ACARA Crash 5: Completing the Squander

The previous A-Crash consisted of everything we could find in the Daft Curriculum on the algebraic treatment of polynomials and polynomial equations. This companion A-Crash consists of everything we could find in the Year 10 draft on the same material. 

CONTENT (Year 10)

expand and factorise expressions and apply exponent laws involving products, quotients and powers of variables. Apply to solve equations algebraically

ELABORATIONS

reviewing and connecting exponent laws of numerical expressions with positive and negative integer exponents to exponent laws involving variables

using the distributive law and the exponent laws to expand and factorise algebraic expressions

explaining the relationship between factorisation and expansion

applying knowledge of exponent laws to algebraic terms, and simplifying algebraic expressions using both positive and negative integral exponents to solve equations algebraically

CONTENT (Year 10 Optional Content)

numerical/tabular, graphical and algebraic representations of quadratic functions and their transformations in order to reason about the solutions of \color{blue}\boldsymbol{f(x) = k}

 ELABORATIONS

connecting the expanded and transformed representations

deriving and using the quadratic formula and discriminant to identify the roots of a quadratic function

identifying what can be known about the graph of a quadratic function by considering the coefficients and the discriminant to assist sketching by hand

solving equations and interpreting solutions graphically

recognising that irrational roots of quadratic equations of a single real variable occur in conjugate pairs

19 Replies to “ACARA Crash 5: Completing the Squander”

  1. I don’t understand the final sentence. The solutions of x^2=4 occur in “conjugate pairs” if we understand “conjugate” as plus/minus sign, yet x is of course rational, making the statement very misleading. Or, perhaps more likely, “imaginary [and not real]” is meant instead of “irrational”, and the coefficients a,b,c in f(x)=ax^2+bx+c are supposed to be real numbers (as is probably taken for granted at this level, but which I think should not be entirely swept under the carpet).

    1. I wasn’t going to comment here, mainly because
      1) I don’t have the stamina, and
      2) there are similarities to things here: https://mathematicalcrap.com/2021/05/21/acara-crash-4-the-null-fact-law/#comment-8703

      Nevertheless … In response to:
      a) CR. The clowns at ACARA probably mean conjugate root pairs. This is what happens when clowns plagiarise jargon from other sources without understanding what it means.

      b) Optional material. It’s a total disgrace that it’s … optional. Who’s the idiot that decided that solving a quadratic equation using the quadratic formula should be optional in Yr 10?

      c) Optional Content Elaborations.
      “connecting the expanded and transformed representations”.
      In response to unavoidable stupidity – WTF??

      1. Thank you John for chipping in on my question in your a). Sorry for asking something possibly rather trivial, but is the term “root” used in secondary education for both the square root and the zeros of an equation f(x)=0 (as is standard in university mathematics)? In any case, no matter which of the two meanings was intended by ACARA here, it does not clear up the issues which I raised in my last posting. Which of course proves, in line with what you say, that the ACARA clowns were able to buy enough lipstick for their doubtlessly handsome salaries.

        1. Hi CR.
          Re: Is the term “root” used in secondary education for both the square root and the zeros of an equation f(x)=0?

          I wouldn’t refer to a ‘square root’ as a ‘root’. I always call a square root a square root. I wouldn’t refer to the solutions of f(x)=0 as zeros or roots. I would call them solutions to the equation. The solutions to the equation are the zeros (or roots) of the function f(x).

          Since I initially misunderstood your question, I initially typed a whole lot stuff (below) that’s irrelevant to your question. But since I made the effort typing it I don’t want to delete it. So for those who are interested:

          You should search up “conjugate root” on the internet.

          “Conjugate root” is typically a phrase you will see when rationalising the denominator of a fraction. For example:

          \displaystyle \frac{3}{1 + \sqrt{2}}

          Multiply the numerator and denominator by the \displaystyle conjugate \displaystyle root of the denominator:

          \displaystyle = \frac{3}{(1 + \sqrt{2})} \times \frac{(1 - \sqrt{2})}{(1 - \sqrt{2})}

          \displaystyle = \frac{3(1 - \sqrt{2})}{1-2}

          \displaystyle = 3(\sqrt{2}-1)

          The interesting question is what is the conjugate root of something like \displaystyle \sqrt{5} - \sqrt{3}. There are two:

          \displaystyle \sqrt{5} + \sqrt{3} and \displaystyle - \sqrt{5} - \sqrt{3}.

          Which one gets used will depend on which one leads to a simpler calculation.

          You might also see the term used at higher levels when finding limits of indeterminant forms such as

          \displaystyle \lim_{x \rightarrow \infty} \left(\sqrt{x^2 - x + 1} - \sqrt{x^2 + x - 2}\right)

          So the context of the question will remove the ambiguity of the meaning.

          The use of the phrase “conjugate pairs” by ACARA is particularly ambiguous since you can obviously get \displaystyle complex conjugate pairs when solving a quadratic equation. It could be argued that the earlier mention of “irrational roots” together with the fact that only real solutions are considered removes this ambiguity. But the ACARA clowns do not deserve the benefit of the doubt.

  2. The disgusting use of grammar is making it painful to read these. It’s repulsive, as in, it is repulsing me from even trying to understand what they are saying. Exhausting. Blech.

