MitPY 13: Trigonometry and Wolfram Alpha

This MitPY comes from frequent commenter, John Friend:

Dear Colleagues,

I gave a CAS-FREE question to my Specialist students whose first part was to solve (exactly) the equation \boldsymbol{\cot 2x = \sec x}. I solved it two different ways and got two different answers that are equivalent. I’ve attached my calculations.

I checked my answers using Mathematica, which lead to my question: Mathematica gives a third different but equivalent answer (scroll down to real solutions). How has Mathematica got this answer?

It may be that Mathematica ‘used’ my Method 2, got my tan answer and then for some arcane reason ‘manipulated’ this answer into the one it finally gives. If so, I can ascribe the answer to a Mathematica quirk. But it may be that Mathematica is using a method unclear to me that leads to its answer. If so, I’m curious.

Any thoughts are appreciated.

Click to access Calculations.pdf

The VCAA Draft and its Third Rail

We’ve looked a little more closely at VCAA’s Draft for the new mathematics VCE subjects. Yes, the time for feedback has ended, unless it hasn’t: the MAV are offering a Zoom session TODAY (Thursday 25/3) for members. God knows how or why. But in any case, it’ll be a while before VCAA cements the thing in place: plenty of time to ignore everyone’s suggestions.

The following are our thoughts on the Draft and Overview. It will be brief and disorganised, since there is no point in doing more; as we wrote, the content doesn’t matter as much as the fact that, whatever content, VCAA will undoubtedly screw it up. Still, there are some clear and depressing points to be made. We haven’t paid much specific attention to what is new nonsense, and what is the same old nonsense; nonsense is nonsense.

GENERAL POINTS ON THE DRAFT

  • The draft looks like a primary school book report. Someone at VCAA really should learn \LaTeX.
  • “Computational Thinking” is meaningless buzzery, and will be endemic, insidious and idiotic. It will poison everything. Every step of Methods and Specialist is subject to the scrutiny of Outcome 3:

“On completion of this unit the student should be able to apply computational thinking and use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in practical situations requiring problem-solving, modelling or investigative techniques or approaches.”

“Statistical functionalities of technology”. And, there’s way more:

“key elements of algorithm design: sequencing, decision-making, repetition, and representation including the use of pseudocode.”

“use computational thinking, algorithms, models and simulations to solve problems related to a given context”

“the role of developing algorithms and expressing these through pseudocode to help determine and understand mathematical ideas and results”

“the purpose and effect of sequencing, decision-making and repetition statements on relevant functionalities of technology, and their role in the design of algorithms and simulations”

“design and implement simulations and algorithms using appropriate functionalities of technology”

This will all be the same aimless, pseudo-exploratory, CAS-drenched garbage that currently screws VCE, but much, much worse. Anybody who signs off on this idiocy should hang their head in shame.

  • CAS shit will now be worse than ever.
  • There should be no CAS exam, at all.
  • There should be no bound notes permitted in any exam.
  • Don’t write “technology”. It is pompous and meaningless. If you mean “CAS” then write “CAS”.
  • SACs have always been shit and will always be shit. The increased weight on them is insane.
  • The statistics is the same pointless bullshit it always was.
  • The presence of “proof” as a topic in Specialist highlights the anti-mathematical insanity of VCAA and ACARA curricula: proof has zero existence elsewhere. Much of what appears in the proof topic could naturally and engagingly and productively be taught at much lower levels. But of course, that would get in the way of VCAA’s constructivist fantasy, now with New and Improved Computational Thinking.

 

MATHEMATICAL METHODS

  • Not including integration by substitution is still and will always be the most stupid aspect of Methods.
  • Dilations must be understood expressed as both “parallel to an axis” and “from an axis”? But not in terms of the direction the damn points are moving? Cute.
  • The definition of independent events is wrong.
  • The demand that, for the composition \boldsymbol{f\circ g}, the range of \boldsymbol{g} must be a subset of the domain of \boldsymbol{f} is as pedantic and as pointless as ever.
  • “literal equations” is the kind of blather that only a maths ed clown could think has value.
  • The derivative of the inverse is still not in the syllabus, and everyone will still cheat and use it anyway.
  • “trapezium rule” is gauche but, more importantly, what is the purpose of teaching such integral approximation here? Yes, one can imagine a reasonable purpose, but we’ll lay odds there is no such purpose here.

