WitCH 52: Lines of Attack

Yes, we have tons of overdue homework for this blog, and we will start hacking into it. Really. But we’ll also try to keep the new posts ticking along.

The following, long WitCH comes from the Cambridge text Mathematical Methods 3 & 4 (including an exercise solution from the online version of the text).

UPDATE (07/02/21)

Commenter John Friend has noted a related question from the 2011 Methods Exam 1. We’ve added that question below, along with the discussion from the assessment report.

 

 

MitPY 11: Asymptotes and Wolfram Alpha

This MitPY comes from frequent commenter, John Friend:

Dear colleagues,

I figured this was as good place as any to ask for help. I’m writing a small test on rational functions. One of my questions asks students to consider the function \displaystyle f(x) = \frac{x^3 + x}{x^2 + ax - 2a} where a \in R and to find the values of a for which the function intersects its oblique asymptote.

The oblique asymptote is y = x - a so they must first solve

\displaystyle \frac{x^3 + x}{x^2 + ax - 2a} = x - a … (1)

for x. The solution is \displaystyle x = \frac{2a^2}{(a+1)^2} and there are no restrictions along the way to getting this solution that I can see. So obviously a \neq -1.

It can also be seen that if a = 0 then equation (1) becomes \displaystyle \frac{x^3 + x}{x^2} = x which has no solution. So obviously a \neq 0.

When I solve equation (1) using Wolfram Alpha the result is also \displaystyle x = \frac{2a^2}{(a+1)^2}. But here’s where I’m puzzled:

Wolfram Alpha gives the obvious restriction a + 1 \neq 0 but also the restriction 5a^3 + 4a^2 + a \neq 0.

a \neq 0 emerges naturally (and uniquely) from this second restriction and I really like that this happens as a natural part of the solution process. BUT ….

I cannot see where this second restriction comes from in the process of solving equation (1)! Can anyone see what I cannot?

Thanks.

MitPY 10: Square Roots

This MitPY comes from a student, Jay:

I have a question relating to polynomial equations. For context I have just finished Y11 during which I completed Further 3&4, Methods 1&2 and Specialist 1&2.

This year during my maths methods class we covered the square root graph, however I was confused as to why it only showed the positive solutions. When I asked about it I was told it was because the radical symbol meant only the positive solution.

However since then I have learnt that the graph of \boldsymbol{y=x^{0.5}} also only shows the positive solution of the square root, while \boldsymbol{y^2=x} shows both. I am quite confused by why they aren’t the same. The only reason that I could think of is that it would mean \boldsymbol{y=x^2} would be the same as \boldsymbol{y^2=x^4}, and while the points (-2,-4) and (2,-4) fit the latter they clearly don’t fit former.

Could you please explain why these aren’t the same?

Secret Methods Business: Exam 2 Discussion

UPDATE (31/12/20) The exam is now online.

This is our post for teachers and students to discuss Methods Exam 2 (not online). There are also posts for Methods Exam 1, Specialist Exam 1 and Specialist Exam 2.

UPDATE (21/11/20) A link to a parent complaining about the Methods Exam 2 on 774 is here.

 

UPDATE (24/11/20 – Corrected) A link to VCAA apparently pleading guilty to a CAS screw-up (from 2010) is here. (Sorry, my goof to not check the link, and thanks to Worm and John Friend.)

 

UPDATE (05/12/2020)

We’ve now gone through the multiple choice component of the exam, and we’ve read the comments below. In general the questions seemed pretty standard and ok, with way too much CAS and other predictable irritants. A few questions were a bit weird, generally to good effect, although one struck us as off-the-planet weird.

Here are our question-by-question thoughts:

MCQ1. A trivial composition of functions question.

MCQ2. A simple remainder theorem question.

MCQ3. A simple antidifferentiation question, although the 2x under the root sign will probably trick more than a few students.

MCQ4. A routine trig question made ridiculous in the standard manner. Why the hell write the solutions to \boldsymbol{\cos 2\theta = b} other than in the form \boldsymbol{\theta = \alpha + k\pi}?

MCQ5. A trivial asymptotes question.

MCQ6. A standard and easy graph of the derivative question.

