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Why is -2 X -3 = 6 ?

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squeeze 26 January 2007 11:34
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Joined14/11/06

I was trying to agree an explanation/justification for this with an intelligent non-mathematician: we weren't satisfied with our answer, can anyone help?

Perhaps asking students to accept this sort of thing, with vague or unconvincing justifications, puts off many bright students, with questioning minds, at an early age?

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HeatherNorth 28 January 2007 19:22
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Joined07/09/06

I think of it like this - all of mathematics is a 'pattern' which can be created - we agree the ground rules of the aspects we are working with, we explore the unknown and we come accross new generalisable patterns - which we can then prove.  This for me is mathematics.  Negative numbers were developed from looking at positive numbers - start counting backwards 5, 4, 3, 2, 1, 0, then as we carry on taking away one we get -1, -2, -3, -4 and so on.  Once we have discovered negative numbers we need to understand how to operate with them.  Again we can start with what we know.  Draw a multiplication grid with -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 along the top and the same down the right hand side.  Fill in the part of the grid that you know for example 1  x 1  = 1.  When you have filled in all the parts you know, use the pattern of the mathematics to find out what you don't know.

Another way in is to look at the history of negative numbers, how they were discovered.  One of our RCs has a brillian history web site on mathematics, there may be something there.  I don't think that we should ever let students 'accept' things - why can't they have the same joy of mathematics in discovering rules and formulas that the original mathematicians have - once they discover things for themselves they will always remember it - that is the beauty of being a mathematician.

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Sistermith 29 January 2007 10:10 - Last edited by Sistermith on 29 January 2007 10:11
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I agree with you Heather, patterns and sequences help demonstrate the 'laws' of negative number (though not 'prove them, which is another matter!) For squeeze's example I would ask pupils to work on this little sequence of calculations:-

2 x - 3 =
1 x - 3 =
0 x - 3 =
-1 x - 3 =
-2 x - 3 =

They should be able to see the sequence of solutions generated. I think this is better than the way I was taught at school: "a plus and a plus is a plus" etc etc!!!

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JoanAshley 29 January 2007 23:36
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Joined15/08/06

Here is a more rigorous explanation.

Because (+2) + (-2) = 0 it follows that

-8 x {(+2) + (-2)} = -8 x { 0 } = 0………….(a)

Multiplying out each term in the curly bracket:

-8 x { (+2) +( -2) } = (-8 x +2) + (-8 x –2)…….(b)

Since we know from (a) that this must be zero,

the second term in (b) must cancel out the first term.

Since the first term, (-8 x+2) = -16,

it follows that the second term, (-8 x –2) = +16

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chrispearce 31 January 2007 15:57
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Too be more rigorous still, you will have to justify "multipying out the bracket" when this includes the product -8 x -2 whose value is in dispute (if it exists at all). I am not sure this would convice a sceptical student.

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richard 31 January 2007 18:00
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I'm thinking out loud here...

3 x 2 = 2+2+2 = 6

3 x -2 = -2 + -2 + -2 = -6

so -3 x -2 = -(-2 + -2 + -2) = --6 = 6 (because we know that if you take negative 6 you're going to be 6 more positive)

I'm not sure whether this clarifies anything or even works!

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squeeze 31 January 2007 20:48
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Lots of interesting ideas, I look forward to trying them out tomorrow and seeing who I can convince. Thank you all for your help and inspiration.

Heather's reply reminded me of a line from Goedel, Escher, Bach by Douglas Hofstadter

"The pattern is everything and everything is the pattern"

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chrispearce 01 February 2007 13:22
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You will have to convince me that -3 x -2 is the same as -(3 x -2)

And suppose I do not believe that --6 is the same as 6?

We could define -2 x -3 to be anything we like. It is because we want the patterns to be pretty and because we want to be able to multiply out brackets for negative numbers in the way we can for "ordinary" counting numbers that we choose the answer to be 6.

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JohnBibby 01 February 2007 14:33 - Last edited by JohnBibby on 01 February 2007 14:34
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Think visually!

On the number line, you know what adding and multiplying does with positive numbers ......? (Ans: adding is translation; multiplying is stretcing from the origin.)

What does ADDING   -1  do? (translate LEFT)

What does MULTIPLYING by  -1 do? (REFLECT)

What happens if you multipliy by  -1  twice?  (Ans: IDENTITY: you are back where you started)

Combine this with the rule that ab=ba and you have

(-3) x (-2) = (-1 x 3) x (-1 x 2) = (-1 x -1) x (3 x 2) = 1x3x2 .....

Does this help?

JOHN BIBBY

PS: Just did a KS2 workshop on "Maths and Archaeology" - interesting collaborative combination of literacy/numeracy/common sense. Please email me on qed@enterprise.net if you'd like more details.

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squeeze 05 February 2007 09:41
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Joined14/11/06

I think he is almost convinced. Thanks all.

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