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.



22 Replies to “WitCH 52: Lines of Attack”

  1. Example 10 is bad enough – why do we need to draw two graphs to see the lines are parallel? The ratio of the coefficients being equal is enough to see this…

    Example 11… parameter? WHY???

    Example 12 is typical enough VCAA, and very typically on Paper 2 in MCQ, so *perhaps* the calculator use is justified… but even if (and that is a *big* if)… what a stupid way to do it.

    1. Thanks, RF. I’d suggest the problems begin even before Example 10. But yeah, Example 10 is not a great start, for the reasons you suggest.

      I can kind of see the point of Example 11: the issue is to somehow name or capture the infinitely many solutions, which is made a bit more explicit with the use of a parameter. But, yes, it’s not like they’ll ever do anything with that, at least not in Methods.

      Example 12 is insane every which way, not least because it’s the example students are instructed to refer to when attempting Exercise 5.

  2. In the top picture that begins with the words “Two distinct straight lines are either parallel or meet at a point”, Case 3 says “Two copies of the same line”. What does that mean…? That three lines exist, an original and two copies? All I see in the equations is one line, and that single line -is- the set of (infinitely many) solutions. Sure, I know what they meant to say, but I think that “there’s only one line here” is simpler to understand than “Two copies of the same line exist here”.

    1. I suspect they meant to say “coincident” as this word is used elsewhere (from memory) but didn’t here for some reason.

      The interchanging of “infinite” and “infinitely many” is also problematic, but this is one of the lesser gripes I have with this portion of text.

      For such a remarkably small idea, it does seem to be high on VCAA’s list of things to test each year!

      1. Thanks, RF. I’m actually curious about the testing of this. Somehow I thought that VCAA regularly tests the 0/1/infinity thing, but I couldn’t actually find it in recent exams. It definitely appears in tests and SACs; is this just because, like me, everybody *thinks* VCAA is gonna test it? Or, did I just not look hard enough?

        As for the “infinite” solutions, you’re just referring to the answer to Exercise 5? The answer is a scrabbled mess from start to end, so I’d put zero weight on the language used there.

        1. I can probably find you 5 or more examples between 2006 and 2018 (so a bit less than once every 2nd year). When the percentage of students getting it right (on MCQs) was less than 50% it seemed to reappear the following year, with minimal changes. After the year (will check the year) 90% or so got the question correct, it went away for a while.

    2. Thanks, Don. Yes, it’s the kind of close-but-not-close-enough sloppiness that always bogs down these texts. It can be tricky to word these things, and one doesn’t want to be too pedantic. For example, it’s natural to talk about starting with two lines which then may turn out to be equal (meaning we really only started with one line). But, however you then want to refer to that in hindsight, it’s definitely not with “two copies”.

    1. Which part of the solution?

      There is a typo in the second and third lines. They have accidentally used the coefficient matrix instead of the determinant of the coefficient matrix.

      1. The matrix determinant method for 2×2 always seems overly complicated to me. Yes, it works, provided you understand why a singular matrix leads to an absence of a unique solution.

        Checking gradients (or ratios of coefficients) just seems a lot more efficient and thus less error-prone.

        1. Hi, RF. Whatever else, if you’re gonna use determinants out of the blue, then it’d be nice if the word “determinant” appeared. And of course it’d be nice to not screw up the matrix notation. Needless to say, the text makes no mention of determinants in the prior Section 2E.

          In fact, the determinant feels a little easier and less error-prone to me than your two methods (and a third variant I prefer). But only a little. So, although I’m fine with teachers/students/texts using it, it’s probably better not to.

          I think the use of the 2 x 2 determinant here has to be compared to the use of the 3 x 3 determinant to check for the independence of vectors. The 2 x 2 determinant above can be easily justified, and amounts to a justified memory device. But for independence, the use of the 3 x 3 determinant is *much* easier than the official curriculum methods, and I doubt if any teachers justify its use (and I doubt few could).

          1. The justification of its use is beyond what can easily be taught ad hoc (this is why I get so damn angry that matrices are not an actual part of the course). It makes me really angry that the Stupid Design allows the brainless application of things like this. If I had my way, I’d delete linear independence from the Stupid Design altogether – I do not see the point of its inclusion.

            I’ve attached a simple justification (for those who are interested) that I give to my students but I don’t go through it in class. It’s simply there for interested students to get some idea of why it works. It’s actually a pretty pointless justification because the key statement cannot be proved without a lot of background matrix theory.

            So I have no problem at all that teachers don’t justify its use – it would be ridiculous to even try. And I think you’re right that few teachers could justify its use – although one could argue that they should not even need to.

            Linear Independence and Determinants

    2. Hi, Terry. Note the typo pointed out by studyroom. Note also it’s not that much of an alternative; it’s a fractionally different method to determine whether the lines are parallel, and that is all.

