Order of Operations

How do we evaluate expressions with multiple operations?

📚

Standard

5.OA.1

📅

Topic

PEMDAS

⏱️

Duration

60m

📋 Standards & Objectives

📋Common Core Standard

5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

🎯Learning Objectives (SWBAT)

Evaluate two-step expressions using PEMDAS
Evaluate three-step and multi-step expressions
Evaluate expressions with nested grouping symbols and exponents

🚀 Why Order Matters!

Look at this expression. What answer do you get?

3 + 4 × 2 = ?

14

If you go left to right:
(3 + 4) × 2 = 14

❌ WRONG!

11

If you multiply first:
3 + (4 × 2) = 11

✅ CORRECT!

Without rules, we'd get different answers! That's why we need the Order of Operations.

📖 Key Vocabulary

📝Expression

A math phrase with numbers and operations — but NO equal sign. Example: 6 + 3 × 2

📝Evaluate

To find the value of an expression by solving it step by step. Example: 6 + 3 × 2 = 12

📝Grouping Symbols

Parentheses ( ), brackets [ ], and braces { } — these tell you what to do FIRST!

📝Exponent

A small number that tells how many times to multiply the base by itself. Example: 2³ = 2 × 2 × 2 = 8

✨ PEMDAS: The Order of Operations

Always evaluate expressions in this order:

PParentheses

EExponents

MMultiply

DDivide

AAdd

SSubtract

Important!

Multiply & Divide are equal partners — work left to right.
Add & Subtract are equal partners — work left to right.

Memory Trick

"Please Excuse My Dear Aunt Sally"
P-E-M-D-A-S

📌 Two-Step Strategy

For expressions with two operations, follow these steps:

How to Evaluate a Two-Step Expression

1

Scan the Expression

Look for parentheses first, then ×/÷, then +/−

2

Do the First Operation

Solve the higher-priority operation and replace it with the result

3

Do the Second Operation

Solve the remaining operation to get the final answer

Key idea: At each step, highlight the operation you're doing, compute it, and rewrite the expression with the result in place.

✏️ Problem 1

👨‍🏫 I Do • Two-Step

(9 − 2) × 5

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: We see parentheses — do those first! Solve (9 − 2).

🔍 Parentheses first! (9 − 2) = 7

2 Replace & simplify: The expression becomes 7 × 5.

3 Multiply: 7 × 5 = 35.

✅ 7 × 5 = 35

✓ Answer

(9 − 2) × 5 = 35

✏️ Problem 2

👨‍🏫 I Do • Two-Step

6 + 7 × 3

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: No parentheses. We see + and ×. Multiplication comes before addition!

🔍 Multiply first! 7 × 3 = 21

2 Replace & simplify: The expression becomes 6 + 21.

3 Add: 6 + 21 = 27.

✅ 6 + 21 = 27

✓ Answer

6 + 7 × 3 = 27

✏️ Problem 3

👥 We Do • Two-Step

30 ÷ 5 × 4

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: We see ÷ and ×. They're equal partners — so we work left to right!

🔍 ÷ and × are equal — go left to right! 30 ÷ 5 = 6

2 Replace & simplify: The expression becomes 6 × 4.

3 Multiply: 6 × 4 = 24.

✅ 6 × 4 = 24

✓ Answer

30 ÷ 5 × 4 = 24

✏️ Problem 4

👥 We Do • Two-Step

5 × (12 − 7)

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: We see parentheses — do those first! Solve (12 − 7).

🔍 Parentheses first! (12 − 7) = 5

2 Replace & simplify: The expression becomes 5 × 5.

3 Multiply: 5 × 5 = 25.

✅ 5 × 5 = 25

✓ Answer

5 × (12 − 7) = 25

✏️ Problem 5

✏️ You Do • Two-Step

8 + (3 × 6)

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: We see parentheses — do those first! Solve (3 × 6).

🔍 Parentheses first! (3 × 6) = 18

2 Replace & simplify: The expression becomes 8 + 18.

3 Add: 8 + 18 = 26.

✅ 8 + 18 = 26

✓ Answer

8 + (3 × 6) = 26

✏️ Problem 6

✏️ You Do • Two-Step

24 ÷ (8 − 2)

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: We see parentheses — do those first! Solve (8 − 2).

