Associative Properties
The Associative Properties state that when adding or multiplying three or more numbers, the grouping of the numbers does not change the result. For addition: (a + b) + c = a + (b + c), and for multiplication: (a x b) x c = a x (b x c). This Grade 7 math skill from Saxon Math, Course 2 is a foundational number property that makes mental arithmetic easier — you can regroup numbers to create convenient sums or products — and it justifies many algebraic simplification steps in equation solving.
Key Concepts
Property $(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$.
Examples $(3 + 5) + 6 = 3 + (5 + 6)$ is true, as $8 + 6 = 3 + 11$, and both equal 14. Use this to simplify: $4 \times (25 \times 9) = (4 \times 25) \times 9 = 100 \times 9 = 900$. $(16 \div 4) \div 2 \neq 16 \div (4 \div 2)$, because $2$ is not the same as $8$.
Explanation Who you 'associate' with first doesn't change the final group! When you are only adding or only multiplying, you can regroup numbers using parentheses to make problems much easier to solve in your head. It’s all about smart teamwork!
Common Questions
What is the Associative Property?
The Associative Property states that changing the grouping of numbers in addition or multiplication does not change the sum or product. Example: (2 + 3) + 4 = 2 + (3 + 4) = 9.
What is the Associative Property of Addition?
For addition, (a + b) + c = a + (b + c). The grouping (which numbers are in parentheses) can be changed without affecting the total sum.
What is the Associative Property of Multiplication?
For multiplication, (a x b) x c = a x (b x c). You can multiply in any grouping and get the same product. Example: (2 x 3) x 4 = 2 x (3 x 4) = 24.
Does the Associative Property apply to subtraction and division?
No. Subtraction and division are not associative: (8 - 3) - 2 = 3, but 8 - (3 - 2) = 7. The grouping matters for these operations, so be careful not to apply the property incorrectly.
When do students learn the Associative Properties?
Associative properties are introduced in Grade 3-4 and reinforced in Grade 7. Saxon Math, Course 2 reviews them in Chapter 3 as part of number property foundations.
How does the Associative Property help with mental math?
You can regroup numbers to find easier calculations. For 14 + 36 + 25, regroup as (14 + 36) + 25 = 50 + 25 = 75 — much faster to compute than left-to-right.
How does the Associative Property differ from the Commutative Property?
Commutative Property changes ORDER of numbers (a + b = b + a). Associative Property changes GROUPING but not order ((a + b) + c = a + (b + c)).