Math Background

Commutative and Associative Properties
Developing the Concept

Once students feel comfortable in recognizing the patterns in the addition table, you can formalize two important properties. Students will use these properties again in multiplication. New names for the properties will be introduced.

Materials: chalkboard and chalk

Preparation: Poster identifying the Commutative Property for Addition, using numerical examples. Poster identifying the Associative Property for Addition, using numerical examples.

Prerequisite Skills and Concepts: Students should know how to add and subtract 2-digit numbers, with regrouping. Students should know the addition and subtraction fact families to 20.

Write the problems 54 + 67 and 67 + 54 on the board in vertical format.

  • We have been investigating patterns in the addition table. Today we will investigate these patterns with other numbers. Look at the problems on the board.
  • Ask: How are the problems alike? (They use the same addends.) How are the problems different? (The addends are in a different order.)
  • Say: I need two volunteers to find the sums on the board.
    Identify two volunteers and have them find the sums.
  • Ask: What do you notice about the sums?
    Students should recognize that the sums are equal.
  • Repeat with several other examples, inviting different volunteers to the board. If necessary, have students work independently at their desks to check the sums on the board.
  • Ask: We found a lot of sums. We used several examples. Why do you think that some sums were equal?
    Students should recognize that the sums were the same when the addends were reversed.
  • Say: The pattern that you have just investigated is called the Commutative Property or Order Property. It states that when the order of the addends is changed, the sum remains the same. Look at this poster.
    Display the poster of the Commutative Property. Have students copy the information on the poster at their desks and write several examples to “prove” that it is correct.
  • Ask: Do you think this property will work for 3-digit addends?
    Invite volunteers to show the class that the property still holds for 3-digit addends.
  • Ask: Do you think this property will work with subtraction?
    Allow ample time for students to think about the question before calling on volunteers. Students will probably say that you cannot subtract a greater number from a lesser number, so the property will not work for subtraction.
  • Ask: How can the Commutative Property help you solve problems?
    Guide students to realize that the Commutative Property is helpful in changing the order of addends. Students may feel it is easier to add 3 to 45 by counting on from 45, instead of adding 45 to 3.
  • Say: Now we will investigate another pattern. Find the sum of 37, 52, and 8.
    Ask several students to share their sums. Wait until the class agrees that the correct sum is 97.
  • Ask: How did you reach 97?
    Allow students to share their responses. Write their strategies and methods on the board during the discussion.
  • Ask: Which method seems easiest?
    Students may feel it is easiest to add 52 and 8 for a sum of 60, then add 60 and 37. This method can be done mentally.
  • Continue with other examples like the following: 73 + 14 + 6, 42 + 8 + 29, or 19 + 25 + 5. Have students orally explain how they found the sum. You may wish to point out that addition is binary, meaning that only two numbers can be added at any one time. Arrange the written responses in an organized fashion on the board.
  • Ask: Did you notice that the way you added the numbers made the sum easier to find? This is a special pattern called the Associative Property or Grouping Property. The wordassociative comes from the word associate, which means “partner” or “friend.” Associates are people who work together. In the Associative Property, numbers are combined by using parentheses. Look at this poster.
    Display the poster of the Associative Property. Have students copy the information on the poster at their desks and write several examples to “prove” that it is correct.
  • Ask: How can the Associative Property help you add numbers?
    Guide students to realize that using both properties allows them to choose which addends to add first. This allows sums to be found mentally. Spend time emphasizing the use of parentheses to show which addends are added first.

Wrap-Up and Assessment Hints
Students need practice with the Commutative and Associative Properties in order to realize how powerful they are for mental math. As you assess students, find out how they are thinking and select a strategy that builds on what they already know. Encourage students to use these patterns and relationships while solving word problems.

Houghton Mifflin Math Grade 3