Order or Operations/Properties of Numbers

1-1/1-2

*Notes 9

Vocabulary 1-1

Numerical expression- made up of numbers and operations.

Order of operations- When simplifying a numerical expression, rules that must be followed so that everyone gets the same answer.

Vocabulary 1-2

Commutative Property- you can add numbers in any order and multiply numbers in any order.

Associative Property- When you add or multiply, you can group the numbers together in any combination.

Identity Property- the sum of 0 and any number is the number and the product of 1 and any number is the number.

Distributive Property- multiply numbers mentally by breaking apart one of the numbers and writing it as a sum or difference.

*Parenthesis

Exponents MultiplyDivide Add Subtract

ORDER OF OPERATIONS

1. Perform operations within grouping symbols.

2. Evaluate powers.

3. Multiply and divide in order from left to right.

4. Add and subtract in order from left to right.

Remember: PEMDAS

Please Excuse My Dear Aunt Sally

Simplify the expression. Use the order of operations to justify your answer.

Additional Example 1A: Using the Order of Operations

3 + 15 ÷ 5

3 + 15 ÷ 5

3 + 3

6

Divide.

Add.

Additional Example 1B: Using the Order of Operations

44 – 14 ÷ 2 · 4 + 6

Divide and multiply fromleft to right.

Subtract and add fromleft to right.

Simplify the expression. Use the order of operations to justify your answer.

Additional Example 1C: Using the Order of Operations

3 + 23 · 5

Evaluate the power.

Multiply.

Add.

Simplify the expression. Use the order of operations to justify your answer.

Simplify the expression.

Additional Example 2A: Using the Order of Operations with Grouping Symbols

42 – (3 · 4) ÷ 6Perform the operation inside the parentheses.

Divide.

Subtract.

When an expression has a set of grouping symbols within a second set of grouping symbols, begin with the innermost set.

Helpful Hint

Additional Example 2B: Using the Order of Operations with Grouping Symbols

[(26 – 4 · 5) + 6]2

[(26 – 4 · 5) + 6]2

[(26 – 20) + 6]2

[6 + 6]2

122

144

The parentheses are inside the brackets, so perform the operationsinside the parenthesesfirst.

Simplify the expression.

*Properties of Numbers

You can use the Distributive Property to multiply numbers mentally by breaking apart one of the numbers and writing it as a sum or difference.

Additional Example 1: Identifying Properties of Addition and Multiplication

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

Check It Out: Example 1

Tell which property is represented.

A. 7 1 = 7

B. 3 + 4 = 4 + 3

C. (5 1) 2 = 5 (1 2)

Additional Example 2: 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)

= 16 + 30

Associative Property.

= 46

Add.

20 9 5 = 20 5 9 Commutative Property.

= 20 (5 9)

= 20 45

Associative Property.

= 900

Multiply.

Check It Out: Example 2A & B

Simplify each expression. Justify each step.

A. 17 + 14 + 3

B. 12 3 5

17 + 14 + 3 = 14 + 17 + 3

12 3 5 = 3 5 12

Additional Example 3: Using the Distributive Property to Multiply Mentally

Use the Distributive Property to find 6(54).

Method 1:

Method 2:

= (6 50) + (6 4)

Rewrite 54 as 50 + 4.

= 300 + 24

= 324

Use the Distributive Property.

Multiply.

6(54) = 6(60 – 6) Rewrite 54 as 60 – 6.

= (6 60) – (6 6)

= 360 - 36

Use the Distributive Property. Multiply.

= 324 Subtract.

Add.

6(54) = 6(50 + 4)

Check It Out: Example 3

Use the Distributive Property to find 8(19).

Method 1:

Method 2:8(19) = 8(20 – 1)

8(19) = 8(10 + 9)