Order of Operations (BODMAS)

Let’s Start with a Question! 🤔

What is 3 + 4 × 2? Is it 14 (if you add first) or 11 (if you multiply first)? Without a standard rule, different people would get different answers! That’s exactly why mathematicians around the world agree on the Order of Operations - a set of rules that ensures everyone calculates expressions the same way, every time.

What Is the Order of Operations?

The Order of Operations is a set of rules that tells us which calculations to do first when an expression has multiple operations. In the UK, we remember it by the acronym BODMAS (or BIDMAS):

B - Brackets (also called parentheses) O - Orders (powers, indices, exponents, roots) D - Division M - Multiplication A - Addition S - Subtraction

(In some countries, they use PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction - it’s the same thing!)

The Critical Rule: Work from Top to Bottom!

Operations higher in BODMAS are done before operations lower down:

1. First: Do anything inside Brackets
2. Second: Calculate Orders (powers like 5² or roots like √16)
3. Third: Do Division and Multiplication (left to right)
4. Fourth: Do Addition and Subtraction (left to right)

Important note: Division and Multiplication have equal priority - do them left to right. Same for Addition and Subtraction - left to right!

Why Does This Matter?

Without BODMAS, the expression 3 + 4 × 2 could mean:

• Add first: (3 + 4) × 2 = 7 × 2 = 14 ❌
• Multiply first: 3 + (4 × 2) = 3 + 8 = 11 ✓

BODMAS says: multiplication before addition, so the answer is 11!

The Order of Operations ensures:

• Everyone gets the same answer
• Calculators and computers work correctly
• Mathematical communication is clear and consistent

Teacher’s Insight 👨‍🏫

Here’s what I’ve learned from teaching thousands of students: The most common mistake isn’t forgetting BODMAS - it’s working strictly left to right, like reading a sentence! When my students understand that maths isn’t read like English (left to right), but has its own priority system, everything clicks.

My top tip: Don’t just memorize “BODMAS” - understand the reasoning! Brackets come first because they let us override the usual order. Multiplication/division come before addition/subtraction because they’re “stronger” operations (they combine numbers more fundamentally). Think of BODMAS as traffic rules for mathematics - they prevent crashes and confusion!

Key Vocabulary

• BODMAS/BIDMAS/PEMDAS: Acronyms for remembering the order of operations
• Brackets (Parentheses): Symbols ( ) that group operations together
• Orders (Indices/Exponents): Powers like 3² = 3 × 3 = 9
• Expression: A mathematical phrase with numbers and operations
• Operation: A mathematical action: +, -, ×, ÷
• Priority: Which operation should be done first
• Left to right: When operations have equal priority, work from left to right

Understanding Each Step of BODMAS

B - Brackets First! ( )

Brackets have the highest priority. Always calculate what’s inside brackets before anything else.

Why? Brackets let you override the normal order. They’re like saying “do this part first!”

Example: (5 + 3) × 2

• Brackets first: 5 + 3 = 8
• Then multiply: 8 × 2 = 16

O - Orders (Powers and Roots)

Orders (also called indices or exponents) are calculated next.

Examples:

• 5² = 5 × 5 = 25
• 2³ = 2 × 2 × 2 = 8
• √16 = 4

In expressions: 2 + 3²

• Orders first: 3² = 9
• Then add: 2 + 9 = 11

D & M - Division and Multiplication (Equal Priority!)

Division and multiplication have the same priority. When both appear, work left to right.

Example: 12 ÷ 3 × 2

• Left to right: 12 ÷ 3 = 4
• Then: 4 × 2 = 8
• NOT 12 ÷ 6 = 2!

A & S - Addition and Subtraction (Equal Priority!)

Addition and subtraction also have equal priority. Work left to right.

Example: 10 - 3 + 2

• Left to right: 10 - 3 = 7
• Then: 7 + 2 = 9
• NOT 10 - 5 = 5!

Worked Examples

Example 1: Basic BODMAS

Problem: 3 + 4 × 2

Solution: 11

Detailed Explanation:

• According to BODMAS, multiplication comes before addition
• First, multiply: 4 × 2 = 8
• Then add: 3 + 8 = 11
• If we mistakenly worked left to right: 3 + 4 = 7, then 7 × 2 = 14 (WRONG!)