MCQ7. A nice chain rule question. It’s easy, but we’re guessing plenty of students will screw it up.

MCQ8. A routine and routinely depressing binomial CAS question.

MCQ9. A routine transformation of an integral question. Pretty easy with John Friend’s gaming of the question, or anyway, but these questions seem to cause problems.

MCQ10. An unusual but OK logarithms question. It’s easy, but the non-standardness will probably confuse a number of students.

MCQ11. A standard Z distribution question.

MCQ12. A pretty easy but nice trigonometry and clock hands question.

MCQ13. The mandatory idiotic matrix transformation question, made especially idiotic by the eccentric form of the answers.

MCQ14. Another standard Z distribution question: do we really need two of these? This one has a strangely large number of decimal places in the answers, the last of which appears to be incorrect.

MCQ15. A nice average value of a function question. It can be done very quickly by first raising and then lowering the function by \boldsymbol{a} units.

MCQ16. A routine max-min question, which would be nice in a CAS-free world.

MCQ17. A really weird max-min question. The problem is to find the maximum vertical intercept of \boldsymbol{f(x) = -log_e(x+2)}. It is trivial if one uses the convexity, but that is far from trivial to think of. Presumably some Stupid CAS Trick will also work.

MCQ18. A somewhat tangly range of a function question. A reasonable question, and not hard if you’re guided by a graph, but we suspect students won’t do the question that well.

MCQ19. A peculiar and not very good “probability function” question. In principle the question is trivial, but it’s made difficult by the weirdness, which outweighs the minor point of the question.

MCQ20. All we can think is the writers dropped some acid. See here.

 

UPDATE (06/12/2020)

And, we’re finally done, thank God. We’ve gone through Section B of the exam and read the comments below, and we’re ready to add our thoughts.

This update will be pretty brief. Section B of Methods Exam 2 is typically the Elephant Man of VCE mathematics, and this year is no exception. The questions are long and painful and aimless and ridiculous and CAS-drenched, just as they always are. There’s not much point in saying anything but “No”.

Here are our question-by-question thoughts:

Q1. What could be a nice question about the region trapped between two functions becomes pointless CAS shit. Finding “the minimum value of the graph of \boldsymbol{f'} ” is pretty weird wording. The sequence of transformations asked for in (d) is not unique, which is OK, as long as the graders recognise this. (Textbooks seem to typically get this wrong.)

Q2. Yet another fucking trig-shaped river. The subscripts are unnecessary and irritating.

Q3. Ridiculous modelling of delivery companies, with clumsy wording throughout. Jesus, at least give the companies names, so we don’t have to read “rival transport company” ten times. And, yet again with the independence:

“Assume that whether each delivery is on time or earlier is
independent of other deliveries.”

Q4. Aimless trapping of area between a function and line segments.

Q5. The most (only) interesting question, concerning tangents of \boldsymbol{p(x) = x^3 +wx}, but massively glitchy and poorly worded, and it’s still CAS shit. The use of subscripts is needless and irritating. More Fantasyland computation, calculating \boldsymbol{b} in part (a), and then considering the existence of \boldsymbol{b} in part (b). According to the commenters, part (d)(ii) screws up on a Casio. Part (e) could win the Bulwer-Lytton contest:

“Find the values of \boldsymbol{a} for which the graphs of \boldsymbol{g_a} and \boldsymbol{g_b},
where \boldsymbol{b} exists, are parallel and where \boldsymbol{b\neq a}

We have no clue what was intended for part (g), a 1-marker asking students to “find” which values of \boldsymbol{w} result in \boldsymbol{p} having a tangent at some \boldsymbol{t} with \boldsymbol{x}-intercept at \boldsymbol{-t}. We can’t even see the Magritte for this one; is it just intended for students to guess? Part (h) is a needless transformation question, needlessly in matrix form, which is really the perfect way to end.

Secret Methods Business: Exam 1 Discussion

UPDATE (31/12/20) The exam, and all the exams, are now online. (Thanks to Red Five for the flag.)

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.

