Welcome to Bearing, Direction, and Position!
Have you ever wondered how an airplane pilot finds an airport in the middle of a foggy night, or how a ship captain sails across thousands of kilometres of open ocean without any road signs? They use the language of bearing, direction, and position!
In this chapter, we explore how mathematicians, explorers, and navigators describe exactly where an object is and which way it is heading. Don't worry if this seems new or challenging at first—we will build the ideas step-by-step, starting with turns you already know and moving to precise mathematical bearings.
Note: For pinpointing points on a map using pairs of numbers, make sure to explore our related chapter on Coordinates on a grid.
---Part 1: Finding Our Way — Direction and Turns
Before we can use precise numbers, we need to understand how we describe directions in the real world.
1. The Compass Rose
You may already be familiar with the four main directions, known as the cardinal points:
• North (N): Points straight up towards the North Pole.
• East (E): Points to the right (where the Sun rises).
• South (S): Points straight down towards the South Pole.
• West (W): Points to the left (where the Sun sets).
Memory Trick: To remember the order going clockwise around a compass, use a silly sentence like: "Never Eat Soggy Waffles" or "Naughty Elephants Squirt Water"!
Between these four main points are the intercardinal points:
• North-East (NE): Halfway between North and East.
• South-East (SE): Halfway between South and East.
• South-West (SW): Halfway between South and West.
• North-West (NW): Halfway between North and West.
2. Turns as Rotations
In mathematics, an angle is a measure of a turn or rotation around a fixed point. Think of the hands of an analogue clock moving around the face:
• A quarter turn clockwise is a right angle: \(90^\circ\).
• A half turn clockwise forms a straight line: \(180^\circ\).
• A three-quarter turn clockwise is: \(270^\circ\).
• A full turn all the way back to the start is: \(360^\circ\).
Key Takeaway: Compass directions match these turns perfectly. If you face North and turn a quarter turn to the right (clockwise), you face East (\(90^\circ\)). Turn another quarter turn, and you face South (\(180^\circ\)).
---Part 2: What is a Bearing?
Saying "sail north-east" is helpful, but what if an island is just slightly to the left of north-east? If a pilot is off by even a tiny bit, they could miss their runway by miles! To solve this, navigators invented bearings.
A bearing is a precise direction given as an angle in degrees.
The Three Golden Rules of Bearings
To avoid any confusion across the world, mathematicians and navigators follow three golden rules:
1. Always measure from North: You always imagine standing at the starting point, facing directly North.
2. Always measure Clockwise: You turn to the right, just like the hands on a clock.
3. Always write with Three Figures (Digits): Every bearing must have three digits. If the angle is less than \(100^\circ\), put one or two zeros in front!
Let's look at why the three-figure rule is so important:
• If an angle is \(45^\circ\), its bearing is written as \(045^\circ\) (pronounced "zero-four-five degrees").
• If an angle is \(5^\circ\), its bearing is written as \(005^\circ\) (pronounced "zero-zero-five degrees").
• If an angle is \(230^\circ\), it already has three digits, so it is written as \(230^\circ\).
Did you know? The three-digit rule was created so that radio signals between ships and pilots would never be misunderstood. If a pilot hears "four-five", static might have cut off a number! If they hear "zero-four-five", they know they received the complete message.
Bearings of the Main Compass Points
• North (N): \(000^\circ\) (or \(360^\circ\) for a full turn)
• North-East (NE): \(045^\circ\)
• East (E): \(090^\circ\)
• South-East (SE): \(135^\circ\)
• South (S): \(180^\circ\)
• South-West (SW): \(225^\circ\)
• West (W): \(270^\circ\)
• North-West (NW): \(315^\circ\)
Key Takeaway: A bearing tells you which direction to face. It always starts at North, turns clockwise, and uses three digits.
---Part 3: Measuring and Drawing Bearings
To measure or draw a bearing accurately on a map or diagram, we use a protractor.
Step-by-Step: Measuring the Bearing of Point B from Point A
Always pay close attention to the word "FROM". The word from tells you where you are standing!
