(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:
CONTENT
identify and describe the properties of prime and composite numbers and use to solve problems and simplify calculations
ELABORATIONS
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 
as 
which can be rearranged to simplify calculation to 
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.
The concept of a prime number is not defined. Instead we are told that “a prime number has two unique factors of one and itself and hence 1 is not a prime number”. I would define the concept this way.
A positive integer,
, is a prime number if 
, and the only positive divisors of 
are 
and 1 (Hardy and Wright, 4th ed., p. 2).
The condition that
is part of the definition of a prime number rather than a consequence.
This is important because definitions given by curriculum authorities are regarded as sacrosanct. They are repeated by textbooks and passed on to the students.
Am I being too picky here?
Is 1 being composite just a consequence of the fundamental theorem of arithmetic?
No, Craig. 1 is not composite. It’s something else. But Terry’s very important point is, whatever 1 is or isn’t amounts to a definition. Once you’ve defined things, then you can think about stating (and, but no one ever does, proving) the FTA.
No, you are not being too picky. Regarding the first elaboration, you’re not being picky enough.
Thankyou, Glen (I think …!)
1) I think Wikipedia does a pretty good job of reminding us why ACARA’s definition given in the first elaboration is bollocks: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic
2) I’m assuming the point of the second elaboration is to notice that numbers (apart from 2) ending in either an even number or 0 are not prime, and that numbers ending in 5 are not prime. Maybe some genius will notice that numbers with a digital sum of 9 are divisible by 9 and hence not prime. Maybe ACARA thinks some genius will spot a pattern in the prime numbers … What SHOULD be elaborated here but is NOT is that 2 is the first prime number and is the only even prime number.
(Actually, the 5th elaboration suggests what patterns might be intended).
3) It’s a great relief to see that prime numbers can be used “when necessary” when factorising a composite number. This is where the Fundamental Theorem of Arithmetic should have been mentioned and the reason why 1 is DEFINED to not be a prime number. ACARA love jargon, this was a chance to use some that actually makes sense.
4) “using spread sheets to list all of the numbers that have up to three factors using combinations of only the first three prime numbers”
All of the numbers …? How absurd. To paraphrase a famous movie line, “You’re gonna need a bigger spreadsheet.” How about all of them up to 99, for example. A reasonable finite ceiling should have been stated. And why use a spreadsheet to do this? Gratuitous use of technology.
“recognise any emerging patterns, making conjectures and experimenting with other combinations” Aha, let’s be mathematical explorers … I don’t know what ACARA has in mind here. Perhaps there will be a young Dirk Hartog who conjectures the Fundamental Theorem of Arithmetic …? (Actually, the 5th elaboration suggests what discoveries might be intended).
5) “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”
So these sorts of divisibility tests (https://nrich.maths.org/1308#:~:text=Since%201000%3D%208%5Ctimes125%2C,that%20is%20divisible%20by%208.) are what our intrepid explorers are meant to discover …?
I don’t see how this is an example of “understanding that if a number is divisible by a composite number then it is also divisible by the prime factors of that number”. But at least it suggests/clarifies what are intrepid explorers are meant to discover when they set off in their boats and on their horses.
I think it’s terrific that tests of divisibility are included (well … implied), but it’s stupid to expect the average student to discover some of these tests (such as the test for divisibility by 8). Divisibility tests should have been stated as a very clear ‘Elaboration’.
And would it have killed ACARA to include the fact that there are an infinite number of prime numbers (but maybe that’s a discovery intended via spreadsheet in Elaboration 4 *snort*). And surely Euclid’s magnificent proof of this should get be mentioned. I think that, carefully done and with explicit instruction, this is accessible to the average Grade 6 student and I think it could really spark the imaginations of the brightest.
Anyway, another spectacular fail by the ACARA clowns and their Svengali. (And wtf … no Indigenous Cultures references …??)
By the way … It occurs to me that we’re doing a lot of ACARA’s work for them. Maybe there should be a copyright placed on the content of Marty’s blogs?
Tests for divisibility.
PS The comment that follows the article “New Maths” is interesting too.
1979-function
Thanks, Terry. The late, great Function.
Thanks, John. I tidied your link to the Function archive. (The archive exists courtesy of the Evil Mathologre.)
In keeping with cultural sensitivity, I shall be putting on a sombrero, poncho, and a large fake moustache. Most of the points I would have made have been already stated. But my main concern is the utter god-awful, incompetent sloppiness of it all. Just where some precision would have been necessary – and indeed given us a vague hope that somebody, somewhere, may have a few working neurons – we see this garbage.
