At a glance. One lesson of about 60 minutes. It assumes the ratios and bearings from Years 9 and 10. Students derive h = d ÷ (1/tan a − 1/tan b), use it, then add the components of a journey and find a distance and a back bearing.
Calculator: GDC optional. A calculator is used for the ratios and the inverse tangent, so the label is GDC optional. A scientific calculator does the same work, and the closed section covers storing the two cotangents.
What students will learn
Know
- Two sight lines to one top give two right-angled triangles with the same height.
- A leg of length r on bearing θ has east part r sin θ and north part r cos θ.
- A back bearing differs by 180°.
Understand
- The walk between two points is the difference of the two distances to the foot, so the cotangents are subtracted, not the angles.
- Splitting every leg into components makes a journey of any length a single right-angled triangle.
Do
- Derive and use the two-sighting height formula.
- Add the components of a multi-leg journey and find a distance and a bearing.
- State why a formula fails or is unreliable for particular inputs.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using sine, cosine, tangent and bearings in the right quarter (questions 1 to 9).
- Criterion B. Investigating patterns: the first figure. Students describe what happens to the height as the two angles move together, and why.
- Criterion C. Communicating: a labelled diagram with the walk, both angles and the unknown, and the algebra that links them (worked examples, faded example).
- Criterion D. Applying mathematics in real-life contexts: the minaret, the ferry and the lookouts on a shore.
The one thing to do with the figure
Make students predict, then reveal. Before touching the first figure, ask: is the tower taller or shorter than the walk? Then let them move the sliders and say which setting makes the height most sensitive.
In the second figure, set both legs to the same bearing and check the totals are just 16 km. Then set them 180° apart and say what is left.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the minaret and the moat. Collect votes and reasons. |
| 12 min | First figure; key idea derivation; settle the hook; what-if. |
| 10 min | Second figure; worked examples 2 and 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to why angles do not subtract. |
| 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 | 15 cm; 5.6 m; 270°. |
|---|---|
| Worked examples | 54.6 m; 11.5 km on 150°; undefined, as the denominator is zero. |
| Faded example | 34.1 m. |
| Something to think about | A lot: about 165 m against about 141 m. |
| 1. Setting up a two-sighting height | B. A subtracts the angles; C subtracts tangents instead of cotangents; D takes 35 m as each distance to the foot. |
| 2. Height with an eye level | 53.6 m. 80 tan 33° + 1.6. |
| 3. East part of a leg | 15.9 km. 18 sin 62°. |
| 4. Height from two sightings | 85 m. 120 ÷ (cot 18° − cot 31°). |
| 5. Distance of a two-leg journey | 11.4 km. √(49 + 81). |
| 6. Back bearing of a journey | 259°. 79.3° + 180°. |
| 7. A point seen from two ends of a base line | 179 m. 150 ÷ tan 40°. |
| 8. A distance after a two-sighting height | 30.4 m. h ≈ 26.42, then h ÷ tan 41°. |
| 9. Reversing a bearing | C. A forgets to subtract 360°; B is 360° − 312°; D turns a quarter-turn. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Subtracted the angles, subtracted the tangents, or treated 35 m as each distance. | Draw both triangles with the same height and mark the 35 m as the gap on the ground, not a side of either triangle. |
| 52 or 50.4 in question 2 | Left out the eye height, or subtracted it. | Ask: is the top of the tower above or below her eyes? Where does the triangle start? |
| 8.5 or 33.9 in question 3 | Found the north part, or used tangent with the hypotenuse. | Draw north first and shade the angle between north and the leg. |
| 72 or 520 in question 4 | Treated the walk as the distance to the foot, or subtracted the angles. | Write the two distances to the foot using h and subtract those. |
| 16 or 5.7 in question 5 | Added the lengths, or took a leg as a hypotenuse. | Ask what angle the two legs make at the turning point. |
| 79 or 101 in question 6 | Gave the bearing from the start, or subtracted 180°. | Mark both ends on a sketch and say whose bearing is asked for. |
| 126 or 358 in question 7 | Multiplied by tan, or used the whole 300 m. | Ask where the boat sits relative to the midpoint, and which side the 40° is next to. |
| 40.3 or 65.4 in question 8 | Used 35 m as the height, or gave the further car. | The gap between the cars is not the height. Find the height first. |
| A, B or D in question 9 | Did not reduce 492°, used 360° minus the bearing, or turned a quarter-turn. | Say aloud: a reverse is a half-turn, 180°. |
Questions worth asking
- Why do we subtract the cotangents rather than the angles? Listen for “the angles belong to different triangles; it is the distances along the ground that subtract”.
- Why does the height become unreliable when the two angles are nearly equal? Listen for “the denominator is close to zero, so a small error is magnified”.
- Why does a bearing use sine for the east part when the angle is measured from north? Listen for “the east part is opposite the angle from north”.
- How do you know which quarter of the compass the finish is in? Listen for “the signs of the east and north totals”.
Differentiation
A way in for anyone stuck
- Give a labelled two-triangle template with the walk marked as the gap between two points on the ground.
- Provide a table with columns leg, length, bearing, east part, north part, and let students fill in then add.
- Restrict to the first four questions at first, using whole-number lengths and angles.
Stretch
- Show that h = d sin a sin b ÷ sin(b − a), by writing the two cotangents over a common denominator and using the sine of a difference.
- A ship sails on three legs and finishes at its start. Explain what must be true of the east parts and the north parts.
Students learning in English as an additional language
“Bearing” is a direction measured clockwise from north, always with three figures. “Component” means one part of a whole; say “the east component of the journey” while pointing along the east line. “Back bearing” is the bearing for coming back.
| Word | Plain meaning | Example |
|---|---|---|
| angle of elevation | the angle measured up from a level line to a line of sight | looking up at the top of a minaret |
| angle of depression | the angle measured down from a level line to a line of sight; it equals the angle of elevation from the ground (alternate angles) | looking down from a bridge at a boat |
| bearing | a direction measured clockwise from north, written with three figures | 090° is due east |
| component | the part of a journey in one direction, such as the east part | walking 5 km east and 3 km north has components 5 and 3 |
| back bearing | the bearing for the return trip, 180° from the outward bearing | the way home from a landmark |
| cotangent | one divided by the tangent of an angle | 1 ÷ tan 40° is the cotangent of 40° |
Sentence starters for explaining aloud or in writing:
- The two sight lines make two triangles with the same …
- The east part of this leg is … because the angle is measured from …
- The finish is in the … quarter because the east total is … and the north total is …
Two things not to say
“Just subtract the angles.” The angles are in different triangles.
“Bearings are an angle from the horizontal.” 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.
- ElevationA 20 m pole casts a shadow 14 m long. Find the angle of elevation of the sun, to the nearest degree.
55°. - Earlier work (Year 9)A right-angled triangle has hypotenuse 17 cm and one side 8 cm. Find the other side.
15 cm.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- BearingsA walker goes 5 km on 090° then 12 km on 180°. How far is she from the start?
13 km. - Back bearingThe bearing of B from A is 075°. Give the bearing of A from B.
255°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningExplain why walking a longer distance between the two sightings, with the same angles, gives a taller tower.
The height is proportional to the walk, h = d ÷ (cot a − cot b). - OpenTwo sightings give 18° and 25° with a walk of 100 m. A second surveyor finds 18° and 26°. By how much do the heights differ?
About 107 m against about 97 m, so roughly 10 m.
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 clinometer made from a protractor and a weighted string works in a school yard. The With a GDC section is optional.