Multiplying Integers: Writing Multiplication as Addition or Subtraction

Starter: In your team, talk about why the sum of two negative numbers is negative, but the product of two negative numbers is positive; for example, show why 5(-3) = -15 and why -5(-3) = 15. Use paper and pencil if necessary.

Explore: How are multiplying and dividing integers similar to adding and subtracting integers?

1) Remember the rules for multiplying integers.
     a)  When the factors are in the same direction from zero, the                  product will be positive.
          2(3) = 6                -2(-3) = 6
     b)  When the factors are in opposite directions from zero, the                product will be negative.
         2(-3) = -6              -2(3) = -6
2) Remember why the rules work.
     a)  2(3) = 3+3 = 6                -2(-3) = -(-3)-(-3) = 6   
     b)  2(-3) = -3+(-3) = -6        -2(3) = -3-3 = -6
3) Remember that division is the inverse of multiplication, so the same rules apply:
     a)  When the divisor and dividend are in the same direction                    from zero, the quotient will be positive.
          6/2 = 3                -6/(-3) = 2 
     b)  When the divisor and dividend are in opposite directions                  from zero, the quotient will be negative.
          6/(-3) = -2              -6/2 = -3

Practice: 

In your teams, determine whether each product or quotient will be positive or negative, and then calculate the product or quotient. For multiplication exercises, re-write the problem as addition or subtraction to prove your products are correct, and decide how to prove your quotients are correct.

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