Order of Operations

What is Order of Operations?

Order of Operations tells us the correct order to solve math problems that have more than one operation.

Without a set order, people might get different answers to the same problem!

Example: 6 + 3 × 2 = ?

• If you add first: 6 + 3 = 9, then 9 × 2 = 18 ❌
• If you multiply first: 3 × 2 = 6, then 6 + 6 = 12 ✓

The correct answer is 12! We need rules to know which way is right.

PEMDAS - The Order

We use PEMDAS to remember the order:

P - Parentheses ( ) E - Exponents (powers like 3²) M - Multiplication D - Division A - Addition S - Subtraction

Memory tricks:

• "Please Excuse My Dear Aunt Sally"
• "Purple Elephants March Down Avenues Smoking"

The Rules

Rule 1: Do operations inside Parentheses first Rule 2: Then do Exponents (powers) Rule 3: Then Multiply and Divide from LEFT to RIGHT Rule 4: Finally Add and Subtract from LEFT to RIGHT

Important: Multiplication and Division are equal (do left to right) Important: Addition and Subtraction are equal (do left to right)

Step-by-Step Examples

Example 1: Simple Problem

Problem: 8 + 5 × 2

Step 1: Check for parentheses - None ✓ Step 2: Check for exponents - None ✓ Step 3: Multiply: 5 × 2 = 10 Step 4: Add: 8 + 10 = 18

Answer: 18 ✓

Example 2: With Parentheses

Problem: (6 + 2) × 3

Step 1: Parentheses first: 6 + 2 = 8 Step 2: Multiply: 8 × 3 = 24

Answer: 24 ✓

Example 3: Multiple Operations

Problem: 20 - 3 × 4 + 6

Step 1: Multiply: 3 × 4 = 12 Step 2: Now we have: 20 - 12 + 6 Step 3: Subtract and add from left to right:

• 20 - 12 = 8
• 8 + 6 = 14

Answer: 14 ✓

Example 4: With Division

Problem: 12 ÷ 3 + 4 × 2

Step 1: Multiply AND divide (left to right):

• 12 ÷ 3 = 4
• 4 × 2 = 8

Step 2: Now we have: 4 + 8 Step 3: Add: 4 + 8 = 12

Answer: 12 ✓

Example 5: Everything Together

Problem: 5 + (8 - 3) × 2² - 1

Step 1: Parentheses: 8 - 3 = 5 Step 2: Exponents: 2² = 4 Step 3: Now we have: 5 + 5 × 4 - 1 Step 4: Multiply: 5 × 4 = 20 Step 5: Now we have: 5 + 20 - 1 Step 6: Add and subtract (left to right):

• 5 + 20 = 25
• 25 - 1 = 24

Answer: 24 ✓

Common Mistakes

❌ Mistake 1: Doing operations from left to right without following PEMDAS

Wrong: 6 + 3 × 2 = 9 × 2 = 18 ✗ Right: 6 + 3 × 2 = 6 + 6 = 12 ✓

❌ Mistake 2: Forgetting parentheses change everything

Wrong: 2 × (3 + 4) = 6 + 4 = 10 ✗ Right: 2 × (3 + 4) = 2 × 7 = 14 ✓

❌ Mistake 3: Doing addition before multiplication

Wrong: 5 + 2 × 3 = 7 × 3 = 21 ✗ Right: 5 + 2 × 3 = 5 + 6 = 11 ✓

Left to Right Rule

When operations are EQUAL in order, go from LEFT to RIGHT:

Example: 12 ÷ 3 × 2

• Multiply and divide are equal
• Go left to right: 12 ÷ 3 = 4, then 4 × 2 = 8 ✓

Example: 10 - 3 + 2

• Subtract and add are equal
• Go left to right: 10 - 3 = 7, then 7 + 2 = 9 ✓

Why Parentheses Matter

Parentheses can completely change an answer!

Without parentheses: 3 + 4 × 2 = 3 + 8 = 11 With parentheses: (3 + 4) × 2 = 7 × 2 = 14

Same numbers, different answer!

Practice Strategy

When solving:

Step 1: Underline or circle parentheses - do these first! Step 2: Look for exponents - do these next Step 3: Find all × and ÷ - do left to right Step 4: Find all + and - - do left to right Step 5: Check your work!

Real-World Applications

Shopping: You buy 3 notebooks at $2 each plus a $5 backpack. Cost = 3 × $2 +$5 = $6 +$5 = $11 ✓

Cooking: Recipe says: (2 cups sugar + 1 cup) × 2 (to double it) = 3 cups × 2 = 6 cups ✓

Using Parentheses to Show Work

You can add parentheses to show what you did:

Problem: 10 + 2 × 5

Show your work: = 10 + (2 × 5) [multiply first] = 10 + 10 [then add] = 20 ✓

Exponents Review

Exponents show repeated multiplication:

• 2³ = 2 × 2 × 2 = 8
• 5² = 5 × 5 = 25
• 3² = 3 × 3 = 9

Do these AFTER parentheses but BEFORE multiply/divide!

The No-PEMDAS Problems

Some problems don't need PEMDAS because they only have one operation:

• 5 + 3 + 2 = 10 (just addition)
• 20 - 5 - 3 = 12 (just subtraction, left to right)
• 4 × 2 × 3 = 24 (just multiplication)

Checking Your Work

Method 1: Do it again following PEMDAS carefully Method 2: Use a calculator (it follows PEMDAS!) Method 3: Ask: Does my answer make sense?

Tips for Success

✅ Write out each step ✅ Cross out operations as you do them ✅ Use parentheses to show your thinking ✅ Double-check you followed PEMDAS order ✅ Practice, practice, practice!

Summary

Remember PEMDAS:

1. Parentheses first
2. Exponents second
3. Multiply and Divide from left to right
4. Add and Subtract from left to right

The Golden Rule: Follow the order every time, and you'll get the right answer! ✓