 

WitCH 46: Paddling in the Gene Pool

The question below is from the first Methods exam (not online), held a few days ago, and which we’ll write upon more generally very soon. The question was brought to our attention by frequent commenter Red Five, and we’ve been pondering it for a couple days; we’re not sure whether it’s sufficient for a WitCH, or is a PoSWW, or is just a little silly. But, whatever it is, it’s pretty annoying, so what the hell.

WitCH 44: Estimated Worth

This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, Estimating the area under a graph. (It is followed by 21B, Finding the exact area: the definite integral.)

We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.

MAV’s Trials and Tribulations

Yeah, it’s the same joke, but it’s not our fault: if people keep screwing up trials, we’ll keep making “trial” jokes. In this case the trial is MAV‘s Trial Exam 1 for Mathematical Methods. The exam is, indeed, a trial.

Regular readers of this blog will be aware that we’re not exactly a fan of the MAV (and vice versa). The Association has, on occasion, been arrogant, inept, censorious, and demeaningly subservient to the VCAA. The MAV is also regularly extended red carpet invitations to VCAA committees and reviews, and they have somehow weaseled their way into being a member of AMSI. Acting thusly, and treated thusly, the MAV is a legitimate and important target. Nonetheless, we generally prefer to leave the MAV to their silly games and to focus upon the official screwer upperers. But, on occasion, someone throws some of MAV’s nonsense our way, and it is pretty much impossible to ignore; that is the situation here.

As we detail below, MAV’s Methods Trial Exam 1 is shoddy. Most of the questions are unimaginative, unmotivated and poorly written. The overwhelming emphasis is not on testing insight but, rather, on tedious computation towards a who-cares goal, with droning solutions to match. Still, we wouldn’t bother critiquing the exam, except for one question. This question simply must be slammed for the anti-mathematical crap that it is.

The final question, Question 10, of the trial exam concerns the function

\color{blue}\boldsymbol{f(x) =\frac{2}{(x-1)^2}- \frac{20}{9}}

on the domain \boldsymbol{(-\infty,1)}. Part (a) asks students to find \boldsymbol{f^{-1}} and its domain, and part (b) then asks,

Find the coordinates of the point(s) of intersection of the graphs of \color{blue}\boldsymbol{f} and \color{blue}\boldsymbol{f^{-1}}.

Regular readers will know exactly the Hellhole to which this is heading. The solutions begin,

Solve  \color{blue}\boldsymbol{\frac{2}{(x-1)^2}- \frac{20}{9} =x}  for  \color{blue}\boldsymbol{x},

which is suggested without a single accompanying comment, nor even a Magrittesque diagram. It is nonsense.

It was nonsense in 2010 when it appeared on the Methods exam and report, and it was nonsense again in 2011. It was nonsense in 2012 when we slammed it, and it was nonsense again when it reappeared in 2017 and we slammed it again. It is still nonsense, it will always be nonsense and, at this stage, the appearance of the nonsense is jaw-dropping and inexcusable.

It is simply not legitimate to swap the equation \boldsymbol{f(x) = f^{-1}(x)} for \boldsymbol{f(x) = x}, unless a specific argument is provided for the specific function. When valid, that can usually be done. Easily. We laid it all out, and if anybody in power gave a damn then this type of problem could be taught properly and tested properly. But, no.

What were the exam writers thinking? We can only see three possibilities:

a) The writers are too dumb or too ignorant to recognise the problem;

b) The writers recognise the problem but don’t give a damn;

c) The writers recognise the problem and give a damn, but presume that VCAA don’t give a damn.

We have no idea which it is, but we can see no fourth option. Whatever the reason, there is no longer any excuse for this crap. Even if one presumes or knows that VCAA will continue with the moronic, ritualistic testing of this type of problem, there is absolutely no excuse for not also including a clear and proper justification for the solution. None.

What of the rest of the MAV, what of the vetters and the reviewers? Did no one who checked the trial exam flag this nonsense? Or, were they simply overruled by others who were worse-informed but better-connected? What about the MAV Board? Is there anyone at all at the MAV who gives a damn?

*********************

Postscript: For the record, here, briefly, are other irritants from the exam:

Q2. There are infinitely many choices of integers \boldsymbol{a} and \boldsymbol{b} with \boldsymbol{a/\sqrt{b}} equal to the indicated answer of \boldsymbol{-2/\sqrt{3}}.