One example of the sloppiness is in the last paragraph:
“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…”
I may be old-fashioned, but in my day 4 was not a prime number. (Of course the supreme sloppiness is in the definition of primality, which must include being greater than 1. This is not a consequence, as has already been pointed out, but part of the definition.) Who writes this stuff, and where do they park their brains when they do?
I also question the pedagogical notion of exploring prime/composite by trial division. Yes, this can be used, but surely a better approach, at least at first, would be to play around with the sieve of Eratosthenes. This is not only good fun, but is quite easy – you don’t need a calculator; you can just write down a list of integers and then cross them out by simple counting. You can use trial division for testing whether a larger number is prime or composite, but a bit of sieving gives a nice sense of prime vs composite right at the start.
And as well this would be an excellent place to at least state – and maybe give a sort of rough hand-wave-y proof – of the fundamental theorem of arithmetic. In fact, if prime numbers are going to be part of the curriculum, then this theorem should be named. This then leads back to the necessary definition of prime numbers as being greater than 1.
F*ck.
Sorry, Alasdair, I forgot my role here. I should’ve waited for your inevitably excellent shredding. And damn it for my missing the 4 is prime thing.
I would have posted yesterday, but I was too tired, and jeez this stuff makes my brain hurt. It’s such drivel. And the thing is, these are all small things – but as you have said on numerous occasions, details matter. How on earth does ACARA expect mathematics to be taught when they themselves are so sloppy, ill-informed, lazy, and downright incompetent?
Yes AtA, a very nice pick-up on the 4. My eyes were glazing over by that stage and just read it as divisible by 8 therefore divisible by factors of 8, viz 2 and 4.
And yes, for many reasons the sieve of Eratosthenes is a much better tool to use than some gratuitous ‘digital tool’. It would be a worthwhile activity to ask students to research Eratosthenes, a remarkable polymath.
I think it’s very possible that the ACARA clowns have never heard of the sieve of Eratosthenes. Or of Eratosthenes. It’s hard to believe otherwise when the sieve gets no mention at all (unless it’s considered too old school by that bunch of giblet heads).
I suspect the Svengali Defence will be used by ACARA at some stage.
This struck a nerve as I am currently tutoring a Year 5 student, who I will call Ramanujanino. She is absorbing maths like a sponge and will soon be beyond me; she had heard about negative numbers and used the internet to get the idea.
With regard to primes I started by defining “breakable” numbers as counting numbers that can be written as a product of smaller numbers. (Perhaps not a good term as breakable suggests addition rather than multiplication.) Unbreakable numbers are the rest. Then “prime” numbers are the unbreakable ones, excluding 1. Why exclude 1? Well mathematicians have decided life is easier if 1 is not a prime. No big deal. Unlike Alasdair, my memory from schooldays was that 1 was a prime – I don’t think I have been irredeemably scarred by this. Ramanujanino concurs.
Tom, it’s not a huge deal if 1 is prime or not. Prior to around 1900, there was no consensus. (I didn’t realise you were that old.) The issue is having a clear and appropriate definition of 1.
It seems that the decision took a while to arrive at Marong State School number 400. Mail was even slower at the high school that my wife attended in the 50s. The teacher was explaining the number pi and like all good teachers wanted some class participation. So she had the class work out pi to 20 places – using long division on 22/7. They discovered an interesting pattern in the digits!
Yes I think it is beneficial for students to learn that these definitions are not god given. If we include 1 as a prime then many theorems need a coda “except for the case n=1”.
Do we need the modern definition? I suppose, if we are following Alasdair and quoting the fundamental theorem of arithmetic, then yes we do. I am not averse to changing definitions as the sophistication level increases – but I suspect that is anathema to you Marty? Can your word “appropriate” allow for changes.
Tom, why on Earth would you think I’d be against changing definitions? I’m only against stupidity.
Patterns in
are indeed interesting.
2009a-Xsection
Sorry Marty
Nice paper Terry
It would make for an excellent class discussion to start defining primes roughly as “any whole number which can’t be factored into smaller whole numbers”, which might lead into a discussion of whether 1 was or was not a prime. This might lead then into a discussion (in a manner, of course, appropriate to the age group) of the fundamental theorem of arithmetic, and what we mean by a mathematical “definition”. And in fact, in a well regulated discussion about mathematical definitions, the student may learn a great deal more than going through the current or proposed dusty curriculum.
I’m reminded of Donald Knuth’s discussion of why 0^0 should be defined to be 1. He claims that the function x^0 is too important to be arbitrarily restricted, whereas the function 0^x is relatively unimportant.