    If this is a tactic, a ploy to wear down the critical reader in this war of attrition… well played ACARA. Well played.

    1. Hi Glen. Well said. I couldn’t quite put my finger on my lack of desire to post a comment. You’ve hit the nail right on the head.

      I don’t think it’s a deliberate tactic. I think it’s just a bunch of self-indulgent clowns suffering from the Dunning-Kruger Effect who are trying look smart and trying to make rotting fish (mutton is too flattering) look like lamb (the lamb in this case being the Singaporean curriculum).

      And it doesn’t help when you have a simpleton like de Carvalho who, apparently, is so easily sucked in to advocating this bullshit. He should go back to his Petri dish and get his sign.

  3. “explaining the relationship between factorisation and expansion”

    I would have thought that a(b+c) = ab + ac is Year 8 content, at the latest.

    1. Indeed. And your comment is equally applicable to:

      “apply exponent laws involving products, quotients and powers of variables.”

      Who knows what this means? I don’t. In the context of quadratics functions it’s clearly limited to \displaystyle 5 \times x \times x = 5x^2 and \displaystyle 3 \times x = 3x. Correct me if I’m wrong, but that is (or \displaystyle was) Yr 7 level stuff.

      But we all know how much repetition there is in the mathematics curriculum. Eg. Pythagoras’ Theorem is done in Yr 8, Yr 9 and Yr 10 and there are only marginal differences in its treatment for each year level. Trigonometry is done in Yr 9, Yr 10, Yr 11, again with only marginal differences in its treatment for each year level.

      More importantly, it’s all about the ‘optics’: All the decent Yr 10 stuff is in the Optional Content, so you need something to make the ‘Core’ content look better than it really is.

      What’s needed is a public forum where Rachel Whitney-Smith (Curriculum Specialist Mathematics Curriculum, ACARA) can be asked the tough questions. Be pinned down on the details. Elaborate on the elaborations. It looks like she’s the one pulling the strings on the mathematics curriculum and duping *ahem* I mean briefing de Carvalho. Her LinkedIn profile makes interesting reading. It should come as no surprise that Whitney-Smith has a vested interest in the “Inquiry Approach”.

      The mickey-mouse consultation the MAV is facilitating comes nowhere close to this and is nothing more than a PR stunt by ACARA, from what I can see. And the MAV gets to tick the box. I’d love to see some outrage from the MAV over this Daft Curriculum, but we all know that will never happen.

      1. ACARA is holding public fora/consultations/somethings. There is one this coming Thursday, through the MAV. Needless to say, and if only as a strategic matter, direct confrontation of ACARA should only involve criticism, not abuse.

        1. Yes, the MAV get to tick a box and pat itself on the back for living up to its mission statement. The MAV will be Switzerland as always rather than a genuine voice for teachers.

          ACARA have complete control over what feedback is given, just like VCAA did. An old trick to blunt inconvenient and serious criticism. I’ll be surprised if any meaningful criticism gets air-time.

          It looks like a simple PR stunt to me, fool the rubes into thinking that what they have to say actually matters. ACARA does not want feedback from teachers, it wants lemmings to pat them on the back.

          1. It’s possible. I don’t know the structure of the forum. But my understanding is that it is supposed to include an opportunity for public questions and public response.

        2. Planning on asking a few things? You know, in a public forum, I’m not quite sure what I’d ask to get across even a fraction of my thoughts on this mess.

          1. I’m not an MAV member. But in any case, you have a very good point.

            If you ask a specific question then it feels like nitpicking and doesn’t raise the systemic awfulness. But a more general question is easily pushed away with a general, hand-waving answer. So, what could one *quickly* ask/say to focus proper attention on the draft’s awfulness?

            1. OK, here are some suggestions:

              Two broad questions that come to mind are:
              1) What are the primary sources/references informing the content of the Daft Curriculum? Please provide a list.
              2) What are the clear mathematical points of difference between the Year 9 Algebra and the NON-optional Year 10 Algebra?

              You could also ask some specific questions that are representative of the systemic awfulness. Some questions from each Algebra year level. For example:

              Year 9 Algebra:
              1) How is a Yr 9 student expected to solve a quadratic equation algebraically using a sequence of inverse operations?
              2) How do you solve a quadratic equation graphically?
              3) What is the area model and how exactly does it correctly teach expanding and factorising?
              4) What non-trivial mathematics is learnt from “systematically explor[ing] factorisation from x^2 + bx + c where one of b or c is fixed and the other coefficient is systematically varied.”?

              Year 6 Algebra:
              1) What is meant by a pattern growing multiplicatively? Give some examples. What are the primary sources/references for this term?
              2) What can “Fibonacci patterns in shells” possibly mean?

              I suppose the question
              Who is the Svengali pulling the puppet strings on all this bullshit?
              is a bit too sensitive to ask.

  4. In the optional content for Year 10 preparing students for Years 11 and 12, there is no reference to Indigenous cultures. Why not?

    1. Answer to TM’s question: Because clearly such a reference would be too gratuitous and contrived even for the VCAA clowns.

      But another question that could be asked is:
      Where are these mostly gratuitous and contrived references to Indigenous cultures coming from?

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