Think about it: The order matters! Always check the operations and follow BODMAS.

Example 2: Brackets Change Everything

Problem: (8 + 2) × 3

Solution: 30

Detailed Explanation:

• Brackets come first in BODMAS!
• Calculate inside brackets: 8 + 2 = 10
• Then multiply: 10 × 3 = 30
• Compare to: 8 + 2 × 3 = 8 + 6 = 14 (different answer without brackets!)

Think about it: Brackets override the usual order - they’re the “boss” of BODMAS!

Example 3: Multiple Operations

Problem: 20 - 3 × 4 + 8 ÷ 2

Solution: 12

Detailed Explanation:

• Step 1: Identify operations: subtraction, multiplication, addition, division
• Step 2: Do multiplication and division first (left to right):
  • 3 × 4 = 12
  • 8 ÷ 2 = 4
• Step 3: Expression becomes: 20 - 12 + 4
• Step 4: Do addition and subtraction (left to right):
  • 20 - 12 = 8
  • 8 + 4 = 12

Think about it: Handle multiply/divide in one pass, then add/subtract in another!

Example 4: Brackets Within Expressions

Problem: 5 × (3 + 2) - 8

Solution: 17

Detailed Explanation:

• Step 1: Brackets first: (3 + 2) = 5
• Step 2: Expression becomes: 5 × 5 - 8
• Step 3: Multiplication before subtraction: 5 × 5 = 25
• Step 4: Finally subtract: 25 - 8 = 17

Think about it: Work through BODMAS step by step, simplifying as you go!

Example 5: Powers (Orders)

Problem: 2 + 3² × 4

Solution: 38

Detailed Explanation:

• Step 1: Orders first: 3² = 3 × 3 = 9
• Step 2: Expression becomes: 2 + 9 × 4
• Step 3: Multiplication before addition: 9 × 4 = 36
• Step 4: Finally add: 2 + 36 = 38
• Common mistake: 2 + 3 = 5, then 5² = 25, then 25 × 4 = 100 (WRONG!)

Think about it: Powers are calculated before multiplication, and both before addition!

Example 6: Division and Multiplication Together

Problem: 24 ÷ 4 × 2

Solution: 12

Detailed Explanation:

• Division and multiplication have equal priority
• Work left to right: 24 ÷ 4 = 6
• Then: 6 × 2 = 12
• Common mistake: 4 × 2 = 8, then 24 ÷ 8 = 3 (WRONG - not left to right!)

Think about it: Equal priority means left to right - don’t look ahead!

Example 7: Complex Expression

Problem: 4 × (7 - 3) + 5² ÷ 5

Solution: 21

Detailed Explanation:

• Step 1: Brackets: (7 - 3) = 4
• Step 2: Orders: 5² = 25
• Step 3: Expression becomes: 4 × 4 + 25 ÷ 5
• Step 4: Multiplication and division (left to right): 4 × 4 = 16, then 25 ÷ 5 = 5
• Step 5: Expression becomes: 16 + 5
• Step 6: Addition: 16 + 5 = 21

Think about it: Complex expressions are just multiple steps of BODMAS - take it slowly!

Common Misconceptions & How to Avoid Them

Misconception 1: “Work strictly left to right”

The Truth: You don’t read maths like English! Operations have different priorities that override left-to-right order.

How to think about it correctly: First identify which operations are present, then follow BODMAS order, not reading order!

Misconception 2: “Multiplication always comes before division”

The Truth: Multiplication and division have equal priority. When both appear, work left to right.

How to think about it correctly: The “D” and “M” in BODMAS are on the same level. Neither comes first!

Misconception 3: “Brackets only affect what’s directly inside”

The Truth: Calculate everything inside brackets first, but then that answer interacts with the rest of the expression according to BODMAS.

How to think about it correctly: Brackets give you a sub-answer, which you then use in the larger expression.

