Properties of multiplication

The properties of multiplication are five rules that are always true. They make the times tables smaller to learn and big multiplications easy to break up.

The five properties at a glance

Commutative property: the turn-around rule

You can swap the factors and the product stays the same: 3 × 8 = 8 × 3 = 24. Turn an array on its side and you see why: the same dots, counted a different way.

Why it matters: there are 100 facts from 1 × 1 to 10 × 10, but because of this rule only about 55 are different. Once your child knows 4 × 9, they know 9 × 4.

Associative property: the grouping rule

With three numbers, you can multiply any two first: (2 × 5) × 7 = 2 × (5 × 7). Both are 70. Pick the pair that is easiest: 2 × 5 = 10, then 10 × 7 = 70.

It also explains tricks: 4 × 6 = (2 × 2) × 6 = 2 × (2 × 6), which is “double, then double again”, the trick for the 4 times table.

Distributive property: the splitting rule

You can split one factor into parts, multiply each part, and add the answers: 6 × 7 = 6 × 5 + 6 × 2 = 30 + 12 = 42. This is the most useful property of all. It turns hard facts into easy ones and is how children multiply big numbers: 23 × 4 = 20 × 4 + 3 × 4 = 80 + 12 = 92.

Try it with the array splitter: a red line cuts the array into two easier arrays.

In grades 6 and 7 the same rule works with letters: 3(x + 4) = 3x + 12. See the distributive property with variables.

Identity property (× 1) and zero property (× 0)

Times 1: one group of 8 is just 8, so 8 × 1 = 8. Times 0: 8 groups of nothing is nothing, so 8 × 0 = 0. These two are easy to mix up with addition (8 + 0 = 8), so they have their own lesson: multiplying by 0 and 1.

Using the properties together

Real calculations often use two or three properties at once. Here are three to try with your child:

• 25 × 7 × 4: turn it around to 25 × 4 × 7, then group: (25 × 4) × 7 = 100 × 7 = 700.
• 8 × 27: split 27 into 20 and 7: 8 × 20 + 8 × 7 = 160 + 56 = 216.
• 5 × 13 × 2 × 0: one factor is 0, so the answer is 0 without any other work.

Asking “which rule makes this easier?” before calculating is a habit that pays off all the way to algebra.

Where this fits: the first steps in multiplication

1. What is multiplication
2. Repeated addition
3. Arrays
4. Multiplication properties (this page)
5. Multiplying by 0 and 1
6. Multiplication strategies
7. The times tables
8. Practice the facts

What to say to your child

More worked examples

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

1. Which property: 5 × 9 = 9 × 5?
2. Which property: 7 × 1 = 7?
3. Fill in: 6 × 8 = 6 × 5 + 6 × __
4. Work out 4 × 2 × 5 the easy way.
5. Which property: (3 × 2) × 4 = 3 × (2 × 4)?
6. Use splitting: 9 × 7
7. 12 × 0
8. Use splitting: 13 × 4

Common mistakes

How to teach it at home

Lego arrays

Build a 4 × 6 rectangle of bricks and split it into 4 × 5 and 4 × 1.

Spot the easy pair

Give three numbers like 2, 7, 5 and ask which pair to multiply first.

Turn-around race

Say a fact; your child says its turn-around and the product.

By grade

• Grade 3 (3.OA.B.5): apply the commutative, associative and distributive properties as strategies.
• Grade 4: use the distributive property to multiply multi-digit numbers.
• Grade 6: write the properties with letters, for example a × b = b × a, and expand brackets like 3(x + 4) = 3x + 12 (see the distributive property with variables).

More for each grade: Grade 3 math and the CBSE class 3 maths guide; Grade 4 math and the CBSE class 4 maths guide.

Worksheets and practice

Multiplication properties worksheet: printable PDF with an answer key. · Make a fresh test

Multiplication practice · times tables · multiplication chart