The Distributive Property

The statement

Write the gift-bag total the two ways you counted it:

Two words from the properties lesson make this easier to talk about. The numbers being added inside the parentheses, $10$ and $3$, are the terms of that sum. The $6$ multiplying them is a factor, and since it stands outside the parentheses we can call it the outside factor. The rule is that the outside factor multiplies every term, and then you add the products.

Here is the same sentence in letters, so that you can put your own numbers in:

In it, $a(b + c)$ means $a$ times the quantity $(b + c)$, and $ab$ is shorthand for $a \times b$. The outside factor $a$ is distributed across the terms $b$ and $c$, which is where the property gets its name. In this lesson the letters stand for whole numbers, and later on the same rule will hold for fractions, decimals, and negative numbers too.

Why it is true: the area of a split rectangle

The reason the rule holds is a picture you can measure. One word first. The area of a shape is the number of squares of side $1$ needed to tile it, with no gaps and no overlaps. Tile a rectangle $3$ tall and $9$ wide that way and it holds $3$ rows with $9$ squares in each row. That is why its area is the height times the width, $3 \times 9 = 27$. Counting rows of squares is all the geometry this argument needs; a later chapter takes area up in its own right.

Now cut that rectangle’s width into $4$ and $5$. Nothing has been added or taken away, so the area is still $27$. It now sits in two pieces: a $3$ by $4$ piece of area $12$, and a $3$ by $5$ piece of area $15$. And $12 + 15 = 27$.

Nothing in that argument depended on the numbers $3$, $4$, and $5$. Put the same picture in letters. A rectangle $a$ tall and $b + c$ wide has area $a(b + c)$. One vertical cut splits the width into a piece of length $b$ and a piece of length $c$. That gives an $a$ by $b$ piece of area $ab$, and an $a$ by $c$ piece of area $ac$. The whole is still the sum of its parts:

A rectangle’s area does not care where you chop its width, so multiplying the whole width at once must equal multiplying the pieces and adding.

More than two terms is no different. Cut the width twice instead of once, and the outside factor reaches all three pieces:

Checking straight through, $3 \times 16 = 48$ as well.

In the picture above the cut is drawn for you. In the rectangle below you set the width and the height, and the readout does the cutting. It splits the width down the middle and reports both piece areas beside the whole.

Set the height to $4$ and the width to $7$. Then change the width to $9$, and after that raise the height. Each time, add the two piece areas and compare that total with the area of the whole rectangle. They match every time, because $a(b + c)$ and $ab + ac$ count the same unit squares.

Distributing over subtraction

The rule works the same way when the parentheses hold a difference instead of a sum:

For now, keep the first number at least as large as the one being subtracted, so that the difference inside is not negative.

Picture a rectangle of width $b$ with a strip of width $c$ removed from the end. The full $a$ by $b$ area is $ab$, the removed $a$ by $c$ strip is $ac$, and what is left is $ab - ac$. For example, with $a = 4$, $b = 8$, and $c = 3$:

Again both routes agree. The outside factor still reaches every term inside; only the sign joining the two products changes from a plus to a minus.

Mental math: distribution is the trick behind it

Distribution turns one hard multiplication into two easy ones. Split a number that is awkward to multiply into a friendly sum or difference, usually around a round number like $10$ or $100$, then distribute.

Take $6 \times 13$. The number $13$ is just $10 + 3$, and multiplying by $10$ or by $3$ is easy:

That is the same $78$ from the opening. This split is exactly how people multiply in their heads without realizing they are using a named property. When a number sits just below a round one, split it as a difference instead:

Every split gives the right answer, so choose the one that leaves you the easiest work. You want two products you can do at a glance, and an addition or subtraction you can finish in your head.

Factoring: running the rule backward

Read the rule from right to left and it says $ab + ac = a(b + c)$. Start with two products that plainly share a factor:

The shared $4$ has moved outside a single pair of parentheses. Pulling out a shared factor like that is called factoring. It is the distributive property traveled in the opposite direction, so each move undoes the other.

Usually the shared factor is hidden, because the terms are written as plain numbers. To factor $12 + 18$, look for the largest number that divides both terms. That number is called their greatest common factor.

Finding it does not take a special technique yet. When both terms are above zero, no shared factor can be bigger than the smaller term. So start at that term and count downward, stopping at the first number that divides both. For $12 + 18$ the smaller term is $12$, and $12$ itself does not divide $18$, nor does $11$, $10$, $9$, $8$ or $7$. Then $6$ divides $12$ and divides $18$, so the search stops there and $6$ is the greatest common factor. Counting down works because it reaches the largest candidate first. For numbers too big to search this way, the Factors and Multiples chapter later gives a faster method.

Both $12$ and $18$ are divisible by $6$, and nothing larger divides both, so rewrite each term and take the $6$ out front:

You can always check a factoring by distributing back: $6(2 + 3) = 12 + 18$, which matches. Pull out the greatest common factor, not just any common factor. Taking out only $3$ would give $3(4 + 6)$, which is true but not finished, because $4 + 6$ still share a factor of $2$.

Common mistakes