Order of Operations

PEMDAS — The Rules That Keep Math in Order

📚

Subject

Math

⏱️

Duration

60+ min

🎯

Standard

5.OA.A.1

📋 Standards & Objectives

📜Standards

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

5.OA.A.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

🎯SWBAT

Evaluate numerical expressions using the order of operations (PEMDAS)
Use parentheses, brackets, and braces to control the order of computation
Identify and correct common order-of-operations errors
Write and interpret numerical expressions without evaluating them

📖 Key Vocabulary

📝Expression

A number sentence with operations but no equal sign. An expression is like a math puzzle waiting to be solved.

8 + 3 × 2 is an expression — it has numbers and operations, but no equal sign.

(15 − 7) ÷ 4 is also an expression. Once you solve it, you get a value.

📝Evaluate

To find the value of an expression by solving it step by step. When you evaluate, you do the math!

When your teacher says "evaluate 6 + 2 × 5," she's asking you to solve it using PEMDAS to get the answer: 16.

To evaluate an expression, follow the order of operations one step at a time.

📝Parentheses

Curved symbols ( ) that group parts of an expression together. Whatever is inside parentheses gets solved first!

In (9 − 2) × 5, the parentheses tell you to subtract 9 − 2 = 7 first, then multiply: 7 × 5 = 35.

Without parentheses, 9 − 2 × 5 = 9 − 10 = −1. The parentheses completely change the answer!

📝Operation

A math action you perform on numbers: addition (+), subtraction (−), multiplication (×), or division (÷).

In 12 + 8 ÷ 4, there are two operations: addition and division.

The order of operations tells you which math action to do first so everyone gets the same answer.

📝Grouping Symbols

Symbols that group parts of an expression: parentheses ( ), brackets [ ], and braces { }. Always solve from the inside out!

In 6 × [8 − (3 + 1)], the grouping symbols nest inside each other — solve the innermost ( ) first, then the [ ].

Grouping symbols are like Russian nesting dolls — start with the smallest one inside and work your way out.

🚀 Why Does Order Matter?

Same numbers, VERY different answers!

Look at this expression. Two students solved it differently. Who's right?

Student A

3 + 4 × 2

= 7 × 2

= 14

"I went left to right!"

Student B

3 + 4 × 2

= 3 + 8

= 11 ✓

"I multiplied first!"

Without rules, the same expression gives different answers. That's why we need PEMDAS!

💡 Test-Taking Tip

💡 Order of Operations: Do You Need Parentheses? Then Start There.

Before you start calculating, scan the entire expression. Find parentheses or grouping symbols first. Then look for × and ÷. Save + and − for last. This prevents the #1 mistake on the SBA: solving left-to-right without checking operations.

How to use it: Every time you see a multi-step problem, circle the parentheses first, then underline × and ÷. That tells you your first two moves.

👨‍🏫 The PEMDAS Rule

The order we follow EVERY time

P

Parentheses

( ) [ ] { }

E

Exponents

x²

M

Multiply

×

D

Divide

÷

A

Add

+

S

Subtract

−

M and D are solved together, left to right. A and S are solved together, left to right. They're partners, not a strict sequence!

📓 PEMDAS Notes

Write this in your notebook!

Key Terms

PEMDAS

Order of Operations

Notes

PEMDAS = Parentheses, Exponents, Multiply, Divide, Add, Subtract

Rule 1: Always solve parentheses / grouping symbols first.

Rule 2: Multiply and Divide go together — left to right.

Rule 3: Add and Subtract go together — left to right.

⚠️ Common mistake: Multiply does NOT always come before Divide. They're equal — whichever comes first from left to right wins!

👨‍🏫 I Do: Two-Step

Multiply before you add

Evaluate this expression:

Step 1: Find × or ÷ first

5 + 7 × 3

5 + 21

Step 2: Now add

5 + 21 = 26

Even though + comes first when reading left to right, × beats + in PEMDAS. Always scan for × and ÷ before + and −.

👨‍🏫 I Do: Parentheses First

Grouping changes everything

Evaluate:

Step 1: Solve inside ( ) first

( 12 − 4 ) × 6

8 × 6

Step 2: Multiply

8 × 6 = 48

Parentheses are the boss! Without them, 12 − 4 × 6 = 12 − 24 = −12. With them, the answer is 48.

👨‍🏫 I Do: Divide Before Subtract

Same level? Go left to right!

Evaluate:

Step 1: Find ÷ (it beats − in PEMDAS)

24 ÷ 6 − 3

4 − 3

Step 2: Subtract

4 − 3 = 1

Division and subtraction are not on the same level. ÷ is in the M/D tier. − is in the A/S tier. Always do M/D before A/S.

✅ Quick Check

What operation do you do FIRST?

10 + 15 ÷ 5 × 2

❌ Addition is in the A/S tier — do it last!

✅ Correct! ÷ and × are in the M/D tier. 15 ÷ 5 comes first (left to right), then × 2.

