Multiplying and Dividing With Negative Integers

You've begun connecting multiplying and dividing negative integers to what your students already know. Now, you will have your students investigate multiplying two negative numbers through the use of patterns and summarize all the rules for multiplying and dividing with negative integers.

Materials: two large posters which will be used to summarize the rules for multiplying and dividing with negative integers.

• Post the following table for students to consider:

  3 4	=	12
  2 4	=	8
  1 4	=	4
  0 4	=	0
  –1 4	=	–4
  –2 4	=	–8
  –3 4	=	–12

• Ask: What is happening in the table above as we go down from one row to the next?
  As the number being multiplied by 4 decreases by 1, the product decreases by 4.

• Now, post this table:

  3		(–4)	=	–12
  2		(–4)	=	–8
  1		(–4)	=	–4
  0		(–4)	=	0

• Ask: Look at the table. What is changing in each row?
  Students should say that the number being multiplied by ^–4 is decreasing by one in each row and that the product is increasing by 4 each time.

• Ask: Who can tell me what the next row should be?
  Students will probably say ^–1 (^–4) = 4. Have students write down the next two rows in the pattern.
  ^–2 (^–4) =  8
  ^–3 (^–4) = 12

• Ask: In the above examples, when you multiplied a negative integer by a negative integer, what was the sign of the product? (positive)
  Who can state a rule about multiplying two negative integers?
  Students should say, "The product of two negative integers is a positive integer."

• Say: Let's summarize the rules for multiplying with negative integers.
  

• Say: Fill in a large poster with a chart similar to the one below stating the rules for multiplying integers.
  
  

• Say: Let's fill in a chart for dividing two integers.
  Begin to fill in the chart below by looking back at the multiplication chart.
  

  Wrap-Up and Assessment Hints
  Understanding that addition and subtraction are inverse operations and that multiplication and division are inverse operations are important concepts in justifying the rules for operations with negative integers. When doing subtraction, adding the opposite of the number to be subtracted is the same thing as going in the opposite direction of the number being subtracted. The rules for dividing with negative integers are basically the same as for multiplying with negative integers. Namely, if the two numbers being divided (or multiplied) are of the same sign, the quotient (or product) is positive. If the two numbers being divided (or multiplied) are opposite in sign, the quotient (or product) is negative.