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

Yikes. If you are serious your maths teacher wasn't exactly interested in encouraging intellectual curiosity!

Minus times minus is always a plus
The reason for this we never discuss

• while banging a rhythm on the table with your hands for emphasis. Well, that's how I was taught.

I would imagine working for NASA could be pretty exciting.

Don't panic, I promise it's not bullshit.

"Times minus three" Or × -3 means taking that thing away three times.

Just like ×3 means adding that thing on, three times.

5 × 3 means add 5, three times.

Multiplying is like adding, but several times. That's why we call it "times".

It's all very well sighing, but it was a pretty daft comment demonstrating a complete lack of understanding of even the most basic principles of mathematics.

Of course. But if we write 0 as its equivalent expression of 1-1 or 2-2 or anything of that sort and attempt a multiplication we can immediately see that if negative x positive is negative then it follows that negative x negative must be positive.

I imagine that as a fundamental arithmetic rule there are limitless ways to demonstrate it.

Multiplication can be thought of as repeated addition, (or subtraction if you are adding an IOU/negative number)

Therefore 2x3 is starting from zero on the number line and moving two units along it 3 times, in the positive direction.
2x (-3) is starting from zero on the number line and moving plus two units along it 3 times, in the negative direction.
(-2)x(-3) is starting from zero on the number line and subtracting three (IOUs) groups of -2, so you end up at plus six.

I don't know. I think explicitly teaching the idea that maths is about taking some initial assumptions and basically forming a step by step logical argument to show that something is true makes it a lot more comprehensible. Sooner or later you will get to a point where maths gets sufficiently abstract that real world analogies will break down or become too cumbersome to be useful.

We teach the scientific method of seeking knowledge through experiment. So why not explain the mathmatical one?

@BathsAreBliss that's also very good.

Tbf 6th form have never actually asked me to justify why two negatives make a plus but they do frequently forget...

But surely in this, once you have got to: 1-1 =0 (3rd line) it doesn’t work after that? Because 0 multiplied by anything is 0. So yes, you could multiply both sides by the same number and it would still be 0, but you could multiply both sides by different numbers and it would also still be 0.

Eg

0 = 0

zx0 = y x0 does not prove that z=y

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.

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.

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!

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

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

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

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 🤣

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

you could also think of it in terms of language:

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

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.

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

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).

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!

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.

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