School Maths: 12 divided by zero = 12?!

[WeBuiltThisBuffetOnSausageRoll]

My DS came home yesterday quite sad and frustrated because he other classmates had lost marks in a school maths (his best subject) test for not getting the 'correct' answer that '12 divided by zero = 12'.

His reaction, upon coming home, was to look up the expected result of dividing by zero on several reputable maths sites and, as expected, none of them gave the answer that 'X divided by zero = X'. They all backed up his (and my) reckoning that the only possible correct answers could be 'undefined/impossible', 'infinity' or (possibly, at a real outside semantic push) 'zero'.

Thankfully, the teacher raised it the following day (I don't know if she had looked into it herself - it was a centrally-set test - after seeing the pattern of usually-able children unexpectedly all getting it 'wrong') and re-instated all the lost marks; but I'm still baffled as to how anybody could arrive at that answer in the first place, and it's bugging me!

Suppose the sum had been the simpler '12/2=6', the reckoning process would mean that you could have 12 apples and remove 2 apples 6 times, thus ending up with zero apples afterwards (as a valid 'checksum'); equally with '12/12=1', you could remove 12 apples once and again end up with the sum-validating 0 apples; but if 12/0=12 were true, you could thus remove 0 apples 12 times and be left with 0 apples - but of course, you wouldn't have 0 apples left after that: you would have 12 apples left; indeed, just as you would have if you removed 0 apples a thousand billion times!

Was this just a brain-fart by a tired maths test-setter - and one that wasn't immediately obviously wrong to a maths teacher yearning for half-term, who initially insisted that it was right when it was queried - or is there some kind of maths/philosophy train of reasoning that any boffins out there know of by which you could legitimately justify/argue that 'X/0=X' can indeed be correct, the same as 'X-0=X' naturally would be?! It doesn't really matter, obviously, but it's still irritating me a bit!

Cannot divide by zero

Divide by half and suddenly you've got double. Eh??

Because you aren't halving, you're dividing by half. It's a totally different concept.

Once we get into fractions, my sweets and children analogy gets a bit horrible and even more prone to misunderstanding, as practical examples of theoretical equations will tend to do. Probably stops working on any level, really, and actually I will struggle to explain it in these terms...

But in essence, it's about needing two halves to make a whole. So if I divide my 12 sweets by half a child (sorry, I told you it gets horrible!), it becomes 24 sweets as I need his other half to make a whole child.

I think this is one that really does need to be explained theoretically!

I was taught some ridiculously complicated thing about flipping the fraction upside down and have totally forgotten how to do it.

It shouldn't really have been that complicated...just flip the second fraction and then multiply. It's essentially about multiplication and division cancelling each other out.

@Ndd135632 I think it’s because people are thinking practically not mathematically.

If you are standing in your kitchen with 12 jaffa cakes and 6 kids and thinking “right, that is 2 each”, then all the kids declare they hate Jaffa cakes and no one takes one, you are left with the 12 Jaffa cakes all for yourself.

Which is a win 😃 and the type of scenario mums deal with daily- it’s easy to see why people would think 12 automatically.

The mum here is an undefined quantity.

😀

I think it’s because people are thinking practically not mathematically.

That's what it all boils down to.

But I think it helps to start off learning the concepts practically before you go into pure theory. In real life, maths is almost always practical.

I agree. She's shown how your handle mistakes well.

I'm wondering if this is the same people who wrote the assessment paper my DS (9) did the other week. He came home upset because he'd lost a mark - he said the question was 65÷10=? , only it didn't specify how the answer was to be given, so he'd answered 6.5 or 6 1/2 or 6 remainder 5. All the same number, in different formats.
The class were told afterwards (generally) that they couldn't give three different answers, with an example of three different numbers given. I'm going to take it up with them because they're either being unfair in not recognising that all answers given were correct, expecting him to mind-read, or there was a clue in the question he hasn't been taught to look for.
But 12 can't be divided by 0, that's for sure!

The mum here is an undefined quantity.

The mum neither knows or cares what this means, because she's about to eat the Jaffa Cakes. Problem solved!

