Commutative property of multiplication
Theory
The commutative property of multiplication is a Year 3 maths skill: you can multiply two numbers in either order and the product stays the same. An easy name for it is the turn-around fact. This page uses arrays to show why \(3 \times 7 = 7 \times 3\), and how turn-around facts make times tables quicker to learn. Curriculum: VC2M3N05 / AC9M3N04.
Multiplication has a special rule: the order of the two numbers does not change the answer. This is called the commutative property. An easier name for it is the turn-around fact.
An array is a set of dots or objects in equal rows. An array of 3 rows of 7 gives \(3 \times 7 = 21\). Turn that array a quarter turn and the rows become columns: now it shows 7 rows of 3, which is \(7 \times 3 = 21\). It is the same dots, only turned, so the total does not change.
Because the answer is the same, you can start with either number. That means every multiplication fact you learn comes with a free turn-around fact, so \(5 \times 7\) and \(7 \times 5\) are really one fact, not two.
The turn-around rule works for every multiplication. Swap the two numbers and the product does not change.
How to use a turn-around fact
- Write the multiplication. Note the two numbers being multiplied, such as \(7 \times 2\).
- Swap the order. Put the second number first to make the turn-around fact: \(7 \times 2 = 2 \times 7\).
- Keep the product. The answer is the same, so use whichever order is easier to count. Two 7s (\(7 + 7 = 14\)) is quicker than seven 2s.
| \(5\text{ rows of }6\) | \(=\) | \(5 \times 6 = 30\) |
| \(6\text{ rows of }5\) | \(=\) | \(6 \times 5 = 30\) |
So \(5 \times 6 = 6 \times 5\).
| \(9 \times 4\) | \(=\) | \(4 \times 9\) |
| \(9 \times 4\) | \(=\) | \(36\) |
It is the turn-around fact, so the product is the same, \(36\).
| \(3 \times 9\) | \(=\) | \(9 \times 3\) |
| \(\text{missing}\) | \(=\) | \(3\) |
The turn-around swaps the order, so the missing number is \(3\). Both sides equal \(27\).
| \(7 \times 2\) | \(=\) | \(2 \times 7\) |
| \(2 \times 7\) | \(=\) | \(7 + 7 = 14\) |
Counting two 7s is quicker, so \(7 \times 2 = 14\).
Common pitfalls
Frequently asked questions
What is the commutative property of multiplication?
The commutative property means you can multiply two numbers in either order and get the same product. For example, \(3 \times 7 = 21\) and \(7 \times 3 = 21\).
What is a turn-around fact?
A turn-around fact is the same multiplication with the two numbers swapped. \(5 \times 7\) and \(7 \times 5\) are turn-around facts, and both equal 35.
Does the order matter in multiplication?
No. In multiplication the order of the two numbers does not change the answer. An array of 3 rows of 7 has the same total as 7 rows of 3, just turned.
Does the commutative property work for division and subtraction?
No. It works for multiplication and addition, but not for subtraction or division. \(12 - 4\) is not the same as \(4 - 12\), and \(12 \div 4\) is not the same as \(4 \div 12\).
How do turn-around facts help with times tables?
Every fact you learn gives you its turn-around fact for free. If you know \(4 \times 8 = 32\), then you also know \(8 \times 4 = 32\), so you learn half as many facts.
Is 3 times 7 the same as 7 times 3?
Yes. Both equal 21. The picture is different, because \(3 \times 7\) shows 3 rows of 7 and \(7 \times 3\) shows 7 rows of 3, but the total number is the same.