Odd and even numbers

[user789653241]

Ds(9, yr5) asked me if decimal numbers can be defined as even or odd. I said no, but haven't got a clue why.

Can anyone give us some explanations please?

noblegiraffe and how do you know 4 isn't a range of numbers rounded to 1s.f. ?

I know a threeit isn't as useful as a twoit, but the reason a twoit is useful only actually comes from the twoness of it, and not anything special about odd/even - knowing that dividing by 2 provides information to help you, and that also doing it by 3,5 etc. will give you more.

What I mean is, why teach about odd/even special - when it's not that special - when teaching about the maths and the shortcuts help more, as then you understand the shortcuts.

On the simplification thing for example, knowing your twoits will help you 25% of the time, adding in your threeits and it brings it up to 33% of the time, fiveits add another a few percent etc.

Of course, this requires deeper understanding of the underlying maths, so may not be appropriate for all.

sir I've been thinking a bit more about this

I'm thinking the level of usefulness of the concept of odd and even is a bit the same as why we have a square and not a regular quadrilateral. So simply, it comes up enough that having a word for it is worth while. So 'even' is easier to say than 'is divisible by two' and 'odd' is easier than 'is not divisible by two'.

I wonder if we had 3 hands and 3 legs we would have a proper word for threeit? So does odd and even come up in every day life and hence need its own name because we like pairing things off?

It's because ALL decimal numbers can be divided by 2 to get another decimal number, so it's not an interesting property for them.

Some whole numbers can (even), and some can't (odd), so it is an intersting question for them.

how do you know 4 isn't a range of numbers rounded to 1s.f. ?

Good question. You don’t! But to see the number 4.0 written in that way with a deliberate zero after the decimal point suggests rounding is likely as that’s probably the only time(?) you’d leave it there and not drop it.

What I mean is, why teach about odd/even special - when it's not that special

Because it’s useful. We have way more focus on ten-its because that’s the basis of the entire number system, then twos. 5s get a look-in because they’re also easy, then 3s. Probably then 9,6,4, maybe 8 if you’re hard enough.

The main time I hear 4.0 is on American films as in 'has a four point oh grade point average' which I believe to be top marks but otherwise have no idea what it means.

This odd/even stuff is interesting. We teach it the concept to quite young children, but it only becomes useful quite a few years later, and only in a 'is this divisible by 2' way rather than because of any 'feature' of oddness or evenness.
Sadly DD2, 13, still hasn't grasped odd/even very well (despite being reasonably capable in other ways, e.g. algebra).

Obviously some labels apply to any representation, such as ‘surd’ - however it’s written it still can’t be written as a fraction. But other labels don’t, like ‘decimal fraction’. Are you sure that ‘integer’ does? Does it really apply to 4.0 when 4.0 may represent the integer 4, but may also represent a range of numbers rounded to 2s.f.?

Aargh, typed a long reply which I seem to have forgotten to post. You can label/describe a representation, but it's important not to confuse that with a property of a number itself.

"surd" I don't know about - the term popped up briefly as a topic heading in A-level maths, never seen it referred to before or since. How is it defined exactly? An irrational number which can be written as an integer root of an integer?

"decimal fraction" - as with "fraction", I think it would be normal to use this to describe a representation of a number, rather than trying to make it a property of the number itself.

Back to integers. A label that depends on how a real number is represented does not properly define a subset of real numbers. To identify a subset you need a rule that says for any real number, is it in the subset or not? And in the real numbers 4=4.0=3.9999.... = 40/10 are all the same number, so any well defined rule will have to give the same answer for all of them.

And yes it really is important that the integers are seen as a subset of the reals. At least I hope that's obvious. We mix up the notation so much all the time without even thinking about it. If someone says "1.4 x 3" we don't want to have to stop to ask if they mean multiply by 3.0000000.... or add it to itself 3 times ... or just to not let them because 3 is in the integers and 1.4 is in the real numbers and they're not in the same mathematical structure to multiply.

I don't like the 3.0 represents a range of numbers approach. Surely you din into your students that no it doesn't, you must always write "to 1 dp" if you have rounded an answer or your answer is incorrect. If you want 3.0 to represent a range, not only is it not an integer, it's also not a real number. And you can't even do addition with the things: 3.0 + 1.0 would be - er - maybe 3.9, or 4.1, or possibly even 4.0.

