## Operations with Negative Numbers

• What do you mean by a directed distance?
A directed distance is a distance in a particular direction. The numeral indicates how far to move; the positive or negative sign indicates which direction to move. Numbers on a number line can be thought of as directed distances. Thus, when adding integers, think of the second number as a directed distance to go from the first number.
• Why do we change the sign of the second number and add when we are subtracting integers?
By having students review subtraction involving two positive numbers on the number line, you can point out that +4 – +1 was equal to +3. They were at +4 on the number line and went the opposite direction of the directed distance +1, which is the same thing as doing +4 + (1). Similarly if your students were to do the subtraction 6 – (+4), they would start at 6 and go in the opposite direction of +4, which would be like adding 4. This, 6 – +4 is equivalent to 6 + 4.
• Why is the product of two negative numbers positive?
There are many ways to help students realize that the product of two negative numbers is positive. One way is to make use of a table of patterns. Another way to justify this is to use repeated addition. If +3 (2) means to add 2 three times, then 3 (2) means to subtract 2 three times. Thus, 0 – (2) – (2) – (2) = 2 + 2 + 2, since when subtracting signed numbers you change the sign of the number being subtracted and add.