Multiplying and Dividing with Negative Numbers

You've begun connecting multiplying and dividing negative numbers 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 numbers.

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

• 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 multiplying -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 number times a negative number, what was the sign of the product? (Positive)
  Who could state a rule about multiplying two negative numbers?
  Students should say, "The product of two negative numbers is a positive number."
• Say: Let's summarize the rules for multiplying with negative numbers.
  
• Say: Fill in a large poster with a chart similar to the one below stating the rules for multiplying integers.
  

  First Factor	Second Factor	Product
  Positive	Positive	Positive
  Positive	Negative	Negative
  Negative	Positive	Negative
  Negative	Negative	Positive
  Summary Table for Multiplying Positive and Negative Numbers

• 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.

  Dividend	Divisor	Quotient
  Positive	Positive	Positive
  Positive	Negative	Negative
  Negative	Positive	Negative
  Negative	Negative	Positive

  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 numbers. 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 numbers are basically the same as for multiplying with negative numbers. Namely, if the two numbers being divided (or multiplied) are of the same sign, the quotient (product) is positive. If the two numbers being divided (multiplied) are opposite in sign, the quotient (product) is negative.

• Introducing the Concept
• Developing the Concept

• Introducing the Concept
• Developing the Concept