justification was that it’ll teach me not to do that again in the VCE exams (whoops.)

But beyond anecdotal evidence, consider these exam report lines:

(a) “ A reminder that ‘show that’ questions require a reasoned argument. The answer is given and students are required to provide a detailed progression to the answer.” – 2020/2021 Methods exam 1

(b) “Where questions require a given result to be shown (for example, Question 6a.), it is important that students present sufficient evidence of the working leading to the given result for marks to be awarded.” -2020 Spesh exam 1

(c) “There were four questions (Questions 2ai., 3aii., 5a. and 5c.) for which students needed to show that a given result was reached. In these cases, steps that led to the given result needed to be clearly and logically set out to attract full marks.” -2022 spesh exam 2

And there are plenty of others which I didn’t include because they’re all the same: ‘reaching a given result’, ‘progression to the answer’.

If they want you to show LHS=RHS, the ‘answer’ or ‘result’ refers to the RHS. (And the LHS, I suppose, is a question). The ’Question 6a’ referenced in line (b) is such an equation.

This is also why you can’t ‘show that the roots of f(x) are a and b’ by substituting f(a)=0 and f(b)=0. They want the same structure as if you were finding the roots of f(x); ‘progressing to the answer’, as they say.

All of the ‘show’ questions are supposed to be answered as if they were ‘find’ questions. Maybe this is too cynical but I wonder if they do it to give them the opportunity to nitpick? For a ‘find’ question, regardless of convention/method, if a student gets the right answer, you can’t dock marks (except for units/form, I guess). But if that proof of the student’s ability to find the answer isn’t there…

1. marty says:

Thanks, student. I think it’s probably too theoretical. It’s really a matter of looking for specific VCE “show that” questions, where LHS = THING = RHS is natural (and valid), and seeing what VCAA indicates about such a question.

Of course I’m not questioning that VCAA tends to be idiotic on this, as your roots of a polynomial example exemplifies. But a general idiocy is not proof of the specific idiocy we’re discussing.