Indeed. And the rest of the packet!

so he'd answered 6.5 or 6 1/2 or 6 remainder 5.

The third one (6 r 5) isn't the same number as the first two. Which is probably why they are not allowed to give multiple answers. Because you could just write down loads of possible answers in the hope that one is right.

It was sort of a joke, but this is a thread about maths...

@Theshadowsthecurtainsmake It's easy to see how people make the mistake when you put it like that. Whereas it should really be more like. I can try and repeatedly give the 12 Jaffa cakes to children that hate them and they will always refuse and I can continue that for infinity/an undefined number of times they still won't take them. Let's face it I'm going to have to buy chocolate digestives instead.

I’d bet good money that the test setter didn’t actually think that 12/0=12. The “0” will have been the error - it should have been a “1”.

Interestingly, the proof of 2=1 posted up thread requires a division by zero, which is impossible (hence being able to “prove” something which isn’t true). It’s a good example of why older students need to learn proofs.

One of my fave mathematical conundrums is trying to do the division 0 / 0 because it has three answers:
a) zero - because 0 / x = 0,
b) infinity- because x / 0 = infinity
c) 1 - because x / x = 1

cakeorwine

Well that’s blown my mind. I’d make a witty comment about NHS beds if I could, but it’s too depressing.

It's a mistake. The teacher is probably cringing that they did it. I've used resource without checking every question because you simply don't have time to read through it! It's embarassing if something like that happens, by the imports thing is that it was acknowledged and corrected.

@HoboHotel it's a correct answer though, and the problem is still that he didn't know which form the answer should take. They've taught all 3, so should have made it clear which type to give or given him the mark. Fair enough if he'd given one correct and two incorrect answers!

When dividing by fractions I find it useful to think of pizzas.

If I had 12 pizzas, how many 1/2 pizzas are there? 24

12 pizzas how many 1/3 pizzas? 36

So 12 divided by 1/2 is the same as 12 x 2 and 12 divided by 1/3 is the same as 12 x 3

which is where the flipping over comes from.

12 pizzas how many 0 pizzas? Doesn't make sense because you can't have a 0 pizza, so can't be done (is undefined).

Sometimes the answer is taken as infinity because if you think about 12 pizzas how many tiny tiny slices of pizza? The answer approaches infinity as the slices get smaller and smaller (ie as they approach zero pizza in each slice although they'll never actually get there actually or would not be a slice of pizza!)

Questions like dividing by 0 and multiplying by 0 are on the Year 2 and Y6 SATS papers so if the OP is talking about her son doing that (which I suspect she is seeing as he is in Y6), the Year 6 teacher cannot mark it correct if the mark scheme says differently.

Yeah. If you don’t deal with theoretical numbers/maths it just isn’t going to occur to you that in this scenario you are not real, and the Jaffa cakes are only sort of real until no kids want them and then they cease to exist.

I'm 40's.

I remember being taught at primary that anything divided by or multiplied by 0 = 0.

Anything adding or minusing 0 = the original number.

So I guess with the actual answer being infinity it was close enough 🤣

Obviously I've known the actual maths since secondary!

Not sure I'm onboard with those who say they'd get people sacked for this. Bit extreme.

The reason you have option of remark for xmas and are encouraged to for very close boundaries is because human error is natural.

I think the teacher handled this well and those students would have learnt more than just the maths answer from the experience.

Divide by half and suddenly you've got double. Eh??

In this scenario it is useful to remember that division is also 'how many x go into y?' Like 42 divided by 7 is "how many 7s are there in 42? 6"

So 1 divided by 1/2 is "how many halves are there in one?"

How many half pizzas are there in a whole pizza? Two. You've got double.

How many half pizzas are there in 12 pizzas? Each pizza has two halves, so you have 12 lots of 2 halves = 24 halves in 12 pizzas.
12 divided by a half is 24.

<high five to other mathematicians teaching by pizza>

If you do any calculation an infinite number of times, then the answer also has to be infinity.

@noblegiraffe ,

I always find it fascinating how deep simple Maths concepts can get.

OTOH, sometimes I think pupils need to be given solid method first and deep understanding later.

it is amazing how many A level physics students are ‘fluent’ in Maths.