Math Core

What if you forget a fact like $6 \times 8$ ? You don't have to be stuck. You can break apart one factor into two easier numbers, solve two small facts, and add. This is called the distributive property.

Break apart an array

Here is an array with 6 rows of 8.

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Each row of 8 is split into 5 blue and 3 orange.

Blue part: 6 rows of 5 is $6 \times 5 = 30$ .
Orange part: 6 rows of 3 is $6 \times 3 = 18$ .
All together: $30 + 18 = 48$ .

So $6 \times 8 = 48$ . Splitting the array did not change the number of squares. It just made them easier to count.

Break apart a rectangle

You can draw the same idea as a rectangle. This one is 6 units tall and 8 units wide. A line splits the width into 5 and 3.

A 6 by 8 rectangle split into a 6 by 5 part (left) and a 6 by 3 part (right)

We write the break apart with parentheses. The parentheses show which numbers go together:

6 \times 8 = 6 \times (5 + 3) = (6 \times 5) + (6 \times 3) = 30 + 18 = 48

The distributive property

You can break one factor into two parts. Multiply the other factor by each part, then add the two answers.

6 \times (5 + 3) = (6 \times 5) + (6 \times 3)

Choosing how to break it apart

Break a factor into numbers whose facts you know well. Breaking off a 5 or a 2 is often a great choice.

$7 \times 6$ : break 7 into $5 + 2$ . $\;(5 \times 6) + (2 \times 6) = 30 + 12 = 42$ .
$4 \times 9$ : break 9 into $5 + 4$ . $\;(4 \times 5) + (4 \times 4) = 20 + 16 = 36$ .

Worked example: Break apart the 8

Find $7 \times 8$ by breaking 8 into $5 + 3$ .

7 \times 8 = (7 \times 5) + (7 \times 3) = 35 + 21 = 56

$7 \times 8 = 56$ .

Worked example: Break apart the 9

Find $3 \times 9$ by breaking 9 into $4 + 5$ .

3 \times 9 = (3 \times 4) + (3 \times 5) = 12 + 15 = 27

$3 \times 9 = 27$ .

Worked example: Numbers bigger than 10

Find $4 \times 12$ .

Break 12 into $10 + 2$ . Both of those facts are easy.

4 \times 12 = (4 \times 10) + (4 \times 2) = 40 + 8 = 48

$4 \times 12 = 48$ .

Common mistake

Multiply both parts. A common mistake is $6 \times (5 + 3) = 30 + 3$ . The 3 must be multiplied by 6 too: $6 \times 3 = 18$ , so the answer is $30 + 18 = 48$ .

Tip

Check your work by breaking apart a different way. For $6 \times 8$ , try $8 = 4 + 4$ : $24 + 24 = 48$ . Same answer!

Practice

Practice 1

To find $6 \times 7$ , Ana writes $(6 \times 5) + (6 \times 2)$ . What is $6 \times 2$ ?

Type a number.

Practice 2

Find $4 \times 8$ by breaking 8 into $5 + 3$ .

Type a number.

Practice 3

Which is equal to $8 \times 7$ ?

Practice 4

Find $6 \times 9$ by breaking 9 into $5 + 4$ .

Type a number.

Practice 5

Fill in the missing number: $5 \times 8 = (5 \times 5) + (5 \times \underline{\quad})$ .

Type a number.

Practice 6

Find $3 \times 12$ by breaking 12 into $10 + 2$ .

Type a number.

Practice 7

A garden has 7 rows of carrots. Each row has 9 carrots. Use the distributive property to find how many carrots there are.

Type a number.