Misconception 4: “You must do all multiplications before any additions”

The Truth: You do any multiplications/divisions that appear, then move to additions/subtractions. But if there are multiple of each, you work left to right within that priority level.

How to think about it correctly: Complete one priority level before moving to the next, but within each level, work left to right.

Common Errors to Watch Out For

Memory Aids & Tricks

BODMAS Rhyme

“Ben’s Old Dirty Monkey Ate Six Bananas” Or make up your own silly sentence using B-O-D-M-A-S!

Visual: The BODMAS Staircase

Think of operations as a staircase:

🔝 Brackets (highest priority)
  ↓ Orders
    ↓ Division & Multiplication
      ↓ Addition & Subtraction (lowest priority)

Always start at the top!

The “Underline” Method

When solving complex expressions:

1. Underline all brackets - do these first
2. Circle all powers - do these next
3. Box multiplication and division - do left to right
4. Finally, do addition and subtraction left to right

Remember: DM and AS are Partners!

• D and M hold hands (equal priority, left to right)
• A and S hold hands (equal priority, left to right)

Calculator Check

Modern calculators follow BODMAS automatically. Use one to check your answers!

Practice Problems

Easy Level (Basic Order)

1. 5 + 3 × 2 Answer: 11 (multiply first: 3 × 2 = 6, then add: 5 + 6 = 11)

2. 12 ÷ 4 + 2 Answer: 5 (divide first: 12 ÷ 4 = 3, then add: 3 + 2 = 5)

3. (6 + 4) ÷ 2 Answer: 5 (brackets first: 6 + 4 = 10, then divide: 10 ÷ 2 = 5)

4. 10 - 2 × 3 Answer: 4 (multiply first: 2 × 3 = 6, then subtract: 10 - 6 = 4)

Medium Level (Multiple Operations)

5. 15 - 3 × 3 + 6 Answer: 12 (multiply: 3 × 3 = 9, then left to right: 15 - 9 + 6 = 12)

6. 4 × (7 - 3) + 5 Answer: 21 (brackets: 7 - 3 = 4, multiply: 4 × 4 = 16, add: 16 + 5 = 21)

7. 20 ÷ 4 × 2 Answer: 10 (left to right: 20 ÷ 4 = 5, then 5 × 2 = 10)

8. 3² + 4 × 2 Answer: 17 (power: 3² = 9, multiply: 4 × 2 = 8, add: 9 + 8 = 17)

Challenge Level (Complex Expressions)

9. 5 × (3 + 2²) - 10 ÷ 2 Answer: 30 (brackets: 2² = 4, 3 + 4 = 7; multiply & divide: 5 × 7 = 35, 10 ÷ 2 = 5; subtract: 35 - 5 = 30)

10. 100 - (6 + 2) × 5 + 3² Answer: 69 (brackets: 6 + 2 = 8, orders: 3² = 9, multiply: 8 × 5 = 40, then: 100 - 40 + 9 = 69)

Real-World Applications

Shopping with Discounts 🛍️

Scenario: You buy 4 items at £15 each and 3 items at £8 each, then use a £20 discount voucher. What’s your total?

Solution:

• Expression: 4 × 15 + 3 × 8 - 20
• Multiply first: 60 + 24 - 20
• Left to right: 84 - 20 = £64

Why this matters: BODMAS ensures you multiply quantities by prices before applying discounts!

Calculating Total Cost 💰

Scenario: Concert tickets are £30 each, plus a £5 booking fee per ticket, for 3 people. Total cost?

Solution:

• Expression: 3 × (30 + 5)
• Brackets first: 30 + 5 = 35
• Multiply: 3 × 35 = £105
• Without brackets: 3 × 30 + 5 = 95 (WRONG - that’s only 1 booking fee!)

Why this matters: Brackets ensure you add the booking fee to each ticket before multiplying by the number of people!

Recipe Adjustments 👨‍🍳

Scenario: A recipe uses 2 cups of flour plus 3 cups of sugar, doubled, then you remove 1 cup. How much total?

Solution:

• Expression: 2 × (2 + 3) - 1
• Brackets: 2 + 3 = 5
• Multiply: 2 × 5 = 10
• Subtract: 10 - 1 = 9 cups

Why this matters: BODMAS ensures you double the combined amount, not each ingredient separately!

