Main Idea and New Vocabulary Key Concept Commutative

Main Idea and New Vocabulary Key Concept: Commutative Properties Key Concept: Associative Properties Key Concept: Number Properties Example 1: Identify Properties Example 2: Find a Counterexample Example 3: Use Mental Math Example 4: Simplify Algebraic Expressions Example 5: Simplify Algebraic Expressions

• Identify and use mathematical properties to simplify algebraic expressions. • property • counterexample • simplify

Identify Properties Name the property shown by the statement (3 • m) • 2 = 2 • (3 • m). Answer: The grouping of the numbers and variable did not change, but their order did. This is the Commutative Property of Multiplication.

Name the property shown by the statement 4 + (8 + x) = (4 + 8) + x. A. Associative Property of Addition B. Associative Property of Multiplication C. Commutative Property of Addition D. Commutative Property of Multiplication

Find a Counterexample State whether the following conjecture is true or false. If false, provide a counterexample. Subtraction of whole numbers is associative. Write two subtraction expressions using the Associative Property. ? (12 – 5) – 3 = 12 – (5 – 3) 4 ≠ 10 State the conjecture. Simplify. Answer: The conjecture is false. We found a counterexample: (12 – 5) – 3 ≠ 12 – (5 – 3). So, subtraction is not associative.

State whether the following conjecture is true or false. If false, provide a counterexample. Division of whole numbers is associative. A. true B. false; (16 ÷ 8) ÷ 2 ≠ 16 ÷ (8 ÷ 2)

Use Mental Math GARDENS In a museum garden, a decorative pool containing fish is 2 feet deep, 17 feet long, and 5 feet wide. Use mental math to find the volume of water in the pool. Write an expression for the volume. You can rearrange the numbers using the properties of math. Look for products that are multiples of 10. 2 × 17 × 5 = 2 × 5 × 17 = (2 × 5) × 17 = 10 × 17 = 170 Commutative Property Associative Property Multiply. Simplify. Answer: The volume of the water is 170 cubic feet.

ROCKS A landscaper bought four rocks to use in a landscaping project. The rocks weighed 15 pounds, 22 pounds, 8 pounds, and 25 pounds. Use mental math to find the total weight of the rocks. A. 50 pounds B. 60 pounds C. 70 pounds D. 80 pounds

Simplify Algebraic Expressions Simplify the expression 6 + (d + 8). Justify each step. Answer: 6 + (d + 8) = 6 + (8 + d) Commutative Property of Addition = (6 + 8) + d Associative Property of Addition = 14 + d Simplify.

Simplify the expression 5 m + (3 + 11 m). Name two properties you can use to justify the steps. A. 5 m + 14; Commutative (+) and Associative (+) B. 11 m + 8; Commutative (×) and Associative (×) C. 16 m + 3; Commutative (+) and Associative (+) D. 16 m; Commutative (×) and Associative (×)

Simplify Algebraic Expressions Simplify the expression a • (9 • b). Justify each step. Answer: a • (9 • b) = (a • 9) • b Associative Property of Multiplication = (9 • a) • b Commutative Property of Multiplication = 9 ab Simplify.

Simplify the expression 7 • (t • 2). Name two properties you can use to justify the steps. A. 9 t; Commutative (×) and Associative (×) B. 14 t; Commutative (×) and Associative (×) C. 14 + t; Commutative (+) and Associative (+) D. 7 t + 2; Commutative (×) and Associative (+)