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Grade 6
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Introducing the Concept

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 times 4 mean as a repeated addition? (4 + 4 + 4)
  • Ask: Using that definition, what would 3 times (negative4) mean? [(negative4 + (negative4) + (negative4)]
    So what does 3 times (negative4) equal? (negative12)
  • Say: Do the following at your desks and then we'll compare answers: 6 times (negative2); 4 times (negative4); and 8 times (negative5).
    After comparing and writing the answers on the board, go over any questions they may have.
    Write the following on the board: 42 times 23 = 23 times 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 negative5 times 6 equal? Explain.
    Students should say that negative5 times 6 must equal 6 times (negative5) and 6 times (negative5) = negative30, since negative5 added sitimes times equals negative30. Place the following on the board for the students to do at their desks: negative7 times 8; negative5 times 3; and negative6 times 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 times 9 = 36, then we could also write 9 times 4 = 36, 36 divided by 9 = 4 and 36 divided by 4 = 9. If that is the case, then what multiplication and division sentences could you write for the multiplication sentence 8 times (negative4) = negative32?
    Hopefully, the students will write negative4 times 8 = negative32, negative32 divided by (negative4) = 8 and negative32 divided by 8 = negative4. If they have trouble, show them.
  • Say: Now write down the four sentences in each of the following families: 6 times (negative3); negative7 times 2; and negative5 times 4.
    Place each example on the board. Group the cases of dividing a negative number by a positive number together: negative32 divided by 8; negative18 divided by 6; negative14 divided by 2; and negative20 divided by 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.
         Adding and Subtracting
       with Negative Numbers
       Multiplying and Dividing
       with Negative Numbers

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