Finally, on a side note, and commenting on John Friend’s comment about Euclid’s proof of the infinity of primes – did you know that since the Greeks had no concept of infinity, what Euclid actually proved was that there was no largest prime? The two are equivalent, but it’s a nice historical note, don’t you think?
(Another proof, beyond school but lovely nonetheless, is that the series of the reciprocals of the primes diverges.)
Yes, very nice. And very interesting. There is so much potential richness in this topic. What a shame that in practice we get so much crap.
I know, and it’s just so depressing. I mean, there’s richness in all mathematical topics; all that’s needed is a bit of sense to bring them out. But instead we get this superficial crap. And sometimes as well as being superficial it’s downright wrong. A shame indeed.
And even worse, you don’t need much pre-requisite maths (in the sense of algebra etc.) to understand and enjoy this stuff, to learn a lot and have your imagination and sense of wonder ignited. This could be an amazing topic, with something in it for every student to enjoy – from weakest to strongest.
It truly is just disgustingly awful, isn’t it? They don’t even manage to hide it with fancy language or jargon. Rotten on the surface with agonisingly painful expression, and then straight through to the core.
I don’t think the “first three primes” activity has been bashed
enough yet… the tech angle has been mentioned, but I disagree that it is a reasonable use of time. Even with paper. Because what kind of “emergent pattern” can a child or anyone really observe in this:
6, 10, 15, 30
That’s FOUR NUMBERS. If you say “well we can use them more than once”, and “well we don’t need to restrict ourselves to the first three”, etc. OK, that’s fine, but that’s *not* what they wrote.
Although, with this mess of expression, another interpretation of this is that the *factors* themselves could be 2, 3, 5, 6, 10, 15, or 30. OK, it’s a generous interpretation, and students would struggle to write down the result of even taking just three of these factors. A spreadsheet would be able to do it, but will the teacher explain the algorithm used, and why it gives all the factors? As far as I’m aware this combinatorial discussion is not only out of place and distracting from the topic at hand (being primes…) but is likely not appropriate for this year level. (Aside: There are ZERO hits for “combinatorial” in the ACARA document, and also ZERO hits for “Pascal”. Maybe one of our helpful teachers here can enlighten me as to when precisely students should be taught about the coefficients of binomial expansions… I’ve had enough of looking at that ACARA abomination for today.)
Alright, even still, using the spreadsheet or if the teacher writes the sequence of numbers on their smart board or whatever, what kind of pattern should students see in the result? That’s the point I’m trying to get at. Any kind of “pattern recognition” should be done first and foremost in a proper Sieve of Eratosthenes fashion (also mentioned by others). This crippled semi-functional distracting version of the sieve exercise is at best wasteful of class time.
I feel icky. Thanks for posting Marty and thanks for commenting everyone.
Thanks for making you feel icky? You’re most welcome. Please comment on the word search post, indicating Pascal etc, and I’ll add to the boxes.
Glen, I’m a bit puzzled by your criticism of the three-primes activity, or at least the activity I have in mind.
If you take three primes and list all the no-repeat-use products, then you’ll get all the factors of the largest number. That’s not super-deep, but it seems natural and reasonable.
Of course, if that is what ACARA intends then they should have just damn well said it. And the spread sheeet is really idiotic. And the “emerging patterns” is really idiotic.
But are you criticising a de-ACARA-ed activity, such as I’ve indicated above?
Yes, Glen. More bashing is definitely warranted.
“list all of the numbers that have up to three factors using combinations of only the first three prime numbers”
makes no sense to me.
So do they mean “up to three [prime] factors”? As Glen says, this gives 2, 3, 5, 6, 10, 15, 30. I don’t think there’s any difficulty getting all of these numbers, it just requires a systematic approach. It certainly encourages the careful and methodical student, regardless of ability. I certainly wouldn’t advocate the gratuitous use of a spreadsheet or any other digital technology to do this.
If this is what is meant, then the question is:
What does
“recognise any emerging patterns, [make] conjectures and [experiment] with other combinations”
actually mean? Call me thick, but I don’t see any ’emerging pattern’, I have no conjectures to offer, and I don’t know what “experimenting with other combinations” means. Does it mean use the first four prime factors? Does it mean use powers of the first three prime factors? Is it deliberately vague teachers can interpret it however they please?
I think ACARA use the word ‘experiment’ when they have no clue whatsoever as to what their statements mean. Which is often. Just experiment, explore, inquire. Let’s all go on a mathematical adventure. Where? Who cares? Let’s just go. Courtesy of ACARA Svengali Adventure Tours.