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Question from Brett, a parent:

Someone asked a question about multiplication and division of two negative numbers yielding a positive result here:
http://mathcentral.uregina.ca/qq/database/QQ.09.99/butler1.html I was not fully happy with the explanation b/c I want to give me daughter a real-world example and I can't seem to find one.

The following illustrates why multiplying negative numbers has become difficult to explain:

2 X 2 = 4

----(-4)---(-2)---0---2---4
In this example we start with 2 and then want 2 more of them.
When we move across the number line from 2 to our answer, which is four, we have moved only 2 units to the right.

-2 X -2 = 4

----(-4)---(-2)---0---2---4
In this example we start with -2 and then want -2 more of them.
When we move across the number line from -2 to our answer, which is four, we have moved 6 units to the right.

How can the phenomenon of multiplying two negative numbers being more powerful than multiplying two positive numbers be explained?

Thank you,

-Brett

Hi Brett.

Let's say I have a box of wild strawberries for lunch and every minute I am eating two strawberries (I guess I'm a slow eater). I've been eating for awhile and at 12:30, I count how many strawberries are left: there are 30 left.

How many strawberries are left at 12:35? I am removing 2 per minute, so that means (-2 strawberries per minute) times 5 minutes: -2 x 5 = -10. So I have ten less than I had at 12:30; in other words, twenty left.

How many strawberries did I have at 12:29? That was one minute BEFORE I did my total, so I have (-2 strawberries per minute) times -1 minutes: -2 x -1 = +2. So I have two MORE than I had at 12:30; in other words, 30+2 = 32 left.

What about at 12:15? That's fifteen minutes before my total, so it is -15 minutes. -15 x -2 = +30. So I had 30 + 30 = 60 strawberries then.

I could also use this idea to figure out when I started eating if I know how many strawberries I started with. If I started with 70 strawberries, then I ate 40 of them before 12:30 (that is 70 - 30). So I can divide: 40 strawberries divided by (-2 strawberries per minute) = -20 minutes. So I started eating 20 minutes before I counted the strawberries at 12:30, that is, I started eating at 12:10.

A "number" (in the everyday use of the word) consists of two parts: a magnitude (size) and a sign (direction). Multiplying anything by a negative amount flips the sign and reverses the direction. As you
indicates, this is often seen as a powerful effect on the first number.

However, it doesn't have to be seen that way. The sign sometimes doesn't matter so much. For example, if you are in a canoe and your fellow paddler isn't sitting in the middle, you don't really care whether she is sitting 10 cm right of center (+10) or 10 cm left of center (-10). But if she is 30 cm away from the center in either direction, you'll both be wet. Sometimes the sign doesn't matter, but just the magnitude. So in this case multiplying by a negative (switching sides) isn't really more powerful than multiplying by a
positive.

Hope these example helps,
Stephen.

 

Brett,

This is a difficult question and I'm not sure I can be of any help. I think the real question is "Can you think of a situation outside of school where you would need to multiply a negative by a negative?" Our basic concept of multiplication is either pooling, putting together 3 collections of 5 objects each to obtain 15 objects, or repeated addition, adding 5 objects to a box 3 times. Stephen's example builds on this concept of multiplication being repeated addition.

In the example Chris develops in http://mathcentral.uregina.ca/qq/database/QQ.09.99/butler1.html there is a different model of multiplication. The objects are directed line segments (Chris calls them arrows). Each arrow has a number attached to it. The size of the number gives the length of the arrow and the sign is positive if the arrow points to the right and negative if the arrow points to the left. For example 2 is a arrow which is 2 units long and points to the right. You can multiply an arrow by a number, for example 3 × 2 and you get the arrow 6 which is 6 units long and points to the right. If you multiply by a negative number the arrow reverses direction so -5 × 2 = -10 , the arrow that is 10 units long and points to the left. Also 5 × -2 = -10 and -5 × -2 = 10 (-2 points to the left and multiplying by -5 reverses its direction).

This might sound very abstract but it's not really. Suppose you have a car moving along a straight road. Attached to the car is an arrow. The length of the arrow tells you the speed at which the car is moving and the arrow points in the direction the car is moving. We represent the arrow by a number whose size is the speed at which the car is moving and the sign tell us the direction, positive if the car is moving to the right and negative if the car is moving to the left. Notice that I haven't said anything about where the car is on the road (its position), I am only interested in its speed and direction.

I hope this helps,
Harley

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