How to Teach the Properties of Operations
"The commutative property of multiplication states that..." — and your child's eyes glaze over. But these properties are not abstract rules. They describe things your child already knows intuitively. The goal is to name what they already do.
Commutative property: order does not matter
Addition: 3 + 5 = 5 + 3 Multiplication: 4 × 7 = 7 × 4
Your child already knows this. "Three plus five is the same as five plus three." The property just gives this fact a name.
Does NOT work for: subtraction (5 - 3 ≠ 3 - 5) or division (12 ÷ 4 ≠ 4 ÷ 12).
Associative property: grouping does not matter
Addition: (2 + 3) + 4 = 2 + (3 + 4) = 9 either way Multiplication: (2 × 3) × 4 = 2 × (3 × 4) = 24 either way
This means you can group numbers however you want when adding or multiplying. Useful for mental math: 17 + 8 + 3 is easier as 17 + (8 + 3) = 17 + 11 = 28.
Does NOT work for: subtraction or division.
Distributive property: multiply through parentheses
a × (b + c) = a × b + a × c
6 × (10 + 4) = 6 × 10 + 6 × 4 = 60 + 24 = 84
This is the property behind multi-digit multiplication, the area model, and mental math strategies.
Identity property: adding 0 or multiplying by 1 changes nothing
Addition: 5 + 0 = 5 (zero is the additive identity) Multiplication: 5 × 1 = 5 (one is the multiplicative identity)
Inverse property: undo an operation
Addition: 5 + (-5) = 0 (a number plus its opposite equals zero) Multiplication: 5 × (1/5) = 1 (a number times its reciprocal equals one)
This connects to fact families and equation solving.
Key Insight: Do not start by teaching property names. Start by observing: "Hey, does 4 × 7 give the same answer as 7 × 4? Try a few more. Does this always work?" Let your child discover the pattern, then give it a name. Discovery first, vocabulary second.
Why these properties matter
They are not just vocabulary words for tests. They are tools:
- Commutative: Rearrange numbers to make computation easier (1 + 9 + 3 → 1 + 3 + 9 → 4 + 9 = 13? Or 1 + 9 + 3 → 10 + 3 = 13. Easier.)
- Associative: Regroup to create friendly numbers
- Distributive: Break hard multiplications into easy ones
- Inverse: Solve equations by "undoing" operations
Common mistakes
Applying commutative property to subtraction: 7 - 3 ≠ 3 - 7. This property only works for addition and multiplication.
Memorizing names without understanding: They can say "commutative" but cannot explain what it means. Always connect to examples.
Thinking the distributive property means "multiply everything": 2 × (3 × 4) ≠ (2 × 3) × (2 × 4). Distribution works over addition, not over multiplication.
The properties of operations are names for things your child already does: swapping the order of addends, regrouping numbers, and breaking apart multiplications. Teach through discovery and examples, then attach the vocabulary. When your child uses these properties naturally in mental math, they have internalized the structure of arithmetic.
If you want a system that builds these properties into computation practice — so your child uses them fluently, not just memorizes them — that is what Lumastery does.