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Algebraic Properties Lesson 6.06

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After completing this lesson, you will be able to say: I can generate equivalent expressions using the algebraic properties.

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Algebraic properties Algebraic properties are operational rules you can use to create equivalent expressions. Equivalent expressions have the same value when simplified.

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Commutative Property The Commutative Property of Addition says that the order of the addends in a sum does not matter. The Commutative Property of Multiplication says that the order of the factors in a product does not matter.

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Commutative Property – Examples Sarah drove 30 miles to the beach and m extra miles to the store. The total distance can be represented as 30 + m or m + 30. The sum is the same either way because of the commutative property The total amount of gear needed for the beach day can be represented as 5 + (x + 2). You can see it as the sum of 5 and another quantity x + 2. According to the commutative property of addition, you can change the order around of the two addends. 5 + (x + 2) = (x + 2) + 5 The sum is the same either way because of the commutative property of addition.

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Associative Property The Associative Property of Addition says that the grouping of the addends in a sum does not matter. The Associative Property of Multiplication says that the grouping of the factors in a product does not matter.

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Associative Property - examples Sarah swam 30 minutes in the morning, 45 minutes at lunch, and x minutes in the afternoon. The total amount she swam can be represented with the expression 30 + (45 + x) or (30 + 45) + x because of the associative property. Sarah bought 2 cans of sunscreen for some friends, f, and each can cost $4. Sarah can represent the total cost of the sunscreen as (f ⋅ 2) ⋅ 4 or f ⋅ (2 ⋅ 4) because of the associative property.

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Identity Property The Identity Property of Addition states that adding 0 to a number does not change the identity (or value) of the number. The Identity Property of Multiplication states that multiplying a number by 1 does not change the identity (or value) of the number Zero is the identity element for addition because zero has no effect on the value in a sum. One is the identity element for multiplication because it has no effect on the value in a product. Because of the identity properties, you can manipulate the expression to suit your needs while maintaining equality between two expressions.

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Caution with the properties The commutative property does not work for subtraction or division. This means you cannot reverse the order of a subtraction or division expression and keep the same value. The associative property does not work if the expression contains more than one operation The associative property does not work if the expression contains subtraction or division. This means you cannot move the parentheses around on a subtraction or division expression and always keep the same value

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Distributive Property Rule Distributive Property (with a sum): a(b + c) = a ⋅ b + a ⋅ c Distributive Property (with a difference): a(b − c) = a ⋅ b − a ⋅ c The distributive property says that any number multiplied to a sum or difference of two or more numbers is equal to the sum or difference of the products. The property allows you to rewrite a product as a sum or difference to suit your needs without changing the value of the expression

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Distributive Property Caution In order to apply the distributive property, a factor must be multiplied by a sum or difference. You cannot use the distributive property on algebraic expressions that contain only one operation. 3(4 ⋅ 5 ⋅ 6) ≠ 3(4) ⋅ 3(5) ⋅ 3(6) Also, the distributive property does not work when dividing a quantity by a sum or difference

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Distributive Property - example Apply the distributive property to generate an equivalent expression for 4(x + 10) Step 1: Multiply the first term in the parentheses by the factor. In this example, remember there is an invisible 1 as the coefficient for the x variable. Step 2: Bring over the mathematical operation in the parentheses. Step 3: Multiply the second term in the parentheses by the factor, and simplify. Therefore, 4(x + 10) is equivalent to 4x + 40 because of the distributive property.

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Try it Apply the distributive property to generate an equivalent expression for 6(3n + 5 + 2c).

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Check your work Multiply the first term in the parentheses by the factor, and bring over the first mathematical operation in the parentheses. Multiply the second term in the parentheses by the factor, and bring over the second mathematical symbol. Multiply the third term in the parentheses by the factor, and simplify. Therefore, 6(3n + 5 + 2c) is equivalent to 18n + 30 + 12c because of the distributive property

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Visual Models Using models can help you understand how the distributive property works and also help you simplify expressions more easily Sarah and her friends laid down two beach blankets. The blankets are the same length but of different widths. The distributive property can be used to find the total area of the two blankets together. The length of both blankets is 75 inches. The width of the orange blanket is unknown, and the width of the green blanket is 52 inches. Lets see how we can simplify this problem using both a visual model and the distributive property

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Visual Models Sarah and her friends laid down two beach blankets. The blankets are the same length but of different widths. The distributive property can be used to find the total area of the two blankets together. The length of both blankets is 75 inches. The width of the orange blanket is unknown, and the width of the green blanket is 52 inches. Remember the area of a rectangle = l ⋅ w. Together, the blankets form a rectangle that has a length of 75 inches. The width of the rectangle is the sum of each blanket’s width, which is x + 52 Using the area formula, the area of the rectangle can be found by multiplying the two sides, which is represented as the expression 75(x + 52)

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Using visual models to create equivalent expressions Using the area formula, the area of the rectangle can be found by multiplying the two sides, which is represented as the expression 75(x + 52) Can you create an equivalent expression to represent the area of the two blankets? Therefore, the expression 75(x + 52) is equivalent to the expression 75x + 3,900 because of the distributive property

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Try it Rewrite the equivalent expression for 5(3x – 7) using the distributive property

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Check your work This is a product of a number and a difference. You know the equivalent expression will be the difference of products, according to the distributive property. 5(3x − 7) = 5 ⋅ 3x – 5 ⋅ 7 When simplified, the expression becomes 15x − 35

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Now that you completed this lesson, you should be able to say: I can generate equivalent expressions using the algebraic properties.

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