Q3. This is not, or at least should not be, a Methods question. Integrals of the form \boldsymbol{\int\!\frac{f'}{f}\ }  with \boldsymbol{f} non-linear are not, or at least are not supposed to be, examinable.

Q4. The writers do not appear to know what “hence” means. There are, once again, infinitely many choices of \boldsymbol{a} and \boldsymbol{b}.

Q5. “Appropriate mathematical reasoning” is a pretty fancy title for the trivial application of a (stupid) definition. The choice of the subscripted \boldsymbol{g_1} is needlessly ugly and confusing. Part (c) is fundamentally independent of the boring nitpicking of parts (a) and (b). The writers still don’t appear to know what “hence” means.

Q6. An ugly question, guided by a poorly drawn graph. It is ridiculous to ask for “a rule” in part (a), since one can more directly ask for the coefficients \boldsymbol{a}, \boldsymbol{b} and \boldsymbol{c}.

Q7. A tedious question, which tests very little other than arithmetic. There are, once again, infinitely many forms of the answer.

Q8. The endpoints of the domain for \boldsymbol{\sin x} are needlessly and confusingly excluded. The sole purpose of the question is to provide a painful, Magrittesque method of solving \boldsymbol{\sin x = \tan x}, which can be solved simply and directly.

Q9. A tedious question with little purpose. The factorisation of the cubic can easily be done without resorting to fractions.

Q10. Above. The waste of a precious opportunity to present and to teach mathematical thought.

UPDATE (28/09/20)

John (no) Friend has located an excellent paper by two Singaporean maths ed guys, Ng Wee Leng and Ho Foo Him. Their paper investigates (and justifies) various aspects of solving \boldsymbol{f(x) = f^{-1}(x)}.

Bernoulli Trials and Tribulations

This one feels relatively minor to us. It is, however, a clear own goal from the VCAA, and it is one that has annoyed many Mathematical Methods teachers. So, as a public service, we’re offering a place for teachers to bitch about it.*

One of the standard topics in Methods is the binomial distribution: the probabilities you get when repeatedly performing a hit-or-miss trial. Binomial probability was once a valuable and elegant VCE topic, before it was destroyed by CAS. That, however, is a story is for another time; here, we have smaller fish to fry.

The hits-or-misses of a Binomial distribution are sometimes called Bernoulli trials, and this is how they are referred to in VCE. That is just jargon, and it doesn’t strike us as particularly useful jargon, but it’s ok.** There is also what is referred to as the Bernoulli distribution, where the hit-or-miss is performed exactly once. That is, the Bernoulli distribution is just the n = 1 case of the binomial distribution. Again, just jargon, and close to useless jargon, but still sort of ok. Except it’s not ok.

Neither the VCE study design nor, we’re guessing, any of the VCE textbooks, makes any reference to the Bernoulli distribution. Which is why the special, Plague Year formula sheet listing the Bernoulli distribution has caused such confusion and annoyance:

Now, to be fair, the VCAA were trying to be helpful. It’s a crazy year, with big adjustments on the run, and the formula sheet*** was heavily adapted for the pruned syllabus. But still, why would one think to add a distribution, even a gratuitous one? What the Hell were they thinking?

Does it really matter? Well, yes. If “Bernoulli distribution” is a thing, then students must be prepared for that thing to appear in exam questions; they must be familiar with that jargon. But then, a few weeks after the Plague Year formula sheet appeared, schools were alerted and VCAA’s Plague Year FAQ sheet**** was updated:

This very wordy weaseling is VCAA-speak for “We stuffed up but, in line with long-standing VCAA policy, we refuse to acknowledge we stuffed up”. The story of the big-name teachers who failed to have this issue addressed, and of the little-name teacher who succeeded, is also very interesting. But, it is not our story to tell.

 

*) We extend our standard apology to all precious statisticians for our language.

**) Not close to ok is the studied and foot-shooting refusal of the VCAA and textbooks to use the standard and very useful notation q = 1 – p.

***) Why on Earth do the exams have a formula sheet?

****) The most frequently asked question is, “Why do you guys keep stuffing up?”, but VCAA haven’t gotten around to answering that one yet.