This is another quick one, but it keeps the bullets flying while we prepare a more substantial post for tomorrow(ish). It can be considered a companion to the previous ACARA Crash a Content-Elaborations for Year 8 Number.
CONTENT
recognise and investigate irrational numbers in applied contexts including certain square roots and π
ELABORATIONS
recognising that the real number system includes irrational numbers which can be approximately located on the real number line, for example, the value of π lies somewhere between 3.141 and 3.142 such that 3.141 < π < 3.142
using digital tools to explore contexts or situations that use irrational numbers such as finding length of hypotenuse in right angle triangle with sides of 1 m or 2 m and 1 m or given area of a square find the length of side where the result is irrational or the ratio between paper sizes A0, A1, A2, A3, A4
investigate the Golden ratio as applied to art, flowers (seeds) and architecture
For approximately locating numbers on the number line, rationals would do nicely. Irrationals in applied contexts remind me of the famous exercise on mazes (Exercise 49 in the following pdf):
Click to access chpt_02_properties_of_real_numbers.pdf
I don’t remember which philospher once said that if you can’t say something in a clear and intelligible manner, you should shut the fuck up. But it bears repeating.
Jesus. That’s right up there with the cheese.
One version of the “shut up” is Tom Lehrer:
Speaking of love, one problem that recurs more and more frequently these days, in books and plays and movies, is the inability of people to communicate with the people they love: husbands and wives who can’t communicate, children who can’t communicate with their parents, and so on. And the characters in these books and plays and so on, and in real life, I might add, spend hours bemoaning the fact that they can’t communicate. I feel that if a person can’t communicate, the very least he can do is to shut up.
“1m or 2m and 1m”
Boole would have a few issues with this clause…
…or say it reduces to 1m.
I think Boole turned over in his grave, looked at that sentence, and went back to dead.
Nowhere does it mention an actual definition of irrational number …
is irrational? Why is 
irrational but 
is not …?
And what are these certain square roots that have to be recognised? Should a student recognise that
And how the fuck are students meant to
irrational numbers …? What does “investigate irrational numbers in applied contexts” mean? If we go to the punchline (last sentence), it looks like it means to go on an irrational scavenger hunt.
And the following is as bloated a piece of nonsensical blather as ever I’ve seen:
“using digital tools to explore contexts or situations that use irrational numbers such as finding length of hypotenuse in right angle triangle with sides of 1 m or 2 m and 1 m or given area of a square find the length of side where the result is irrational or the ratio between paper sizes A0, A1, A2, A3, A4”
Maybe ACARA should investigate the use of punctuation and how to write a sequence of coherent sentences.
And then we’re back to looking for the Golden ratio in nature. So that’s what investigate means – go on a nature hunt for irrational numbers.
The Con-tent and Elaborations are one huge irrational number. I don’t have the words to adequately say what a gigantic load of shit this is.
The quality of ACARA’s mathematics curriculum lies somewhere between utter dogshit and fucking atrocious such that utter dogshit < ACARA's mathematics curriculum < fucking atrocious.
One of the few pleasing things about all this is that people are coming around to the idea that my very foul language is entirely appropriate.
Can we just remove everything except for “recognise an irrational number”?
Do the even have a notion of “real number system” at this point? Do they know about the continuum? About inequalities? Can they see that if
for all 
then 
? If it will make ACARA happy they can have 2.5 minutes of the lesson to apply this to show that 0.999999… = 1.
Yes, pie in the sky. I know. I just can’t bring myself to process this stupidity. The great utility of mathematics is to find the length of the diagonal on a random funny sized bit of paper. I know the students will think so highly of math class after that. Fucking nuts.
“Now year 8’s, today we are going to prove that the rational numbers – which as you know are countable – are in fact dense over the reals, which as you also know, are uncountable. This actually follows as a simple corollary from the standard topology induced by the absolute value metric, and it in turn forms the basis for the Dedekind cut.”
