Mixed Operations: Whole Numbers - Sixth Grade

Complete Notes & Formulas

The Four Operations

Operation	Symbol	Result	Example
Addition	+	Sum	15 + 8 = 23
Subtraction	−	Difference	23 − 8 = 15
Multiplication	× or •	Product	6 × 7 = 42
Division	÷ or /	Quotient	42 ÷ 6 = 7

1. Order of Operations (PEMDAS)

What is PEMDAS?

PEMDAS is an acronym that helps remember the correct order to perform operations in mathematical expressions.

PEMDAS

P - Parentheses ( ) [ ] { }

E - Exponents (powers, roots)

M - Multiplication (left to right)

D - Division (left to right)

A - Addition (left to right)

S - Subtraction (left to right)

Memory Trick: "Please Excuse My Dear Aunt Sally"

Also known as: BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Important Rules

1. Multiplication and Division have equal priority - work left to right

2. Addition and Subtraction have equal priority - work left to right

3. Always do operations inside parentheses first

4. Work from left to right when operations have equal priority

2. Evaluate Numerical Expressions

Example 1: Basic Order of Operations

Problem: Evaluate 12 + 8 × 3

Step 1: Identify operations: Addition and Multiplication

Step 2: Multiplication comes before addition

8 × 3 = 24

Step 3: Now add

12 + 24 = 36

Answer: 36

Common Mistake: 12 + 8 = 20, then 20 × 3 = 60 ✗ (Wrong!)

Example 2: With Parentheses

Problem: Evaluate (12 + 8) × 3

Step 1: Parentheses first!

(12 + 8) = 20

Step 2: Now multiply

20 × 3 = 60

Answer: 60

Example 3: Multiple Operations

Problem: Evaluate 24 ÷ 6 + 3 × 5 − 2

Step 1: Division and Multiplication (left to right)

24 ÷ 6 = 4

Expression becomes: 4 + 3 × 5 − 2

Step 2: Continue with multiplication

3 × 5 = 15

Expression becomes: 4 + 15 − 2

Step 3: Addition and Subtraction (left to right)

4 + 15 = 19

19 − 2 = 17

Answer: 17

Example 4: With Exponents

Problem: Evaluate 5 + 2³ × 3

Step 1: Exponents first

2³ = 8

Expression becomes: 5 + 8 × 3

Step 2: Multiplication

8 × 3 = 24

Step 3: Addition

5 + 24 = 29

Answer: 29

3. Word Problem Keywords

Operation	Keywords/Phrases
Addition	Sum, Total, Plus, Combined, Altogether, Increase, More than
Subtraction	Difference, Minus, Less, Decrease, Fewer, Remain, Left, How many more
Multiplication	Product, Times, Each, Per, Of, At this rate, Double, Triple
Division	Quotient, Divide, Split, Share, Per, Each group, Average, Distribute

4. Single Operation Word Problems

Steps to Solve Word Problems

Step 1: Read the problem carefully

Step 2: Identify what you need to find

Step 3: Look for keywords to determine the operation

Step 4: Write the equation

Step 5: Solve and check if the answer makes sense

Example: Addition Word Problem

Problem: A store sold 345 shirts on Monday and 428 shirts on Tuesday. How many shirts did they sell in total?

Keyword: "total" → Addition

345 + 428 = 773

Answer: 773 shirts

5. Multi-Step Word Problems

What are Multi-Step Problems?

Multi-step problems require more than one operation to solve. You must solve the problem in steps, using the answer from one step in the next step.

Example 1: Two-Step Problem

Problem: Maria bought 5 boxes of pencils with 12 pencils in each box. She gave away 18 pencils to her friends. How many pencils does she have left?

Step 1: Find total pencils (multiplication)

5 × 12 = 60 pencils

Step 2: Subtract pencils given away

60 − 18 = 42 pencils

Answer: 42 pencils

Example 2: Three-Step Problem

Problem: A school has 3 grades with 4 classes in each grade. Each class has 25 students. How many students are in the school?

