Mathematics • Year 9 • Unit 3 • Lesson 17
True Bearings in the Real World
Apply true bearings to authentic navigation contexts: air-traffic control, marine charts, orienteering, drone surveying, and emergency return routes. Then explain why three-digit bearings are the international standard.
1. Word problems
Each problem uses the true-bearing form from Lesson 17 (three digits, clockwise from N, 000-360°) and the compass ↔ true conversions or the reverse-bearing rule (± 180°). Show your working — a single final answer with no working only earns half marks.
1.1 — Air-traffic control. A pilot calls in: "Heading two-five-zero, requesting clearance." Air-traffic control needs to know the compass-form equivalent for a backup paper chart.
(a) Convert the true bearing 250° to a compass bearing.
(b) State the reverse bearing the pilot would use if asked to turn around. 3 marks
1.2 — Marine chart. A captain's old paper chart shows the heading to Coffs Harbour as S25°E. Their new digital chart-plotter only accepts true bearings.
(a) Convert S25°E to a true bearing (3 digits).
(b) After arriving, the captain wants the bearing for the return trip to the starting port. State both the true bearing AND the compass bearing for that return. 4 marks
1.3 — Orienteering checkpoints. An orienteering course gives directions as true bearings: start → checkpoint A is 015°; checkpoint A → checkpoint B is 290°.
(a) Convert both bearings to compass form.
(b) On the way back, the runner must reverse each leg in turn (B → A → start). State the true bearings of the two return legs. 4 marks
1.4 — Drone surveying. A surveying drone is programmed in compass form. The site plan lists three legs: N20°E, S50°E, S15°W.
(a) Convert each leg to a true bearing (three-digit format) so the drone software can accept them.
(b) Which of the three legs is closest to "due east"? Justify in one sentence. 3 marks
1.5 — Emergency return. A hiking app records the outward route as four short legs: 040°, 110°, 180°, 305°. The hiker hits the "emergency return" button — the app must reverse all four bearings and display them in the opposite order (so the hiker walks them back in reverse).
(a) Find the reverse of each bearing.
(b) Write out the four return-trip bearings in the order the hiker would walk them (i.e. the reverse of leg 4 first, then leg 3, etc.). 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A pilot has heard the bearing as "forty-five degrees", but they cannot tell if their colleague meant 045° or 145° or 245° or 345° — they all sound similar in a noisy cockpit. Explain in your own words: (i) what RULE about true bearings prevents this confusion when written down, (ii) why three digits is safer than fewer, and (iii) why bearings are always measured CLOCKWISE from North (rather than anti-clockwise or from some other axis). Refer to "international convention" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Air-traffic control (true 250°)
(a) 250 is in the SW quadrant (180-270). θ = 250 − 180 = 70. Compass: S70°W.
(b) Reverse = 250 − 180 = 070° (already three digits — pad if needed).
The reverse takes you due opposite — turning around exactly.
1.2 — Marine chart (S25°E)
(a) SE quadrant: true = 180 − θ = 180 − 25 = 155°.
(b) Return bearing = 155 + 180 = 335° true. 335 is in NW quadrant: θ = 360 − 335 = 25. Compass: N25°W.
Sanity check: outward was 25° off the south axis toward E; the return is 25° off the north axis toward W — exactly opposite. ✓
1.3 — Orienteering (015° and 290°)
(a) 015° in NE: compass = N15°E. 290° in NW: θ = 360 − 290 = 70. Compass: N70°W.
(b) Reverses: 015 + 180 = 195°; 290 − 180 = 110°. The runner walks 110° first (B → A), then 195° (A → start).
1.4 — Drone surveying
(a) N20°E → 020°; S50°E → 180 − 50 = 130°; S15°W → 180 + 15 = 195°.
(b) Due east is 090°. The closest of the three is 130° (S50°E) — it's 40° off east, whereas 020° is 70° off and 195° is 105° off (closer to "south").
1.5 — Emergency return (legs 040, 110, 180, 305)
(a) Reverses: 040 + 180 = 220°; 110 + 180 = 290°; 180 + 180 = 360 → 000°; 305 − 180 = 125°.
(b) Reverse-of-leg-4 first: 125°; then reverse-of-leg-3: 000°; then leg-2 reverse: 290°; then leg-1 reverse: 220°.
2.1 — Why three digits + clockwise (sample response)
The rule that prevents the pilot's confusion is the three-digit format: written down, "045", "145", "245" and "345" are all visually distinct, whereas "45 degrees" alone is ambiguous in speech. Three digits is safer because it forces you to specify the FULL angle, all the way around the circle from 000 to 360 — there's no "is it the small one or the big one?" question left. Bearings are measured CLOCKWISE from North because that's the international convention shared by aviation, marine, military and survey worlds, so a heading reported in one country is read the same way in any other; mixing in counter-clockwise or using a different axis (e.g. measuring from East) would mean two bearings written the same way could point opposite ways depending on which system the reader assumed.
Marking: 1 mark for naming the three-digit rule; 1 for explaining why three digits is unambiguous; 1 for explaining the clockwise-from-N convention; 1 for mentioning international consistency / shared standard.