🔍 Parentheses first! (8 − 2) = 6

2 Replace & simplify: The expression becomes 24 ÷ 6.

3 Divide: 24 ÷ 6 = 4.

✅ 24 ÷ 6 = 4

✓ Answer

24 ÷ (8 − 2) = 4

✨ Multi-Step Expressions

Now we'll tackle expressions with three or more operations. The same PEMDAS rules apply — just more steps!

What Changes?

With more operations, you may need to evaluate multiple operations at the same level (like both × and ÷) before moving to the next level. Always work left to right within each level!

Grouping Inside Grouping

Sometimes there's an operation inside parentheses that needs PEMDAS too! Like 3 × (10 + 24 ÷ 8) — inside the parentheses, divide before you add!

📌 Multi-Step Strategy

How to Evaluate Multi-Step Expressions

1

Parentheses

Solve everything inside ( ) first — apply PEMDAS inside too!

2

× and ÷

Do all multiplication and division, left to right

3

+ and −

Do all addition and subtraction, left to right

4

Rewrite

After each operation, rewrite the expression with the new value

✏️ Problem 7

👨‍🏫 I Do • Three-Step

(5 − 3) + 9 ÷ 3

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first: Solve (5 − 3) = 2. Expression becomes: 2 + 9 ÷ 3

🔍 Parentheses! (5 − 3) = 2

2 Division next: Solve 9 ÷ 3 = 3. Expression becomes: 2 + 3

🔍 Division before addition! 9 ÷ 3 = 3

3 Add: 2 + 3 = 5

✅ 2 + 3 = 5

✓ Answer

(5 − 3) + 9 ÷ 3 = 5

✏️ Problem 8

👨‍🏫 I Do • Three-Step

20 − 7 + 6 × 2

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: No parentheses. We see −, +, and ×. Multiplication first! Solve 6 × 2 = 12. Expression becomes: 20 − 7 + 12

🔍 Multiply first! 6 × 2 = 12

2 Left to right: Solve 20 − 7 = 13. Expression becomes: 13 + 12

🔍 + and − are equal — go left to right! 20 − 7 = 13

3 Add: 13 + 12 = 25

✅ 13 + 12 = 25

✓ Answer

20 − 7 + 6 × 2 = 25

✏️ Problem 9

👥 We Do • Three-Step

3 × (10 + 24 ÷ 8)

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first! Inside the ( ) we see + and ÷. Division first inside the parentheses: 24 ÷ 8 = 3. Inside becomes: 10 + 3

🔍 Inside parentheses: divide first! 24 ÷ 8 = 3

2 Finish parentheses: 10 + 3 = 13. Expression becomes: 3 × 13

🔍 Finish the parentheses: 10 + 3 = 13

3 Multiply: 3 × 13 = 39

✅ 3 × 13 = 39

✓ Answer

3 × (10 + 24 ÷ 8) = 39

✏️ Problem 10

👥 We Do • Three-Step

11 × 8 − 56 ÷ 7

Evaluate the expression

💡 Step-by-Step Evaluation

1 Scan: No parentheses. We see × and ÷ (equal level) and −. × and ÷ first, left to right! Solve 11 × 8 = 88.

🔍 Multiply first (left to right): 11 × 8 = 88

2 Continue × / ÷ level: Solve 56 ÷ 7 = 8. Expression becomes: 88 − 8

🔍 Then divide: 56 ÷ 7 = 8

3 Subtract: 88 − 8 = 80

✅ 88 − 8 = 80

✓ Answer

11 × 8 − 56 ÷ 7 = 80

✏️ Problem 11

✏️ You Do • Three-Step

4 × (15 − 9) + 6

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first: Solve (15 − 9) = 6. Expression becomes: 4 × 6 + 6