❌ × and ÷ are the same level — go left to right. ÷ comes first!

❌ There's no subtraction here — but good news, you don't do + first either!

👨‍🏫 I Do: Three Steps

More operations = more steps

Evaluate: 20 − 4 × 3 + 8

Step 1: Find × (M/D tier)

20 − 4 × 3 + 8

Step 2: Rewrite with the product → then A/S left to right

20 − 12 + 8

Step 3: Subtract first (left to right), then add

8 + 8 = 16

👨‍🏫 I Do: ( ) + × −

Parentheses, multiply, then subtract

Evaluate: 9 + 6 × (8 − 5)

Step 1: Parentheses first!

9 + 6 × ( 8 − 5 )

9 + 6 × 3

Step 2: Multiply (M/D tier)

9 + 6 × 3 = 9 + 18

Step 3: Add

9 + 18 = 27

📓 Three-Step Strategy

Write this in your notebook!

Key Terms

Three-Step

Evaluate

Notes

To evaluate multi-step expressions:

1. Solve parentheses first (inside out)

2. Do × and ÷ next, left to right

3. Do + and − last, left to right

Example: 9 + 6 × (8 − 5) → 9 + 6 × 3 → 9 + 18 → 27

🔄 What We Learned → What's Next

✅So Far

We can evaluate expressions with parentheses ( ) using PEMDAS — up to 3 steps.

🚀Next Up

What happens when there are MULTIPLE layers of grouping symbols? Brackets [ ] and braces { } join the party!

👨‍🏫 Grouping Symbols

( ) → [ ] → { } — Work inside out!

( )

Parentheses

Innermost — solve first

[ ]

Brackets

Middle layer

{ }

Braces

Outermost — solve last

Think of grouping symbols like Russian nesting dolls. Start with the smallest one inside and work your way out. Always solve the innermost group first!

👨‍🏫 Inside Out Example

Brackets and parentheses together

Evaluate: 5 × [8 − (2 + 3)]

Step 1: Innermost ( ) first

5 × [ 8 − ( 2 + 3 ) ]

5 × [ 8 − 5 ]

Step 2: Now solve inside [ ]

5 × [ 8 − 5 ] = 5 × 3

Step 3: Multiply

5 × 3 = 15

🎯 SBA-Style Practice

Multiple Choice

Evaluate the expression:

18 + ( 9 − 3 ) ÷ 2 × 5

A60

❌ This comes from adding 18 + 9 first (left to right), ignoring PEMDAS.

B33

✅ Correct! (9−3)=6, then 6÷2=3, then 3×5=15, then 18+15=33. PEMDAS done right!

C48

❌ This comes from doing (9−3)÷2=3 but then adding 18+3=21 before multiplying by 5. Must do all M/D first!

D21

❌ This comes from doing (9−3)=6, ÷2=3, then 18+3=21 and forgetting the ×5 entirely.

💬 Turn & Talk

🤔Discuss with a Partner

A friend says, "I always do multiplication before division because M comes before D in PEMDAS." Is your friend correct? Why or why not?

Sentence starter: "My friend is ___ because M and D are actually ___."

👥 We Do: Two-Step

Let's solve together!

AProblem 1

14 − 8 ÷ 2

What do you do first?

Step 1: 8 ÷ 2 = 4 (M/D first)

Step 2: 14 − 4 = 10

BProblem 2

( 16 + 4 ) ÷ 5

What do you do first?

Step 1: (16 + 4) = 20 (Parentheses!)

Step 2: 20 ÷ 5 = 4

👥 We Do: Three-Step

Work through it together

AProblem 3

15 − 3 × 4 + 7

Step 1: 3 × 4 = 12

Step 2: 15 − 12 = 3

Step 3: 3 + 7 = 10

BProblem 4

( 7 + 5 ) × 3 − 16

Step 1: (7 + 5) = 12

Step 2: 12 × 3 = 36

Step 3: 36 − 16 = 20

👥 We Do: Grouping Symbols

Inside out!

Evaluate: 4 × [10 − (3 + 1)]

What goes first? What goes second? What goes last?

4 × [ 10 − ( 3 + 1 ) ]

Step 1: (3 + 1) = 4 — innermost first

Step 2: [10 − 4] = 6 — now the brackets

Step 3: 4 × 6 = 24 — multiply last

✅ Quick Check

True or False?

In the expression 30 − [6 × (2 + 3)], you should multiply 6 × 2 first.

❌ Nope! The parentheses (2 + 3) are innermost — solve them first to get 5, THEN multiply 6 × 5 = 30.

✅ Correct! Inside out: (2 + 3) = 5 first, then 6 × 5 = 30, then 30 − 30 = 0.

🎯 Wrong Answer Analysis

Learning from mistakes

Evaluate: 8 + 6 × 3 − 4

❌ Student A

8 + 6 = 14

14 × 3 = 42

42 − 4 = 38

"I went left to right!"

Student A added before multiplying. Addition is A/S tier — it comes LAST. Must do 6 × 3 first!

