How do I explain to a Y7 why when you multiply two negative numbers it becomes a positive?

[loveyouradvice]

Just that really - I know it does, but can't explain why!

Hoping @noblegiraffe and others might know!!

Today I learned it is possible to round in different directions?

(Finance director for 20 years)

By which I mean round to even etc, not the halves go up rule

Me! I’m curious!!

@modgepodge What is a weird number?

70 is a weird number. The factors of 70 (excluding 70 itself) are 1, 2, 5, 7, 10, 14, 35. These add to 74, greater than the number itself, which means that 70 is an "abundant" number. However, it is not possible to find a set of them which adds to 70, unlike most other abundant numbers.

Weird numbers are rare, and all the ones discovered so far are even. Whether any odd weird numbers exist is one of the unsolved mysteries of maths.

This seems quite straightforward to me.

Say I have ten apples.

If I give two to someone else that is -2 so 8 apples left. If I do so twice that is 2 x -2 so -4 = 6 apples left.

If they were to decide they prefer bananas and give me back the 2 x 2 apples that I gave to them this would reverse what we just did above so is -2 (the reversal) x -2 apples, hence it is has the effect of +4 (i.e. 2 x 2 has the same effect as -2 x -2 because the deduction from my number of apples when I gave the person 4 is being reversed with them giving the 4 apples back to me and me now having 10 again in total).

Oh gosh, I am starting to see what I struggled with maths at school, because don’t understand this at all. I am sure it is a great explanation as everyone says, but I can’t make any sense of it. Honestly it’s awful, gives me flashbacks to so many Maths lessons where the teacher’s voice just seemed to be getting further and further away!

Thanks both.
it makes more sense. But agree still don’t get why it’s a change maker.

The mind boggles!

That seems like a good way to understand it intuitively. However generally in maths I find I really like and am very comfortable with the idea that it isn't necessary to understand something in the same way as one does in other contexts. If you can prove / show by logical argument that a statement is true then it is true.

Also you might not have time to delve into every proof or explanation of why something is true. So we can look at the proof of pythagorous' theorem and that is interesting. However later if we forget the proof that is fine. We can just remember the formula, know that it has been proven to be true, and use it in a mathematical argument to deduce that other things are true.

It's a form of abstraction so that our brains don't explode. Maths became for me much easier when I started to think about it in this way.

Exactly. If you multiply a positive number by a negative number it has a negative effect because you’re effectively deducting the amount several times (however many times the positive multiplier number is).

Doing the opposite (i.e. changing the multiplier from a positive to negative number) will have the opposite effect. 🤷🏻‍♀️

Yes, I think especially when it comes to more complex maths you can’t relate a proof some kind of “real world scenario” like this: it becomes more about logical inference.

I do think, though, that actually understanding the most basic concepts like this one which form the foundations is important - rather than just learning them as “rules” - because they are so intuitive and obviously true and are demonstrable with simple examples. This kind of level of understanding is quite necessary to have a basic level of functional mathematical skill that is needed for everyday life. It’s worrying that a YR7 wouldn’t yet have been taught this at school in a way they can grasp!

So remember for multiplication the order isn’t important (3 lots of 2 apples is the same as 2 lots of 3 apples). So you can make any number the “change maker” if that works for you. So -6x5 you can rewrite as 5x-6 to make 5 the main number and -6 the change make if that order works for you.

Posts like that are quite flabbergasting aren’t they!?

Yes, but I think it just reflects on the way maths is taught early in school and a general view (even in primary school teachers) that maths is "hard". It's so common to hear people say "oh, I can't really do maths" but you never really hear "oh, I can't really do reading"...

I find it quite sad, really, that people aren't given the grounding to understand maths and then it does just seem like a huge list or arbitrary rules to memorise.

I taught SEND maths catch up in KS2 primary for years, and there are a remarkable number of young people who have not been taught the very basics of maths so don't really understand numbers at all.

Also if you are struggling to visualise this rule then you can prove it through logical inference from the otherarithmetic rules in a couple of steps. For me personally I am quite happy to understand things in this way and it feels very different from just accepting them as facts that must be learnt.

.... please tell me I'm not the only one that this is news to?!??

I'm pretty sure I never, ever knew that 🙈

Ye gods. I'm still trying to work out how this could possibly work, and I got nuthin'.

(I'm an otherwise vaguely functioning member of society I promise.)

So, what does -2 x -2 =? And whyyyyy/how?!?

I got an A in my maths A level and I admit I never understood this. My maths teachers were never able to clearly explain why. In the end, I just accepted it (which offended my scientifically inclined brain) and got on with giving the required answers to exam questions.

Because if you do -2 x 2 you are taking away 2, twice, so that is -4.

If you do opposite, so -2 x -2, then that is “eliminating” the taking away of 2 twice, i.e. adding 4 back. You “cancel” two subtractions of two so it is +4.

I think part the problem with this is that really there’s less ‘why’ to it than people who think they don’t understand it imagine there is.

Or, if you do -3 x 2 you are taking away 3, twice, so -3 and -3 again is -6.

If you do opposite, so -3 x -2, then that is “eliminating” the subtraction of 3 that you have done twice, i.e. adding 6 back to “reverse” -3 and -3. Effectively you “cancel” 2 subtractions of 3 so it is +6.

I think @JustHereForthePIPis right that it must be people not understanding the very most basic idea of what numbers are and the basic operations because if you do then the reason why two negatives multiplied is a positive number is very obviously self-evident.

That would seem to indicate an enormous widespread failing in the teaching of the most simple maths in early primary school years, which is pretty depressing.

You, ma’am, are a bloody genius.

Try this explanation, but a bit beyond yr 7.

My maths teacher explained it by writing it out like a double entry bank statement with a negative and a positive side (she was hot on real life examples), So if my salary was £200 in the positive side for the month and each week I took out £50 from the negative side then I'd expect to have zero left. (ie 50 x-4 = -£200 and 200-200 =0) If I didn't realise that the bank also took a £1 transaction charge for each withdrawal then the bank had taken away an extra £1x-4 = -£4) from the negative side . If I then complained to the bank that they had not told me about the transaction charges and they agree to refund the 4 lots of -£1 then they will take away from the negative side of the bank account so -£1x-4 - which means that £4 will go to the positive side of the bank account - so +£4. Its easier to follow if its written out with debit and credit sides.

🫠 brain + maths = malfunctioning