ACARA Crash 4: The Null Fact Law

Well, the plan to post each day lasted exactly one day.* We have an excuse,** but we won’t make excuses. We’ll try to do better.

This A-Crash consists of two Content-Elaboration combos for Year 9 Algebra.


expand and factorise algebraic expressions including simple quadratic expressions


recognising the application of the distributive law to algebraic expressions

using manipulatives such as algebra tiles or an area model to expand or factorise algebraic expressions with readily identifiable binomial factors, for example, \color{blue}\boldsymbol{4x(x + 3) = 4x^2 +12x} or \color{blue}\boldsymbol{(x + 1)(x + 3) = x^2 + 4x + 3}

recognising the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions including the use of digital tools to systematically explore factorisation from \color{blue}\boldsymbol{x^2 + bx + c} where one of \color{blue}\boldsymbol{b} or \color{blue}\boldsymbol{c} is fixed and the other coefficient is systematically varied

exploring the connection between exponent form and expanded form for positive integer exponents using all of the exponent laws with constants and variables

applying the exponent laws to positive constants and variables using positive integer exponents 

investigating factorising non-monic trinomials using algebra tiles or strategies such as the area model or pattern recognition


graph simple non-linear relations using graphing software where appropriate and solve linear and quadratic equations involving a single variable graphically, numerically and algebraically using inverse operations and digital tools as appropriate


graphing quadratic and other non-linear functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs

using graphs to determine the solutions to linear and quadratic equations

representing and solving linear and quadratic equations algebraically using a sequence of inverse operations and comparing these to graphical solutions

graphing percentages of illumination of moon phases in relationship with Aboriginal and Torres Strait Islander Peoples’ understandings that describe the different phases of the moon


*) Luckily, 1 is a Fibonacci number.

**) “Burkard, please put down the whip.”

27 Replies to “ACARA Crash 4: The Null Fact Law”

  1. *Sigh* What a disaster. Where to start. Perhaps a few thoughts on each line. There’ll be plenty left for others because I just don’t have the stamina.

    I have no problem with Content 1:
    “expand and factorise algebraic expressions including simple quadratic expressions.”

    I have no problem with sentence 1 of Elaboration 1:
    “recognising the application of the distributive law to algebraic expressions.”

    That’s where it should have stopped. Unfortunately it did not.
    We get sentence 2. I’m predicting lots of textbook questions along the lines of:
    ‘The area of a rectangular mat is 4x^2 +12x. Find the lengths of its sides.’
    We might even get cheese.

    We get sentence 3. I don’t actually know what’s intended here. Maybe a ‘digital tool’ gets used to ‘discover’ that:
    1) a quadratic doesn’t always have real factors…? Or maybe
    2) the factors don’t always involve integers …?
    But I’m only guessing what’s intended. And if I’m guessing right, the words ham-fisted, clumsy and gratuitous use of technology come to mind.

    Sentences 4 and 5: I have no idea. I need an elaboration of the elaboration. (Luckily there’ll be lots of snake-oil sellers only too happy to oblige).

    Sentence 6: Do they mean non-monic quadratic trinomial? Or is something like 2x^3 - 3x + 1 suddenly fair game? Fancy jargon (for Yr 9 teachers eyes only?) – I wonder how many teachers will understand it? The writers don’t seem to. And I see we’re back to rectangular mats again.

    At least there’s no gratuitous Aboriginal and Torres Strait Islander Peoples’ elaboration. (But then I kept reading, unfortunately)

    Content 2 …

    Line 1: “graph simple non-linear relations …”
    looks OK. Hopefully the elaborations tell us what is meant by simple (Spoiler alert: they don’t).
    “… using graphing software where appropriate”
    And there it is – the mandatory tugging of the forelock to technology.

    Line 2: “… and solve linear and quadratic equations involving a single variable graphically, numerically and algebraically …”
    How do you solve a quadratic equation graphically? I know how you can approximately solve, but the statement implies solving exactly. And solving numerically …? Does this mean:
    1) using Newton’s method (or some other numerical algorithm)? Or does it mean
    2) use a digital tool?
    Or does it mean get approximate solutions using 1) and/or 2)?

    Line 3: “… using inverse operations and digital tools as appropriate”
    By using inverse operations, does ACARA mean first complete the square and then make x the subject?
    OK, I see solving numerically probably does mean using a ‘digital tool’.

    Elaborations 2: …
    I can’t bring myself to comment any further, except to ask what the fuck does
    “graphing percentages of illumination of moon phases in relationship with Aboriginal and Torres Strait Islander Peoples’ understandings that describe the different phases of the moon”

    Acknowledgement is one thing, being gratuitous is a very different thing.

    The snake-oil sellers are rubbing their hands in glee. Not to mention the textbook publishers.

    1. John, the answer to your “inverse operations” question is, with 100% certainty, not completing the square.

      1. Yeah, I already knew the answer when I asked.

        It would be helpful if there was an Elaboration on what “a sequence of inverse operations” is meant to mean in this context … The ACARA clowns are simply writing down a shit-load of plagiarised jargon that they don’t understand but think sounds good. And de Carvalho is happy to be its advocate.

