Lesson 5 ~30 min Unit 1 · Number +85 XP

Order of Operations

BODMAS — the universal agreement for which operation comes first. Get this right and every calculation follows.

Today's hook: If you type 2 + 3 × 4 into a calculator, you better hope it knows BODMAS. If it doesn't, it'll tell you 20 instead of 14. Who decides the order? We all do.

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Think First

warm-up

Before you read on — quickly: What is 4 + 3 × 2? Try it mentally, then explain why you got that answer.

Record your answer in your workbook.

The Big Idea

+5 XP

BODMAS (Brackets, Orders/Indices, Division and Multiplication, Addition and Subtraction) is the agreed order of operations. Without it, everyone would get different answers for the same calculation.

B = Brackets (highest priority). O = Orders/Indices (powers, roots). DM = Division & Multiplication (equal priority, left to right). AS = Addition & Subtraction (equal priority, left to right). Brackets always win. They break the normal order.

Brackets > Orders > ×÷ > +−

Brackets always win

They override everything else. Always do brackets first.

× and ÷ are best friends

Same priority. Work left to right.

+ and − are also best friends

Same priority. Work left to right.

What You'll Master

objectives

Know

The BODMAS order and what each letter means
That × and ÷ have equal priority
That + and − have equal priority

Understand

Why brackets override everything
Why left-to-right matters for same-priority ops
How calculators use BODMAS

Can Do

Evaluate any expression with multiple operations
Insert brackets to make an equation true
Spot and fix BODMAS errors

Words You Need

vocabulary

BODMASThe order of operations: Brackets, Orders, Division, Multiplication, Addition, Subtraction.

BracketsSymbols ( ) that group operations. Always evaluate inside brackets first.

IndicesPowers and roots. 2³ means 2 × 2 × 2. Also called "orders" in BODMAS.

EvaluateFind the numerical value of an expression by performing the operations.

ExpressionA combination of numbers and operations without an equals sign.

Nested bracketsBrackets inside brackets. Work from the innermost outwards.

Spot the Trap

heads-up

Wrong: 2 + 3 × 4 = 20. You did + before ×. BODMAS says × comes before +. Correct: 2 + 12 = 14.

Right: BODMAS: × and ÷ come before + and −. 2 + 3 × 4 = 2 + 12 = 14.

Wrong: 18 ÷ 3 × 2 = 3. You did 3 × 2 = 6 first, then 18 ÷ 6. But × and ÷ are equal priority — work left to right: 18 ÷ 3 = 6, then 6 × 2 = 12.

Right: Same priority operations: work left to right. 18 ÷ 3 × 2 = 6 × 2 = 12.

What is BODMAS?

+5 XP

BODMAS is the agreed order for performing operations. It ensures everyone gets the same answer. Each letter represents a priority level.

B = Brackets: ( ), [ ], { }. Do the innermost first. O = Orders: powers (2³) and roots (√9). DM = Division and Multiplication: work left to right. AS = Addition and Subtraction: work left to right. If two operations have the same priority, the one on the left goes first.

Brackets > Orders > ×÷ > +−

Memorise BODMAS

Write it at the top of every page until it's automatic.

Left to right tiebreak

Same priority? The leftmost operation wins.

One step per line

Never skip steps. Show every operation.

Brackets First

+5 XP

Brackets are the highest priority. They override the normal order. Nested brackets: start from the innermost and work outward.

For 2 × (3 + 4)²: brackets first: 3 + 4 = 7. Then orders: 7² = 49. Then multiply: 2 × 49 = 98. For nested brackets like [10 − (2 + 3)] × 2: innermost first: 2 + 3 = 5, then 10 − 5 = 5, then 5 × 2 = 10.

Innermost brackets first, work outward

Start from inside

Innermost bracket first, then work outward.

Fraction bars = brackets

A fraction bar groups numerator and denominator like brackets.

Rewrite after each bracket

Simplify one bracket, rewrite, then do the next.

Common Tricks

+5 XP

Watch out for these common BODMAS traps: implied multiplication, negative signs as brackets, and calculator pitfalls.

Implied multiplication: 2(3+4) means 2 × (3+4) = 14. The bracket still has priority. Negative signs: −3² = −9 (the square applies to 3 only, then the minus). But (−3)² = 9 (the bracket makes the minus part of the base). Calculator caution: Basic calculators may not follow BODMAS. Use a scientific calculator or work it out yourself.

Brackets change everything — use them wisely

−a² vs (−a)²

−3² = −9 (power first). (−3)² = 9 (bracket first).

Use a scientific calc

Basic calculators may give wrong answers. Always check mentally.

Add brackets generously

When in doubt, add brackets. They make order explicit.

Watch Me Solve It · 3 examples

Watch Me Solve It · Basic BODMAS

+15 XP per step

PROBLEM

Evaluate 12 + 8 ÷ 2 × 3.

1
Division first (leftmost × or ÷)
8 ÷ 2 = 4
÷ and × have equal priority. 8 ÷ 2 is leftmost, so do it first.
2
Multiplication next
4 × 3 = 12
Now expression is 12 + 12. The × is the only remaining high-priority op.
3
Addition last
12 + 12 = 24
+ is lowest priority. Add the two 12s together.

Answer24

Watch Me Solve It · Brackets and powers

+15 XP per step

PROBLEM

Evaluate (5 + 3)² − 4 × (10 − 6).

1
Brackets first
(5 + 3) = 8, (10 − 6) = 4
Expression becomes: 8² − 4 × 4
2
Orders (powers) next
8² = 64
Expression becomes: 64 − 4 × 4
3
Multiplication, then subtraction
4 × 4 = 16, then 64 − 16 = 48

Answer48

Watch Me Solve It · Nested brackets

+15 XP per step

PROBLEM

Evaluate [18 − (4 + 2) × 2] ÷ 3 + 5².

