Commutativity and associativity guarantee that all 12 ways of doing this sum give the same answer—so it does not matter which one I do. (For adding four numbers, there are 120 (!) conceivable different schemes, all of which again give the same result.) This flexibility is very useful when students do computations. For example, 1+8 can be found by thinking of it as 8+1 and then just recalling the next whole number after 8. The standard procedures for doing multidigit arithmetic also heavily exploit commutativity and associativity. However, the flexibility permitted by these rules also greatly increases the challenges of teaching arithmetic. When there are several ways to do a calculation, it is virtually certain that students will produce the answer more than one way. A teacher must therefore have a sufficiently flexible knowledge of arithmetic to evaluate the various student solutions, to validate the correct ones, and to correct errors productively.
The commutative and associative laws also hold for multiplication (see Box 3–1). The commutativity of multiplication by 2 is also reflected in the equivalence of the two definitions of even number typically offered by children. The “fair share” definition says that a number is even if it can be divided into two equal parts with nothing left over (which may be written as 2×m); the “pairing” definition says that a number is even if it can be divided into pairs with nothing left over (m×2).
In addition to these two laws for each operation, there is a rule, known as the distributive law, connecting the two operations. It can be written symbolically as a×(b+c)=a×b+a×c.
An example would be 2×(3+4)=2×7=14=6+8=2×3+2×4. A good way to visualize the distributive law is in terms of the area interpretation of multiplication. Then it says that if I have two rectangles of the same height, the sum of their areas is equal to the area of the rectangle gotten by joining the two rectangles into a single one of the same height but with a base equal to the sum of the bases of the two rectangles:
The basic properties of addition and multiplication of whole numbers are summarized in Box 3–1.