Operations and Order of Operations

This lesson covers the four arithmetic operations (addition, subtraction, multiplication and division) including working with negative numbers, and the correct order in which operations must be carried out. The Edexcel GCSE Mathematics (1MA1) specification requires fluency in these skills, which appear throughout both Foundation and Higher tiers.

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Key Vocabulary

Term	Meaning
Sum	The result of addition
Difference	The result of subtraction
Product	The result of multiplication
Quotient	The result of division
BIDMAS	Brackets, Indices, Division, Multiplication, Addition, Subtraction
Integer	A whole number (positive, negative or zero)
Inverse operation	The operation that "undoes" another (e.g. subtraction undoes addition)

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The Four Operations

Addition and Subtraction of Integers

When adding or subtracting, use a number line or column method.

Column Addition Example: 4,587 + 2,946

  4587
+ 2946
------
  7533

Carry the 1 where column totals exceed 9.

Column Subtraction Example: 5,032 - 1,478

  5032
- 1478
------
  3554

Use borrowing (exchange) where needed.

Multiplication of Integers

Use long multiplication or grid method.

Worked Example 1 (Grid Method)

Calculate $47 \times 36$47×36

	40	7
30	1200	210
6	240	42

Total: 1200 + 210 + 240 + 42 = 1692

Division of Integers

Use short division (bus stop method) or long division.

Worked Example 2

Calculate $756 \div 12$756÷12

Using short division: $756 \div 12 = 63$756÷12=63

Check: $63 \times 12 = 756$63×12=756 ✓

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Operations with Negative Numbers

Rules for Addition and Subtraction

Operation	Rule	Example
Adding a positive	Move right on number line	-3 + 5 = 2
Adding a negative	Move left on number line	4 + (-7) = 4 - 7 = -3
Subtracting a positive	Move left on number line	2 - 8 = -6
Subtracting a negative	Move right on number line	-3 - (-5) = -3 + 5 = 2

Key Rule: Two adjacent signs combine: + + = +, + - = -, - + = -, - - = +

Worked Example 3

Calculate -8 - (-3) + (-2)

• -8 - (-3) = -8 + 3 = -5
• -5 + (-2) = -5 - 2 = -7

Answer: -7

Rules for Multiplication and Division

Signs	Result	Example
Same signs $(+ \times +$(+×+ or $- \times -)$−×−)	Positive	$(-4) \times (-3) = 12$(−4)×(−3)=12
Different signs $(+ \times -$(+×− or $- \times +)$−×+)	Negative	$5 \times (-6) = -30$5×(−6)=−30

The same rules apply to division:

• $(-20) \div (-4) = 5$(−20)÷(−4)=5
• $(-20) \div 4 = -5$(−20)÷4=−5

Worked Example 4

Calculate $(-3) \times 4 \times (-2)$(−3)×4×(−2)

• $(-3) \times 4 = -12$(−3)×4=−12
• $(-12) \times (-2) = 24$(−12)×(−2)=24

Answer: 24

Edexcel Exam Tip: Questions involving negative numbers are common on Paper 1 (non-calculator). Take extra care with double negatives — a very common source of lost marks.

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BIDMAS — Order of Operations

When a calculation contains more than one operation, you must follow the correct priority order.

B — Brackets (work out what's inside brackets first) I — Indices (powers and roots) D — Division M — Multiplication A — Addition S — Subtraction

Important: Division and Multiplication have equal priority — work from left to right. Similarly, Addition and Subtraction have equal priority — work from left to right.

Worked Example 5

Calculate $3 + 4 \times 2$3+4×2

• Multiplication before addition: $4 \times 2 = 8$4×2=8
• Then: 3 + 8 = 11

Answer: 11 (NOT 14)

Worked Example 6

Calculate $(5 + 3)^{2} \div 4 - 1$(5+3)2÷4−1

Step 1 (Brackets): 5 + 3 = 8 Step 2 (Indices): $8^{2} = 64$82=64 Step 3 (Division): $64 \div 4 = 16$64÷4=16 Step 4 (Subtraction): 16 - 1 = 15

Answer: 15

Worked Example 7

Calculate $20 - 12 \div 3 + 2 \times 5$20−12÷3+2×5

Step 1: Division: $12 \div 3 = 4$12÷3=4 Step 2: Multiplication: $2 \times 5 = 10$2×5=10 Step 3: Left to right: 20 - 4 + 10 = 26

Answer: 26

Worked Example 8

Calculate $48 \div (2 + 6) \times 3$48÷(2+6)×3

Step 1 (Brackets): 2 + 6 = 8 Step 2 (left to right, D/M equal priority): $48 \div 8 = 6$48÷8=6, then $6 \times 3 = 18$6×3=18

Answer: 18

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Using BIDMAS with Negative Numbers

Worked Example 9

Calculate $(-2)^{3} + 4 \times (-3)$(−2)3+4×(−3)

Step 1 (Indices): $(-2)^{3} = -8$(−2)3=−8 Step 2 (Multiplication): $4 \times (-3) = -12$4×(−3)=−12 Step 3 (Addition): -8 + (-12) = -8 - 12 = -20

Answer: -20

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Inserting Brackets to Make Statements True

A common Edexcel question type asks you to insert brackets to make an equation correct.

Worked Example 10

Insert one pair of brackets to make this statement true: $3 + 5 \times 2 - 1 = 8$3+5×2−1=8

Without brackets: 3 + 10 - 1 = 12 ✗

Try: $3 + 5 \times$3+5× (2 - 1) $= 3 + 5 \times 1 = 3 + 5 = 8$=3+5×1=3+5=8 ✓

Answer: $3 + 5 \times (2 - 1) = 8$3+5×(2−1)=8

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Common Mistakes and Misconceptions

Mistake	Correction
Calculating left to right ignoring BIDMAS	Always apply operations in the correct priority order
Thinking $\times$× comes before $\div$÷	They have EQUAL priority — work left to right
Thinking + comes before -	They have EQUAL priority — work left to right
$(-2)^{2} = -4$(−2)2=−4	$(-2)^{2} = (-2) \times (-2) = 4$(−2)2=(−2)×(−2)=4. Note: $-2^{2} = -(2^{2}) = -4$−22=−(22)=−4
Subtracting a negative makes it "more negative"	Subtracting a negative is the same as adding: 5 - (-3) = 8

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Practice Problems

1. Calculate: -5 + 8 - (-3) - 4
2. Calculate: $(-6) \times (-4) \div 8$(−6)×(−4)÷8
3. Calculate: $3 + 2 \times (8 - 5)^{2}$3+2×(8−5)2
4. Calculate: $100 - 48 \div (2 + 6) \times 3$100−48÷(2+6)×3
5. Calculate: $(-3)^{2} - (-2)^{3}$(−3)2−(−2)3
6. Insert brackets to make this true: $8 - 3 \times 2 + 1 = 11$8−3×2+1=11
7. Calculate: $5 \times 4 - 18 \div 3 + 2$5×4−18÷3+2
8. Explain why $(-1)^{100} = 1$(−1)100=1 but $(-1)^{99} = -1$(−1)99=−1.

Answers