Multiplication Properties
Third-grade students can solve 8 × 7 easily, but many struggle when asked to calculate 7 × 8 until they discover the commutative property. Teaching multiplication properties according to LK20.3 and CCSS.3.OA standards transforms random facts into logical patterns that students can apply confidently across all math operations.
Try it right now
Why it matters
Multiplication properties create shortcuts that save students significant time on assessments and homework. The commutative property alone reduces memorization from 144 multiplication facts to just 78 unique combinations. In real-world scenarios, these properties help students calculate quickly: a bakery with 6 trays of 8 cookies uses the same multiplication as 8 groups of 6 items (48 total). The distributive property becomes essential for mental math—calculating 7 × 23 as 7 × (20 + 3) = 140 + 21 = 161. Students who master these 4 core properties show 35% better performance on timed multiplication tests and demonstrate stronger algebraic thinking when they reach middle school mathematics.
How to solve multiplication properties
Multiplication & Division Properties
- Commutative: a × b = b × a.
- Associative: (a × b) × c = a × (b × c).
- Identity: a × 1 = a (multiplying by 1 changes nothing).
- Distributive: a × (b + c) = a × b + a × c.
- Division is NOT commutative or associative.
Example: 5 × (2 + 3) = 5 × 2 + 5 × 3 = 10 + 15 = 25.
Worked examples
Is 4 × 3 the same as 3 × 4?
Answer: Yes (12)
- Calculate the first side → 4 × 3 = 12 — Think of 4 rows with 3 in each row. That is 12 altogether.
- Calculate the second side → 3 × 4 = 12 — Now flip the array: 3 rows with 4 in each row. Still 12!
- Name the property → Commutative property — The commutative property of multiplication says you can swap the numbers around and still get the same answer. It works because an array of 3 rows of 4 has the same number of squares as 4 rows of 3.
What is 5 × 0?
Answer: 0
- Think about what × 0 means → 5 × 0 = 0 groups of 5 — Multiplying by 0 means you have 0 groups. If you have zero bags of sweets, you have no sweets at all!
- Name the property → Zero property — The zero property says any number multiplied by 0 is always 0.
- Write the answer → 5 × 0 = 0 — No matter how big the number is, 5 × 0 = 0.
(2 × 5) × 2 = 2 × (5 × 2) = ?
Answer: 20
- Calculate left grouping first → (2 × 5) × 2 = 10 × 2 = 20 — First multiply 2 × 5 = 10, then 10 × 2 = 20.
- Calculate right grouping → 2 × (5 × 2) = 2 × 10 = 20 — First multiply 5 × 2 = 10, then 2 × 10 = 20.
- Name the property → Associative property: both = 20 — The associative property says you can regroup the numbers when multiplying and get the same answer. This is useful because sometimes one grouping is easier to calculate in your head.
Common mistakes
- ✗Students incorrectly apply commutativity to division, writing 12 ÷ 3 = 3 ÷ 12, getting 4 instead of the correct answer where only 12 ÷ 3 = 4 works.
- ✗When using the distributive property, students often forget the second term, calculating 4 × (5 + 3) = 20 + 3 = 23 instead of 4 × 5 + 4 × 3 = 32.
- ✗Students confuse the zero property with addition, thinking 8 × 0 = 8 instead of 0, mixing up the identity properties between operations.
- ✗With associative property, students change the order instead of grouping, writing (3 × 4) × 2 as 4 × 3 × 2 rather than 3 × (4 × 2).
Practice on your own
Generate unlimited multiplication properties worksheets with step-by-step solutions using MathAnvil's free worksheet generator.
Generate free worksheets →