UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products (e.g., 3(4 + 5) = 3 x 4 + 3 x 5, 5 x (3 + 7) = (5 x 3) + (5 x 7); or (2 x 3) + (2 x 5) = 2 x (3 + 5). The distributive property can be used to simplify expressions (e.g., 9 x 23 = 9(20+3) =180+ 27 = 207; or 5 x 19 = 5(10 + 9) = 50 + 45 = 95). All students should Understand that the distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. Understand that the distributive property with whole numbers helps with understanding mathematical relationships. Understand when and why the distributive property is used. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Investigate and recognize the distributive property of whole numbers, limited to multiplication over addition using diagrams and manipulatives. Investigate and recognize an equation that represents the distributive property, when given several whole number equations, limited to multiplication over addition. 6.19 The student will investigate and recognize a) the identity properties for addition and multiplication; b) the multiplicative property of zero; and c) the inverse property for multiplication. UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS
7.16 The student will apply the following properties of operations with real numbers: a) the commutative and associative properties for addition and multiplication; b) the distributive property; c) the additive and multiplicative identity properties; d) the additive and multiplicative inverse properties; and e) the multiplicative property of zero. UNDERSTANDING THE STANDARD (Background Information for Instructor Use Only) ESSENTIAL UNDERSTANDINGS ESSENTIAL KNOWLEDGE AND SKILLS The commutative property for addition states that changing the order of the addends does not change the sum (e.g., 5 + 4 = 4 + 5). The commutative property for multiplication states that changing the order of the factors does not change the product (e.g., 5 · 4 = 4 · 5). The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3]. The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 · (4 · 3) = (5 · 4) · 3]. Subtraction and division are neither commutative nor associative. The distributive property states that the product of a number and the sum (or difference) of two numbers equals the sum (or difference) of the products of the number and each other number [e.g., 5 · (3 + 7) = (5 · 3) + (5 · 7), or 5 · (3 – 7) = (5 · 3) – (5 · 7)]. Identity elements are numbers that combine with other numbers without changing the other numbers. The additive identity is zero (0). The multiplicative identity is one (1). There are no identity elements for subtraction and division. The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., 5 + 0 = 5). Why is it important to apply properties of operations when simplifying expressions? Using the properties of operations with real numbers helps with understanding mathematical relationships. The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to Identify properties of operations used in simplifying expressions. Apply the properties of operations to simplify expressions.