        Clown: “And then we’ll toss in a bit about solving quadratic equations algebraically using a sequence of inverse operations”.
        de Carvalho: “That sounds amazing, Bozo. You’re really nailing it. What a fantastic curriculum we’re going to have. Singapore, eat my shorts! Now, where’s my sign.”

        Actually, I think it might mean solving something like 2x - 3 = 0 (after having used the Null Factor Law). And they haven’t lied, because you do indeed need to use a sequence of inverse operations to do this.

        1. I think you’re giving them too much credit. I think they just figured that somehow solving a quadratic must be an inverse thing.

  2. Area model?!? How the fuck does anyone actually think that is a real, useful method of factorising expressions.

    Look, sir, I factorised it! A rectangle with sides of length f(x) and 1.

    I bailed after that. Might look more later…

  3. I am confident that we agree that to educate all our students about Aboriginal and Torres Strait Islander Peoples is a worthwhile goal. However, lacing every subject with these references might not be the way to go. It might be better to have some compulsory subjects in the curriculum dedicated to this goal.

    1. Terry, I share your confidence. I have no problem with worthwhile references, so allow me to re-phrase:

      “However, lacing every subject with [gratuitous and forced] references might not be the way to go”.

      I find the gratuitous elaborations more insulting than none at all.

      Do we know any of the clowns on the ACARA panel writing this stuff ? Is there an Aboriginal or Torres Strait Islander on or advising the panel? ACARA are getting all its bullshit from somewhere … Where? Who/what are the primary sources?

      1. I don’t think it matters. As I wrote in my Dark and Stormy Night post, the Aboriginal stuff is ridiculous, but not as ridiculous, nor nearly as damaging than the more general content.

        Don’t be Duttons. There is much, much worse in the above than the Aboriginal nonsense.

        1. Hopefully I’ve touched on some of the worse-ness you mention in my very first comment. But I think the Aboriginal stuff does matter. And I still want to know:

          Who’s on the review panel? Who/what are the primary sources being used by the panel? I think this matters a lot.

          The identities of the people on the teacher training review panel are known – loudly trumpeted for all to hear. And yet the curriculum review panel is anonymous. We have a right to know who the clowns are that are serving up all this bullshit.

          Educating students about Aboriginal and Torres Strait Islander Peoples and their contributions is worthwhile. I think it’s important to know whether representatives of these cultures are having an active input into this Daft Curriculum. One of the worse-nesses for me is that this doesn’t seem to be the case. Unsurprising, since it is clear that ACARA is not using experts of any sort, whether they be mathematicians, Aboriginal and Torres Strait Islander Peoples, or competent and experienced practising teachers.

          1. Yes, the major points are 1) the Curriculum is shit; 2) some people wrote this shit; 3) some people approved this shit. I agree, the answers to 2) and 3) should be publicly available.

  4. Most of the major points have been raised. But I’m curious about this elaboration:

    “recognising the relationship between expansion and factorisation and identifying algebraic factors in algebraic expressions including the use of digital tools to systematically explore factorisation from x^2 + bx + c where one of b or c is fixed and the other coefficient is systematically varied”

    My question is the pedagogy behind this. In fact, changing just one coefficient (especially the coefficient of x) without changing the other can lead into a nice algebraic mess unless you know what you’re doing – which clearly these clowns don’t. Surely you’d want to spend time ensuring that students had a good understanding of the relationships between roots and coefficients – but treating coefficients on their own seems a bit pointless. Indeed, much of the behavior of a quadratic is related to its discriminant, which is of course a function of all coefficients. What on earth will students learn (even with using “digital tools”) by playing around with just one of them?

    Then there’s the sop to Aboriginal and Torres Strait Islanders. Here I fully agree with Terry. It’s vital that we bring them, their histories, and their culture (the oldest continuously maintained culture on the planet, and by a huge margin) more fully into our national consciousness. But I ask – really – what does this have to do with the learning of mathematics? It’s another example of a contrived “application”, but this time overlaid with a thick layer of smug condescension and – it seems to me – plenty of patronizing.


    1. Indeed. Nothing can be gained from “systematically explore factorisation from x^2 + bx + c where one of b or c is fixed and the other coefficient is systematically varied.”

      Nada. Bupkis. Zilch.

      This is the ignorant sort of ‘Inquiry Approach’ bullshit that is being pushed by Rachel Whitney-Smith (Curriculum Specialist Mathematics Curriculum, ACARA). And de Carvalho is too busy looking for his sign to realise.

      You need either the discriminant or an understanding that x^2 + bx + c = \left(x - \frac{b}{2}\right)^2 + c - \frac{b^2}{4}. Neither are in the ACARA Yr 9 Daft Curriculum. And neither are in the ‘Core Yr 10’ ACARA Daft Curriculum (they are only optional at the Yr 10 level).

