Properties of operations chart
The properties of operations are the rules that let you add or multiply in a different order or grouping and still get the same answer. This chart lists all eight, each with what it says and an example in numbers.
Free printable PDF, 1st grade to 6th grade.
One page, black and white, prints the same every time.
What is on the sheet
| Property | What it says | Example |
|---|---|---|
| Commutative property of addition | You can add in any order. | 3 + 5 = 5 + 3 |
| Commutative property of multiplication | You can multiply in any order. | 4 × 6 = 6 × 4 |
| Associative property of addition | You can group the addends any way. | (2 + 3) + 4 = 2 + (3 + 4) |
| Associative property of multiplication | You can group the factors any way. | (2 × 3) × 5 = 2 × (3 × 5) |
| Identity property of addition | Adding 0 leaves a number the same. | 9 + 0 = 9 |
| Identity property of multiplication | Multiplying by 1 leaves a number the same. | 9 × 1 = 9 |
| Zero property of multiplication | Multiplying by 0 always gives 0. | 9 × 0 = 0 |
| Distributive property | Multiplying a sum is the same as multiplying each part and adding. | 3 × (4 + 2) = 3 × 4 + 3 × 2 |
Why the properties help
Each property is a shortcut a child already half knows. The commutative property means 9 + 2 can be counted on from 9, and that learning 3 × 8 also gives 8 × 3. The associative property lets a child regroup 4 × 7 × 5 as 7 × (4 × 5), which is 7 × 20. The distributive property splits a hard fact into two easy ones: 7 × 8 is 7 × 5 plus 7 × 3, so 35 + 21, which is 56.
Subtraction and division have none of the first four. 5 − 3 is not 3 − 5, and 12 ÷ 2 is not 2 ÷ 12, which is a good question to ask once a child knows the chart.
When each one is taught
First grade uses the commutative and associative properties to add (CCSS 1.OA.B.3). Third grade uses them, and the distributive property, to multiply (3.OA.B.5). Sixth grade writes them with letters, as in a × (b + c) = a × b + a × c, and uses them to show that two expressions are equal (6.EE.A.3 and 6.EE.A.4).
Common questions
- What is the difference between commutative and associative?
- Commutative is about order: the numbers swap places. Associative is about grouping: the numbers stay in order and the parentheses move.
- Does the distributive property work with subtraction?
- Yes. 6 × (10 − 1) = 6 × 10 − 6 × 1, which is 60 − 6, or 54. That is the quick way to work out 6 × 9.