Common content, so this runs for both courses.
Ask for the angle at the turn before you draw anything. Bearing 062, then 145. Almost the whole room will say 83°, confidently, because subtracting is the obvious move and the answer looks like an angle.
Do not correct it. Step the figure to the third view and let them see the line stop 1.04 km short of the boat. Then step to the fourth and draw the back bearing. The explanation lands because they already own the wrong answer, and 83° and 97° being supplementary is the reason it felt right.
The sentence worth writing on the board is the reason, not the rule: a bearing is measured from north at the point you are standing on, and at the turn you are looking back down the first leg.
| Question | Answer |
|---|---|
| 1. Angle at the turn | 242 − 145 = 97°. |
| 2. Distance from the pier | d² = 98.75, so 9.94 km. |
| 3. Bearing of the boat from the pier | East 9.00, north −4.21, so 115. |
| 4. The student who answered 8.90 | B. They used 83°. |
| 5. Angle of elevation | tan⁻¹(45/120) = 20.6°. |
1 markA labelled diagram with north at every point measured from.
1 markThe angle inside the triangle.
1 markCorrect substitution into the cosine rule or into the components.
1 markThe answer, with any bearing to three figures.
The diagram mark is awarded for the drawing alone. It is the cheapest mark in this sub-topic and the one most often left on the table, because students who are confident with trigonometry skip straight to the rule and then get the angle wrong, which costs them the next two marks as well.
| They wrote | What happened |
|---|---|
| 83° | Subtracted the bearings. The expected answer, and the lesson. |
| 8.90 km | The same error, carried through. Always too small, because cos 83° is positive and cos 97° is not. |
| 9.43 km | Treated the turn as a right angle. Worth praising: it is the right order of magnitude and a sound check. |
| 13 km | Added the legs. They have not drawn anything. |
| 12.77 km | Radians. cos 97 radians is −0.93. |
| 65 | On question 3, the raw calculator output. It is an angle off the south line, not a bearing. |
| 295 | On question 3, the back bearing. They answered the bearing of the pier from the boat. |
| 25 | On question 3, measured from the east line. Bearings start at north. |
| 69.4° | On question 5, from the vertical rather than the horizontal. Note that dividing 120 by 45 gives the same 69.4°, because tan(90 − x) = 1/tan x, so you cannot tell the two slips apart from the number. Ask to see the working. |
"Why do I need two north lines?" Because bearings are not angles between the legs, they are angles from north, and the north line moves with you. Draw the second one and the 97° becomes visible without being explained.
"Can I just always take it off 180?" Here 97 = 180 − 83, and because the north lines are parallel that holds for any right turn whose two bearings differ by less than 180°. It is co-interior angles, not a fluke. But it reverses when the boat turns the other way, and it needs a case split when the difference exceeds 180°, while the back bearing needs none.
"How did it end up south? It never went south." It did, on the second leg: 145 is south of east. The north component of that leg is −6.55, which outweighs the +2.35 from the first. This is the moment to show that components carry the sign and the sign carries the meaning.
| Stage | What to do |
|---|---|
| Demonstrate | Type the two component sums in front of them, 5sin62+8sin145 and 5cos62+8cos145, which give 9.0033 and −4.2059, and stop on the negative. Ask what a negative north component means before you say it. Then tan⁻¹(9.0033/4.2059) gives 65.0, and the point to make is that the machine cannot get any further: it was given two lengths and knows nothing about the map. |
| Where they stick | Writing the 65.0 down as the answer. Also entering a bearing of 062 as 062 and getting 62, which is fine, and then writing the answer back as 62, which is not. |
| The check | Before the cosine rule, ask whether the answer will be more or less than 9.43, the right-angle case. They should say more, because 97° opens the triangle out. Anyone who then writes 8.90 has a check of their own that they ignored. |
Ask which machine each student has before the lesson, not during it. Degrees mode is the setting that ruins this whole topic, and the two machines show it in different corners of the screen.
| Step | What |
|---|---|
| 1 | Ask for the angle at the turn cold. Collect 83° without comment. |
| 2 | Step the figure to the short line. Let the gap do the work. |
| 3 | Back bearings as their own small thing: 062, 300, and the over-360 case. |
| 4 | Return to the turn. 242 − 145. Then the cosine rule. |
| 5 | Components for the bearing back, and the negative north. |
| 6 | Elevation and depression, which are quick and give the room a win. |
| 7 | Close on the four diagram habits. Set one question where the diagram is the only thing asked for. |
Do not teach "the angle is 180 minus the difference" as the method. It is true for a right turn with the bearings less than 180° apart, and it is the wrong way round for a left turn, so a student who learns it has to work out which case they are in before they can use it. The back bearing does not care.
Do not let "bearing" and "angle" be used interchangeably. Insist on three figures for a bearing and a degree sign for an angle, in your own writing as well as theirs. The notation is doing real work in this sub-topic, and students who blur it produce 65 as a final answer.