This was a fun thread to read lazily in bed in the morning after having spent the night up with a poorly child :)

OP I had a thought as to where the original confusion may have come from. You know how children often think 'x10 is easy, you just add a 0 at the end' and this is fine but stops working as soon as you get into decimals. Because it is not actually about 'adding a zero', but rather it is about moving each digit one place to the left in the place value system. Still easy, but now works for decimals too.

Similarly when looking at odd and even numbers, children are often taught that 'you only have to look at the last digit'. If the last digit is odd, the number is odd and vice versa. So we can quickly tell even with very large numbers if they are odd or even. However this is a shorthand, and again doesn't work with decimals. If you learn however that a even number is any number that if divided by 2 results in a 'whole' number, you won't get that confusion.

I do think knowing about odd and even is useful even at KS1 level, rather than just a 'fact' that you learn. It's great for quick answer checks. And it also helps in visualising number bonds to 10, where 'odd' blocks 'slot' into each other etc.

As to why two-its are important enough to be learned/taught but three-its not, I'd say that there is nothing inherent in the numbers to explain this; it is the same sort of question as to why we do maths on a base 10 system. Base 10 - most likely down to having ten fingers. So in the early development of number understanding, 10 was the limit of easily representable amounts. The importance of 2 - not just two hands, two eyes, but also the very clear back/front distinction of human physiology. I do think it's highly likely that all human culture is influenced by having this perspective split of the world into two - before me and behind me. And that it is therefore plausible that 2 matters more to us than other numbers, which hasn't got anything to do inherently with the number two, but rather with us who use the concept. It is us that organise the world very fundamentally into two distinct spheres. Perhaps that's why we tend to 'pair up' and things come in twos in all sorts of situations, which then makes understanding of 2-ness more important to us than understanding of e.g. 3-ness - simply because it has more application (for us 2-part beings).

I'm interested to read this thread (and others like it). There's an important moral, I think, for our dealings with primary school children in particular. It's this: questions like, 'Can a (decimal) fraction be even?', 'Is zero a number?', 'Is 4.0 the same as 4 (and what about 3.999...)?' - questions small children often ask (and there are lots more) - such questions indicate, not that the children asking don't understand, but rather the beginnings of a serious interest in intellectual matters.

Such questions, if encouraged, can lead to a deeper understanding. And we don't really need to understand the answers ourselves to encourage our children to ask such questions. 'Can decimals be even?' -- 'That's an interesting question! I wonder how we might think about answering it?' (Not 'No, it's just a fact that they're not'.)

Too often, ime (as an ex-maths teacher, amongst other things), children's intellectual curiosity about such (deep, interesting, exciting, useful ...) matters gets squashed by being treated as a lack of understanding and/or knowledge rather than what it is, an indication of natural intellectual curiosity.

And, yes, this thread has led to some interesting discussion in its own right. To continue that a little ...

catkind, you say, ' And yes it really is important that the integers are seen as a subset of the reals. At least I hope that's obvious. ' Actually, in certain contexts, it might be important to see Z ({integers}) not as a subset of R ({reals}), but rather as a set isomorphic to a subset of R (where the isomorphism includes all the relations we normally define on Z, of course).

Strictly speaking, Z is not a subset of R, albeit that R contains a proper subset which is a copy of Z. This may be important to emphasise, for instance, if we think of how we define integers (as ordered pairs of members of N ({naturals})), as opposed to R (as Dedekind cuts, say). There is a natural progression from defining natural numbers as sets of sets as (say) Russell did (following Frege, of course), then integers as ordered pairs of naturals, rationals as ordered pairs of integers ... but then reals need a new kind of definition. (We can then go back to ordered pairs of reals to get complex numbers, of course.)

So, N is not a subset of Z, although Z contains an isomorphic copy of N, ... and so on.

My 'strictly speaking' shouldn't be taken too seriously, however. Another reason for distinguishing Z from the subset of R it copies is that we might want carefully to beware of begging ontological questions that 'strictly speaking' may also incline us to beg.

