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

Some common stats packages do this. I’ve attached a screenshot of the question I asked, and the AI result.

It’s also in a parliamentary explainer here.

It is similar to repeatedly subtracting negative numbers

3 take away negative three is 6
3 - (-3) = 6
3 - - 3 = 6

So starting at zero: -(-3) - (-3) - (-3) - (-3) - (-3) - (-3)

Brackets are just there to be clear.

It seems like the best way to teach this is to look at adding and subtracting negative numbers and then look at the multiplication as repeated addition and subtraction.

And if anyone still needs convincing you can quickly infer it from the other arithmetic rules

2 - 2 = 0

Multiply both sides by negative 1 to get
(2 x -1) + (-2 x -1) = 0
-2 + (-2 x -1) = 0

Add two to both sides
-2 x -1 = 2

OK, now do the same for -6 x -3

Because of BODMAS

Omg it makes sense at last!!!!! What a roller coaster of learning today has been ❤️

Yep!

Multiplication is continuous addition for positive numbers and continuous subtraction for negative numbers.
6 x 3 =18
3+3+3+3+3+3
6x -3 =-18
-3-3-3-3-3-3
we would show children using a number line with 0 in the middle and show the jumps.

(Oh, and many secondary curriculum designers completely failed to notice the 2014 revision of the Primary Maths curriculum. So they spend Y7 re-teaching concepts that are actually done to death, while berating the ‘terrible teaching’ of things like statistics that have in fact been removed from the Primary curriculum entirely. A local Y5-Y8 Maths working group was a revelation, and lesson observations between the primaries and secondaries involved were eye-opening!)

In many ways, the current primary curriculum for Maths is ‘too big’. You can get a 3 in Maths GCSE answering only questions that explicitly test the primary Maths curriculum, and a set of examples proved a few years ago that the GCSE questions testing this curriculum were simpler than the corresponding ones in Y6 SATs.

This is exacerbated by the fact that able primary mathematicians have traditionally been accelerated through the curriculum to higher years, rather than exploring the large hinterland of Maths that simply isn’t on the curriculum (as an able primary mathematician in the 70s, I explored formal logic; topology; statistics, via a series of assorted books from the back of the classroom cupboard!)

A slimmed down primary curriculum with a lot more explanation and consolidation and (yes) memorisation of key facts like number bonds etc would build stronger roots (while alienating parents who want their children to ‘do the written methods’ as ‘that is proper maths).

What we have instead is White Rose. 47 different explanations of every concept, delivered at a speed that varies arbitrarily between snail-like and Concorde, via 25 page PowerPoints.

Nail on the head. I do think it's poor teaching at primary level, often by teachers who havn't a degree in Maths (maybe not even A level in Maths) so are "teaching" without knowing the subject properly themselves. Hence why it seems some teachers "teach" pupils to simply learn things rather than explaining them, which explains some of the comments above about being "made up rules". When you have primary school teachers with that attitude, you've no hope of them actually teaching the subject properly.

Maths is almost unique in that if you don't "get" the basics, you'll never succeed in later years. It's like the roots of a tree. The "Bit above ground" is never going to flourish if the roots aren't complete and strong.

We've had various comments on here over the years from secondary Maths teachers bemoaning the standard of primary teaching and saying that they have to spend virtually the entire first year of secondary teaching the basics that should have been taught at primary, before they can start working through the secondary curriculum.

It's a national disgrace really that so many people have the attitude of "I can't do Maths, me!" and are almost proud to say it!

🫠 brain + maths = malfunctioning

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.

Try this explanation, but a bit beyond yr 7.

You, ma’am, are a bloody genius.

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.

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 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.

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 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.

.... 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?!?

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.

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.

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

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.

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!