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

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

6. 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
C = 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

7. Tell which property is represented. A. 7  1 = 7
B = 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

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

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

10. 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. 

11.  The Distributive Property:

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

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

14. 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

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

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

17. 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