Do Now Simplify. 1. 2 + 5  3 – 7 2. 5(3 – 1) ÷ (3 + 2) 1. 2 + 5  3 – 7 2. 5(3 – 1) ÷ (3 + 2) 3. (4 + 1)2 – 8 ÷ 2 4. 12 ÷ 3  6 – 20 10 2 21 4

Goal: Learn how to identify properties of numbers and use them to simplify numerical expressions.

Tell which property is represented. A. (2  6)  1 = 2  (6  1) Example A Tell which property is represented. A. (2  6)  1 = 2  (6  1) B. 3 + 0 = 3 C. 7 + 9 = 9 + 7 (2  6)  1 = 2  (6  1) The numbers are regrouped. Associative Property 3 + 0 = 3 One of the factors is 0. Identity Property 7 + 9 = 9 + 7 The order of the numbers is switched. Commutative Property

Tell which property is represented. A. 7  1 = 7 B. 3 + 4 = 4 + 3 C. (5  1)  2 = 5  (1  2) Extra Examples 7  1 = 7 One of the factors is 1. Identity Property 3 + 4 = 4 + 3 The order of the numbers is switched. Commutative Property (5  1)  2 = 5  ( 1  2) The numbers are regrouped. Associative Property

Example: Using Properties to Simplify Expressions Simplify each expression. Justify each step. A. 21 + 16 + 9 B. 20  9  5 21 + 16 + 9 = 16 + 9 + 21 Commutative Property. = 16 + (9 + 21) Associative Property. = 16 + 30 Add. = 46 20  9  5 = 20  5  9 Commutative Property. = 20  (5  9) Associative Property. = 20  45 Multiply. = 900

Example: Using Properties to Simplify Expressions Simplify each expression. Justify each step. A. 17 + 14 + 3 B. 12  3  5 17 + 14 + 3 = 14 + 17 + 3 Commutative Property. = 14 + (17 + 3) Associative Property. = 14 + 20 Add. = 34 12  3  5 = 3  5  12 Commutative Property. = 3  (5  12) Associative Property. = 3  60 Multiply. = 180

The distributive property involves the operations of multiplication and addition or multiplication and subtraction. When we use the distributive property, we are multiplying each term inside the parentheses with the term outside of the parentheses. 

 The Distributive Property:

Example: Using the Distributive Property to Multiply Mentally 6(54) = 6(50 + 4) Rewrite 54 as 50 + 4. Use the Distributive Property. = (6  50) + (6  4) = 300 + 24 Multiply. = 324 Add.

Use the distributive property to simplify. Example B Use the distributive property to simplify.   2(x + 5)   3(y - 6) 2x + 10 3y - 18

Use the distributive property to simplify. 1) 3(x + 7) 6) x(a - m) ax + mx 3x + 21 2) 2(a + 4) 7) 4(3 - r) 2a + 8 12 - 4r 3) 7(8 + m) 8) 2(x + 8) 2x + 16 56 + 7m 4) 3(4 + a) 9) 7(2m + 3y + 4) 12 + 3a 14m + 21y + 28 5) (3 - k)5 10) (6 + 2y + a)3 15 - 5k 18 + 6y + 3a

Lesson Quiz Identity Property Associative Property Tell which property is represented. 1. 17  1 = 17 2. (12 + 14) + 5 = 12 + (14 + 5) 3. 2  16 = 16  2 Identity Property Associative Property Commutative Property

Lesson Quiz 1. Identify the property that is represented. (21 + 16) + 7 = 21 + (16 + 7) A. Identity Property B. Associative Property C. Commutative Property D. Distributive Property

Lesson Quiz 2. Identify the property that is represented. 17 · 9 = 9 · 17 A. Associative Property B. Commutative Property C. Distributive Property D. Identity Property