## Operations With Integers: When Students Ask

**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. Thus,^{-}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 is to use repeated addition. If^{+}3 x (^{-}2) means to add^{-}2 three times, then^{-}3 x (^{-}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.**How can you show multiplication of two negative integers?**

You can think of the negative sign as the factor^{-}1.

So^{-}3 x^{-}2 = (^{-}1)(3) x (^{-}1)(2)

= 6 x (^{-}1)(^{-}1)

= 6 x 1 = 6

Each factor of^{-}1 changes the sign of the answer.