Bearings
Three-figure bearing notation, back bearings, and the angle between two bearings
About two lessons of 45 to 60 minutes
North is always 000°, and every direction is a number between 0 and 360
A bearing describes a direction as a single angle, measured CLOCKWISE from north, and always written with exactly three digits: 009° for a direction just east of north, 090° for due east, 270° for due west. Ships, planes and hikers all use bearings instead of vague words like 'north-east-ish', because a bearing is exact and never ambiguous.
Two bearing skills come up again and again on GCSE papers. The BACK bearing (the direction back the way you came) is always exactly 180° different from the bearing you started with. And when two directions are both measured from the same point, the ANGLE between them is simply the difference between the two bearings (adjusted to stay under 180°, since the angle between two directions is never reflex).
- A hiker's bearing of 072° from base campwritten as 072°, always 3 digits, never 72°
- A boat's bearing back to harbourif the outward bearing was 072°, the bearing back is 072° + 180° = 252°
- Two planes seen from the same control towerthe angle between their bearings is just the difference between the two bearing numbers
- Orienteering: a bearing of 000° is due north, 180° is due southbearings run the full clockwise circle from 000° to 359°
What students will be able to do
Students will write an angle as a three-figure bearing, find the back bearing of a given bearing (the return direction), and find the angle between two bearings measured from a common point, including after a stated clockwise turn.
- I can write any angle measured clockwise from north as a three-figure bearing (with leading zeros where needed).
- I can find a back bearing: add 180° if the given bearing is less than 180°, or subtract 180° if it is 180° or more.
- I can find the angle between two bearings measured from the same point by finding their difference (adjusted so it is never more than 180°).
- I can find a new bearing after a stated clockwise turn by adding the turn (and subtracting 360° if needed).
Standards this unit teaches
- GCSE Geometry and measures #15UK GCSE Mathematics (DfE, England)Bearings
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Geometry and measures" section, "Mensuration and calculation", item 15, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings". This item is entirely standard type: Foundation content for every GCSE student, not just the underlined-and-assessed subset. This unit builds the fully computable half of the item (three-figure bearing notation, back bearings, and the angle between two bearings taken from a common point); the "measure with a physical protractor" half has no computed, never-wrong answer key, the same reasoning that already excludes ruler/compass constructions across this codebase's UK Key Stage 3 units.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Bearing
- a direction measured as an angle CLOCKWISE from north, always written as three digits (e.g. 007°, 090°, 315°)
- Three-figure bearing
- the standard way to write a bearing, always exactly 3 digits, using leading zeros for angles under 100° (e.g. 8° is written 008°)
- Back bearing
- the bearing of the return direction; always exactly 180° different from the original bearing
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Writing and reading three-figure bearings
ConcreteA bearing is always written with exactly three digits, even when the angle itself is small. An angle of 8° clockwise from north is NOT written '8°'; it is written '008°'.
Bearings run the full circle clockwise from north (000°) through east (090°), south (180°), west (270°), and back to north (360° = 000°).
Write an angle of 45° clockwise from north as a three-figure bearing, and state which compass direction it is closest to.
- 45° needs one leading zero to reach three digits: 045°.
- 045° is exactly halfway between north (000°) and east (090°), i.e. north-east.
Answer: 045°
- Why does a bearing of 90° need to be written as '090°' rather than just '90°'?
- What bearing is due south?
2. Back bearings, and the angle between two directions
AbstractA back bearing is always exactly 180° away from the original bearing, since 'facing the opposite way' means turning through a straight angle. When two bearings share the same starting point, the angle between the two directions is their difference.
The bearing of B from A is 060°. The bearing of A from B (the back bearing) is 060° + 180° = 240°. If the original bearing had been 300° instead (180° or more), you would SUBTRACT 180° instead: 300° - 180° = 120°.
From a control tower O, plane A has bearing 050° and plane B has bearing 200°. Find the angle between the two bearings (angle AOB), and find the bearing of plane C, which is 40° clockwise from plane A as seen from O.
- Difference between the bearings: |200° - 050°| = 150°. Since 150° is 180° or less, the angle AOB is simply 150°.
