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!!

Embarrassingly I have no idea, but we remember it as - and a - put together is a +

I have tried to understand it, maybe someone will be able to finally explain it on here.

There's a basic explanation where you fill out a times tables grid and just continue the patterns backwards, but I like this explanation of it being like a child walking backwards on the number line while facing in the negative direction.

I don’t think there is a reason: it’s just a made up rule. Interested to see if anyone comes up with anything.

What?!? Maths isn't just arbitrary rules made up to confuse school children and test their memories!!!

I know. But this particular rule ……..

OMG I love the baby explanation, and I’m definitely going to need it to explain to my DD soon.

thanks @noblegiraffe !!

@RunsWithDinosaurs
Some of it is though…like why 65 rounds to 70 not 60, and why we do the multiplication first in 6+4x5, for example. It’s just conventions of how we record stuff rather than inherent maths I think…

the explanation involving the baby walking is the best explanation I’ve seen for multiplying negative numbers. I too was taught to think of 2 - signs making a + but that doesn’t actually explain WHY. And I like to know why!

I was taught, rightly or wrongly, that because a negative was the opposite to a positive, if you multiplied a negative by another negative you were applying two opposite transactions consecutively, which meant you ended up back with a positive. Effectively, one negative took you through 180 degrees, and another negative took you another 180 degrees back to a positive. But, despite having an O level, AO level and A levels in maths and further maths that’s the best I can manage!

Are they struggling to understand or remember? find it helps to take it down to using plus or minus one as one of the numbers.
-32x1 is obviously -32
So -32x-1 is +32.
What else could it be? You know it isn't -32 because that's - 32x1.

An enemies enemy is my friend (- x - = +)
a friends enemy is my enemy (+ x - = -)
an enemies friend is my enemy (- x + = -)
a friends friend is my friend ( + x + = +)

I appreciate this doesn’t help with “why” but I’ve been a qualified accountant for 10+ years and I recite this often 😂

You don't explain it. You teach the rule then they apply it. Life's too short to use up brain cells on an explanation for that one. . . There are far more exciting things to think about!

I don't know if this is correct, but I think of it as being like a double negative in the English language. So, you could say, 'I don't like this', and then 'I don't not like this' and the second one would mean that you do like it, because you don't 'not' like it. Sorry, I don't think I'm explaining that well, but to me, that sort of explains why two negatives make a positive.

Here’s how I explain it:
3 x 2 =6: like being given 3 lots of £2.
3 x -2 =-6: given 3 lots of £2 bills.
-3 x 2 = -6: 3 lots of £2 taken away,
-3 x -2 =6: 3 lots of £2 bills taken away. In other words, refunded. You have £6 more than you did before.

I can’t think how to explain it with division, though.

Ok, say you have -2 and you triple it (multiply it by 3) you get -6 of course. -2 x 3 = -6. i.e if you have lost two apples three times you have lost 6 apples in total.

Each minus swaps direction…

So if you multiply by -3 instead of +3 it goes the other direction so -2 x -3 = 6.
i.e. each of the three times you didn’t lose 2 apples, you found 2 apples (-2 x -1).

One minus sends you backwards but going minus/backwards (changing direction) after already being backwards means going forwards (you turn around twice).

So -2 x -3 x -4 is -24
and -2 x -3 x -4 x -5 is 120

@modgepodge wrote "Some of it is though…like why 65 rounds to 70 not 60..... It’s just conventions of how we record stuff rather than inherent maths I think…

It is a sort of convention, but it's not the only one. If 65 is always rounded up, then if you do that with a lot of numbers the rounding "errors" accumulate and the resulting total (or product, etc) would be misleading.

There are ways round this. One convention is to "round even", so 5.5 to the nearest whole number rounds up to 6 (the closest even number), but 6.5 also rounds to 6. Similarly both 55 and 65 would round to 60.

In GCSE and A level exams, if I remember correctly and things haven't changed since I marked papers, any similar method of rounding is acceptable because Mathematicians can argue it's correct.
So, if the answer to a calculation is 6.5 and you're supposed to round it to the nearest whole number, they would expect you to round it to 7 but you'd get the mark if you wrote 6.
However, to the nearest whole number, a number just more than 6.5 (eg 6.0000001) always rounds to (r.t.) 7 and a number just below 6.5, (eg 6.499999) r.t. to 6.
You can't say 6.499999 r.t. 6.5 and then round that 6.5 to 7. It's only exactly 6.5, or 65, or 650 etc, where it's acceptable to round up. I think.

Ooooh @noblegiraffe I do like the baby steps method, but I'm reading this after a night out so will immediately forget it!
I usually use the table that noble mentioned above

If you think of zero as a pivot, each time you multiply by a negative it flips to the other side.

In my head at least. Maths is all machines in my head.

😂

I was very good at Maths at school and drove my year 8 teacher up the wall with my complete refusal to believe any of his explanations for this particular problem! In the end my Grandpa came up with an explanation with a diagram that I finally accepted, but I can't remember what it was

I’ve read that baby explanation 3x now and I still don’t get it. I know the rule but realise i don’t understand why as a result of this thread.

Argh no! I would never have known to do the multiplication first… I would have made = 50. Why do we do the multiplication first?! Why can’t the sequence be done left to right like reading the letters to make a word? Whyyyy….

My teacher in Upper school explained as the following:

If a positive thing happens to a person behaving positively that’s a positive

If a negative thing happens to a person behaving negatively that’s a positive

If a negative thing happens to a person behaving positively that’s a negative

If a positive thing happens to a person behaving negatively that’s a negative

That’s always stuck with me and is how I remember it.

If numbers are a scale that can go one of two directions, -7 take away 7 is -14, because you are removing 7 from it, which makes it go negative. -7 minus -7 make it become positive, because you are removing negative 7, so you're essentially taking the minus away and the scale becomes positive.

Does there need to be a special way? Two negatives make a positive has always stuck with me.

This is about multiplication, though.