AUTORIZACIONES DE PROSPECCIÓN Y EXPLORACIÓN

The distributive law connects addition and multiplication. It states that for any whole numbers m, n, and , the following is valid:

2.4. The Distributive Law 45

Mathematicians love economy in notation (the use of notation was itself a response to the need for economy), so they devised a convention to eliminate the multiple parentheses:

In an expression such as mn + m, we always multiply the num- bers mn and m first before doing the addition.

With this understood, we can rewrite the above equality as follows. For any whole numbers , m, and n,

(2.6) m(n + ) = mn + m.

Because multiplication is commutative, this could equally well be written as

(n + )m = nm + m.

We remark that the notational convention of omitting the parentheses is just a convention; it has no mathematical substance. In school mathemat- ics, this convention goes under the heading of order of operations and, unfortunately, it sometimes assumes a central position in the curriculum. This is wrong because, while children need to learn to observe conventions, no convention of any kind should be elevated to a position of prominence in any discipline, least of all in mathematics.

We will need a geometric model for the distributive law when we come to fractions, so we initiate the discussion of this model for whole numbers. It requires the area model for multiplication. We begin with a definition of area that is adequate for the present need. A square with length 1 on each side is called a unit square. The area of the unit square is by definition equal to 1. We say a collection of rectangles {R_j} tile or pave a given rectangle R if, by combining the Rj’s together we get the whole rectangle

R, and if the Rj’s intersect at most along their boundaries. With all this terminology in place, the area of a general rectangle is by definition the number of unit squares required to pave that rectangle. (Remember that we are dealing only with whole numbers at this point and therefore the lengths of all rectangles are whole numbers.) For example, 3× 5 is the area of a rectangle with vertical length 3 and horizontal length 5 because there are three rows of five unit squares in the rectangle and the area of the latter is therefore 5 + 5 + 5 (unit squares):

The same reasoning shows:

The product mn (m and n being whole numbers) is the area of a rectangle with “vertical” length m and “horizontal” length n.

It may be mentioned that the area model of multiplication is in fact the mathematical underpinning of the base ten blocks manipulatives.

We now use the area model of multiplication to interpret the distributive law. This is best done through an explicit example. If m = 3, n = 2 and  = 4 in m(n + ) = mn + m, then 3(2 + 4) is the area of the following rectangle with “vertical” length 3 and “horizontal” length 6:

On the other hand, 3× 2 is the area of the “left” rectangle and 3 × 4 is the area of the “right” rectangle. Thus 3×(2+4) = (3×2)+(3×4). Again, the essence of this picture is unchanged when 2, 3, and 4 are replaced by other triples of numbers.

The distributive law generalizes to more than three numbers. For exam- ple,

m(a + b + c + d) = ma + mb + mc + md

for any whole numbers m, a, b, c, and d. This can be seen by applying the distributive law (equation (2.6)) twice and making liberal use of Theo- rem 2.1, as follows.

m(a + b + c + d) = m (a + b) + (c + d) = m(a + b) + m(c + d) = (ma + mb) + (mc + md) = ma + mb + mc + md.

Activity. One can use the distributive law to multiply a two-digit number by a one-digit number using mental math. For example, to compute 43× 6, we break up 43 into (40 + 3) so that 43×6 = (40+3)×6 = (40×6)+(3×6), and the last is just 240 + 18 = 258.

40 6

3

240 18

Therefore, 43× 6 = 258. Following this example, use mental math to compute: (a) 24× 8, (b) 53 × 7, (c) 39 × 6, (d) 79 × 5, (e) 94 × 9, (f) 47 × 8.

2.4. The Distributive Law 47

As a typical application of the distributive law, we bring closure to the assertion in section 1.7, All about Zero, on page 32 by providing an explanation for

N × 10k equals the whole number obtained from N by attaching

k zeros to the right of the last digit of N ,

where N is any positive whole number. It suffices to consider N = 372 and k = 4 in order to simplify the exposition, as the reasoning is completely general. Thus,

372× 104 = (3× 102) + (7× 101) + 2× 104

= (3× 102)× 104+ (7× 101)× 104+ (2× 104) (dist. law) = (3× 106) + (7× 105) + (2× 104) (Theorem 2.2, eq. (1.4)) = 3720000.

We have now made the formal acquaintance of the five laws of arithmetic, but did you notice that, with the exception of the distributive law, each of them is about one operation alone, be it + or ×? The distributive law is the only law that involves both + and×:

The distributive law is the glue that binds addition + and multiplication × together.

For example, if you have forgotten the definition of multiplication, the dis- tributive law can remind you that multiplication is repeated addition, e.g., 3× 7 = (1 + 1 + 1) × 7 = 7 + 7 + 7. Yet, despite its obvious importance, it seems to be the least understood of the five laws among students. This could be partly due to the lack of insistence by teachers that students learn it. Therefore, let us begin by convincing you that the distributive law is important. Then there may be a better chance for your students to learn this law.

The distributive law (equation (2.6)) says not only that “the left side of (equation (2.6)) is equal to the right side” (i.e., m(n + ) is equal to mn + m), but also that “the right side of (equation (2.6)) is equal to the left side” (i.e., mn + m is equal to m(n + )). In other words, while it is true that 35× (74 + 29) is equal to (35 × 74) + (35 × 29), it is equally, if not more important to recognize that (35× 74) + (35 × 29) is also equal to 35× (74 + 29). As a simple demonstration, consider the straightforward computation

(35× 74) + (35 × 29) = 2590 + 1015 = 3605.

However, if we use the distributive law, we can actually do this computation by mental math, as follows. We have (35× 74) + (35 × 29) = 35 × (74 + 29), but 74 + 29 = 103 = (100 + 3). So we get 35×(100+3) = 3500+105 = 3605.

But the main point of this remark is that knowing that mn+m is equal to m(n + ) could be the difference between success and failure in a logical argument. See, for example, Chapter 6 when we discuss the multiplication algorithm, and Chapter 29 when we discuss the multiplication of rational numbers. It may also be mentioned that, in algebra, this way of using the distributive law underlies the skill of “collecting like terms”.

Moral: Be sure you know that mn + m equals m(n + ). Activity. If a, b, c, d are whole numbers such that a + c = b + d = 11, what is ba + bc + da + dc?