Commutative & Associative Properties - Sixth Grade Mathematics

The commutative and associative properties allow algebraic expressions to be reordered and regrouped, providing the theoretical justification for simplifying expressions and combining like terms.

The Commutative Properties in Algebra

The commutative property states that changing the order of the terms does not change the result.

Commutative Property of Addition:         a + b = b + a
                                          3x + 7 = 7 + 3x

Commutative Property of Multiplication:   a * b = b * a
                                          x * 5 = 5 * x = 5x

Subtraction and Division are NOT Commutative

You cannot commute subtraction or division: Subtraction: x - 4 is NOT equivalent to 4 - x. If x = 10, 10 - 4 = 6, but 4 - 10 = -6. Division: x / 2 is NOT equivalent to 2 / x. If x = 4, 4 / 2 = 2, but 2 / 4 = 0.5.

The Associative Properties in Algebra

The associative property states that when three or more terms are added or multiplied, the grouping of the terms in parentheses can be altered without affecting the result.

Associative Property of Addition:         (a + b) + c = a + (b + c)
                                          (2x + 5) + 8 = 2x + (5 + 8) = 2x + 13

Associative Property of Multiplication:   (a * b) * c = a * (b * c)
                                          4 * (3x) = (4 * 3) * x = 12x

Simplifying Expressions Using Commutative and Associative Laws

Together, these properties allow you to collect like terms and simplify coefficients.

Simplify: 5x + 9 + 4x + 3

Step 1: Apply Commutative Property of Addition to reorder terms:
        5x + 4x + 9 + 3

Step 2: Apply Associative Property of Addition to group like terms:
        (5x + 4x) + (9 + 3)

Step 3: Combine terms:
        9x + 12.

Simplifying Coefficient Multiplications

To simplify an expression like 6(4x): Apply the Associative Property of Multiplication: (6 * 4)x. Multiply the constants: 24x. The expression 6(4x) simplifies cleanly to 24x.

Exercises

Exercise 1. Identify which property is demonstrated in each algebraic statement: (a) 8x + 14 = 14 + 8x; (b) 5 * (2y) = (5 * 2) * y; (c) (3a + 7) + 4 = 3a + (7 + 4).

Exercise 2. Use the associative and commutative properties to simplify: 7 + 4x + 15 + 8x. Show each transformation step.

Exercise 3. Explain why the statement (x - 5) - 2 = x - (5 - 2) is false by testing x = 10.

Solutions

Solution to Exercise 1. For (a) 8x + 14 = 14 + 8x: Commutative Property of Addition (order of addends changed). For (b) 5 * (2y) = (5 * 2) * y: Associative Property of Multiplication (grouping of factors changed). For (c) (3a + 7) + 4 = 3a + (7 + 4): Associative Property of Addition (grouping of addends changed).

Solution to Exercise 2. Start with 7 + 4x + 15 + 8x. Step 1 (Commutative Property of Addition): Reorder terms so like terms are adjacent: 4x + 8x + 7 + 15. Step 2 (Associative Property of Addition): Group like terms: (4x + 8x) + (7 + 15). Step 3: Combine terms: 12x + 22.

Solution to Exercise 3. Test x = 10 in both sides of the equation. Left side: (10 - 5) - 2 = 5 - 2 = 3. Right side: 10 - (5 - 2) = 10 - 3 = 7. Because 3 does not equal 7, the statement is false. This demonstrates that subtraction is not associative; changing the grouping parentheses altered the final value.