The word “equivalent” is one of the most useful in mathematics and one of the most abused in mathematics education. The implication is that this one is not all ACARA’s fault. Nonetheless, the fact that it was predictable that ACARA would make a mess of it doesn’t alter the fact that ACARA made a mess of it. So, here we are. And a warning: this is a long post; there seemed no way around it.
This one has no deeper meaning, or at least not much. On a recent excursion with the little monsters to Electronic Noise Hell, we noticed the sign above, which didn’t seem quite right. Teachers may enjoy having their students get puzzled and/or unpuzzled. (The very friendly service girl was determinedly not puzzled, except by our puzzlement.)
OK, following up on our previous post, we have another fraction question. This one is not new, has appeared in the comments of a previous post, and many readers will have heard us bang on about it. Nonetheless, given the discussion on the previous question, and given the possibility that some new readers might not have yet read Marty’s Collected Sermons, it seems worthwhile giving the question its own post. Here it is:
![\[\color{Mulberry}\boldsymbol{(-1)^\frac26 \overset{?}{=} (-1)^\frac13}\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-b13eb8594441b1389288385f86679aa5_l3.svg "Rendered by QuickLaTeX.com")
The following are Year 10 Number-Algebra content-elaborations in the current curriculum:
CONTENT
Apply the four operations to simple algebraic fractions with numerical denominators
ELABORATIONS
expressing the sum and difference of algebraic fractions with a common denominator
using the index laws to simplify products and quotients of algebraic fractions
CONTENT
Solve linear equations involving simple algebraic fractions
ELABORATIONS
solving a wide range of linear equations, including those involving one or two simple algebraic fractions, and checking solutions by substitution
representing word problems, including those involving fractions, as equations and solving them to answer the question
And what does the draft curriculum do with these?
Removed
And, why?
Not essential for all students to learn in Year 10.
God only knows how one develops fluency with expressions that cease to exist.
Whatever the merits of undertaking a line by line critique of the Australian Curriculum, it would take a long time, it would be boring and it would probably overshadow the large, systemic problems. (Also, no one in power would take any notice, though that has never really slowed us down.) Still, the details should not be ignored, and we’ll consider here one of the gems of Homer Simpson cluelessness.
In 2010, Burkard Polster and I wrote an Age newspaper column about a draft of the Australian Curriculum. We focused on one line of the draft, an “elaboration” of Pythagoras’s Theorem:
recognising that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds
Though much can be said about this line, the most important thing to say is that it is wrong. Seven years later, the line is still in the Australian Curriculum, essentially unaltered, and it is still wrong.
OK, perhaps the line isn’t wrong. Depending upon one’s reading, it could instead be meaningless. Or trivial. But that’s it: wrong and meaningless and trivial are the only options.
The weird grammar and punctuation is standard for the Australian Curriculum. It takes a special lack of effort, however, to produce phrases such as “right-angled triangle calculations” and “generate results”. Any student who offered up such vague nonsense in an essay would know to expect big red strokes and a lousy grade. Still, we can take a guess at the intended meaning.
Pythagoras’s Theorem can naturally be introduced with 3-4-5 triangles and the like, with integer sidelengths. How does one then obtain irrational numbers? Well, “triangle calculations” on the triangle below can definitely “generate” irrational “results”:
Yeah, yeah, 
is not a “surd”. But of course we can replace each by √7 or 1/7 or whatever, and get sidelengths of any type we want. These are hardly “triangle calculations”, however, and it makes the elaboration utterly trivial: fractions “generate” fractions, and irrationals “generate” irrationals. Well, um, wow.
We assume that the point of the elaboration is that if two sides of a right-angled triangle are integral then the third side “generated” need not be. So, the Curriculum writers presumably had in mind 1-1-√2 triangles and the like, where integers unavoidably lead us into the world of irrationals. Fair enough. But how, then, can we similarly obtain the promised (non-integral) fractional sidelengths? The answer is that we cannot.
It is of course notable that two sides of a right-angled triangle can be integral with the third side irrational. It is also notable, however, that two integral sides cannot result in the third side being a non-integral fraction. This is not difficult to prove, and makes a nice little exercise; the reader is invited to give a proof in the comments. The reader may also wish to forward their proof to ACARA, the producers of the Australian Curriculum.
How does such nonsense make it into a national curriculum? How does it then remain there, effectively unaltered, for seven years? True, our 2010 column wasn’t on the front of the New York Times. But still, in seven years did no one at ACARA ever get word of our criticism? Did no one else ever question the elaboration to anyone at ACARA?
But perhaps ACARA did become aware of our or others’ criticism, reread the elaboration, and decided “Yep, it’s just what we want”. It’s a depressing thought, but this seems as likely an explanation as any.