The Distributive Property Section 1-7 Part 2

Goals Goal To use the Distributive Property to simplify expressions. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Term Constant Coefficient Like Terms

The Distributive Property The process of distributing the number on the outside of the parentheses to each term on the inside. a(b + c) = ab + ac and (b + c) a = ba + ca a(b - c) = ab - acand (b - c) a = ba - ca Example 5(x + 7) 5 ∙ x + 5 ∙ 7 5x + 35

Two ways to find the area of the rectangle As a wholeAs two parts Geometric Model for Distributive Property

Two ways to find the area of the rectangle As a wholeAs two parts same

Find the area of the rectangle in terms of x, y and z in two different ways. x yz As a wholeAs two parts

Your Turn: Find the area of the rectangle in terms of x, y and z in two different ways. x yz As a wholeAs two parts same xy + xz

Write the product using the Distributive Property. Then simplify. 5(59) 5(50 + 9) 5(50) + 5(9) Rewrite 59 as Use the Distributive Property. Multiply. Add. Example: Distributive Property with Mental Math You can use the distributive property and mental math to make calculations easier.

9(48) 9(50) - 9(2) 9(50 - 2) Rewrite 48 as Use the Distributive Property. Multiply. Subtract. Write the product using the Distributive Property. Then simplify. Example: Distributive Property with Mental Math

8(33) 8(30 + 3) 8(30) + 8(3) Rewrite 33 as Use the Distributive Property. Multiply. Add. Write the product using the Distributive Property. Then simplify. Your Turn:

12(98) 1176 Rewrite 98 as 100 – 2. Use the Distributive Property. Multiply. Subtract. 12(100 – 2) 1200 – 24 12(100) – 12(2) Write the product using the Distributive Property. Then simplify. Your Turn:

7(34) 7(30 + 4) 7(30) + 7(4) Rewrite 34 as Use the Distributive Property. Multiply. Add. Write the product using the Distributive Property. Then simplify. Your Turn:

Find the difference mentally. Find the products mentally. The mental math is easier if you think of $11.95 as $12.00 – $.05. Write as a difference. You are shopping for CDs. You want to buy six CDs for $11.95 each. Use the distributive property to calculate the total cost mentally. 6(11.95) = 6(12 – 0.05) Use the distributive property. = 6(12) – 6(0.05) = 72 – 0.30 = The total cost of 6 CDs at $11.95 each is $ M ENTAL M ATH C ALCULATIONS S OLUTION

Definition Term – any number that is added or subtracted. –In the algebraic expression x + y, x and y are terms. Example: –The expression x + y – 7 has 3 terms, x, y, and 7. x and y are variable terms; their values vary as x and y vary. 7 is a constant term; 7 is always 7.

Definition Coefficient – The numerical factor of a term. Example: –The coefficient of 3x 2 is 3.

Definition Like Terms – terms in which the variables and the exponents of the variables are identical. –The coefficients of like terms may be different. Example: –3x 2 and 6x 2 are like terms. –ab and 3ab are like terms. –2x and 2x 3 are not like terms.

Definition Constant – anything that does not vary or change in value (a number). –In algebra, the numbers from arithmetic are constants. –Constants are like terms.

The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms. 4x – 3x + 2 Like terms Constant Example:

A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1. 1x 2 + 3x Coefficients Example:

Like terms can be combined. To combine like terms, use the Distributive Property. Notice that you can combine like terms by adding or subtracting the coefficients. Keep the variables and exponents the same. = 3x Distributive Property ax – bx = (a – b)x Example 7x – 4x = (7 – 4)x Combining Like Terms

Simplify the expression by combining like terms. 72p – 25p 47p 72p and 25p are like terms. Subtract the coefficients. Example: Combining Like Terms

Simplify the expression by combining like terms. A variable without a coefficient has a coefficient of 1. Write 1 as. Add the coefficients. and are like terms. Example: Combining Like Terms

Simplify the expression by combining like terms. 0.5m + 2.5n 0.5m and 2.5n are not like terms. Do not combine the terms. Example: Combining Like Terms

Caution! Add or subtract only the coefficients. 6.8y² – y² ≠ 6.8

Simplify by combining like terms. 3a. 16p + 84p 16p + 84p 100p 16p + 84p are like terms. Add the coefficients. 3b. –20t – 8.5t 2 –20t – 8.5t 2 20t and 8.5t 2 are not like terms. –20t – 8.5t 2 Do not combine the terms. 3m 2 + m 3 3m 2 and m 3 are not like terms. 3c. 3m 2 + m 3 Do not combine the terms. 3m 2 + m 3 Your Turn:

S IMPLIFYING BY C OMBINING L IKE T ERMS Each of these terms is the product of a number and a variable.terms +–3y2y2 x +–3y2y2 x number +–3y2y2 x variable. +–3y2y2 x –1 is the coefficient of x. 3 is the coefficient of y 2. x is the variable. y is the variable. Each of these terms is the product of a number and a variable. x2x2 x2x2 y3y3 y3y3 Like terms have the same variable raised to the same power. y 2 – x 2 + 3y 3 – – 3x 2 + 4y 3 + y variablepower.Like terms The constant terms –5 and 3 are also like terms.

Combine like terms. S IMPLIFYING BY C OMBINING L IKE T ERMS 4x – x 2 = (8 + 3)x Use the distributive property. = 11x Add coefficients. 8x + 3x = Group like terms. Rewrite as addition expression. Distribute the –2. Multiply. Combine like terms and simplify. 4x 2 – x = 3x – 2(4 + x) = 3 + (–2)(4 + x) = 3 + [(–2)(4) + (–2)(x)] = 3 + (–8) + (–2x) = –5 + (–2x) = –5 – 2x

– 12x – 5x + x + 3a Commutative Property Combine like terms. –16x + 3a – 12x – 5x + 3a + x Procedure Justification Simplify −12x – 5x + 3a + x. Justify each step. Your Turn:

Simplify 14x + 4(2 + x). Justify each step. 14x + 4(2) + 4(x) Distributive Property Multiply. Commutative Property of Addition Associative Property of Addition Combine like terms. 14x x (14x + 4x) x + 4x x x + 4(2 + x) StatementsJustification Your Turn:

Joke Time How does the moon get a haircut? Eclipse it. What’s black & white, black & white, black & white, black & white, black & white, black & white? A penguin rolling down a hill. What do you call a fake noodle? An imPASTA.

Assignment 1.7 Pt 2 Exercises Pg. 58 – 60: #8 – 56 even