🔍 Parentheses! (15 − 9) = 6

2 Multiply next: Solve 4 × 6 = 24. Expression becomes: 24 + 6

🔍 Multiplication before addition! 4 × 6 = 24

3 Add: 24 + 6 = 30

✅ 24 + 6 = 30

✓ Answer

4 × (15 − 9) + 6 = 30

✏️ Problem 12

✏️ You Do • Three-Step

36 ÷ (3 + 6) − 2

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first: Solve (3 + 6) = 9. Expression becomes: 36 ÷ 9 − 2

🔍 Parentheses! (3 + 6) = 9

2 Divide next: Solve 36 ÷ 9 = 4. Expression becomes: 4 − 2

🔍 Division before subtraction! 36 ÷ 9 = 4

3 Subtract: 4 − 2 = 2

✅ 4 − 2 = 2

✓ Answer

36 ÷ (3 + 6) − 2 = 2

✨ Grouping Symbols & Exponents

Time to level up! Expressions can have nested grouping symbols and exponents.

🔲Three Types of Grouping

( ) [ ] { }

Parentheses • Brackets • Braces

Work from the innermost grouping outward!

🔢Exponents

2³ = 2 × 2 × 2 = 8

The small number (exponent) tells you how many times to multiply the base by itself.

Exponents come after parentheses but before × ÷ + −

📌 Nested Grouping Strategy

Work Inside Out!

{ [ 7 × (4 + 5) ] − 21 } ÷ 6

① Innermost ( ) → ② Middle [ ] → ③ Outer { } → ④ Remaining operations

PEMDAS with Nesting

P — Solve innermost ( ) first
E — Evaluate exponents
MD — Multiply/Divide left to right
AS — Add/Subtract left to right
Then move to the next grouping layer!

Common Exponents

2² = 4    3² = 9    4² = 16
5² = 25    10² = 100
2³ = 8    3³ = 27
4³ = 64    5³ = 125

✏️ Problem 13

👨‍🏫 I Do • Nested Grouping

6 × [8 × 3 − (12 − 8)]

Evaluate the expression

💡 Work Inside Out!

1 Innermost ( ) first: Solve (12 − 8) = 4. Expression becomes: 6 × [8 × 3 − 4]

🔍 Innermost parentheses! (12 − 8) = 4

2 Inside [ ] — multiply first: Solve 8 × 3 = 24. Inside becomes: [24 − 4]

🔍 Inside brackets: multiply before subtract! 8 × 3 = 24

3 Finish brackets: Solve [24 − 4] = 20. Expression becomes: 6 × 20

🔍 Finish brackets: 24 − 4 = 20

4 Multiply: 6 × 20 = 120

✅ 6 × 20 = 120

✓ Answer

6 × [8 × 3 − (12 − 8)] = 120

✏️ Problem 14

👨‍🏫 I Do • Exponents

(20 + 18) − 2³

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first: Solve (20 + 18) = 38. Expression becomes: 38 − 2³

🔍 Parentheses! (20 + 18) = 38

2 Exponents next: Solve 2³ = 2 × 2 × 2 = 8. Expression becomes: 38 − 8

🔍 Exponents! 2³ = 2 × 2 × 2 = 8

3 Subtract: 38 − 8 = 30

✅ 38 − 8 = 30

✓ Answer

(20 + 18) − 2³ = 30

✏️ Problem 15

👥 We Do • Exponents

5² − 3 × (2 + 4)

Evaluate the expression

💡 Step-by-Step Evaluation

1 Parentheses first: Solve (2 + 4) = 6. Expression becomes: 5² − 3 × 6

🔍 Parentheses! (2 + 4) = 6

2 Exponents next: Solve 5² = 5 × 5 = 25. Expression becomes: 25 − 3 × 6

🔍 Exponents! 5² = 5 × 5 = 25

3 Multiply next: Solve 3 × 6 = 18. Expression becomes: 25 − 18

🔍 Multiply before subtract! 3 × 6 = 18

4 Subtract: 25 − 18 = 7

✅ 25 − 18 = 7

✓ Answer

5² − 3 × (2 + 4) = 7

✏️ Problem 16

👥 We Do • Nested Grouping

{[7 × (4 + 5)] − 21} ÷ 6

Evaluate the expression

💡 Work Inside Out!

