GCSE Maths Practice: integers-and-directed-numbers

Question 2 of 10

This problem uses negative numbers to represent movement below sea level. You’ll combine multiple changes to find the final depth.

‹ ›

\( \begin{array}{l}\text{A diver descends 5 m, then 3 m more. What is his depth?}\end{array} \)

Choose one option:

Think of zero as sea level. Downward movement subtracts, upward adds. The total shows position relative to zero.

Understanding Directed Numbers in Context

Directed numbers are used to describe values that have both size and direction. Real-life examples include temperature changes, bank balances, and altitude. In this question, a diver’s movements below sea level are represented by negative numbers because they show direction downwards.

Problem Context

A diver starts at sea level (0 m). He dives down 5 m, then goes 3 m deeper. Each downward movement is represented as a negative change. His total depth is therefore 0 − 5 − 3 = −8 m. The negative sign means he is 8 m below the surface.

How to Approach These Questions

  1. Identify which movements are positive (upward or increase) and which are negative (downward or decrease).
  2. Combine them using addition and subtraction.
  3. The final sign tells you the overall direction relative to the starting point.

Worked Examples

  • A submarine at −60 m rises 15 m → −60 + 15 = −45 m.
  • An elevator starts on floor 0, goes down 3 floors, then up 1 → 0 − 3 + 1 = −2 floors.
  • A hiker 10 m below sea level climbs 12 m → −10 + 12 = 2 m above sea level.

Common Misunderstandings

  • Forgetting that downward means subtracting (negative change).
  • Adding values without considering direction.
  • Interpreting −8 as “8 above” instead of “8 below”.

Real-Life Links

Directed numbers appear in aviation (altitude), science (temperature and pressure), and banking (debits and credits). Understanding their meaning helps interpret data with positive and negative values correctly.

FAQs

  • Q: Why is downward negative?
    A: Because we define upward as the positive direction by convention.
  • Q: Can a negative plus another negative ever become positive?
    A: Only in multiplication or division, not in addition.
  • Q: Does this rule change if we start below zero?
    A: No, it’s the same — you continue counting relative to zero.

Study Tip

When you see a word problem, visualise the number line. Movement upward means add, downward means subtract. This mental image makes directed number questions easy and intuitive in GCSE Maths.