✅ Student B

6 × 3 = 18 (M/D first)

8 + 18 = 26

26 − 4 = 22

"I found × first!"

✅ Student B scanned for M/D first (6 × 3 = 18), then did A/S left to right. 22 is correct!

💡 Click to compare, or use J/K keys

📓 Before You Practice

Write this in your notebook!

Checklist

PEMDAS Steps

My Strategy

Every time I see an expression:

1. Circle any parentheses / grouping symbols

2. Underline × and ÷

3. Solve inside out → M/D left to right → A/S left to right

4. Check: Does my answer make sense?

🔍 You Try: Set A

Two- and three-step expressions

1Two-Step

36 ÷ 4 + 11

36 ÷ 4 = 9, then 9 + 11 = 20

2With ( )

7 × ( 11 − 6 )

(11 − 6) = 5, then 7 × 5 = 35

3Three-Step

25 − 5 × 3 + 2

5 × 3 = 15, then 25 − 15 = 10, then 10 + 2 = 12

4( ) + Three-Step

( 20 − 8 ) ÷ 4 + 9

(20 − 8) = 12, then 12 ÷ 4 = 3, then 3 + 9 = 12

🔍 You Try: Set B

Grouping symbols — inside out!

5Brackets

3 × [ 14 − ( 2 + 6 ) ]

(2+6)=8, [14−8]=6, 3×6 = 18

6Brackets + M/D

40 ÷ [ ( 3 + 7 ) − 2 ]

(3+7)=10, [10−2]=8, 40÷8 = 5

7Multi-Step

12 + 4 × 7 ÷ 2

4 × 7 = 28, 28 ÷ 2 = 14, 12 + 14 = 26

8Challenge

6 × [ 5 + ( 18 ÷ 3 ) ]

(18÷3)=6, [5+6]=11, 6×11 = 66

🎯 SBA-Style Practice

Select TWO Correct Answers

Which TWO number sentences are TRUE?

A8 + 4 × 3 = 20

✅ TRUE — 4 × 3 = 12, then 8 + 12 = 20. PEMDAS correct!

B15 − 6 ÷ 2 = 4.5

❌ FALSE — 6 ÷ 2 = 3, then 15 − 3 = 12 (not 4.5). The trap: (15 − 6) ÷ 2 = 4.5.

C(7 + 3) × 2 = 17

❌ FALSE — (7 + 3) = 10, then 10 × 2 = 20 (not 17). Trap: doing 7 + 3×2 = 7+6 = 13.

D30 ÷ (2 + 3) × 4 = 24

✅ TRUE — (2+3)=5, 30÷5=6, 6×4=24. Inside out, then M/D left to right!

🔍 Make It True Challenge

Add parentheses to make each equation true!

AChallenge 1

5 + 3 × 6 = 48

Where do the ( ) go?

(5 + 3) × 6 = 8 × 6 = 48 ✓

BChallenge 2

24 ÷ 8 − 2 = 4

Where do the ( ) go?

24 ÷ (8 − 2) = 24 ÷ 6 = 4 ✓

CChallenge 3

10 − 4 + 2 × 3 = 24

Where do the ( ) go?

(10 − 4 + 2) × 3 = 8 × 3 = 24 ✓

DChallenge 4

48 ÷ 6 + 2 = 6

Where do the ( ) go?

48 ÷ (6 + 2) = 48 ÷ 8 = 6 ✓

📝 Anchor Chart Recap

Everything you need to remember

1️⃣P — Parentheses

Solve grouping symbols first: ( ) then [ ] then { }. Always inside out!

2️⃣E — Exponents

Solve any exponents next (we'll practice these more soon!).

3️⃣M/D — Multiply & Divide

Same level — go left to right. Whichever comes first wins!

4️⃣A/S — Add & Subtract

Same level — go left to right. Always last!

Remember: Parentheses first → Exponents → × ÷ left to right → + − left to right

📓 Summary Note

Write 1 Sentence

In the bottom of your notebook page, write one sentence explaining what you learned today about the order of operations. Include the word PEMDAS and explain why order matters.

🎫 Exit Ticket

Show what you know — 4 questions!

1️⃣Evaluate

9 + 21 ÷ 7 − 2

21 ÷ 7 = 3, then 9 + 3 = 12, then 12 − 2 = 10

2️⃣SBA Style

What operation do you do first?

6 × [ 4 + ( 10 ÷ 2 ) ]

Divide (10 ÷ 2) — innermost ( ) first! Then [4+5]=9, then 6×9=54

3️⃣Word Problem

Maya bought 3 packs of pencils with 8 pencils each and gave away 5. Write an expression and evaluate.

3 × 8 − 5 = 24 − 5 = 19 pencils

4️⃣🔄 Spiral Review

Round 4.738 to the nearest tenth.

Think: which digit is in the tenths place?

Tenths place is 7. Look at 3 (hundredths) — 3 < 5, round down: 4.7