The Associative Property
The Associative Property states that changing the grouping of addends or factors does not change the sum or product. In Grade 6 Saxon Math Course 1, (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). This means students can regroup numbers to make mental computation easier — for example, computing 4 × (25 × 13) as (4 × 25) × 13 = 100 × 13 = 1,300. The order of numbers stays the same; only the parentheses shift.
Key Concepts
Property For addition: $(a + b) + c = a + (b + c)$ For multiplication: $(a \times b) \times c = a \times (b \times c)$.
Examples $(7 + 3) + 5 = 10 + 5 = 15$ is the same as $7 + (3 + 5) = 7 + 8 = 15$.
$(4 \times 5) \times 2 = 20 \times 2 = 40$ is the same as $4 \times (5 \times 2) = 4 \times 10 = 40$.
Common Questions
What does the Associative Property of Addition say?
(a + b) + c = a + (b + c). Changing which two numbers are grouped first does not change the total sum.
What does the Associative Property of Multiplication say?
(a × b) × c = a × (b × c). The grouping of factors can change without affecting the product.
How does the Associative Property make mental math easier?
You can regroup numbers to create convenient pairs. Example: 17 + (3 + 45) = (17 + 3) + 45 = 20 + 45 = 65.
What is the difference between the Associative and Commutative Properties?
The Associative Property changes grouping (parentheses) while keeping the order the same. The Commutative Property changes the order of the numbers.
Does the Associative Property apply to subtraction?
No. (8 − 3) − 2 = 3, but 8 − (3 − 2) = 7. Subtraction is not associative.