Working With Negative Numbers Quizzes

Introduction

Working with negative numbers is a fundamental skill in GCSE Maths. Negative numbers, also called directed numbers, appear in a variety of contexts such as temperatures below zero, debts, losses, elevations below sea level, and algebraic expressions. Mastering operations with negative numbers allows students to perform addition, subtraction, multiplication, division, and solve real-life problems confidently.

For example, if the temperature drops from +5°C to -3°C, the change can be calculated using negative numbers. Understanding rules for negative numbers ensures accurate calculations and prevents common mistakes in exams and everyday applications.

Core Concepts

Definition of Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-) in front of them.

• Examples: -1, -5, -12, -100
• Zero is neither positive nor negative.

Number Line Representation

Negative numbers are shown to the left of zero on the number line, while positive numbers are shown to the right:

• …, -3, -2, -1, 0, 1, 2, 3, …
• Absolute value represents distance from zero: $$|-4| = 4$$

Adding Negative Numbers

Rules:

• Same signs: add their absolute values and keep the sign
• Different signs: subtract smaller absolute value from larger and keep the sign of the larger

Examples:

• -5 + (-3) = -(5+3) = -8
• -7 + 4 = -(7-4) = -3
• 5 + (-8) = -(8-5) = -3

Subtracting Negative Numbers

Subtracting a number is the same as adding its opposite:

Formula: $$a - b = a + (-b)$$

Examples:

• 5 - (-3) = 5 + 3 = 8
• -4 - (-7) = -4 + 7 = 3
• -6 - 2 = -6 + (-2) = -8

Multiplying Negative Numbers

Rules:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

Examples:

• 3 × -5 = -15
• -4 × -6 = 24
• 7 × 2 = 14

Dividing Negative Numbers

Rules are similar to multiplication:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative

Examples:

• 12 ÷ -3 = -4
• -18 ÷ -6 = 3
• 20 ÷ 5 = 4

Order of Operations with Negative Numbers

Follow BODMAS/BIDMAS rules when calculations include negative numbers, brackets, indices, multiplication, division, addition, and subtraction:

Example:

$$-3 + 5 × (-2)$$

• Step 1: Multiplication first: 5 × (-2) = -10
• Step 2: Addition: -3 + (-10) = -13

Absolute Value and Distance

The absolute value of a number is the distance from zero on the number line:

• |-7| = 7
• |5| = 5

Absolute value is useful when measuring differences or distances regardless of direction.

Negative Numbers in Real-Life Contexts

• Temperature: -5°C indicates 5 degrees below zero
• Finance: £-50 indicates debt
• Altitude: -10 m below sea level
• Science: negative charges or deficits

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Worked Examples

Example 1 (Foundation): Adding negative numbers

Calculate: -8 + (-5)

• Same signs: add absolute values: 8 + 5 = 13
• Keep sign: -13

Example 2 (Foundation): Adding numbers with different signs

Calculate: 7 + (-10)

• Different signs: subtract smaller absolute value from larger: 10 - 7 = 3
• Sign of larger: -10 → result = -3

Example 3 (Higher): Subtracting negative numbers

Calculate: -4 - (-6)

• Subtracting negative → add opposite: -4 + 6 = 2

Example 4 (Higher): Multiplication

Calculate: -3 × -7

• Negative × Negative = Positive
• 3 × 7 = 21
• Answer: 21

Example 5 (Higher): Division

Calculate: -24 ÷ 6

• Negative ÷ Positive = Negative
• 24 ÷ 6 = 4 → -4

Example 6 (Higher): Order of operations

Calculate: -2 + 5 × (-3) - (-4)

• Step 1: Multiplication: 5 × (-3) = -15
• Step 2: Addition/subtraction left to right: -2 + (-15) = -17
• Step 3: Subtract (-4): -17 - (-4) = -17 + 4 = -13
• Answer: -13

Example 7 (Real-life): Temperature change

Temperature drops from +7°C to -5°C. Calculate the change:

• Change = Final - Initial = -5 - 7 = -12
• Temperature decreased by 12°C

Example 8 (Real-life): Bank balance

Bank balance: £-150, deposit £50. New balance?

• -150 + 50 = -100
• Still in debt: £-100

Example 9 (Real-life): Altitude

Sea level is 0 m. Mountain is +350 m, valley is -120 m. Difference in height?

• Difference = 350 - (-120) = 350 + 120 = 470 m

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Common Mistakes

• Confusing signs when adding or subtracting
• Multiplying or dividing negative numbers incorrectly
• Ignoring BODMAS with negative numbers
• Misinterpreting negative results in real-life contexts
• Confusing absolute value with the original negative number

Tips to avoid errors:

• Visualize calculations on a number line
• Always check the sign of each number before operation
• Apply BODMAS carefully for multi-step calculations
• Use absolute values for verification
• Practice real-life examples for better understanding

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Applications

• Temperature: Weather changes above and below zero
• Finance: Tracking debts and credits
• Altitude: Heights above and below sea level
• Algebra: Solving equations with negative values
• Science: Negative quantities in physics or chemistry

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Strategies & Tips

• Memorize rules for adding, subtracting, multiplying, and dividing negative numbers
• Practice using number lines to visualize operations
• Check answers using absolute values or estimation
• Apply rules carefully in multi-step problems
• Use real-life examples to reinforce understanding

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Summary / Call-to-Action

Working with negative numbers is a critical skill in GCSE Maths. By mastering operations, BODMAS, and real-life applications involving negative numbers, students can confidently solve a wide range of problems. Consistent practice ensures accuracy, understanding, and readiness for exam scenarios.

Next Steps:

• Attempt quizzes on negative numbers to reinforce learning
• Practice multi-step problems involving negative numbers
• Apply operations to real-life contexts such as temperature, finance, and altitude
• Challenge yourself with higher-level algebra involving negative numbers

With systematic practice, working with negative numbers becomes intuitive and error-free.