The Distributive Property Section 1-7 Part 2

Goals Goal Rubric To use the Distributive Property to simplify expressions. 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 - ac and (b - c) a = ba - ca Example 5(x + 7) 5 ∙ x + 5 ∙ 7 5x + 35

Geometric Model for Distributive Property 4 5 2 Two ways to find the area of the rectangle. As a whole As two parts

Geometric Model for Distributive Property 4 5 2 Two ways to find the area of the rectangle. As a whole As two parts same

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

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

Example: Distributive Property with Mental Math You can use the distributive property and mental math to make calculations easier. Write the product using the Distributive Property. Then simplify. 5(59) 5(50 + 9) Rewrite 59 as 50 + 9. 5(50) + 5(9) Use the Distributive Property. 250 + 45 Multiply. 295 Add.

Example: Distributive Property with Mental Math Write the product using the Distributive Property. Then simplify. 9(48) 9(50 - 2) Rewrite 48 as 50 - 2. 9(50) - 9(2) Use the Distributive Property. 450 - 18 Multiply. 432 Subtract.

Your Turn: Write the product using the Distributive Property. Then simplify. 8(33) 8(30 + 3) Rewrite 33 as 30 + 3. 8(30) + 8(3) Use the Distributive Property. 240 + 24 Multiply. 264 Add.

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

Your Turn: Write the product using the Distributive Property. Then simplify. 7(34) 7(30 + 4) Rewrite 34 as 30 + 4. 7(30) + 7(4) Use the Distributive Property. 210 + 28 Multiply. 238 Add.

The total cost of 6 CDs at $11.95 each is $71.70. MENTAL MATH CALCULATIONS You are shopping for CDs. You want to buy six CDs for $11.95 each. SOLUTION The mental math is easier if you think of $11.95 as $12.00 – $.05. 6(11.95) = 6(12 – 0.05) Write 11.95 as a difference. = 6(12) – 6(0.05) Use the distributive property to calculate the total cost mentally. Use the distributive property. = 72 – 0.30 Find the products mentally. = 71.70 Find the difference mentally. The total cost of 6 CDs at $11.95 each is $71.70.

Definition Term – any number that is added or subtracted. Example: 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 3x2 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: 3x2 and 6x2 are like terms. ab and 3ab are like terms. 2x and 2x3 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.

Example: 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. Like terms Constant 4x – 3x + 2

1x2 + 3x Example: Coefficients 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. Coefficients 1x2 + 3x

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

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

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

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

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

Your Turn: Simplify by combining like terms. 3a. 16p + 84p 16p + 84p 16p + 84p are like terms. 100p Add the coefficients. 3b. –20t – 8.5t2 –20t – 8.5t2 20t and 8.5t2 are not like terms. –20t – 8.5t2 Do not combine the terms. 3c. 3m2 + m3 3m2 + m3 3m2 and m3 are not like terms. 3m2 + m3 Do not combine the terms.

x is the y is the x2 y3 y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y + – 3 y2 SIMPLIFYING BY COMBINING LIKE TERMS Each of these terms is the product of a number and a variable. Each of these terms is the product of a number and a variable. terms + – 3 y2 x variable. + – 3 y2 x number + – 3 y2 x + – 3 y2 x x is the variable. y is the –1 is the coefficient of x. 3 is the coefficient of y2. variable power. Like terms Like terms have the same variable raised to the same power. y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y The constant terms –5 and 3 are also like terms. x2 y3

8x + 3x = (8 + 3)x = 11x 4x2 + 2 – x2 = 4x2 – x2 + 2 = 3x2 + 2 SIMPLIFYING BY COMBINING LIKE TERMS 8x + 3x = (8 + 3)x Use the distributive property. = 11x Add coefficients. 4x2 + 2 – x2 = 4x2 – x2 + 2 Group like terms. = 3x2 + 2 Combine like terms. 3 – 2(4 + x) = 3 + (–2)(4 + x) Rewrite as addition expression. = 3 + [(–2)(4) + (–2)(x)] Distribute the –2. = 3 + (–8) + (–2x) Multiply. = –5 + (–2x) Combine like terms and simplify. = –5 – 2x

Your Turn: Simplify −12x – 5x + 3a + x. Justify each step. Procedure Justification 1. –12x – 5x + 3a + x 2. –12x – 5x + x + 3a Commutative Property 3. –16x + 3a Combine like terms.

Your Turn: Statements Justification 1. 14x + 4(2 + x) 2. Simplify 14x + 4(2 + x). Justify each step. Statements Justification 1. 14x + 4(2 + x) 2. 14x + 4(2) + 4(x) Distributive Property 3. 14x + 8 + 4x Multiply. 4. 14x + 4x + 8 Commutative Property of Addition 5. (14x + 4x) + 8 Associative Property of Addition 6. 18x + 8 Combine like terms.

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 Exercises Pg. 50: #10 – 64 even