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[quote=Anonymous]With my kids I made a construction paper number line on the floor with a different number on each piece of paper.  You stand on the first number.  The operations sign tells you whether you're walking forwards (addition) or backwards (subtraction) and the sign of the second number tells you which direction to face (negative or positive)

For example:

1 + (+) 2 = 3
 
Stand on 1.  The addition sign says you're walking forward and since you're adding a positive number, you're facing to the positive side.  Two steps up the number line puts you on three.

1 - (+2) = -1

Stand on 1.  The subtraction sign means you're walking backwards.   Since you're subtracting a positive number, you're looking at the positive side.  Facing the positive direction, walking back 2 spaces from 1 will put you on negative one.

1 + (-2) = -1

Stand on 1.   The addition sign means you're walking forward, but the negative sign on the second number means you're facing the negative direction.  Facing the negative side and walking forward 2 spaces will move you in a negative direction 2 spaces, so you'll end up on -1

1 - (-2) = 3

Stand on one.  You're subtracting, so you're walking backwards.  The second number is negative, so you're facing that way.  Facing the negative direction and walking back 2 steps moves you in a positive direction to 3.

If it makes more sense to you, you could have the operation sign tell you which way to face and the sign on the second number tell you if you're walking forwards or backwards.  It's the same either way.

Please note a number without a sign is understood to be positive and can be viewed as added to 0.

Ex.  1 = (+1) = 0 + 1

So when I say 1 + 2, it's the same as saying 0 + (+1) + (+2).  This understanding may help in the next section.

Multiplication

The product of the absolute value tells you how many spaces you walk.  The signs tells you which direction you walk.

2 x 3 = 6
(+2) + (+2) + (+2) = (+6)
It might help to think of 0 as starting point.
0 + (+2) + (+2) + (+2) = (+6)
Facing positive direction, I walk forward 2 spaces, 3 times.

(-2) x (3) = (-6)
(-2) + (-2) + (-2) = (-6)
0 + (-2) + (-2) + (-2) = (-6)
Starting at 0, facing the negative direction (because the 2s are negative), I walk forward (addition) six spaces to -6.

If multiplying by a positive number is repeated addition, multiplying by a negative number can be thought of as repeated subtraction.

(+2) x (-3) = (-6)
0 - (+2) - (+2) - (-2) = (-6)
Starting at 0, I look in a positive direction (positive 2s) and walk backwards to -6.

(-2) x (-3) = (+6)
0 - (-2) - (-2) - (-2) = (+6)
Starting at 0, look negative, and walk backwards (subtraction) 2 spaces, three times to get to number six.

Please note: each time you multiply a negative, you change direction:

No negatives means you're facing positive moving forward.  Answer will be positive.

One negative you're either facing positive moving backwards or facing negative moving forwards.  Either way, answer will be negative.

Two negatives means the answer will cjange direction again.  You're facing negative, walking backwards.  Answer will be positive.

If you multiply 3 negative numbers, the answer will be negative.  You're basically multiplying 2 negative numbers to get a positive number amd then multiplying this new positive number by the third original negative number to get the total answer which will be negative.

If you are multiplying a string of numbers and you have an odd number of negative factors, your product will be negative. If you have an even number of negative factors, your product will be positive.

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