I'm also interested, catkind in your distinction between representations and descriptions of properties. 'Representation' is really a fraught term in this area, with multiple meanings/uses. Think of representation theorems in measurement theory, for instance, or (compare/contrast with) representations of groups on vector spaces. And so on. How do these kinds of representation relate, if at all, to descriptions of properties of what they represent?

I won't develop this latter point. Nor am I suggesting any of this as appropriate for primary school children, their teachers, or their parents. I offer the existence of such discussions, though, as evidence that we should take children's questions seriously in these areas in particular. Don't close them down by telling children there are facts they just have to learn; rather encourage them to keep asking questions whether they get answers or not.

Okay you've out-technicalled me 9toenails

Definitely not talking about representation in any technical sense, I was trying to make a clear distinction between descriptions of the way a real number is shown (recurring decimal, fraction etc) and something that is a properly defined property that a real number may or may not have (is an integer, is rational).

You've gone several steps more technical on the integers as a subset of reals than I did. Which I'll allow but only if you promise never to write down 3 without specifying whether you mean 3.0000000 in R, 3 in Z, 3 in N or 3 as in ordinal (do it 3 times). Or any arithmetic operations without specifying which domain you're applying them in. Deal? If you want to be able to get on with normal arithmetic without a stupid degree of care, I think it may be preferable to identify Z with the image of your isomorphism.

And yes very much agree about asking questions like this being brilliant. Especially with a child gifted in maths like irvine's DS is, you can explore a long way.

catkind Yes, that's a deal . I wholeheartedly agree it may be preferable to identify Z with [its isomorphic image in R] for normal arithmetic. I have had occasion to insist on the difference between, say, the real number 3 and the integer 3; but that has always been in abnormal circumstances. (As my children have often complained to me, my field is charitably to be described as 'niche ' ).

So we are agreed? The integers are a set of whole numbers written as -2 or 4 or 10, and only when they are written as -2, 4 or 10? That it is a set and not a property?

And, yes, catkind, about children 'gifted in maths' and questions. I want to extend the point, though, to take in children who may not turn out to be gifted in maths.

It can happen that a child be taken not to understand something in maths when they are actually stuck on one of these questions. 'Why do we ...?' or 'Why is ...?' can get blocked by insensitive teaching, with adverse effects on the learner. Sometimes, the child in the class who's 'stuck ' is the one who is trying hardest to get to grips with what's going on ... and sometimes that child learns, not (the truth) that she is thinking more than her peers, but rather ends up thinking that she is stupidly missing the point, 'not any good at maths', whatever.

'Brilliant', exploratory questions are the birthright of a majority of our children. Too often these children are put off by insensitive responses to their brilliance. This goes some way to explain why it is the majority of children (and adults!) think themselves below average in maths.

That's what I think, anyway.

You've got a choice noble. Either every time you write down a number or a sum, you specify what domain you're working in, and for e.g. you can't calculate 3 * 1/3 without first applying some mapping to get 3 into the set of rational numbers. Or we agree Z subset of Q subset of R and we can get on with doing arithmetic normally.

Or we could assume R and specify when we're talking about Z?
That's what happens in secondary maths.

Hmm, that's an interesting one 9toenails. I used to find with some students they ask a brilliant question and you try to give them a lovely explanatory answer ... and they look at you like you've grown an extra head. Some students do need a simple world view and find too many alternatives too confusing. So questions are always good, they are part of someone trying to understand, but the degree I'd explore in my answer depends on the student. I also try to make a point of always asking the "stupid" question myself as a grown up, I was too worried about looking an idiot when I was a student and it did me no good at all.

You don't just use real numbers though noble. If you want to be able to say for a real X that X+X+X = 3*X, you need to have 3 both as a counting number and as a real number. Unless you've identified natural-number-3 with real-number-3 you'll need to do some hedging and mapping to make clear what you even mean. That may not be the best example, but we do blithely mix up integers and reals all the time, there is an implicit identification being made.

I'm a bit confused, catkind are you saying that if you treat the integers as a separate set, then you can't add the real numbers which look like integers?

noblegiraffe

So we are agreed? The integers are a set of whole numbers written as -2 or 4 or 10, and only when they are written as -2, 4 or 10? That it is a set and not a property?