Step 1: Find total classes

3 × 4 = 12 classes

Step 2: Find total students

12 × 25 = 300 students

Answer: 300 students

Example 3: Complex Multi-Step

Problem: Jake has $500. He buys 3 books for $15 each and 2 games for $25 each. How much money does he have left?

Step 1: Cost of books

3 × $15 = $45

Step 2: Cost of games

2 × $25 = $50

Step 3: Total spent

$45 + $50 = $95

Step 4: Money left

$500 − $95 = $405

Answer: $405

6. Estimate to Solve Word Problems

Why Estimate?

Estimating helps you check if your answer is reasonable before doing exact calculations.

Example: Estimation

Problem: A restaurant serves 289 customers per day. About how many customers does it serve in 31 days?

Estimate:

289 ≈ 300 (round to nearest hundred)

31 ≈ 30 (round to nearest ten)

300 × 30 = 9,000

Exact calculation:

289 × 31 = 8,959

Answer: About 9,000 customers (estimate) or exactly 8,959

✓ The estimate is close to the actual answer, so it's reasonable!

7. Identify Reasonable Answers

How to Check Reasonableness

1. Estimate the answer using rounding

2. Ask: Is the answer too big or too small?

3. Check: Does the answer make sense in context?

4. Verify: Use inverse operations

Example: Check Reasonableness

Problem: A student calculated 48 × 52 = 249,600. Is this reasonable?

Check: Estimate

48 ≈ 50, 52 ≈ 50

50 × 50 = 2,500

✗ No! 249,600 is NOT reasonable (way too big!)

✓ Correct answer: 48 × 52 = 2,496

8. Identify Mistakes in Order of Operations

Common Errors

Error #1: Working Left to Right Without Following PEMDAS

Expression: 10 + 5 × 2

❌ Wrong: 10 + 5 = 15, then 15 × 2 = 30

✓ Correct: 5 × 2 = 10, then 10 + 10 = 20

Reason: Multiplication comes before addition!

Error #2: Ignoring Parentheses

Expression: 3 × (4 + 6)

❌ Wrong: 3 × 4 = 12, then 12 + 6 = 18

✓ Correct: (4 + 6) = 10, then 3 × 10 = 30

Reason: Do parentheses first!

Error #3: Exponents After Multiplication

Expression: 2 × 3²

❌ Wrong: 2 × 3 = 6, then 6² = 36

✓ Correct: 3² = 9, then 2 × 9 = 18

Reason: Exponents come before multiplication!

Quick Reference: Problem-Solving Strategy

1. READ the problem carefully

2. IDENTIFY what you need to find

3. LOOK for keywords

4. WRITE the equation or expression

5. SOLVE using PEMDAS

6. CHECK if the answer is reasonable

💡 Important Tips to Remember

✓ PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract

✓ Parentheses ALWAYS first!

✓ Multiply and Divide have equal priority - work left to right

✓ Add and Subtract have equal priority - work left to right

✓ Estimate first to check if answer is reasonable

✓ Multi-step problems: solve one step at a time

✓ Look for keywords to identify operations

✓ Check your work using inverse operations

✓ Ask: "Does this make sense?"

✓ Show your work step by step

🧠 Memory Tricks & Strategies

PEMDAS Memory Phrase:

"Please Excuse My Dear Aunt Sally"

Alternative: BODMAS

Brackets, Orders, Division, Multiplication, Addition, Subtraction

MD and AS Go Together:

Multiply/Divide = same level (left to right)

Add/Subtract = same level (left to right)

Word Problem Strategy:

"Draw a picture, find what's hidden, solve step by step!"

Check Reasonableness:

"Round it, estimate it, compare it!"

Multi-Step Problems:

"One step at a time, don't rush the climb!"

Master Mixed Operations! ➕ ➖ ✖️ ➗

Remember PEMDAS and practice daily to build confidence!