Multiplying and Dividing with Negative Numbers

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 addition. Having students explore patterns with negative numbers will also help them justify the rules.

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

• Ask: How is multiplication related to addition? (Multiplication is repeated addition.)
  So what does 3 4 mean as a repeated addition? (4 + 4 + 4)
  
• Ask: Using that definition, what would 3 (4) mean? [(4 + (4) + (4)]
  So what does 3 (4) equal? (12)
  
• Say: Do the following at your desks and then we'll compare answers: 6 (2); 4 (4); and 8 (5).
  After comparing and writing the answers on the board, go over any questions they may have.
  Write the following on the board: 42 23 = 23 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 numbers, what would 5 6 equal? Explain.
  Students should say that 5 6 must equal 6 (5) and 6 (5) = 30, since 5 added si times equals 30. Place the following on the board for the students to do at their desks: 7 8; 5 3; and 6 9.

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

• Ask: We just multiplied a positive number and a negative number. Then we multiplied a negative number times a positive number. 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 number and a negative number?
  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 9 = 36, then we could also write 9 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 (4) = 32?
  Hopefully, the students will write 4 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 (3); 7 2; and 5 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 number by a positive number. What was the sign of the quotient? (Negative)
  Who could state a rule for dividing a negative number by a positive number?
  Students should indicate that a negative number divided by a positive number is negative. Write down the rules for multiplying a positive and a negative, and for dividing a negative by a positive.

• Introducing the Concept
• Developing the Concept

• Introducing the Concept
• Developing the Concept