Integers And Directed Numbers Quizzes

Introduction

Integers and directed numbers are fundamental concepts in GCSE Maths, forming the basis for arithmetic, algebra, and problem-solving. Integers include positive numbers, negative numbers, and zero. Directed numbers are numbers with a positive or negative sign that indicate direction, debt, or temperature below zero. Understanding integers and directed numbers is essential for calculations involving addition, subtraction, multiplication, division, and real-life applications such as temperature changes, bank balances, and altitude differences.

For example, -5 + 3 represents a movement from -5 to 3 on the number line, giving a result of -2. Mastery of integers and directed numbers ensures students can handle negative values confidently, which is vital for GCSE success.

Core Concepts

Definition of Integers

Integers are whole numbers that can be positive, negative, or zero:

• Positive integers: 1, 2, 3, …
• Negative integers: -1, -2, -3, …
• Zero: 0

Directed Numbers

Directed numbers have a positive (+) or negative (-) sign indicating direction, loss, or deficit:

• Positive numbers (gain, above zero): +7, +3
• Negative numbers (loss, below zero): -5, -2

They are often represented on a number line:

• Zero is the central point.
• Numbers to the right of zero are positive.
• Numbers to the left of zero are negative.

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Number Line Representation

The number line is a visual way to represent integers and directed numbers. Moving to the right increases the value, and moving to the left decreases the value:

• Example: $$-3 < 0 < 4$$
• Example: Adding 5 to -2 moves 5 units right: $$-2 + 5 = 3$$

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Absolute Value

The absolute value of a number is its distance from zero on the number line, ignoring the sign. Notation: $$|a|$$

• Example: $$|-7| = 7$$
• Example: $$|5| = 5$$

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Adding Integers

Rules:

• Same sign: Add their absolute values and keep the sign.
• Example: $$-3 + (-5) = -(3+5) = -8$$
• Different signs: Subtract the smaller absolute value from the larger and keep the sign of the larger.
• Example: $$7 + (-4) = 7 - 4 = 3$$
• Example: $$-6 + 2 = -(6-2) = -4$$

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Subtracting Integers

Rule: Subtracting a number is the same as adding its opposite:

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

• Example: $$5 - 8 = 5 + (-8) = -3$$
• Example: $$-3 - (-6) = -3 + 6 = 3$$

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Multiplying Integers

Rules:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Example: $$3 × 4 = 12$$
• Example: $$-3 × -5 = 15$$
• Example: $$-2 × 6 = -12$$

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Dividing Integers

Rules are the same as multiplication:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Example: $$12 ÷ 3 = 4$$
• Example: $$-12 ÷ -4 = 3$$
• Example: $$-15 ÷ 5 = -3$$

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Order of Operations with Integers

Follow BODMAS/BIDMAS rules:

• Brackets → Orders → Division → Multiplication → Addition → Subtraction
• Example: $$-3 + 5 × (-2) = -3 + (-10) = -13$$
• Example: $$(-4 + 6) × 3 = 2 × 3 = 6$$

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Worked Examples

Example 1 (Foundation): Adding integers

Calculate $$-7 + 5$$

Step 1: Different signs → subtract smaller from larger: $$7 - 5 = 2$$

Step 2: Take the sign of larger absolute value: -7 → result = -2

Answer: $$-2$$

Example 2 (Foundation): Subtracting integers

Calculate $$4 - (-3)$$

Step 1: Subtracting negative → add opposite: $$4 + 3 = 7$$

Answer: $$7$$

Example 3 (Higher): Multiplying integers

Calculate $$-6 × -4$$

Rule: Negative × Negative = Positive

Step 1: Multiply absolute values: 6 × 4 = 24

Answer: $$24$$

Example 4 (Higher): Dividing integers

Calculate $$-20 ÷ 5$$

Rule: Negative ÷ Positive = Negative

Step 1: Divide absolute values: 20 ÷ 5 = 4

Answer: $$-4$$

Example 5 (Higher): Using BODMAS

Calculate $$-3 + 6 ÷ -2$$

Step 1: Division first: 6 ÷ -2 = -3

Step 2: Addition: -3 + (-3) = -6

Answer: $$-6$$

Example 6 (Higher): Absolute value

Calculate $$|-8 + 3|$$

Step 1: Add integers: -8 + 3 = -5

Step 2: Absolute value: |-5| = 5

Answer: $$5$$

Example 7 (Real-life application)

Temperature drops from +5°C to -3°C. Calculate the change.

Change = Final - Initial = -3 - 5 = -8

Answer: Temperature decreased by 8°C

Example 8 (Real-life application)

Bank balance: £-120 (debt). Deposit £50. Calculate new balance.

New balance = -120 + 50 = -70

Answer: £-70 (still in debt)

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Common Mistakes

• Confusing signs when adding or subtracting integers
• Incorrectly multiplying or dividing negative numbers
• Forgetting BODMAS when performing multi-step calculations
• Misinterpreting absolute value
• Mixing real-life context with signs incorrectly (e.g., temperature or debt)

Tips to avoid errors:

• Always check the sign of each number before operation
• Use a number line to visualize additions and subtractions
• Apply BODMAS carefully for multi-step problems
• Double-check negative results in real-life context
• Practice absolute values separately to build confidence

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Applications

• Temperature: Positive and negative values indicate above or below zero
• Finance: Bank balances (credits and debits)
• Altitude: Above and below sea level
• Problem-solving: Algebra with positive and negative numbers

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Strategies & Tips

• Visualize problems on a number line to understand direction
• Memorize rules for adding, subtracting, multiplying, and dividing negatives
• Check calculations with absolute values
• Always use parentheses carefully in multi-step operations
• Practice real-life word problems to reinforce understanding

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Summary / Call-to-Action

Integers and directed numbers are essential for GCSE Maths and everyday applications. Mastery of addition, subtraction, multiplication, division, absolute value, and BODMAS with directed numbers ensures confidence in algebra, number problems, and real-life calculations.

Next Steps:

• Attempt quizzes on integers and directed numbers to reinforce learning
• Practice multi-step calculations using BODMAS
• Apply knowledge to real-life scenarios such as temperature changes, debts, and altitude differences
• Challenge yourself with higher-level problems combining multiple operations and integers

Consistent practice and visualization on the number line will ensure confidence and accuracy in all topics involving integers and directed numbers.