ACARA Crash 7: Spread Sheeet

(In keeping with our culturally sensitive ways, the title should be read with a thick Mexican accent.)

We’re working on a non-WitCHlike Crash post, but no way will that be done tonight. Luckily, frequent commenter Glen has flagged some easily postable nonsense, and we can keep the Crash ball rolling.

This A-Crash consists of a Content-Elaboration combo for Year 6 Number:


identify and describe the properties of prime and composite numbers and use to solve problems and simplify calculations


understanding that a prime number has two unique factors of one and itself and hence 1 is not a prime number

testing numbers by using division to distinguish between prime and composite numbers, recording the results on a number chart to identify any patterns

representing composite numbers as a product of their factors including prime factors when necessary and using this form to simplify calculations involving multiplication such as \color{blue}\boldsymbol{15 \times 16} as \color{blue}\boldsymbol{5 \times 3 \times 4 \times 4} which can be rearranged to simplify calculation to \color{blue}\boldsymbol{5 \times 4 \times 3 \times 4 =20 \times 12}

using spread sheets to list all of the numbers that have up to three factors using combinations of only the first three prime numbers, recognise any emerging patterns, making conjectures and experimenting with other combinations

understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number, for example, 216 is divisible by 8 because the number represented by the last three digits is divisible by 8, and hence 216 is also divisible by 2 and 4, using this to generate algorithms to explore


UPDATE (25/05/21)

Thanks, everyone, so far. We’re going nuts with work, so a quick WitCHlike update while the window is open.

0) How can ACARA be so, so, so appallingly bad with their grammar and punctuation? We honestly don’t get it. Is the content descriptor accidentally missing a pronoun, and a comma, and a preposition, or do they genuinely like how it reads?

1) Yes, the free-floating and otherwise irritating “hence”, the fact that “prime” is undefined is appalling. So is using “1” and “one” in the same sentence to refer to the same thing. So is “two unique factors of one and itself and …”.

2) Possibly John’s guess on the second elaboration is correct. What would be focussed and useful is to take a 12 x 12 table of numbers and cross off the multiples (and circle 1). So, you get the kids to do the sieve of Eratosthenes thing, and emphasise the multiples as composites. You know, a clearly expressed investigation, with clear purposes.

3) This is Year 6, and so we’re not so concerned about “Fundamental theorem of arithmetic” not being mentioned here, although of course both existence and uniqueness of the prime factorisation should have been spelled out, even if only as something to “explore”. It’s way too important to be included as just a “by the way” part of a multiplication trick. As a side point, in regard to our previous Crash post, it is notable when and how “Fundamental theorem” first appears.

4) 15 x 16? Really?

5) We’re guessing the spread sheeet activity was intended to mean using each prime at most once. Given these people can’t write, however, it’s only a guess. But if so, that would be a reasonable exercise, IF you ditched the spread sheeet, and IF you repeated the exercise a few times with varying selections of primes. None of which will happen.

6) It is unbelievably stupid to introduce prime stuff in combination with divisibility tricks. The former is, well, fundamental, and the latter is a base ten game.

7) “The number represented by the last three digits”. Of what? Who talks this way? Who talks this way and expects to be understood?

8) What are the other digits of 216?

9) Even if there were other digits, a number ending in 216 is a really stupid choice to demonstrate divisibility by 8. These things matter.

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


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}


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

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.”

PoSWW 17: Blessed are the Cheesemakers

This one is old, which is not in keeping with the spirit of our PoSWWs and WitCHes. And, we’ve already written on it and talked about it. But, as the GOAT PoSWW, it really deserves its own post. It is an exercise from the textbook Heinemann Maths Zone 9 (2011), which does not appear to still exist. (And yes, the accompanying photo appeared alongside the question in the text book.)

PoSWW 5: Intelligence is not a Factor

The following PoSWW comes courtesy of Franz, who states that “when it comes to ‘stupid curricula, stupid texts and really monumentally stupid exams’ no Western country, with the possible exception of the US, is worse than Germany.” We take that as a challenge, and we’re waiting for Franz to back up his crazy-brave claim.

Franz’s PoSWW, however, has nothing to do with Germany. This PoSSW follows on from two of our previous posts, on idiotic questions appearing in New Zealand exams. Franz wrote to us, noting that the same style of question appears in the Oxford Year 8 text My Maths. Indeed, a number of versions of this ludicrous question appear in My Maths, all inventively awful in their own way. The two examples below are enough to give the flavour:

PoSWW 4: Overly Complex

This PoSWW comes courtesy of a smart Year 11 VCE student who, it appears, may be a rich source of such nonsense. It’s an exercise in the Jacaranda text MathsQuest 11, Specialist Mathematics (2019).

To be honest, we’re not sure the exercise below is a PoSWW. It may simply be a minor error, the likes of which are inevitable in any text, and of which it is uninteresting and unfair to nitpick. But, for the life of us, we have no idea what the authors might have intended to ask. Make of it what you will:

UPDATE: For those hoping that context will help make sense of the exercise, the section of the text is an introduction to factoring over complex numbers. And, the text’s answer to the above exercise is A = 2, B = 5, C = -1, D = 2.

Polynomially Perverse

What, with its stupid curriculastupid texts and really monumentally stupid exams, it’s difficult to imagine a wealthy Western country with worse mathematics education than Australia. Which is why God gave us New Zealand.

Earlier this year we wrote about the first question on New Zealand’s 2016 Level 1 algebra exam:

A rectangle has an area of  \bf x^2+5x-36. What are the lengths of the sides of the rectangle in terms of  \bf x.

Obviously, the expectation was for the students to declare the side lengths to be the linear factors x – 4 and x + 9, and just as obviously this is mathematical crap. (Just to hammer the point, set x = 5, giving an area of 14, and think about what the side lengths “must” be.)

One might hope that, having inflicted this mathematical garbage on a nation of students, the New Zealand Qualifications Authority would have been gently slapped around by a mathematician or two, and that the error would not be repeated. One might hope this, but, in these idiot times, it would be very foolish to expect it.

A few weeks ago, New Zealand maths education was in the news (again). There was lots of whining about “disastrous” exams, with “impossible” questions, culminating in a pompous petition, and ministerial strutting and general hand-wringing. Most of the complaints, however, appear to be pretty trivial; sure, the exams were clunky in certain ways, but nothing that we could find was overly awful, and nothing that warranted the subsequent calls for blood.

What makes this recent whining so funny is the comparison with the deafening silence in September. That’s when the 2017 Level 1 Algebra Exams appeared, containing the exact same rectangle crap as in 2016 (Question 3(a)(i) and Question 2(a)(i)). And, as in 2016, there is no evidence that anyone in New Zealand had the slightest concern.

People like to make fun of all the sheep in New Zealand, but there’s many more sheep there than anyone suspects.

UPDATE (04/02/19): An Oxford school text joins in the fun.