or in terms of a rotation (see National Research Council, 2001, Box 3-1). The associative property of multiplication can be illustrated by decomposing a 3-D array (or box shape built of blocks) in different ways (one way is shown in Figure 2-7). The distributive property is illustrated by viewing objects as grouped in two different ways (see National Research Council, 2001, Box 3-1).

The properties of multiplication can be illustrated with arrays and rectangles, and they are also visible in the multiplication tables, which contain many relationships and have important structure. One structural aspect of the multiplication tables is their diagonal symmetry. This diagonal symmetry corresponds with the commutative property of multiplication, namely that a × b = b × a for all numbers a and b. Recognizing this symmetry allows children to learn multiplication facts more efficiently. In other words, once they know the upper right-hand triangular portion of the multiplication tables in around third grade, they can fill in the rest of the table by reflecting across the diagonal (see Figure 2-10).

Patterns associated with horizontal or vertical shifts (slides) can also be seen in the multiplication tables. For example, the entries in two columns are related by the column that is associated with the amount of shift between the columns (see Figure 2-10). This structural relationship corresponds with the distributive property.

Once children encounter measurement situations, the possibility of fractions arises naturally. Fractions can be shown well in the context of

FIGURE 2-10 Symmetry and relationships in the multiplication table.