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!!

Loving the baby stephen @noblegiraffe !!

Most of the replies here, including the baby one, are just ways of remembering the rule. They do t answer why. The reason is per @scalt and @WhatYaGottaDoo: two negatives multiplied make a positive because you’re doing the opposite of negative things multiple times.

2 x -3 =-6, you’re doing a negative thing twice. You’re making it twice as bad.

-2 x -3 =6, you’re undoing a negative thing, twice. You're doing a good thing.

HiRen I think the baby explanation does show why it's positive or negative. The direction baby Stephen is facing in is either positive (right) or negative (left). The steps he takes show the direction he's moving in (forwards or backwards). Whether he overall moves in the positive or negative direction depends on both (e.g. facing left and moving backwards represents negative times negative and that moves him to the right)

Could you relate it to the double negative of English

She did not say she didn't want the cake = she wanted the cake

or my more simple way of remembering it - two negative signs = one positive sign

@Occitane Sorry - you'd already covered the double negative angle

I remember having such difficulty understanding this, then it clicked. You’re multiplying a negative amount by a negative amount so it removes that negative.

Lol of course not. Do you imagine that 2+2=4 is made up just for fun as well.

I wonder if you have similar thoughts about science, history etc or if its just maths that you imagine has arbitrary rules and facts in place just to baffle people.
I suggest you get yourself a copy of this and have a read. There@s another one aimed at parents of teens iirc.
Maths for Mums and Dads : Askew, Mike, Eastaway, Rob: Amazon.co.uk: Books

I saw a brilliant explanation years ago on Khan academy about why negative numbers work the way they do - possibly this video, not sure
Negative symbol as opposite (video) | Khan Academy

I never needed to know why. I just learned the rule and got correct answers. Just teach the rule.

Lol of course not. Do you imagine that 2+2=4 is made up just for fun as well

No (sigh), don’t be so patronising.

Are you happy that 1x1=1?
Are you happy that -1×1=-1?

Ok we can prove in a couple of lines that
-1×-1=1

1 = 1
1-1 = 1-1 (subtract 1 from both sides)
1-1 = 0
-1(1-1) = 0 x -1 (multiply both sides by minus -1)
-1×1 + -1×-1 = 0

If we want the above statement to hold true -1×-1 must equal 1.

In maths it isn't so much about understanding why something is correct by relating it back to real world concepts but by proving it to be true based on logical steps.

You're removing a debt. So you have more money.

Say -2 × -3 = 6...

You have some IOU tickets of £2. 3 of them get taken away. You're now £6 better off.

I think lots of basic maths is best explained using things like money.

My dd is way younger but I'm teaching her division by talking about sharing out treats between her and her brother (and her cousin etc).

Kids have an instinctive understanding of fairness, debt and so on, so tap into that

Nail on the head. Understanding and applying logic is the key to being good at maths.

Just rote learning and remembering rules is why so many are crap at it because they don’t fully understand and can’t cope with something they’ve not seen before.

Whereas someone who understands will be able to have a good attempt at a problem they’ve never seen before.

This has just taken me back 30 years when I was introduced to the same concept at school. I was good at maths (went on to take 1/2 my A-level course on one lesson a week) but could not grasp negatives. My godfather, who was not known for his mathematical prowess, took it upon himself to explain. We both ended up more confused than ever. I think in the end I just accepted it without knowing why, which annoyed me but it let me get on with it. No use to you at all but thanks for the trip down memory lane!

This approach only works well with kids who already are confident with maths.

To a child who "fears" maths, that is a sea of mumbo jumbo (I know it isn't, I mean from their point of view).

Its simply that if you multiply any number by a negative it changes its sign.
intuitively (hopefully)
1x1 = 1
therefore
1x-1 =-1 as the only thing that’s changed is the sign.
By the same logic if the number you multiply by a negative number is already negative and the rule is that multiplying negative numbers changes the sign, then the negative number must become positive. So
-1x-1=1

Draw a quadrant graph and draw a square on the + + or quadrant 1.

Now the x and y axes act like a mirror so if you were to reflect the square in to quadrant 2 or 4 you get a square with 2 negative points and 2 positive points.

You can't directly reflect into quadrant 4, you need to do a 'second reflection' from one of the initial reflections.

It is this double reflection that makes it a +ve. A bit like double negatives in English.

Another way to look at is is that numbers don't have one square root they have two. So the square route of 4 is +2 and -2, so to get back to 4 you multiply one of the roots by itself and you end up with 4.

you could also think of it in terms of language:

not not happy = happy (ie double negative=positive)

But baby Steven has ended up on -6, not 6….. ah I can feel my maths anxiety brewing, I’m off! 😆😭

Say -2 × -3 = 6...
You have some IOU tickets of £2. 3 of them get taken away. You're now £6 better off.

But it’s a times, not a take away. Surely if 3 people pay you £2, then they’re giving you PLUS £2, not negative £2, three times, not negative 3 times??

Ohhh it’s all just bullshit, this is why I’m the thick sister 🤣

As an Economics teacher I really liked @RobinHeartella's explanation! Soon be recycling that at work, thank you.

We do using directed number counters with the zero pairs model. Put the counters up on MathsBot on the big screen and let them drag and drop

@Mumjaro doesn't the fictional person OWE three others £2?

I love that! I was very much a blind rule follower and did well in maths because I didn't ask questions about why! But this is a very cute rule!

Completely agree.

And for the majority of people, mathematics and the use of it in the rest of their lives is completely about real life. So negative x negative= positive needs to be taught as debts being removed (as described above) and not with abstract concepts or, even worse, just learn the rule and blindly accept that negative x negative equals positive because it's a made up rule you just have to know.

But it’s not just a “made up rule” - it can be proved by logic in lots of different ways. Saying it’s a “made up rule” implies it isn’t actually true nor provable.