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!

No. That's just not mathematically correct.

Thank you.

To expand...

"I can't hang them up in wardrobes, m'lady. You haven't got any!" (Impossible.)

"Well then, it's going to take you forever to do it, isn't it?" (Infinity.) "And by the way, how many dresses will be in each wardrobe?"

"What are you talking about? There's no answer to that. There aren't any wardrobes to hang dresses in!" (Indeterminate.) "But I'm holding them here, all 12 of them."

"Are you a wardrobe, then? No. So there aren't 12 dresses in a wardrobe, are there? So whatever the answer is, it can't be 12."

At which point My Lady gets a cyanide cucumber sandwich at tea time...

I’m almost 50 and was taught that anything divided by zero is undefined

how can it be 12 when 12/1 = 12?

Simple answer is that you cannot divide anything by zero as it's not mathematically possible

But "you" are the "group" holding the apples, so there is no "zero". Zero would infer there was no group at all, which is clearly wrong, as you're holding the apples, so you're a group of one. Hence the answer is that 12/0 is impossible, and the rationale that you're still holding the apples is that 12/1 applies, you being the group of one!

I genuinely struggle with maths and threads like this feel as though they are attacking people's intelligence - which I suppose they are. If you have never had a good teacher with the patience to explain things, and never had to apply these principles then it isn't as easy as others find it.

I get what you mean - I feel the same on threads about grammar and sentence structure. I never just “got” it, wasn’t taught it properly and often feel completely stupid when other people talk about it. My primary age nieces know way more than me! I got decent gcse grades in English because I learned the right tricks to get marks, not because I actually understood the damn poetry.

@GimmeBiscuits Your post made me smile... well shudder actually. We had a teacher when I was 9 who used to throw CHAIRS at pupils to reinforce his point. we were absolutely terrified of him. I once remember him yelling at some poor kid "The 4 times table isn't that hard if I chuck 3 chairs and a table at you how many legs are going to hit you?"

No. It was a specific question to a specific poster on their specific choice of words.

Perhaps we should look into Comprehension questions next.

@ReneBumsWombats You would make a great teacher.

With Mathematics, the teacher is everything.
Mathematics almost more than any other subject , if students don't have a natural affinity with the subject.
There was a wonderful teacher we knew, Dr C he could make people understand that 'n' could be any number when teaching basic maths, despite the student's initially puzzled faces.

He was endlessly patient and encouraging with the students who found it tricky, to those who went to Cambridge.

He said ''Many people , especially adults, fear maths as they were belittled or made to feel bad at school.
Dr C got good results with his students.

In theory yes. One is the inverse of the other. But I struggle with the concept of 0

There's a difference between simplifying something and teaching something that's inherently wrong!

If primary pupils are deemed incapable of understanding infinity or a question that's impossible to answer, then surely the correct way of teaching is to avoid questions/examples that give such an incomprehensible answer. It's surely not right to teach a wrong answer which will confuse the pupil for years to come, if not their lifetime!

I'm not sure just what is the worth/point of setting a question such as 12 divided by zero. What possible benefit is there to teach young pupils something that they don't have the knowledge/experience to answer? Isn't it just setting them up to fail?

You have to think of it the other way round. If you have zero apples and multiply that by 12 you still have zero apples

That sounds fairly standard for my school.
We once had a substitute teacher for maths. He came in, sat down and put his feet on the desk and told us to 'just get on with whatever it is'. Someone asked him about the work and he said, "I don't know. Just make something up."

Absolutely agree.
Wonderful Dr. C a maths lecturer used to have to undo a lot of fear around the subject with adult students {over 18's} in particular, because of brutal treatment in the past.
He would get students who needed basic GCSE maths to go on to university ..

One person who 'Dismally failed' {Their own words} at school got A's at GCSE and 'A' level and under Dr C's tutelage.
I remember kids having board rubbers {Wooden blocks with compacted felt, maybe 500grammes in mass} thrown at them, ditto books by the teachers.

Teaching is a real skill.
Not everyone is suited to it.

That makes far more sense.

Yep, I think Maths is the main subject where success is entirely dependant upon the teacher. My DS was a whiz at Maths at primary school, and got pretty close to top marks in year first year (year 7) at secondary. In year 8, he got a crap teacher, and went downhill rapidly, ending the year with around 45% in the end of year exam! We spent that Summer "teaching" him the year 7 content ourselves via CGP and similar books. He was back to 90+ percent in year 8 and ended up with a grade 9 at GCSE and A* Further Maths GCSE, then an A at A level and he's on target for a first degree in Maths at Uni! That just shows how much damage a crap teacher can do!

My OH was barely numerate when we got together, despite having decent grades (A-C) in all other subjects. He says he had the same crap teacher for 3 of his 5 years at secondary school and ended up with a grade U! In his 20's, he went to college to take Maths at evening classes, and secured a grade A, after just a year of 2 hours per week!

We do seem to have major problems with teaching of Maths at both primary and secondary schools. Even at Uni, the teacher isn't always great, my DS has had to do a hell of a lot of "self teaching", and yes, I know that Uni does require a lot of self study, but many times, the course notes and lectures have been ambiguous, or plain wrong!

My understanding of what division means is that you are asking "how many times does that number fit into the number?"

So when you do 12/2 you are asking how many times does 2 go into 12.

You can't ask how many times 0 goes into 12 because 0 is nothing. It's like asking how much of nothing can you fit into a box.

@ReneBumsWombats

Please tell me you are a teacher 🙏!

See, this type of thing baffles me.....

If you have a group of 12 'somethings' and divide it by 2 you have 6 groups of 2 ... if you divide it by 1/2 you have 2 groups of 6.

If you have a group of 12 'somethings' and divide it by 1 you have 12 groups of 1= 12 or 1 group of 12

If you have 12 'somethings' and divide it by nothing (0) you still have the same group of 12 'somethings' you began with and, surely to goodness, the answer is still 12...isn't it????

I actually can (sort of) see the other reasoning, but I would have put 12 in that test.

Maths is definitely not my strongest subject,though. And I really have to sit and study the questions in "Maths, No Problem" which was introduced a year ago at the school I work at, because sometimes it just makes no sense... to me or some of the teachers!

This is why it helps to see division as "how many of x number can you fit into y number" :)

When you frame it that way, it makes sense why number can't be divisible by 0. If you frame it as splitting a tangible number into physical groups, the 0 principle doesn't apply so therefore logically it must be the wrong way of looking at it, if that makes sense.

It's how many 0's can go into 12 so there are an infinite amount of divisions.

How many somethings did you put in the no groups though? It's not about how many you are left with or how many you started with. It's about how many groups you can make and how much you put in each group. If you have 0 groups, you can't put anything in them. If you put 0 in each groups, you could have an infinity of groups.

Don't know if this makes sense.

🤔 yes!!

quickly munches the apples

@Badbadbunny you are absolutely right. I feel like I had a string of ‘I understand it, why don’t you?’ teachers who just never explained anything. I was well behaved and turned up but never learned anything despite doing well in all other subjects. I got a U in my gcse maths.

I presumed it was me and I just couldn’t learn maths.

Then I got a tutor who actually explained things and I passed my resit without issue. I didn’t become a maths wiz but I discovered I was capable of at least understanding stuff and comfortably got a C.

If you have a group of 12 'somethings' and divide it by 2 you have 6 groups of 2 ... if you divide it by 1/2 you have 2 groups of 6.

If you divide 12 by 1/2 you have 24.

So if you divide 12 apples by 1/2 you have 24 piles each with half an apple (you can't really have half a pile, that would be a small pile)

Or of course you could divide by 2 and have 2 groups of 6.