Mind you, I think that demonstrating approximation by rationals is a very good idea; in fact approximations themselves are a very good idea. I just don’t think that this curious half-baked idea from the Lords of Mathematical Wisdom (ACARA) is going to cut it. Also, how are they going to prove that, for example, the square root of any non-square integer is irrational? And then they go for pi. Proving that pi is irrational is non-trivial (see the wikipedia page for a few different proofs). So they’ll just have to say: “And also, year 8’s, pi is irrational, too” and leave that sentence hanging there “like a chrome-plated fart”. [Note: that superb phrase is from Robert Sheckley’s 1950s short story “Cordle to Onion to Carrot”.]
Then – and this is the icing of the cake of fuckwittery – they start blathering on about the golden ratio. The much vaunted notion of this number is somehow the basis for all sorts of things, from classical architecture to the human body, has been conclusively disproved time and again. It’s very sad to see it living on in the one place it can do the greatest damage. I’ll bet, in some ACARA-approved textbook, there’ll be that picture of a golden rectangle and its smaller squares, each with a quarter circle arc in it, and the claim that this shows a “spiral”. Except that of course that it isn’t as it has a discontinuous second derivative…
My brain’s bleeding. This is just too awful.
Indeed. Apparently the golden ratio not only explains everything we see in nature, it also has mystical properties that improve educational standards. We should be calling it the golden bullet.
Someone at ACARA needs the golden ratio shoved right up his/her arse.
ACARA’s daft curriculum is lazy, ignorant and incompetent. I can think of a few VCAA-affiliated egos that might have had a hand in writing it.
Do we know the names on the panel that are writing this blather and what their sources/references are?
I am reminded of the Betrand Russel quote about a blind man in a dark room, looking for a black cat (spoiler alert: the cat isn’t there.)
Except, in the case of ACARA, they not only claim to have found the cat, they have named it “Golden Ratio”.
Maybe Fifi was too subtle?
Nobody mentioned this yet, so I will:
“… irrational numbers which can be approximately located on the real number line …”
If they mean drawing a point, with a pen, on a line, on a piece of paper, then all numbers are approximately located, apart from the two you use to define the position and scale of the line.
If they mean the actual real number line, i.e. the platonic ideal, then irrational numbers are exactly located on the real number line. The exact location of some of can even be constructed with a compass and a straightedge (platonic ones, of course).
If they mean that the exact location of an irrational number can be approximated by bounding it by two rational numbers, then sure. But maybe it’s important to first recognise the above two facts.
On irrational numbers
2015-Beyond_the_rational
Very nice stuff, Terry. And none of it will be seen by more than 0.1% of students.
I’ll make a few notes:
*) Your exercises “Develop ICT capacity”? You’re kidding.
*) You could have included exercises to prove that the rationals are exactly the recurring decimals.
*) Yes, the irrationality of
is well-known (for some value of “known”), but there are much nicer proofs than the standard ones, and these are not so well-known.
*) Ironically, the golden ratio is one of the easiest numbers to prove irrational. This is well within reach of Year 10A.
*) Though we don’t know whether
or 
are irrational, we do know that at least one of them is. The proof of that is (just) within reach of a good Year 12 student, and the underlying idea is (just) within reach of a good Year 10A student.
One of the exercises shows that my calculator does not give me
even when it is asked for it. This develops ICT capacity because it may make some students sceptical about what a calculator tells them.
Once I gave a lecture to undergraduate students at a very famous university where I made this point. I used my calculator to find
and wrote the answer on the blackboard. Then I multiplied the answer by itself in just one line. I had 1.414 etc up to 10 places, and wrote down the same number again, and then slowly multiplied the two numbers together in my head and got the answer to 20 decimal places. The students, and my host, were very impressed that I could do this.
Little did they know that I had the answer written on a strip of paper on the chalk ledge below.
My classroom
trick is much more impressive. It involves a hammer.
Inspired by you