  3. @RF: I have no problem with (and in fact I encourage) solutions given in parametric form. And I’d argue that although “it’s not like they’ll ever do anything with that, at least not in Methods”, some of those students will definitely be doing something with it when they study UMEP Mathematics in Yr 12. Those students should be kept in mind … And all Methods students are certainly familiar with parametric solutions from solving trigonometric equations … However, it should definitely have been stated that \lambda \in R.

    And I have no problem with using the determinant of a matrix provided matrices are actually included as a separate topic in the Study Design. (Once you move to three linear equations in three unknowns it becomes almost essential). But matrices not included. And the situation is even more diabolical in the draft Study Design (discussed below).

    @studyroom: Yes, they should have used the notation of straight lines not square brackets when calculating the determinant. But in fact they should NOT have included the alternative method at all!!:

    Setting up a matrix equation and using the determinant has not been in the Study Design since 2014! This method should have been clearly flagged as using content outside of the Study Design. Even then, it uses so many things outside of the Study Design (not the least WHY det = 0 means that there’s no unique solution) that only an idiot who was trying to show off would include it.

    There’s a much bigger issue here for teachers wanting to provide feedback (in the misguided belief that feedback will actually be considered) on the draft Study Design:

    The fools who wrote this draft have included the ad hoc use of matrices throughout both Methods and Specialist to an even greater extent than previous Study Designs and yet matrices are still not included as a specific topic in the Study Design for either Methods or Specialist. Specialist Maths in particular is a mess:

    Area of Study 5: How can the determinant representation of the cross product be taught when matrices are not part of either the Methods or Specialist syllabus? This is yet another example of ad hoc bits of matrix theory being included (see use of matrices for transformations in Maths Methods) without the formal inclusion of matrices in the syllabus. Ditto for systems of linear equations.

    And how can proof include matrices in Area of Study 1 when matrices as a topic is not explicitly included in the Study Design?

    @Marty: Over the last 10 years, the following methods exams contained questions of the same type as Example 12:

    2011 Exam 1 Q6.
    2012 Exam 2 MCQ 17.
    2014 Exam 2 MCQ 17 (matrix methods no longer on Study Design).

    Some teachers might argue that it hasn’t appeared in the last 6 years and so is over-due … (And I suppose the fact that such questions are in the textbooks keeps it on the teacher radar …)

    1. @Marty: Those questions I referenced are from the *November* VCAA exams. I should also include the *NHT* Exams:

      2017: Exam 2 MCQ 3.
      2019: Exam 2 MCQ 8.

      So the question is still well and truly ‘in the air’ …

      Here’s an interesting dot point from the current Study Design:

      • solution of simple systems of simultaneous linear equations, including consideration of cases where no solution or an infinite number of possible solutions exist (geometric interpretation only required for two equations in two variables).

      And here’s an interesting dot point from the previous Study Design (accreditation period 2006 – 2015):

      • solution of systems of simultaneous linear equations, including consideration of cases where no solution or an infinite number of possible solutions exist …. (familiarity with matrix representation of systems of simultaneous linear equations with up to five equations in five unknowns will be assumed).

      1. Thanks again, JF. Yes, definitely there. Interesting that the 2011 style of question has not reappeared on Exam 1. Maybe too algebraic for the current button-monkey style.

    2. Ah, thanks, JF! It was a question like the 2011 question (basically exercise 5) I had been hunting for, and couldn’t find. This kind of question often appears in tests and SACS, and usually in a way that irritates the hell out of me, as does the 2011 question. I’ll update the WitCH to include it.

      Regarding the parameters, I understand your argument. Even more relevant than your examples is the parametrisation of curves in SM. Still, I’m kind of with RF, that it feels a little gratuitous here.

      Regarding determinants, of course you are correct, that the fundamental problem is the idiotic, half-pregnant incorporation of matrices in the curriculum. Still, given the situation, I don’t see how one can or should rule out the use of determinants.

  4. Thanks JF, I have no issue with parameters being used, but in this case, they just feel a bit too out of place.

    Indeed, when looking at the intersection of planes in 3D space, they are essential, but since Methods goes nowhere near this in any meaningful way…

    Matrices, too, just feel really out of place here given the Methods context. But that is nothing new.

  5. In the “cases” table, unique solution does not necessarily mean one unique solution. This seems to be sloppy use of words.

    There is the “Geometry of Simultaneous … ” title and they do not describe the geometric interpretation of the solutions to a linear system of two variables (line, point, empty set). So title and the content of the exercise do not connect to each other that well.

    Though I am struggling to find anything substantially wrong with that section of the textbook.

      1. Yep okay, just took crap to mean wrong. Well to add to my previous comment, they just needed the table with the different behaviours a linear system can have and add a picture for each. Just seems a little clunky with the first part and the table.

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