1
Innermost bracket
(4 + 2) = 6
Expression: [18 − 6 × 2] ÷ 3 + 25
2
Multiplication inside brackets
6 × 2 = 12
Expression: [18 − 12] ÷ 3 + 25
3
Finish brackets, then ÷, then +
18 − 12 = 6, 6 ÷ 3 = 2, 2 + 25 = 27

Answer27

Common Pitfalls

heads-up

Doing + before ×

2 + 3 × 4 = 14, not 20. × and ÷ always come before + and −. This is the most common BODMAS error.

Fix: Circle the × and ÷ operations first. Do them before any + or −.

Forgetting indices come before ×

2 × 3² = 2 × 9 = 18, not 6² = 36. Powers are calculated before multiplication.

Fix: Orders (powers) are second priority. Calculate them right after brackets.

Ignoring left-to-right for same priority

18 ÷ 3 × 2: left to right gives 12, but doing × first gives 3. Always left to right for same-priority operations.

Fix: When operations have equal priority, the leftmost one goes first. Write one step per line.

Copy Into Your Books

BODMAS Order

B — Brackets (highest)
O — Orders / Indices
D/M — Division & Multiplication
A/S — Addition & Subtraction

Brackets

Always first
Innermost outward
Fraction bars act like brackets

Same Priority

× and ÷: left to right
+ and −: left to right
Never skip steps

Calculator Check

Use scientific calculator
Verify mentally first
Add brackets to be safe

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · BODMAS

4 problems

Four drill problems to master the order of operations. Work each, then reveal the answer.

1 20 − 3 × 4 + 6

× first: 3 × 4 = 12. Then 20 − 12 + 6 = 8 + 6 = 14.14
2 (8 − 3)² ÷ 5 + 2 × 4

Bracket: 8 − 3 = 5. Power: 5² = 25. ÷: 25 ÷ 5 = 5. ×: 2 × 4 = 8. +: 5 + 8 = 13.13
3 50 ÷ 5 + 5 × 2 − 3

÷ and × first (left to right): 50 ÷ 5 = 10, 5 × 2 = 10. Then +: 10 + 10 − 3 = 17.17
4 Insert brackets to make true: 4 + 3 × 5 − 2 = 33

Need 4 + 3 = 7, then 7 × 5 = 35... no. Try (4 + 3) × 5 − 2 = 7 × 5 − 2 = 35 − 2 = 33. Yes!(4 + 3) × 5 − 2 = 33

Complete in your workbook.

Quick Check · 5 questions

6 + 4 × 2 = ?

+10 XP

(3 + 2)² − 4 = ?

+10 XP

18 ÷ 3 × 2 = ?

+10 XP

2 + 3² × 2 = ?

+10 XP

[10 − (2 + 3)] × 2 = ?

+10 XP

Show Your Working · 3 questions

Show Your Working

11 marks total

ApplyMedium4 MARKS

Q6. Evaluate 30 − 4 × (2 + 3)² ÷ 5. Show BODMAS at each step.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. Insert one pair of brackets to make 24 ÷ 3 + 5 × 2 = 6 true.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A student calculates 8 + 4 × 2 as 24. Explain their mistake, show the correct working, and explain why BODMAS gives a different answer.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — 6 + 4 × 2 = 6 + 8 = 14 (× before +).

2. A — (3 + 2)² − 4 = 25 − 4 = 21 (brackets, then power).

3. A — 18 ÷ 3 × 2 = 6 × 2 = 12 (left to right).

4. B — 2 + 3² × 2 = 2 + 9 × 2 = 2 + 18 = 20 (power, then ×, then +).

5. B — [10 − (2 + 3)] × 2 = [10 − 5] × 2 = 5 × 2 = 10.

Show Your Working Model Answers

Q6 (4 marks): Brackets: (2 + 3) = 5 [1]. Orders: 5² = 25 [1]. DM: 4 × 25 = 100, 100 ÷ 5 = 20 [1]. AS: 30 − 20 = 10 [1].

Q7 (3 marks): Place brackets around (3 + 5): 24 ÷ (3 + 5) × 2 [1]. Brackets first: 3 + 5 = 8 [1]. Then left-to-right: 24 ÷ 8 = 3, 3 × 2 = 6 [1].

Q8 (4 marks): Student did (8 + 4) × 2 = 12 × 2 = 24, adding before multiplying [1]. Correct: 8 + 4 × 2 = 8 + 8 = 16 [1]. BODMAS: × and ÷ have higher priority than + and − [1]. The + operation must wait until all × and ÷ are done [1].

Stretch Challenge · +25 XP, +10 coins

The Four Fours Puzzle

Using exactly four 4s and any operations (+, −, ×, ÷, brackets), write expressions for the numbers 1, 2, 3, 4, and 5. For example, (4 + 4) ÷ (4 + 4) = 1. Can you find all five?

Reveal solution

1 = (4 ÷ 4) × (4 ÷ 4) or 4 ÷ 4 + 4 − 4. 2 = (4 ÷ 4) + (4 ÷ 4). 3 = (4 + 4 + 4) ÷ 4. 4 = 4 + (4 − 4) × 4. 5 = (4 × 4 + 4) ÷ 4. There are many solutions for each!

Quick Review

BODMAS

B > O > DM > AS

Brackets

Always first, innermost out

× ÷ before + −

Higher priority operations first

Left to right

Same priority: leftmost first

One step per line

Never skip, always show working

Check

Verify with a different method

Interactive: BODMAS Calculator

Type in expressions and see BODMAS applied step by step with visual highlighting.

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