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Lesson 1 # TAKS CAGrade Standard # Objective Alg 1 1.0 # (#.#)(X) Properties << Intro Bar of Numbers Lesson Title >> Three properties of equality govern the way that variables and quantities can be moved around within an expression. These properties are the commutative property, the associative property, and the distributive property. New Vocabulary • <<Vocab commutative New Word >> property • associative property • distributive property Using the Commutative, Associative, and Distributive Properties The Commutative Property of Addition states that the order in which numbers are added does not change their sum: abba The Commutative Property of Multiplication states that the order in which numbers are multiplied does not change their product: a?bb?a In the equation a(b ⴙ c) ⴝ ab ⴙ ac, the factor a is distributed to each term of the sum (b ⴙ c). The Associative Property of Addition states that the order in which numbers are grouped does not change their sum: (a + b) + c a + (b + c) Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. The Associative Property of Multiplication states that the order in which numbers are grouped does not change their product: (a ? b) ? c a ? (b ? c) The Distributive Property states that multiplying a number by a sum or difference gives the same result as the sum or difference of the products of the number and each term: a(b + c) ab + ac EXAMPLE 1 Name the property that each equation illustrates. 352325 The equation involves addition. The order of the numbers is changed. The equation illustrates the Commutative Property of Addition. 5 ? (3 ? 2) (5 ? 3) ? 2 The equation involves multiplication. The grouping of the numbers is changed. The equation illustrates the Associative Property of Multiplication. 2(3 5) 2 ? 3 2 ? 5 The equation involves multiplication and subtraction. The multiplication is applied after taking the difference on one side, and applied before taking the difference on the other. The equation illustrates the Distributive Property. CA Standards Check 1 Name the property that each equation illustrates. 1a. 7 (2 3) (7 2) 3 CA Standards Review 1b. 6(3) 6(5) 6(3 5) LESSON 1 ■ Properties of Numbers 1 CA Standard Alg 1 1.0 LESSON 1 Closure Properties The real number system is the union of two sets of numbers, rational numbers and irrational numbers. Rational numbers are traditionally classified into three subsets. A rational number is any number that can be written as a fraction where the numerator and denominator are integers and the denominator does not equal zero. This includes all terminating and repeating decimals. The set of counting numbers (also called natural numbers) is a subset of the rational numbers. The set of counting numbers is {1, 2, 3, 4, 5, …}. The set of whole numbers is a subset of the rational numbers. The set of whole numbers is the set of counting numbers and zero. The set of integers is a subset of the rational number. Integers are all whole numbers and their opposites. An irrational number is any number that cannot be written as a fraction where the numerator and denominator are integers. This includes nonterminating and nonrepeating decimals. The Closure Property of Addition for the set of real numbers states that when you add two real numbers, their sum is also a real number. The Closure Property of Multiplication for the set of real numbers states that when you multiply two real numbers, their product is also a real number. Write a valid argument to show that the set of whole numbers is closed for subtraction or use a counterexample to show that it is not. Choose two whole numbers to subtract that have a difference that is a whole number. Subtract 8 from 3. 3 is a whole number. 8 is a whole number. 5 is not a whole number; therefore, the set of whole number is not closed for subtraction. 3 8 5 CA Standards Check 2 2a. Write a valid argument to show that the set of integers is closed for division or use a counterexample to show that it is not. 2b. Write a valid argument to show that the set of counting numbers is closed for multiplication or use a counterexample to show that it is not. 2 LESSON 1 ■ Properties of Numbers CA Standards Review Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. EXAMPLE 2 Name__________________________Class____________Date________ 1 Name the property illustrated by the equation below: (a b) c a (b c) A Commutative Property of Addition B Distributive Property C Associative Property of Addition D Commutative Property of Multiplication 2 Which property does the equation below illustrate? a(b c) ab ac 5 Which expression is an example of commutative property for the expression 16 ? 2? A 4?4?2 B 2(8 1) C 2(8 1) D 2 ? 16 6 Which expression is an example of the associative property for the expression 2 (5 13) A 13 2 5 A Distributive Property B (2 5) 13 B Commutative Property of Multiplication C 20 C Associative Property of Addition D 2 ( 13 5) Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. D Associative Property of Multiplication 3 Which is not a counterexample that shows that the set of whole numbers is not closed for division? 7 Which is a counterexample that shows that the set of counting numbers is not closed for subtraction? A 12 2 10 A 0 12 0 B 1 2 1 B 5 0 is undefined. C 0 1 1 C 3 2 1.5 D 413 D 4 12 13 4 Closure Property of Addition for the set of real numbers states that the sum of two real numbers must be 8 For which operation is the set of integers not closed? A addition B subtraction A a positive number. C multiplication B a rational number. D division C an integer D a real number CA Standards Review LESSON 1 ■ Properties of Numbers 3