1 Innermost ( ) first: Solve (4 + 5) = 9. Expression becomes: {[7 × 9] − 21} ÷ 6

🔍 Innermost parentheses! (4 + 5) = 9

2 Brackets [ ] next: Solve [7 × 9] = 63. Expression becomes: {63 − 21} ÷ 6

🔍 Brackets! [7 × 9] = 63

3 Braces { } next: Solve {63 − 21} = 42. Expression becomes: 42 ÷ 6

🔍 Braces! {63 − 21} = 42

4 Divide: 42 ÷ 6 = 7

✅ 42 ÷ 6 = 7

✓ Answer

{[7 × (4 + 5)] − 21} ÷ 6 = 7

✏️ Problem 17

✏️ You Do • Exponents + Grouping

4² + (36 ÷ [2 + 4])

Evaluate the expression

💡 Step-by-Step Evaluation

1 Innermost [ ] first: Solve [2 + 4] = 6. Expression becomes: 4² + (36 ÷ 6)

🔍 Innermost brackets! [2 + 4] = 6

2 Finish parentheses: Solve (36 ÷ 6) = 6. Expression becomes: 4² + 6

🔍 Finish parentheses! (36 ÷ 6) = 6

3 Exponents: Solve 4² = 4 × 4 = 16. Expression becomes: 16 + 6

🔍 Exponents! 4² = 4 × 4 = 16

4 Add: 16 + 6 = 22

✅ 16 + 6 = 22

✓ Answer

4² + (36 ÷ [2 + 4]) = 22

✏️ Problem 18

🏆 Challenge • All Skills

[3 × (8 − 5)]² ÷ 27

Evaluate the expression — use everything you've learned!

🏆 Challenge — All Skills Combined!

1 Innermost ( ) first: Solve (8 − 5) = 3. Expression becomes: [3 × 3]² ÷ 27

🔍 Innermost parentheses! (8 − 5) = 3

2 Brackets [ ] next: Solve [3 × 3] = 9. Expression becomes: 9² ÷ 27

🔍 Brackets! [3 × 3] = 9

3 Exponents: Solve 9² = 9 × 9 = 81. Expression becomes: 81 ÷ 27

🔍 Exponents! 9² = 9 × 9 = 81

4 Divide: 81 ÷ 27 = 3

✅ 81 ÷ 27 = 3

✓ Answer

[3 × (8 − 5)]² ÷ 27 = 3

💬 Turn & Talk

🤔Discuss with a Partner

Talk with your partner about these questions:

1. Your friend says 8 + 2 × 5 = 50. What mistake did they make? What is the correct answer?

2. Why do mathematicians need an agreed-upon order of operations? What would happen without one?

3. Can you create an expression that uses parentheses, multiplication, AND addition where the answer is 20?

📝 Key Takeaways

📌 PEMDAS Rules

Always follow the order: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right).

🔗 Equal Partners

Multiply & Divide are equal — work left to right. Add & Subtract are equal — work left to right. Don't assume × always comes before ÷!

🔲 Work Inside Out

With nested grouping symbols ( ), [ ], { }, start with the innermost group and work your way outward.

✍️ Rewrite Each Step

After each operation, rewrite the expression with the result. This keeps your work organized and helps you avoid mistakes!

🎫 Exit Ticket

Solve each problem on your whiteboard, then check your answers!

1️⃣Two-Step

15 − 3 × 4

Multiply first: 3 × 4 = 12
Then subtract: 15 − 12 = 3

2️⃣Three-Step

(8 + 4) × 3 − 10

Parentheses: (8+4) = 12
Multiply: 12 × 3 = 36
Subtract: 36 − 10 = 26

3️⃣Exponents + Grouping

3² + [20 − (4 × 3)]

Inner ( ): 4 × 3 = 12
Brackets: 20 − 12 = 8
Exponent: 3² = 9
Add: 9 + 8 = 17

Great Work!

You're an Order of Operations expert now!

🧠

Learned

PEMDAS

✅

Solved

18

⭐

Level

Pro