- Bearing of C: turning clockwise means ADDING the turn: 050° + 40° = 090°.
Answer: Angle AOB = 150°. Bearing of C = 090°.
- Why is the back bearing rule different depending on whether the original bearing is above or below 180°?
- If the difference between two bearings came out as 250°, would that be the actual angle between the two directions? Why or why not?
Common misconceptions and how to address them
MisconceptionThe back bearing is always found by adding 180°, even when that goes over 360°.
Why it happens: Students memorise 'add 180' as the whole rule and forget the subtract-180 case (and the wraparound) for bearings that are already 180° or more.
How to address it: If the bearing is LESS than 180°, add 180°. If it is 180° or MORE, subtract 180° instead (adding would go over 360°, which is not a valid bearing). Either way, the result should still be a sensible bearing between 000° and 359°.
MisconceptionThe angle between two bearings is always just their positive difference, however large.
Why it happens: Students subtract the smaller from the larger without checking whether that difference represents the SHORT way around or the long way around.
How to address it: The angle between two directions is never more than 180° (it is the smaller of the two ways around a full circle). If a straightforward difference comes out over 180°, subtract it from 360° instead to get the actual angle between the two directions.
Guided practice (with answers)
1. Write an angle of 5° clockwise from north as a three-figure bearing.
Answer: 005°
2. The bearing of B from A is 100°. Find the bearing of A from B.
Answer: 280°, because 100° + 180° = 280°.
3. The bearing of B from A is 320°. Find the bearing of A from B.
Answer: 140°, because 320° is 180° or more, so subtract 180°: 320° - 180° = 140°.
4. From a lighthouse, ship A has bearing 030° and ship B has bearing 110°. Find the angle between them.
Answer: 80°, because |110° - 30°| = 80°, which is already 180° or less.
5. A hiker's bearing is 340°. She turns 50° clockwise. Find her new bearing.
Answer: 030°, because 340° + 50° = 390°, and 390° - 360° = 30°, written as 030°.
Independent practice worksheets
Practise writing bearings, finding back bearings, and finding angles between bearings, with computed, never-wrong answer keys.
Differentiation
- Sketch a quick compass (N, E, S, W at 0°, 90°, 180°, 270°) before every problem, to sanity-check whether an answer bearing is roughly in the right region.
- Always write out the FULL three-digit bearing, even in working, to build the habit before it becomes automatic.
- For back bearings, physically turn around (180°) while holding an imaginary compass to feel why the rule works, before applying the +/-180° arithmetic.
- Combine a bearing problem with the sine or cosine rule (noting this moves beyond Foundation tier into Higher-tier territory) to find an actual distance, not just a direction.
- Investigate why a back bearing is always exactly 180° different, using the fact that 'north' at two different points on a small map are parallel lines.
- Pose a three-point bearings problem (A to B, then B to C) and ask students to track the running bearing changes.
Assessment: exit ticket
A three-question exit ticket sampling bearing notation, back bearings, and the angle between two directions.
1. Write an angle of 62° clockwise from north as a three-figure bearing.
Answer: 062°
2. The bearing of B from A is 210°. Find the bearing of A from B.
Answer: 030°, because 210° is 180° or more, so subtract 180°: 210° - 180° = 30°.
3. From a tower, plane A has bearing 040° and plane B has bearing 310°. Find the angle between the two bearings.
Answer: 90°, because |310° - 40°| = 270°, and since 270° is more than 180°, the actual angle is 360° - 270° = 90°.
Teacher notes and timings
- Rough timing: Lesson 1 bearing notation (section 1), Lesson 2 back bearings and angles between directions plus the exit ticket (section 2).
- Curriculum note: DfE GCSE Geometry item 15 is standard type (Foundation for every student), covering both 'measuring with a physical protractor' and 'use of bearings'; this unit deliberately builds only the fully computable half (bearing arithmetic), the same scoping this site already applies to ruler/compass constructions.
- No figure accompanies this unit: no compass/bearing diagram exists in components/MathFigures.tsx or components/StandardFigures.tsx, and inventing one was out of scope for this batch (see the batch 4 PR description). Every answer is still fully computed; sketch a compass rose on the board to accompany the worked examples.