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!

Yes, we were thoroughly conceptually screwed. No wonder maths was hard!

There's a question about filling a bath that I probably still can't do. It required you to theoretically allocate a length to the pipe or something. I remember bending my brain on that one.

What' 0 to the power of 0?

(Runs)

TED talk on it

I remember being surprised when my eldest had several lessons about 0 in his first year at school, until I realised it is not actually a simple concept at all!

If you substitute the x with ‘lots/ groups of’ that helps to visualise it.
so 0x12 would be no lots of 12= nothing
12x0 = 12 lots of nothing= nothing.

I wonder what 12 divided by infinity would be?

I thought the answer was infinity/undefined. But my 10 year old ds who is good at maths would be baffled by that question and not know the answer. I think would also put 0

Any number divided by 0 is undefined !

I think we were just never set any problems with divided by 0 when we were at school (and I did a level also - got an A before days of A*) unless I just forgot.So never really thought about it till work where Excel will give you a #DiV/0 error

I'm in my 30s and was quite good at maths and was taught that you can't divide by 0. I also remember being invited to try this on a calculator - it's interesting because it makes the calculator throw an error, which you basically never see otherwise.

However when I initially read the OP I remembered "12 x 0 = 0" and incorrectly generalised "12 / 0 = 0" before I read down and then realised that I was wrong.

How strange that some people were taught 12 / 0 = 12!

Oh yes, BODMAS! I have absolutely no recollection of learning that at school. It wasn't until one of those maths puzzles was circulating at work where people get different answers and then someone does an eye-roll about lack of basic mathematical knowledge, that I realised there was an order of operations. And then wondered how I'd got through school and Maths O-level being completely oblivious to it. I later learned it was called BODMAS on the Mumsnet talk forums.

Everyday’s a school day…. I’m off to quiz my kids about whether they know this. I didn’t.

This is why my maths and science loving son was so pleased to move to secondary school!

I think the problem at primary is they don't need to be anything like subject matter experts. He quite often had to correct his teachers, some were fine with it, others less so...

I learned division as dividing-into-piles, and I agree that makes it confusing, because you're thinking about something concrete which is actually theoretical. It's probably why I could never get my head around dividing by fractions. Divide by half and suddenly you've got double. Eh??

I didn't learn it as repeated subtraction and I can see how that makes logical sense, but it is really surprising to me. Actually that makes it way clearer - is this how they teach it now? It would be so much easier to divide by fractions with this method. I was taught some ridiculously complicated thing about flipping the fraction upside down and have totally forgotten how to do it.

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

The maths may have been wrong on day 1 but this is a great example set by the teacher with lots of life lessons - everyone makes mistakes, check your assumptions, own up, correct the impacts.

Also well done to your son for checking when he got home.

I suspect the question-setter made a mistake - NOT necessarily a conceptual one, rather typing at speed/under pressure, and writing 12/0 instead of 12/1.

A primary teacher might be marking 34 sets of maths, 34 sets of English and 34 sets of another subject every day - allowing just two minutes for each book, that's 3 hrs 24 minutes, and that's on top of teaching, duties, emails, phone calls, meetings, planning, developing their subject knowledge etc.

I like @Peekingovertheparapet 's response: that said, I think the teacher has handled it well, she noticed an issue, went away to check and has rectified it. At the end of the day we all make mistakes but how we handle them is what really matters. I’d be happy with the way she has role modelled this.

Can I just say although I knew the answer, I never understood the infinity thing until now? Thank you so much for making my day.

Lavender, age 54.

I don’t really get the ‘undefined’ idea. If you graph 1/x, the graph clearly tends to infinity as x tends to zero.

it is only 0/0 which is undefined, as it depends on how the numerator and denominator approach zero. That is the whole basis of differential calculus.

You should check into the infinity hotel

I would have said it was 12! I'm already having to check with friends before I help Dd with maths. She's in yr3. Goodness knows how I got a C at gcse...

It's not obvious to me, especially when dealing with zero, which in my mind isn't really a number so may well behave differently.

I'm in my 40s and did Maths and further maths a levels. Dividing by zero is undefined. That's not a new idea.

Sorry @Nimbostratus100 I was agreeing with you and disagreeing with the posters you replied to.

Yes, they teach it as repeated taking away now. I learned this during lockdown / enforced home schooling.

In terms of dividing by fractions or a number less than 1, think about division as the opposite of multiplication. So multiplying by 2 is the same as dividing by 0.5, and vice versa.

As someone with maths degree… i can confirm you can’t divide by zero! It should just be “error”