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!!

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

@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...

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?

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.

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.

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.

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".

I would imagine working for NASA could be pretty exciting.

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.

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

And that's why so many people "can't do Maths" because they're not taught the logic and proofs. You really can't learn Maths from memorising and rote learning! As we keep seeing almost every year with the Maths GCSE and pupils (and teachers!) complaining about question styles they've not seen before - Duh! that's the whole point - trying to figure something out using logic rather than just glibly remembering how you did something previously. Maths really shouldn't be dumbed down to a test of memory!

Nail on the head there!

Some maths is clearly made up and this is one of those things.

you just need to accept it to pass your GCSE and then you can ignore it.

This.
Another way to think of it is rather than thinking about multiplying two negative numbers, think of it as a “main” number and the “multiplier”. The multipliers job is to change the “main” number. If the multiplier is negative it changes the sign of the main number. So if multiplying +1(main) by -1(multiplier) flips the sign to -ve and gives you -1, by the same logic multiplying -1 by -1 flips the sign and gives you 1.

That's 1994 for you.

And for anyone asking the “what the use of math in later life” question, a common use of this fundamental point is in science and
engineering where there a numerous situations where in order to eliminate non-real numbers from the maths (ie you can have 5 apples but not -5 apples) you use equations that involve a squaring (multiplication) step followed by a reversing square root step. You end up with the same result bits always positive.

Best explanation.

@op I was going to say two negatives make a positive. Ie

you can’t not do that means you CAN do that

That’s just another way to explain the rule,
it doesn’t explain why. In maths there’s no concept of good or bad. As a kid I used to hate when people explained stuff this way because I knew intuitively it didn’t answer the question.

I'm with you on this! (I'm not a mathematician.)

OP might tell the kid to get back to them when they get to the square root of minus one. 😀

The baby one isn't a way of remembering the rule, it's about what the arithmetic means on the number line. Everyone learns addition and subtraction are moving up and down the number line and multiplication is repeated addition. The negatives affect direction and movement.

You say that the reason is that you're doing the opposite of negative things multiple times - that's precisely what the baby is illustrating on a number line, showing the correct answer!

You can also use number lines to illustrate e.g. 3 - - 3 = 6. You stand on 3, you face in the negative direction and then you walk backwards 3 steps, so you arrive at 6.
3 + - 3 =0 because you stand on 3, face in the positive direction and then walk backwards and you end up at 0.

That's brilliant

OK apologies for posting a derail.I’ve always struggled with maths and am really trying to follow this. I’ve enjoyed the discussion and explanations given but I I struggle to remember rules, I need to understand the concepts to make doing maths feel less arbitrary and frightening. Not knowing why you get something wrong or right is scary in life generally, to me anyway. Probably relates to people pleasing in some way.. anyway.

So I’m trying to conceptualise ‘the multiplier as the change-maker’ as this idea resonated with me. But what makes it it true that the multiplier is a number, which can be a positive or minus and then its change made will always be made on the sign of the result? (Hope I understood that correctly).

Also, is the ‘main number’ always going to be the first number that you get to in the question, and multiplier next in the sequence, when reading left to right?

Does this mean that for example -2 x -3 = 6?

I got there by making -2, 3 x more negative, making minus 6. Then I applied the rule that ‘the multiplier is the change maker’ and magically flipped it to a be minus 6. But I don’t know why it is the change-maker so I don’t feel secure in this understanding. Don’t want to magic my way through if it can be avoided.

Ah, I’m too old for number lines! I was taught dinosaur maths by dinosaur teachers and am well on my way to being a dinosaur myself.

I think there’s a really important issue of the social-emotional aspect of maths: seen mainly in girls imo but by no means exclusively. It’s desperately difficult to understated for some brain types, and that feeds into a vicious cycle of confidence and further lack of understanding. Some people just don’t think the way required to understand maths. I’m one of them. I can do it, have learned all the rules, got all the A grades, use it effectively in my daily life etc. But I never understood the “why” of it. I understood the logical steps in solving equations, and understood that at my level maths was absolute so getting the right answer was enough of an answer to why (why? Because it can’t be any other way, this is the way things work). But I never was able to picture it, reason it. Putting it in graph form; putting it in lines and curves etc - these were all just other ways of showing me that this is the way things are because look! They can’t be any other way! But that still didn’t explain why they couldn’t be any other way.

Tbh I’m not that much further along in my understanding as a near-dinosaur now…

Oh! That’s actually very helpful, thanks. I will try and remember that, and the double-negative language example someone else posted.

It’s mad how I get instantly choked up and hot when trying to understand anything maths.

@RunsWithDinosaurs Actually, maths is full of arbitrary rules and procedures, some of which are so deeply entrenched that we don't think about them. Some of them are centuries-old debates. Here are some examples:

• Why do we use ten digits, giving us base ten? It's believed this is because we have ten fingers. But actually, ten is quite inconvenient mathematically, because it can only be divided by 2 and 5, and lots of fractions don't give neat decimals. If you think about it, lots of old systems of weights and measures do not use the awkward 10: ounces in a pound, inches in a foot, and so on. Computers do not count in base 10, preferring base 2, or base 16.
• The ancient Greeks thought that 1 was not a number: it was the "unit" from which all numbers rose. When we say "a number of things", we nearly always mean more than 1.
• By convention, 1 is not a prime number. There are many reasons for this, one of which is "a prime number has exactly two factors".
• Centuries ago, there was debate about whether 0 was a number. A long time ago, in numbers such as 106, a space would be used instead of the digit 0. The Romans did not use 0.
• There are endless arbitrary rules and procedures in statistics, compared with "pure" maths.
• With Roman numerals, the convention of shortening them by writing four as IV instead of IIII was probably not used by the Romans themselves, and did not catch on until the invention of printing.
• There are numbers which are called "weird" numbers, which are rare. Sometimes I tell more able children what "weird" numbers are, and nobody actually knows whether an odd weird number exists, or has been able to prove that none exist. (I'll explain what a weird number is if anyone is curious.)