Math Core

What if you need the area of a rectangle and you don't know the fact right away? You can cut the rectangle into two smaller rectangles with easier facts, then add the areas. This is the distributive property, drawn as a picture.

Split one side

Here is a rectangle 3 units wide and 7 units long.

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The long side is 7 units. Split it into $5 + 2$ . Now there are two smaller rectangles:

The blue part is 3 by 5. Its area is $3 \times 5 = 15$ .
The yellow part is 3 by 2. Its area is $3 \times 2 = 6$ .

Add the parts: $15 + 6 = 21$ . The whole rectangle is 3 by 7, and $3 \times 7 = 21$ . Same answer!

3 \times 7 = 3 \times (5 + 2) = (3 \times 5) + (3 \times 2) = 15 + 6 = 21

A picture with side lengths

A rectangle 4 units wide and 9 units long, cut into a 4 by 5 part (left) and a 4 by 4 part (right).

To find $4 \times 9$ , split the 9 into $5 + 4$ .

\begin{aligned} 4 \times 9 &= (4 \times 5) + (4 \times 4) \\ &= 20 + 16 \\ &= 36 \end{aligned}

The area is 36 square units.

Break apart, then add

You can split one side of a rectangle into two parts. The area of the whole rectangle equals the sum of the areas of the two smaller rectangles.

3 \times (5 + 2) = (3 \times 5) + (3 \times 2)

Good ways to split: use 5s, 10s, or facts you already know well.

Worked example: Use a 5

Find the area of a rectangle that is 6 units by 8 units by splitting the 8.

Split 8 into $5 + 3$ .

$6 \times 5 = 30$
$6 \times 3 = 18$

$30 + 18 = 48$ . The area is 48 square units.

Worked example: Use a 10

A rectangle is 7 feet wide and 12 feet long. What is its area?

Split 12 into $10 + 2$ .

$7 \times 10 = 70$
$7 \times 2 = 14$

$70 + 14 = 84$ . The area is 84 square feet.

Worked example: Put two rectangles together

Two rectangles are both 4 units wide. One is 3 units long and one is 6 units long. They are pushed together side by side. What is the total area?

Add the areas: $(4 \times 3) + (4 \times 6) = 12 + 24 = 36$ .

Or join them first. Together they make one rectangle 4 wide and $3 + 6 = 9$ long, so the area is $4 \times 9 = 36$ . The total area is 36 square units.

Common mistake

Only split one side. Keep the other side the same in both small rectangles. For $6 \times 8$ , both parts must still be 6 wide: $(6 \times 5) + (6 \times 3)$ , not $(6 \times 5) + 3$ .

Tip

There is more than one way to split. For $6 \times 8$ , you could also use $(6 \times 4) + (6 \times 4) = 24 + 24 = 48$ . Every correct split gives the same area.

Practice

Practice 1

A rectangle is 6 by 7. Which shows a correct way to split it?

Practice 2

A rectangle is 4 units wide. Its long side is split into 5 units and 3 units. What is the total area, in square units?

Type a number.

Practice 3

Fill in the missing number: $7 \times 9 = (7 \times 5) + (7 \times \_\_)$ .

Type a number.

Practice 4

Find the area of a rectangle 8 units by 7 units. Split the 8 into $5 + 3$ . What is the area, in square units?

Type a number.

Practice 5

Two rectangles are each 3 units wide. One is 4 units long and the other is 6 units long. They are placed side by side to make one big rectangle. What is the area of the big rectangle, in square units?

Type a number.

Practice 6

Which one is not equal to $5 \times 8$ ?

Practice 7

Use a 10 to help. A wall is 6 feet tall and 13 feet long. What is its area, in square feet?

Type a number.

Practice 8

A patio is a rectangle 6 feet by 9 feet. A 6 by 5 part of it is red brick. The rest is gray stone. What is the area of the gray part, in square feet?

Type a number.