Construction Measurements 📏

Scenario: A room is (5 + 2) metres long and 3 metres wide. What’s the area in square metres?

Solution:

• Expression: (5 + 2) × 3
• Brackets: 5 + 2 = 7
• Multiply: 7 × 3 = 21 m²

Why this matters: Measurements often require adding dimensions before multiplying to find area!

Sports Scoring ⚽

Scenario: In a tournament, you get 3 points for a win and 1 for a draw. Your team has 5 wins, 2 draws, and then loses 4 points for a penalty. What’s your score?

Solution:

• Expression: 5 × 3 + 2 × 1 - 4
• Multiply: 15 + 2 - 4
• Left to right: 17 - 4 = 13 points

Why this matters: Sports calculations follow BODMAS to ensure fair scoring!

Study Tips for Mastering Order of Operations

1. Memorize BODMAS Thoroughly

Make it automatic - you should be able to recite it in your sleep!

2. Write Out Each Step

Don’t try to do everything in your head. Show your working clearly, one step at a time.

3. Use Brackets to Check Understanding

Try adding brackets to show what you did: 3 + 4 × 2 means 3 + (4 × 2), not (3 + 4) × 2

4. Practice with Real Calculators

Use a scientific calculator and compare your manual work with its answers.

5. Highlight Different Operations

Use different colors for different operations to see the order clearly.

6. Start Simple, Build Up

Master two operations before tackling three or four!

7. Create Your Own Problems

Make up expressions and solve them - teaching yourself reinforces learning!

How to Check Your Answers

1. Work through it again: Do the problem a second time independently
2. Use a calculator: Scientific calculators follow BODMAS automatically
3. Try reversing: If 3 + 4 × 2 = 11, then 11 - 8 should equal 3 ✓
4. Check with brackets: Add brackets to show your order: 3 + (4 × 2) = 11
5. Ask “does this make sense?”: If you bought 2 £5 items + £3, would you pay £13 or £16?

Common sense check: The answer should be reasonable. If 5 + 3 × 2 gives you 100, something went wrong!

Extension Ideas for Fast Learners

• Explore nested brackets: 2 × ((3 + 4) × (5 - 2))
• Learn about fraction bars acting as brackets: (3 + 5)/(2 × 4)
• Study how programming languages use parentheses and order
• Investigate expressions with multiple powers: 2³ × 3²
• Practice with algebraic expressions: 3x + 2x × 5
• Explore how the equals sign works with order of operations
• Learn about how different calculators handle order (some don’t follow BODMAS!)

Parent & Teacher Notes

Building Understanding: BODMAS isn’t arbitrary - it reflects mathematical structure. Multiplication is “stronger” than addition because it represents repeated addition.

Common Struggles: If a student struggles with BODMAS, check if they:

• Can identify all the operations in an expression
• Understand what each operation means
• Can work methodically through steps
• Remember the acronym and its meaning

Differentiation Tips:

• Struggling learners: Start with two operations only, use visual aids like the staircase
• On-track learners: Practice with three or four operations, introduce brackets
• Advanced learners: Challenge with nested brackets, algebraic expressions, and real-world problems

Teaching Strategy: Use color-coding! Have students highlight brackets in one color, powers in another, multiplication/division in a third, and addition/subtraction in a fourth. This helps them see the order visually.

Common Mistake Prevention: Students often work left to right because that’s how they read. Repeatedly emphasize: “Maths isn’t English - it has its own rules!”

Real-World Connection: Show how BODMAS appears in spreadsheet formulas, programming, engineering calculations, and financial computations. It’s not just a school rule - it’s how the mathematical world works!

Assessment Tips: Test understanding, not just memorization:

• Can students explain WHY multiplication comes before addition?
• Can they create their own BODMAS problems?
• Can they spot and correct errors in worked examples?

Remember: The Order of Operations is the grammar of mathematics. Just as we need grammar rules to communicate clearly in English, we need BODMAS to communicate clearly in maths. Master these rules, and you can solve any expression with confidence! 🌟