At a glance. One lesson of about 60 minutes. It assumes sine, cosine and tangent. Students write bearings with three figures, find back bearings, split a journey into east and north, and combine two legs.
Calculator: GDC optional. A calculator is used for the ratios, so the label is GDC optional. The optional section explains the signed values some calculators give for angles above 90° and how to read them.
What students will learn
Know
- A bearing is measured clockwise from north and has three figures.
- A back bearing differs from the bearing by 180°.
- East = d sin b and north = d cos b, with negative values meaning west and south.
Understand
- The bearing is not the angle in the triangle; the triangle's acute angle depends on the quadrant.
- Two legs that differ by 90° in bearing form a right angle at the turn.
Do
- Write and read three-figure bearings.
- Find a back bearing.
- Resolve a journey into east and north and combine journeys.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the bearing convention and trigonometry in right-angled triangles (questions 1 to 9).
- Criterion B. Investigating patterns: both figures. Students find how the signs of east and north change with the bearing, and the rule for the quadrants.
- Criterion C. Communicating: a labelled sketch with north lines at every turn, and bearings written with three figures (worked examples).
- Criterion D. Applying mathematics in real-life contexts: the ship, the hikers, the plane and the drone.
The one thing to do with the figure
Make students predict the back bearing. Ask them to say it before touching the figure, and to explain in a sentence. Then test several bearings, including ones above 180°.
In the second figure, hold north at 3 and let students sweep east from −8 to 8. Ask: where does the bearing jump, and why does it go from 306.9° at east −4 to 053.1° at east 4 for the same triangle? (It passes through north, 360° = 000°, never 180°.)
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the lighthouse keeper and the ship. Collect predictions. |
| 12 min | First figure; key idea; settle the hook; the warning card. |
| 10 min | Second figure; the three worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, return to the half turn. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | 225°; 15; tangent. |
|---|---|
| Worked examples | 15.4 km and 19.7 km; 124°; 344°. |
| Faded example | 49.0 km. |
| Something to think about | 135°, and 14.1 km. |
| 1. Back bearing | C. A is 360 − 305; B takes off 90; D adds 180 and does not reduce. |
| 2. Distance east from a bearing | 36.3 km. 40 sin 65°. |
| 3. Back bearing | 58°, written 058°. |
| 4. Bearing from east and north | 61°, written 061°. |
| 5. Distance south from a bearing | 20.4 km. 24 cos 32°. |
| 6. Distance west from a bearing | 68.8 km. 120 sin 35°. |
| 7. Two legs at a right angle | 11.7 km. √136. |
| 8. Two legs, northerly parts | 2.0 km. −2.74 + 4.70. |
| 9. A component from a bearing | B. A is the north part; C uses tan; D is a signed value. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Took 360 off, took off a quarter turn, or did not reduce. | Say: a back bearing is a half turn, 180°, away. Then check it is between 0 and 360. |
| 16.9 or 85.8 in question 2 | Used cosine for east, or tangent. | Mark the east side opposite the bearing angle, then name the ratio. |
| 418 or 122 in question 3 | Did not reduce, or subtracted from 360. | Test: is the answer a possible bearing, and is it 180° from 238? |
| 29 or 119 in question 4 | Measured from east, or took 180 away. | Ask: from which line is the angle measured, north or east? |
| −20.4 or 12.7 in question 5 | Gave a signed value, or gave the east part. | Ask: which compass direction is the question asking for? Then which side of the triangle is that? |
| 98.3 or −68.8 in question 6 | Gave the south part, or a signed value. | Mark which angle is acute from the north-south line, then pick opposite or adjacent. |
| 16 or 4 in question 7 | Added or subtracted the legs. | Draw a north line at the turn and find the angle between the legs: here 90°. |
| 7.4 or 9.2 in question 8 | Ignored signs, or used sine for north. | Find each north part with its sign first, in a table, then add. |
| A, C or D in question 9 | Gave north, used tangent, or gave a signed value. | Ask which compass direction, then name opposite or adjacent from the bearing's acute angle. |
Questions worth asking
- Why must a bearing have three figures? It avoids confusion, and the digits are the angle clockwise from north.
- How can you tell from the bearing alone whether a ship has gone east or west? Listen for “between 0 and 180 is east, between 180 and 360 is west”.
- Why can the calculator give a negative distance for sin 215°? The sign is a direction: west.
- If the legs of a journey meet at 90°, what shortcut do you have? Pythagoras for the straight-line distance.
Differentiation
A way in for anyone stuck
- Give a pre-drawn compass with north lines at start and finish; students add the bearing and mark the triangle.
- Provide a quadrant card: NE = angle, SE = 180 − angle, SW = 180 + angle, NW = 360 − angle.
- Restrict to first-quadrant bearings (000 to 090) until the rule is secure.
Stretch
- A ship sails 20 km on bearing 060° then 20 km on bearing 180°. Find its bearing and distance from the start. (The triangle is isosceles; the distance is 20 km on bearing 120°.)
- Prove that the back bearing is always b + 180° if b < 180° and b − 180° otherwise.
Students learning in English as an additional language
“Bearing” is a direction, not a part of a machine. “Clockwise” is the way the hands of a clock go. “Due” means exactly: “due north” is exactly north. Point to the north line and sweep your arm clockwise as you say each bearing.
| Word | Plain meaning | Example |
|---|---|---|
| bearing | a direction given as the clockwise angle from north | a compass reading of a ship from a port |
| three-figure bearing | a bearing written with three digits | a bearing of 5° written 005° |
| back bearing | the bearing of the start as seen from the finish | a lighthouse seen from the boat |
| clockwise | turning the way clock hands turn | from north towards east |
| due | exactly in the direction named | due south |
| component | one part of a movement, along one direction | the east part of a journey |
Sentence starters for explaining aloud or in writing:
- The acute angle in my triangle is … because …
- The bearing is … because the point is in the … quadrant.
- The back bearing is … because it differs by …
Two things not to say
“Just use SOH CAH TOA.” Ask: from which line is the angle measured?
“Bearings are angles from the x-axis.” They are from north, clockwise.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Back bearingThe bearing of a buoy from a pier is 072°. What is the bearing of the pier from the buoy?
252°. - Earlier work (Year 10)Round 87.549 to three significant figures.
87.5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA trawler leaves Kochi and sails 30 km on a bearing of 300°. How far north of Kochi is it, to 1 decimal place?
15.0 km. - Working backA cyclist rides 12 km east and 7 km south. Find her bearing from the start to the nearest degree.
120°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why sin 215° is negative and what the negative sign means on a map.
The ship is west of the start, and sine measures the east part. - OpenA ship sails 15 km on bearing 070° and then on bearing 160° until it is due east of the start. How far did it sail on the second leg?
About 5.5 km.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? A school field and a hand compass give real bearings. The With a GDC section is optional.