      1. JF, maybe keep to the bashing of ACARA, rather than the sniping at individuals (says the guy who walloped De Carvalho). No question that De Carvalho’s fronting for this stuff, for whatever reason, is disgraceful. But we don’t know who wrote this awful stuff, and my main interest at this moment is the documentation of this awful stuff. Later we can quibble over should be first against the wall.

        1. There can be little doubt that Whitney-Smith is the Svengali behind this mess. So credit where credit is due.

          The fact that de Carvalho said things like:
          “No doubt some will argue the proposed revisions don’t go far enough, while others will say they go too far, …”
          shows some insight. But he’s drank so much of the Whitney-Smith Kool-Aid that there’s no going back.

          de Carvalho’s big mistake was in not getting an external, independent audit of this mess before fronting it and proclaiming that everything is awesome. He should go back to the Petri dish and look for his sign.

          de Carvalho is a dupe, It’s time the real Svengali behind this mess gets some credit. ACARA is not the problem, the clowns inside it calling the shots are the problem.

          1. John, I genuinely question some of what you’ve written there, but it doesn’t matter.

            I’ll leave these comments up but no more, please. Unless there is a relevant public statement by someone, please stick to criticising “ACARA” and the draft curriculum. I’m more than happy to marshal a firing squad later on.

            1. OK. Fair enough. Will do. Maybe the (ir)responsible Svengali’s name will get mentioned during the MAV session on Thursday and then we’ll all know for certain who is pulling the strings.

          2. The controversial things aside, it really is a pet peeve of mine (usually it comes up in science media stories) when this kind of phrasing (“some say too far, others say not enough”) is used to describe something multi-dimensional. It implies a kind of single-dimensionality to the subject matter — here, the curriculum — which is very misleading. I hate it.

            1. Yes, it’s really mealy-mouthed. It makes a complex, multi-dimensional discussion look completely one-dimensional. It’s classic politic-speak, making excuses for a piece of shit from the get-go.

    2. I’m fairly certain the only thing they had in mind was varying the c (and not the b, despite what they have written) and then leading the students on to remark that the number of roots goes from two, to one, to zero, or, from zero, to one, or to two. The students may discover that for c sufficiently negative, there are two, and between two and zero there is always a one.

      I’m not excusing it in any way, of course. Quite the opposite. The true intent should also be wacked.

  5. The aboridginal references are important as they more clearly demonstrate the lack of academic and professional standards of the authors. No one should publish minority cultural information without references and context. Further without the comparison to similar historical knowledge of majority groups it risks presenting the minorities as unsophisticated. For example compare it to the beliefs of the typical European circa 1770. Take for example the belief that that all land animals are descendants of those saved from a flood on board a large boat.

    While the rest is just obviously bad to anyone you knows some mathematics the cultural references should be obviously bad to everyone.

  6. In respect of Aboriginal references, I do know that Prof Chris Matthews at UTS, who is the founder of ATSIMA, the Aboriginal and Torres Strait Islander Mathematics Alliance (and he himself is a Noonuccal/Quandamooka man), is involved with ACARA in some way. So maybe I should eat my words. But I still think it’s silly and a bit pointless, bunging indigenous stuff on top of a curriculum for which it is basically irrelevant.

    1. His involvement might simply have been to be asked:

      Give us every possible conceivable link to Aboriginal and Torres Strait Islander mathematics.

      He might have provided a comprehensive list with a disclaimer that a lot of the stuff on it was a bit of a stretch. Which would not have mattered one iota to the clowns wanting to display their woke politics in the curriculum.

  7. An arithmetic aside:
    I turned the TV on a moment ago (5.31 PM). It was on Channel 9 Millionaire Hot Seat.
    I didn’t change to Channel 2, as was my original intent. I was mesmerised by the following question and the contestant’s reaction to it:

    Which of these equations yields the lowest value?

    A: 2 – 1
    B: 1 – 2
    C: (-2) – 1
    D: (-2) + 1

    Overlooking the fact that these arithmetic expressions were called equations, it was like watching a car crash in slow motion. The poor sap finally passed on the question after about 30 seconds of stammering and prevaricating. It was both mesmerising and painful to watch.

    Maybe calling them equations (rather than simply asking “Which of the following has the lowest value?) confused the contestant. Maybe the extraneous use of brackets muddied the waters. Maybe being filmed in a studio before an audience under time pressure flustered the poor sap and his brain just froze and could not calculate. Maybe there was similar effect to having to kick a goal after the siren in the final quarter of a Grand Final to win the game.

    But the fact is that this poor sap could not answer this simple arithmetic question. This sort of dyscalculia seems very common. Is the curriculum and how it’s taught significantly contributing to this? Will it get worse?

    1. I’ve often seen people struggle with the idea of “lower” when thinking of negative numbers. They are happy enough to say -2 < -1, but if you say in words "is minus 2 lower than minis 1?" some will pause, and others will totally freeze. Yes, IMO the education is at fault.

      1. The humble number line (together with the humble unit circle) is still my good friend, even when teaching Specialist Mathematics.

        I can’t see the number line mentioned in the Daft Curriculum, but I haven’t searched for it forensically (for some reason the thought is very unappealing) and I sometimes struggle to find the cheese in the fridge.

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