Math Core

Is 3 groups of 5 the same as 5 groups of 3? The groups look different, but the totals are the same. This handy fact is called the commutative property, and it cuts the number of facts you need to learn almost in half.

Turn the array

Look at this array. It has 2 rows of 4.

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That is $2 \times 4 = 8$ .

Now turn the array on its side. It has 4 rows of 2.

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That is $4 \times 2 = 8$ .

We did not add or take away any squares. We only turned the array. So the total stays the same:

2 \times 4 = 4 \times 2

The commutative property of multiplication

You can multiply two numbers in either order. The product is the same.

3 \times 5 = 5 \times 3

Both equal 15.

Check it with skip counting

$3 \times 5$ : count by fives 3 times: 5, 10, 15.
$5 \times 3$ : count by threes 5 times: 3, 6, 9, 12, 15.

Same product! The groups are different, but the total is the same.

Why it helps

If you know one fact, you know its turn-around fact too. Know $2 \times 9 = 18$ ? Then you also know $9 \times 2 = 18$ .

It also lets you pick the easier way to count. $8 \times 2$ means counting by twos 8 times. But $2 \times 8$ is just $8 + 8 = 16$ . Faster!

Worked example: Use a fact you know

You know that $5 \times 6 = 30$ . What is $6 \times 5$ ?

By the commutative property, $6 \times 5 = 5 \times 6$ .

So $6 \times 5 = 30$ .

Worked example: Pick the easier order

Find $7 \times 2$ .

Counting by twos 7 times works: 2, 4, 6, 8, 10, 12, 14.

But you can switch the order: $7 \times 2 = 2 \times 7$ . Two groups of 7 is $7 + 7 = 14$ .

Either way, $7 \times 2 = 14$ .

Worked example: Fill in the blank

Find the missing number: $4 \times 9 = 9 \times \underline{\quad}$ .

The same two factors must be on both sides, just in a different order. One side has 4 and 9. The other side has 9, so the missing factor is 4.

$4 \times 9 = 9 \times 4$ .

Common mistake

Switching the order keeps the total the same, but the groups change. 3 bags of 5 apples is not the same picture as 5 bags of 3 apples. Both have 15 apples, though!

Also, this only works for multiplication (and addition). It does not work for subtraction: $5 - 3$ is 2, but $3 - 5$ is not 2.

Tip

Stuck on a fact? Try turning it around. The turn-around fact might be one you already know.

Practice

Practice 1

$3 \times 4 = 12$ . What is $4 \times 3$ ?

Type a number.

Practice 2

Find the missing number: $6 \times 7 = 7 \times \underline{\quad}$ .

Type a number.

Practice 3

Which is the turn-around fact for $2 \times 8 = 16$ ?

Practice 4

This array shows $3 \times 6$ . What does it show when you turn it on its side?

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Practice 5

Find $10 \times 2$ . Use the easier order if you like.

Type a number.

Practice 6

Which is true?

Practice 7

Tom has 3 boxes with 8 crayons in each. Mia has 8 boxes with 3 crayons in each. How many crayons does Mia have?

Type a number.

Practice 8

Which does not have a product of 18?