Math Background

Lesson: Adding and Subtracting With Negative Numbers
Developing the Concept

Materials: Large number line up front for the class to see; student number lines

  • Say: We've been learning how to add and subtract with positive and negative numbers on a number line. Do the following additions on your number line: -3 + (-4); -6 + (-3); -2 + (-4).
  • Ask: Is there anything that is the same about all of the answers?
    Students will respond that all the answers are negative.
  • Ask: Do you think that when we add two negative numbers, the sum will always be negative?
    The answer is yes.
    Have students give some examples to support their answers, such as walking backward 3 steps, then backward 4 steps more, or borrowing 5 dollars and then borrowing 3 dollars more.
  • Ask: When we add two positive numbers, do we always get a positive number? (Yes)
    If students ask about the sign of the sum when adding a positive and a negative number, you can tell them that it will vary depending on the numbers being added. If your students understand the basic concepts, point out that the number which is farther from zero determines the sign of the answer.
  • Ask: If the sum of two negative numbers is -8, what could those two numbers be?
    If the sum of a positive and a negative number is -3, what could those two numbers be?
    If the sum of a positive and a negative number is +4, what could those two numbers be?
    A good homework project would be to find all the pairs of numbers between -8 and 8 whose sum is -3.
  • Ask: How does the sum of 5 + (-9) compare to the sum of -9 + 5?
    How does the sum of 6 + (-4) compare to the sum of -4 + 6?
    What property do these illustrate?
    Make sure that students recognize that the Commutative Property of Addition is true for integers. Similarly, emphasize that the Associative Property of Addition is true for integers.
  • Have students do the following subtraction problems by using their number lines.
     -7 − (+2) and -7 + (-2)
     -3 − (+4) and -3 + (-4)
     -6 − (+1) and -6 + (-1)
    and
     -5 − (+4) and -5 + (-4).
  • Ask: What was true about the answers for each pair of problems?
    It should be stated by students or by you that subtracting a positive number from a negative number always gives a difference that is negative and that subtracting a positive number is like adding the opposite of that number.
  • Ask: If the difference between a positive number and a negative number is -7, what could those two numbers be?
    (Possible answers: -5 − (+2); -6 − (+1); -3 − (+4))
    If the difference between a positive and a negative number is -4, what could those two numbers be?
    (Possible answers: -3 − (+1); -2 − (+2))

Wrap-Up and Assessment Hints
Students should be allowed to create and use number lines during assessment. You may want to encourage more advanced students to practice adding and subtracting with negative numbers without using a number line. You might also give students problems with negative numbers that test their understanding of the properties of addition.

Houghton Mifflin Math Grade 5