## Lesson: Multiplying and Dividing With Negative Integers Introducing the Concept

In order to help your students understand and remember the rules for multiplying and dividing integers, you should connect the rules to three key ideas: (1) multiplication can be thought of as repeated addition; (2) multiplication is a commutative operation; and (3) division is the inverse operation of multiplication. Having students explore patterns with negative integers will also help them justify the rules.

Prerequisite Skills and Concepts: Students should be familiar with the idea that multiplication can be thought of as repeated addition and that division is the inverse operation of multiplication.

So what does 3 x 4 mean as a repeated addition? (4 + 4 + 4)
• Ask: Using that definition, what would 3 x (-4) mean? [(-4 ) + (-4) + (-4)]
So what does 3 x (-4) equal? (-12)
• Say: Do the following at your desks and then we'll compare answers: 6 x (-2); 4 x (-4); and 8 x (-5).
After comparing and writing the answers on the board, go over any questions students may have.
Write the following on the board:

42 x 23 = 23 x 42.

• Ask: What property does that illustrate? (Commutative)
And what does the Commutative Property tell us? (It doesn't matter what order we multiply the two factors, the products will be the same.)
• Ask: Since the Commutative Property also holds when multiplying negative integers, what would -5 x 6 equal? Explain.
Students should say that -5 x 6 must equal 6 x (-5) and 6 x (-5) = -30, since -5 added six times equals -30. Place the following on the board for the students to do at their desks: -7 x 8; -5 x 3; -6 x 9.

After comparing and writing the answers on the board and going over any questions, have students generalize about multiplying a positive integer by a negative integer in either order, negative times positive or positive times negative.

• Ask: We just multiplied a positive integer and a negative integer. Then we multiplied a negative integer by a positive integer. What was the sign of the product in each case?
The students should say that the answer was always negative. If they don't, point out to them that it was.
• Ask: What rule could we state about multiplying a positive integer and a negative integer?
When multiplying two numbers, one positive and the other negative, the product will be negative.
• Ask: What is the relationship between multiplication and division? (They are inverses of one another; that is, division “undoes” multiplication.)
• Say: Since 4 x 9 = 36, then we could also write 9 x 4 = 36, 36 ÷ 9 = 4, and 36 ÷ 4 = 9. If that is the case, then what multiplication and division sentences could you write for the multiplication sentence 8 x (-4) = -32?
Hopefully, students will write -4 x 8 = -32, -32 ÷ (-4) = 8 and -32 ÷ 8 = -4. If they have trouble, show them.
• Say: Now write down the four sentences in each of the following families: 6 x (-3); -7 x 2; and -5 x 4.
Place each example on the board. Group the cases of dividing a negative number by a positive number together: -32 ÷ 8; -18 ÷ 6; -14 ÷ 2; and -20 ÷ 4.
• Ask: In the problems listed, we divided a negative integer by a positive integer. What was the sign of the quotient? (negative)
Who can state a rule for dividing a negative integer by a positive integer?
Students should indicate that a negative integer divided by a positive integer is negative. Write down the rules for multiplying a positive and a negative, and for dividing a negative by a positive.