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One of my friends teaches mathematics in primary school. He was asked a question that Why $(-1)\cdot (-1)=1$ ... At higher level we can answer this question saying that it is a definition and we want different things such as associative and distributive law to hold but how we convince a primary level student.

I don't want any proof as I know very well how to prove..I want an intuition for a primary level student. I know how to prove it.

Rayees Ahmad
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  • Duplicate – DHMO Apr 14 '17 at 13:25
  • See for example: https://math.stackexchange.com/questions/539351/how-to-prove-1-cdot-1-1-based-only-on-the-field-axioms – Michael McGovern Apr 14 '17 at 13:26
  • I think a young student will recognize that multiplying a positive number by $-1$ has the effect of "flipping" that number across the origin on the number line. It is a matter of convincing the student that multiplying a negative number has (or should have) the same effect. – Umberto P. Apr 14 '17 at 13:45

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For a primary school perspective, I like to imagine it as an enlargement of some kind. For example, picture the number line. Then $1 \times 2 = 2$ since $2$ doubles $1$ --- you can see this because $1$ moves up the number line from $0$ twice as much. Multiplication by $-1$ is a reflection in the number line at zero --- every positive number jumps over zero to get to it's negative counterpart. Now, if you reflect $-1$ in the same way, what do you get? You're back to $1$!

Bilbottom
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  • You might want to say that multiplying by $-1$ turns you around rather than reflects to be even simpler. Maybe talk about a car going forwards and backwards. – badjohn Apr 14 '17 at 13:52
  • @badjohn I agree, that could also be useful --- I just know that for me, reflections are easier to picture. Both interpretations work, it's just whichever works for the recipient! – Bilbottom Apr 14 '17 at 14:03
  • I suspect that you are slightly past primary school age. I guessed that "turn around" would be simpler since a child can actually do that. "Reflect yourself" would be hard to obey. Feedback from the OP after he tries would be interesting. – badjohn Apr 14 '17 at 14:06
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I guess these students would agree that
$0\times (-1)= 0$
$1\times (-1)=-1$
$2\times (-1)=(-1)+(-1)=-2$
$3\times (-1)=(-1)+(-1)+(-1)=-3$
and so on. (If not, they should go over addition first).

Now we can observe that a pattern is formed. If we go one row up in this list, we substract $1$ from the left factor and add one to the result. We want this pattern to keep existing if we move to negative factors. Therefore it is easy to see that we must have
$(-2)\times (-1) = 2$
$(-1)\times (-1)=1$
[the rest of the list I wrote above. If it's written on a blackboard you just add it to the existing list.]

35T41
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Not so much a proof as a recipe, but it proved simple to understand and apply, at least for my childen:

Explain the action of a negative sign as a indication to switch a + to a -, and a - to a +. It is simple enough to see that two switches return to the initial ‘state’.

Then explain that multiplying two numbers, negative or positive, can be broken into:

  • finding the absolute value of the answer;
  • finding the sign of the answer.

The first step is the product of the absolute values of the terms: $1\cdot1=1$

The second step is done by finding out if we have an even or an odd number of negative signs in total among the terms: $(-1)\cdot(-1)$ → 2 negative signs, so it is even in this case, so back to initial positive value (since the absolute value was used in step 1).

It shortcuts a few explanations, but it is very simple and easy to memorize.

  • Why can the negative signs be gathered at the beginning? What is the meaning behind this? I don't think it's a good explanation. – 35T41 Apr 14 '17 at 14:02
  • Will that feel better? –  Apr 14 '17 at 15:04
  • Yes, but I still think it is too technical rather than intuitive. But that's just my opinion. – 35T41 Apr 14 '17 at 15:17
  • I'll agree it may not work for everyone, but the technicality is fairly simple and I found understanding sometimes come after fiddling around with ready-made recipes. It was especially true for me when I was a kid: I could get to the why from the how. –  Apr 14 '17 at 15:28