Operations & Algebraic Thinking

Properties of Multiplication

Using the commutative, associative, and distributive properties as strategies.

Intermediate15 min lesson2 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Multiplication Facts Fluency

Rules That Make Multiplication Flexible

The strategies from the last lesson — reordering factors, breaking a fact apart — aren't just tricks. They work because multiplication follows three consistent rules, called properties, that hold true for every multiplication problem.

Definition — Commutative, Associative, and Distributive Properties

The commutative property says factors can be multiplied in any order without changing the product. The associative property says, when multiplying three or more factors, they can be grouped in any way without changing the product. The distributive property says a factor can be split into two parts, each multiplied separately and then added.

Worked Example — Using the Commutative Property

If 7 × 3 = 21, what is 3 × 7? By the commutative property, multiplying the same two factors in either order gives the same product: 3 × 7 = 21 too.

Worked Example — Using the Distributive Property

Find 7 × 8 by breaking 8 apart into 5 + 3. Multiply each part separately: 7 × 5 = 35 and 7 × 3 = 21. Add the two results: 35 + 21 = 56. So 7 × 8 = 56, found using only easier facts.

7 \times 8 = 7 \times (5 + 3) = (7 \times 5) + (7 \times 3) = 35 + 21 = 56

Tip

The distributive property is especially useful for a fact you don't know yet: break the harder factor into a 5 or a 10 plus a small leftover amount, since facts with 5 and 10 are usually the easiest to recall.

Common Mistakes

Splitting a factor using the distributive property but forgetting to add the two partial products back together at the end.
After multiplying each part separately, always add the two partial products — the distributive property isn't finished until that final addition happens.
Assuming the properties also apply to division, such as expecting 12 ÷ 4 to equal 4 ÷ 12.
The commutative property only applies to multiplication and addition. Division depends on the order of its numbers, so 12 ÷ 4 and 4 ÷ 12 are different values.

Key Takeaways

The commutative property lets factors be multiplied in any order.
The associative property lets three or more factors be grouped in any way when multiplying.
The distributive property splits a factor into parts, multiplies each part separately, and adds the results.

Summary

These three properties explain why multiplication strategies work and give you flexible ways to find any fact. The next unit turns from multiplication and division back to place value, using it to round numbers and add or subtract with confidence.

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