Step 1: Find your starting point. If the question says "the bearing of B from A", you are standing at A.
Step 2: Draw a North line. At point A, draw a straight vertical line pointing directly upwards towards the top of the page. Label it with an N and an arrow.
Step 3: Connect the points. Draw a light straight pencil line connecting point A to point B.
Step 4: Align your protractor. Place the centre hole of the protractor directly over point A. Line up the \(0^\circ\) line with your North line.
Step 5: Measure clockwise. Follow the numbers around in a clockwise direction until you reach the line pointing to B.
Step 6: Write as three digits. For example, if the protractor reads \(68^\circ\), write the bearing as \(068^\circ\).
What if the angle is greater than \(180^\circ\)?
Don't worry if the turn goes past a straight line! A semi-circular protractor only measures up to \(180^\circ\), but you have two easy ways to solve this:
• Method 1 (Adding from South): A straight line pointing South is \(180^\circ\). Measure the extra clockwise turn past the South line, and add it to \(180^\circ\).
Example: If the direction is \(50^\circ\) past South, the bearing is \(180^\circ + 50^\circ = 230^\circ\).
• Method 2 (Subtracting from a full turn): Measure the smaller anti-clockwise gap back up to North, and subtract that angle from \(360^\circ\).
Example: If the line is \(40^\circ\) before reaching North again, the bearing is \(360^\circ - 40^\circ = 320^\circ\).
Key Takeaway: Always place your protractor at the "FROM" location, keep the \(0^\circ\) mark pointing North, and count clockwise.
---Part 4: Understanding Position and Perspective
Direction depends entirely on where you are standing. This is an important conceptual idea in shape and space: position is relative to a reference point.
Comparing "A from B" and "B from A"
Imagine two friends, Maya and Liam, standing in an open field:
• Maya looks at Liam. Liam is directly to her East (bearing \(090^\circ\)).
• Liam looks back at Maya. Which way is he looking? He is looking directly to his West (bearing \(270^\circ\))!
Even though they are looking at each other along the very same line, their bearings are completely different because their reference points (their starting positions) are different.
Concept: Opposite Directions (Half-Turns)
Because looking back the other way is a complete reversal in direction, it is a half-turn of \(180^\circ\):
• If the bearing from A to B is less than \(180^\circ\), the return bearing from B to A is: Original Bearing \(+ 180^\circ\).
Example: If bearing of B from A is \(070^\circ\), the bearing of A from B is \(070^\circ + 180^\circ = 250^\circ\).
• If the bearing from A to B is greater than \(180^\circ\), the return bearing from B to A is: Original Bearing \(- 180^\circ\).
Example: If bearing of B from A is \(310^\circ\), the bearing of A from B is \(310^\circ - 180^\circ = 130^\circ\).
Key Takeaway: When you reverse your journey, you turn by a half turn (\(180^\circ\)).
---Part 5: Common Mistakes and How to Avoid Them
Mistake 1: Measuring anti-clockwise.
Why it happens: Protractors have two sets of numbers (inner and outer scales), so it is easy to read the wrong scale.
How to avoid it: Always check your direction against the compass. If your line points South-East, the angle must be between \(090^\circ\) and \(180^\circ\). If your protractor shows \(45^\circ\), you read the wrong scale!
Mistake 2: Forgetting the leading zero.
Why it happens: Writing \(65^\circ\) feels natural because we write normal angles that way.
How to avoid it: Count the digits! If there are only two digits, place a zero at the front: \(065^\circ\).
Mistake 3: Starting at the wrong point.
Why it happens: Reading "the bearing of Town P from Town Q" quickly and putting the protractor on P.
How to avoid it: Circle the word FROM. The letter after "from" is where your North line and protractor must go!
Quick Summary Checklist
• North Line: Always points straight up from the reference point.
• Clockwise: Always turn to the right.
• Three Digits: Angles under \(100^\circ\) get leading zeros (e.g., \(035^\circ\)).
• From vs To: Always measure at the location you are travelling from.
• Back Bearings: Add or subtract \(180^\circ\) to find the reverse path.