We’ve posted on the general nature of PISA’s mathematics questions here and here, and the main point is the sheer awfulness of what is being tested. One question, however, seemed worthy of special note. The following is the first of the PISA 2012 test questions included in this document of past questions, followed by a guide to its grading.

# Tag: Geometry

## WitCH 9: A Distant Hope

This WitCH (as is the accompanying PoSWW) is an exercise and solution from Cambridge’s *Mathematical Methods Units 1 and 2*, and is courtesy of the Evil Mathologer. (A reminder that WitCH 2, WitCH3, Witch 7 and WitCH 8 are still open for business.)

## Update

As Number 8 and Potii pointed out, notation of the form *AB* is amtriguous, referring in turn to the line through *A* and *B*, the segment from *A* to *B* and the distance from *A* to *B*. (This lazy lack of definition appears to be systemic in the textbook.) And, as Potii pointed out, there’s nothing stopping *A* being the same point as *C*.

And, the typesetting sucks.

And, “therefore” dots suck.

## PoSWW 2: Take a Hike

This Proof of Stupidity Without Words comes courtesy of a smart Year 11 VCE student. It’s an exercise in the Jacaranda text *Maths Quest 11 Mathematical Methods* (2019). (Just to clarify, the stupidity obviously contains plenty of words; it’s the *proof* of stupidity that requires no words.)

## Obtuse Triangles

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.