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This educational resource focuses on the essential properties of real numbers, including commutative, associative, identity, and zero product properties. It also emphasizes the importance of equivalent expressions, deductive reasoning, and counterexamples in mathematical problem-solving. By completing statements with equalities and exploring practical examples, students will enhance their mental math skills and understanding of algebraic principles. The assignment includes exercises for reinforcing these concepts and ensuring a solid grasp of real number properties.
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Warm Up • Tell whether each expression is equal by completing each statement with = or ≠. • 1.) 34 + 12 _?_ 12 + 34 • 2.) 100 – 1 _?_ 1 – 100 • 3.) 0 + 180 _?_ 180 • 4.) 18 ÷ _?_ 1
Objectives • Identify and use properties of real numbers.
Vocabulary • Equivalent expressions – two algebraic expressions that have the same value for all values of the variable(s). • Deductive reasoning – the process of reasoning logically from given facts to a conclusion. • Counterexample – an example showing that a statement is false.
Properties of Real Numbers Commutative Properties of Addition and Multiplication • Changing the order of the addends does not change the sum. • Changing the order of the factors does not change the product.
Properties Cont. Associative Properties of Addition and Multiplication • Changing the grouping of the addends does not change the sum. • Changing the grouping of the factors does not change the product.
Properties Cont. Identity Properties of Addition and Multiplication • The sum of any real number and 0 is the original number. • The product of any real number and 1 is the original number.
Properties Cont. Zero Product Property of Multiplication • The product of a and 0 is 0. Multiplication Property of -1 • The product of -1 and a is –a.
Example 1 – Identifying Properties • Which property is illustrated by each statement? • A.) 31 * 0 = 0 • B.) (x + 3) + 6 = x + (3 + 6) • C.) 5x + 0 = 0
Example 2 – Using Properties for Mental Calculations • A movie ticket costs $6.75. A drink costs $1.90. Popcorn costs $2.25. What is the total cost for a ticket, a drink, and popcorn? Use mental math.
Example 3 – Writing Equivalent Expressions • What is the simplified form of each expression? • A.) 4(8n) • B.) (3 + 5b) + 7 • C.)
Example 4 – Using Deductive Reasoning and Counterexamples • Is the statement true or false? If false, give a counterexample. • A.) For all real numbers a and b, a – b = b – a. • B.) For all real numbers a, b, and c, a x b x c = a x c x b.
Assignment • Pg. 26 – 27 • (7 – 18 all, 20 – 30 even, 32, 33)
