and of multiplication, the associative properties of addition and multiplication, and the distributive property of multiplication over addition. The commutative properties of addition and multiplication state that

The associative properties of addition and multiplication state that

The distributive property states that

Each property can be illustrated by moving and reorganizing objects, sometimes also by decomposing and recomposing a grouping, and sometimes even in terms of symmetry.

The report Adding It Up: Helping Children Learn Mathematics has a good discussion and an illustration of the commutative and associative properties of addition, the commutative and associative properties of multiplication, and the distributive property (National Research Council, 2001, Chapter 3 and Box 3-1). The commutative property of addition is illustrated by switching the order in which two sets are shown. The commutative property is especially useful in conjunction with counting on strategies for solving addition problems (see Chapter 5 for further discussion of children’s problem-solving strategies for addition and subtraction). For example, instead of counting on 6 from 2 to calculate 2 + 6, a child can switch the problem to 6 + 2 and count on 2 from 6. The associative property involves starting with three separate sets, two of which are close together, separating the two that are close together, and moving one of those sets to reassociate with the other set. The associative property of addition is used in make-a-ten methods, when one number is decomposed so that one of the pieces can be recomposed with another number to make a group of 10.

Early experiences with properties of addition then extend to multiplication in third and fourth grade. The commutative and associative properties of multiplication and the distributive property are essential to understanding relationships among basic multiplication facts and to understanding multidigit multiplication and division. For example, knowing that 3 × 5 = 5 × 3 and that 7 × 8 can be obtained by adding 5 × 8 and 2 × 8 lightens the load in learning the multiplication tables. The commutative property of multiplication is illustrated by decomposing a rectangular array in two different ways: by the rows or by the columns (as shown in Figure 2-6)



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