Grade 2
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Introducing the Concept

Commutative and Associative Properties

When the children worked with addition facts last year, they recognized that fact families included both addition and subtraction facts. They also learned that by reversing the order of the addends, they could form related addition facts with the same sum. Revisit this pattern to help prepare children for addition and subtraction with regrouping.

Materials: large addition table showing sums from 0 to 20, child's hat and coat, highlighter

Preparation: Make a large addition table for the class to see. Addends should be from 0 to 10.

Prerequisite Skills and Concepts: Children should know basic addition facts to 20. See Grade 1: Addition and Subtraction Facts to 20.

Display the addition table for the class to see.

  • Say: Today we are going to investigate some patterns. I need a volunteer to put on this coat and then to put on this hat.
    Choose a volunteer. Hand the coat to the child to put on. Then hand the hat to the child to put on. Structure the scene to make sure children know that the coat is first and the hat is second.

  • Say: Notice how our volunteer looks, dressed in this coat and hat. Now I ask that my volunteer remove the coat and hat.
    Dramatically take the coat and hat from the child. Place the clothing on a chair.

  • Say: Now I'd like that same volunteer to put on this hat and then to put on this coat.
    Hand the hat to the child to put on. Hand the coat to the child to put on.

  • Ask: What do you notice about the way the volunteer looks, dressed in the hat and coat?
    Children should respond that the volunteer looks the same with the coat and hat as with the hat and coat.

  • Ask: Did you notice that our volunteer looked the same? Why was that? Would our volunteer look the same if I asked him/her to put on sneakers first, then socks? Or, would it look different if the socks were put on first, then the sneakers?

  • Lead children to realize that for certain items, order is important. Putting on a hat, then a coat results in the same appearance as putting on a coat, then a hat. However, putting on shoes, then socks is not the same as putting on socks, then shoes. Elicit from children other examples from everyday life that result in the same outcome, regardless of order. Then elicit examples that result in different outcomes when the order is reversed.

  • Ask: Look at the chart displayed in front of the room. How were the answers found?
    Children should recognize the chart as an addition table. Be sure they realize that the sums are found by adding the number at the top of the column to the number at the left of its row. Have children demonstrate with several examples. Emphasize the sums that are identical to the addends.

  • Ask: What special sums are highlighted on the diagonal?
    Highlight the diagonal with sums of 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20. Children should recognize that the sums are doubles.

  • Ask: Look at the sums of 8. Notice their locations. What patterns do you see with other sums? Children should recognize that the same sums are above the diagonal as below it.

  • Ask: What other number "matches" can you find?
    As children announce pairs of numbers with the same sum, write their corresponding equation on the board. Call upon as many children as possible.
    5 + 3 = 3 + 5
    6 + 1 = 1 + 6
    9 + 0 = 0 + 9

  • Ask: How can you describe the pattern in these sentences?
    Children should realize that the addends are reversed, but the sum remains the same.

  • Ask: Do you think the pattern will work for subtraction?
    Allow ample time for children to think about the question. Write their responses on the board without evaluation. Then help the class draw the conclusion that the numbers in subtraction cannot be reversed.
    Addition and Subtraction with Regrouping:

    Commutative and Associative Properties:

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