That's almost right, but not quite. The integers form a set, sure enough. But they form a set however we write them, although we may be led to think of other sets if we write them different ways. For instance the integer -2 is in the set of integers whether we write it like that or as '-2.0', or as 'minus two', whatever. Writing the integer -2 using the signs ('using the representation', perhaps?) '-2.0' might mislead us into thinking we mean, not the integer -2, but rather the real number -2.

The integers form a set, the set of those things which have the property of being an integer. That looks a bit circular unless we can replace 'being an integer' with some other defining property of integers. (Most popular current one is the property of being a set of certain ordered pairs of natural numbers - being the (infinite) set {, , , ...} is the property -2 has that makes it a member of the set of integers. (Think 1-3=-2; 2-4=-2; 3-5=-2 ... etc.) That's what -2 is, in our sense: that infinite set of pairs of natural numbers.)

Sets and properties, another interesting (and fraught!) topic for philosophers of mathematics (and hence pedagogues? I think so .) Not all properties can define a set of the things that have that property. Check out 'Russell's Paradox', if you don't know of it. (And prepare to have your mind boggled!)

It's all a bit counting angels on pinheads isn't it? Practically, if kids are sat in a GCSE exam and the question is:
x is an integer such that -2

catkind

...the degree I'd explore in my answer depends on the student

• We'd hope all teachers do that, I guess. Good teachers manage discrimination in that sense better than not-so-good teachers. And, sure enough, some questions do simply indicate a gap in comprehension and can be answered shortly and appropriately to fill the gap. I suppose, though, I'm advocating a policy that assumes children's questions are at least prima facie 'good ', if not 'brilliant '. Let the teacher feel foolish for attempting an over-ambitious answer, rather than the child being made to feel foolish in asking a question.

... , I was too worried about looking an idiot when I was a student and it did me no good at all.

• Yes, again. So (do you agree ...?) teachers should try to build an atmosphere where asking 'stupid' questions is encouraged rather than deprecated; 'stupid' questions are often the most brilliant when children ask them. (Or when someone like Albert Einstein or Emmy Noether does!) Treat them as 'brilliant' even if to you they're the most banal or/and stupid.

noblegiraffe

Angels on pinheads, yes, it can seem like that. And I'm not suggesting primary schools should teach children about the hierarchy of sets or axioms of comprehension, anything like that. But there are some important ideas around this area that have bothered the greatest thinkers in the world for thousands of years. ('Is zero a number?', 'How can there be a number which isn't a fraction?!', to name but two.) When children - as they are prone to do, children being the most curious and interested-in-the-world creatures in the known universe - when children ask these stupid questions, it's worth knowing a little about the way these 'greatest thinkers' have thought (and still are thinking) about such matters, just so we can wholeheartedly see these childish 'stupid' questions for the deep, interesting, yes, 'brilliant ' questions they actually are.

(Sorry, into rant mode there for a moment!)

' Is 12/4 an integer? ' Yes. ' And if yes, is 2-1 an integer? ' Yes, again.

Why would anyone think differently? (I'm not being disingenuous here, btw. This is a real question.) Do you think 12/4 isn't an integer? Why? What about 2-1? Why isn't that the same as 1 (which is an integer, surely?)

Absolutely agreed 9toenails. "There's no such thing as a stupid question" used to be something of a motto of mine when I was teaching.

I guess my point when talking about giftedness was that OP's DS is a child with whom I would grab a question like that and go exploring interesting stuff like we have in this thread, and isn't that fun! With other kids it'd be more of a straightforward answer like some people gave on the first page.

They aren't going to get marks for 1.9 recurring or 4/2
Well yes, and when you're asked 2+2 you don't get marks for the answer 3+1 even though perfectly correct and lacking any potential type discrepancy. There's a degree of expectation that you express the answer in the simplest possible form when you're answering exam questions.

How about 'is 1 a prime?' 'Is a cylinder a prism?' 'is 4.5 closer to 4 or 5?' Are all arguments I've had in my classroom this year!

I would say that 2-1 is a calculation, but that the answer is an integer.