Integers And Directed Numbers Quizzes

GCSE Integers and Directed Numbers Quiz (Foundation) – 10 Practice Questions with Answers

Difficulty: Foundation

Curriculum: GCSE

Start Quiz

GCSE Integers and Directed Numbers Quiz (Higher) – 10 Exam-Style Negative Number Questions

Difficulty: Higher

Curriculum: GCSE

Start Quiz
Integers And Directed Numbers featured image

Visual overview of Integers And Directed Numbers.

Introduction

Integers and directed numbers underpin arithmetic, algebra, and problem-solving across GCSE Maths. Integers are whole numbers (… −3, −2, −1, 0, 1, 2, 3 …). Directed numbers carry a sign to show direction or context (gain/loss, above/below zero). Mastering them lets you add, subtract, multiply, and divide confidently in real-life settings such as temperature changes, bank balances, and altitude.

For example, \(-5 + 3\) moves three units to the right from \(-5\) on the number line, giving \(-2\). Comfort with negative values is essential for GCSE success.

Core Concepts

Definition of Integers

  • Positive integers: 1, 2, 3, …
  • Negative integers: −1, −2, −3, …
  • Zero: 0
Tip: Integers have no fractional or decimal parts.

Directed Numbers

Directed numbers include a + or − sign to indicate direction, change, profit/loss, or above/below zero.

  • Positive (gain/above zero): \(+7, +3\)
  • Negative (loss/below zero): \(−5, −2\)

On a number line: zero is central; right is positive, left is negative.

Number Line Representation

A number line makes the structure of integers visible. Moving right increases value; moving left decreases value.

  • \(−3 < 0 < 4\)
  • Adding 5 to \(-2\): move 5 right → \(-2 + 5 = 3\)

Absolute Value

The absolute value \(|a|\) is the distance from zero, ignoring sign.

  • \(|−7| = 7\)
  • \(|5| = 5\)
Think: distance can’t be negative.

Adding Integers

  • Same sign: add absolute values, keep the sign. Example: \(−3 + (−5) = −8\)
  • Different signs: subtract the smaller absolute value from the larger; keep the sign of the larger absolute value. Examples: \(7 + (−4) = 3\), \(−6 + 2 = −4\)
Number-line check: different signs = a “tug-of-war” towards the stronger (larger magnitude).

Subtracting Integers

Subtracting is adding the opposite:

\(a − b = a + (−b)\)

  • \(5 − 8 = 5 + (−8) = −3\)
  • \(−3 − (−6) = −3 + 6 = 3\)

Multiplying Integers

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

Examples

\(3 × 4 = 12\)\;|\;\(−3 × −5 = 15\)\;|\;\(−2 × 6 = −12\)

Dividing Integers

Same sign rules as multiplication:

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

Examples

\(12 ÷ 3 = 4\)\;|\;\(−12 ÷ −4 = 3\)\;|\;\(−15 ÷ 5 = −3\)

Order of Operations with Integers

Follow BIDMAS/BODMAS: Brackets → Indices → Division/Multiplication → Addition/Subtraction.

  • \(−3 + 5 × (−2) = −3 + (−10) = −13\)
  • \((−4 + 6) × 3 = 2 × 3 = 6\)
Tip: Do multiplication/division before addition/subtraction, even with negatives.

Worked Examples

Example 1 (Foundation): Adding integers

Compute \(−7 + 5\).

Different signs → \(7 − 5 = 2\), keep sign of larger magnitude (7 is negative) → \(−2\).

Example 2 (Foundation): Subtracting integers

\(4 − (−3) = 4 + 3 = \boldsymbol{7}\)

Example 3 (Higher): Multiplying integers

\(−6 × −4 = \boldsymbol{24}\) (negative × negative = positive)

Example 4 (Higher): Dividing integers

\(−20 ÷ 5 = \boldsymbol{−4}\) (negative ÷ positive = negative)

Example 5 (Higher): Using BIDMAS

\(−3 + 6 ÷ −2\): do division first → \(6 ÷ −2 = −3\); then add → \(−3 + (−3) = \boldsymbol{−6}\)

Example 6 (Higher): Absolute value

\(|−8 + 3| = |−5| = \boldsymbol{5}\)

Example 7 (Real life): Temperature

From \(+5^\circ\mathrm{C}\) to \(−3^\circ\mathrm{C}\): change \(= −3 − 5 = \boldsymbol{−8^\circ\mathrm{C}}\) (an 8°C decrease).

Example 8 (Real life): Bank balance

Debt £−120, deposit £50 → new balance \(= −120 + 50 = \boldsymbol{−70}\) (still in debt).

Common Mistakes

  • Dropping signs when adding/subtracting.
  • Assuming negative × negative is negative (it’s positive!).
  • Ignoring BIDMAS in multi-step problems.
  • Treating \(|a|\) as “make it positive and keep going” without finishing inside first.
  • Mismatching context (e.g., reporting a “−8°C change” as “−8°C temperature”).
How to avoid: Circle each sign before computing; convert subtraction to “add the opposite”; and sanity-check with a number line or quick estimate.

Applications

  • Temperature: above/below zero.
  • Finance: credits (positive) and debits (negative).
  • Altitude: above/below sea level.
  • Algebra: simplifying expressions with negative coefficients.

Strategies & Tips

  • Sketch a quick number line for tricky additions/subtractions.
  • Memorise the sign rules for × and ÷.
  • Rewrite subtraction as “add the opposite.”
  • Handle brackets and indices first (BIDMAS) in multi-step questions.
  • Use absolute value to check distances or sizes regardless of sign.

Summary / Call-to-Action

Confident use of integers and directed numbers unlocks success across GCSE topics. Practise sign rules, number-line reasoning, absolute value, and BIDMAS so negative values feel routine rather than risky.

  • Attempt interactive quizzes on integer operations.
  • Mix real-life contexts (temperature, money, altitude) into practice.
  • Push into multi-step problems combining several operations.