 

SPECIALIST MATHEMATICS

  • The killing of mechanics is a crime.
  • The inclusion of logic and proof and the discrete topics could be good. But it won’t be. It will be shallow and formulaic and algorithmised, and graded in a painfully pedantic manner. Just imagine, for example, how mathematical induction will be assessed on exams: “Students often wrote \boldsymbol{n} instead of \boldsymbol{k}. Students should be aware of the proper use of these variables.”
  • There is no value here in “proof by contrapositive”, and it is confusing. Proof by contradiction suffices.
  • They’re really including integration by parts? Incredible.
  • The inclusion of cross products and plane equations makes some sense.

WitCH 58: Differently Abled

Like the previous post, this one comes from Maths Quest Mathematical Methods 11, and is most definitely a WitCH. It can also been seen as a “contrast and compare” with WitCH 15.

Subsection 13.2.5, below, is on “differentiability”. The earlier part of chapter 13 gives a potted, and not error-free, introduction to limits and continuity, and Chapter 12 covers the “first principles” (limit) computation of polynomial derivatives. We’ve included the relevant “worked example”, and the relevant exercises and answers.

WitCH 57: Tunnel Vision

The following is just a dumb exercise, and so is probably more of a PoSWW. It seems so lemmingly stupid, however, that it comes around full cycle to be a WitCH. It is an exercise from Maths Quest Mathematical Methods 11. The exercise appears in a pre-calculus, CAS-permitted chapter, Cubic Polynomials. The suggested answers are (a) \boldsymbol{y = -\frac{1}{32}x^2(x-6)} , and (b) 81/32 km.

VCAA’s Draft Feedback Due TODAY

We’ve been remiss in not writing further on VCAA’s draft for the new mathematics VCE subjects. It’s just, for reasons we’ll explain briefly here and flesh out elsewhere, we’ve struggled to face up to this new nonsense.

But, feedback is due TODAY (midnight? – see links below), and we really oughta say something. So, here are our brief thoughts and then, after that, why we believe none of it really matters:

  • “Computational thinking and algorithms” is pure snake oil.  Inevitably, it will be nothing but wafer-thin twaddle for the training of data monkeys.
  • The increased weight on these meaningless, revolting SACs is insidious.
  • If we read it correctly, more weight will be placed on the non-CAS Methods/Specialist exams; it is not remotely close to enough, but it is good.
  • Statistics was and is and will always be an insane topic to emphasise in school.
  • Foundation Mathematics: Who Cares?
  • General Mathematics: Who Cares?
  • Mathematical Methods: same old swill.
  • The deletion of mechanics from Specialist Mathematics is criminal, but the topic had already been so bled to meaningless that it hardly matters.
  • In principle, the inclusion in SM of “logic” and “proof and number” and “combinatorics” is a good thing. We’ll see.
  • Similarly, in principle the making of SM12 presumed knowledge for SM34 is good; in practice, it is almost certainly bad. Currently, a good teacher at a good school will take the freedom in SM12 to go to town, to show their students some genuine mathematics and real mathematical thought. In the future, that will be close to impossible, and SM12 will likely become as predictable and as dull as MM12 (and MM34 and SM34).

And now, why doesn’t any of it matter? Because, fundamentally it doesn’t matter what you teach, it matters how you teach. What matters is the manner in which you approach your subject and your students, and none of that will change in other than a microscopic manner. Nothing in VCAA has changed, nothing in the general culture of Victorian education had changed. So, why the Hell would twiddling a few dials on utterly insane subjects assessed in an utterly insane manner make any meaningful difference?

Everything VCAA touches, they will turn to shit. That will continue to be true until there is a fundamental cultural shift, in VCAA and generally.

I hate this place.

LINKS

  • The current (pre-COVID) study design (pdf) is here.
  • The draft for the new study design (word) is here.
  • The key changes overview (work) is here.
  • The link for feedback (until March 9, 2021) is here.

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?