The shape of the lesson
By the end of the lesson, learners can
- Drawing to scale: In the lab, choose a scale and draw the plan of a room.
- Three-figure bearings: In the bearings tab, drag point B around point A and read its bearing.
- Journeys on a scale drawing: Many questions combine both ideas: draw each leg of a journey to scale on the correct bearing, then measure the final distance and bearing from your drawing.
- Stand at the point after 'from': that is where the north line goes.
- Turn clockwise from north to the line joining the points.
- Three figures: \(045^\circ\), not \(45^\circ\).
- Back bearing: add or subtract \(180^\circ\).
- Scales: map to real, multiply; real to map, divide; convert units last.
One picture every idea returns to
A bearing is an angle measured CLOCKWISE from NORTH, written with three figures (\(065^\circ\), not \(65^\circ\)). The bearing back the other way differs by exactly \(180^\circ\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, North, then clockwise.. A map shrinks real distances by a fixed scale, and a bearing fixes a direction with three digits measured clockwise from north. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Drawing to scale | In the lab, choose a scale and draw the plan of a room. Project tab 3 of the lab (Constructions lab, scale drawings) and let a student drive it while the class predicts. Model the worked example Map distance to real distance on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Three-figure bearings | In the bearings tab, drag point B around point A and read its bearing. Project tab 4 of the lab (Constructions lab, bearings) and let a student drive it while the class predicts. Model the worked example Back bearings on the board, one line at a time. Your turn: The bearing of a lighthouse L from a boat B is \(204^\circ\). Find the bearing of the boat from the lighthouse. | Pairs |
| 21 to 30 min | Teach and check: Journeys on a scale drawing | Many questions combine both ideas: draw each leg of a journey to scale on the correct bearing, then measure the final distance and bearing from your drawing. Model the worked example A two-leg journey on the board, one line at a time. Your turn: Point D is due west of point C. Write down the bearing of D from C and the bearing of C from D. | Numbered Heads Together |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 16 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Journeys on a scale drawing, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| With a ratio scale both sides are in the SAME unit. 1: 25 000 does not mean 1 cm to 25 000 m. | Multiply first to get centimetres, then convert to metres or kilometres. |
| Read the question's word order carefully. 'The bearing of A from B' means stand at B. | Measuring at the wrong point gives the back bearing, which is wrong by exactly \(180^\circ\). |
| Draw a NEW north line at every turning point. | Measuring the second bearing from the first leg's line, instead of from north, is the classic error. |
Scale drawings and bearings: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
Write each direction as a three-figure bearing: (a) east (b) south (c) north-west.
A map has a scale of 1 : 25 000.
(a) Two villages are 8.4 cm apart on the map. Find the real distance in km.
(b) A lake is 3 km long. How long is it on the map?
The bearing of B from A is \(118^\circ\). Find the bearing of A from B.
The bearing of X from Y is \(237^\circ\). Find the bearing of Y from X.
A walker goes from A to B on a bearing of \(050^\circ\), then from B to C on a bearing of \(140^\circ\).
Find angle ABC, showing your reasoning.
A boat leaves port P and sails 12 km due east to Q, then 16 km due north to R.
(a) Make a scale drawing using a scale of 1 cm to 2 km.
(b) Use your drawing to find the distance PR in km.
(c) Measure the bearing of R from P.
(d) Find the bearing on which the boat must sail to return directly from R to P.
Answer key and marking notes
- (a) \(090^\circ\)
- (b) \(180^\circ\)
- (c) \(315^\circ\)
- (a) \(8.4 \times 25\,000 = 210\,000\) cm \(= 2.1\) km.
- (b) \(3\) km \(= 300\,000\) cm; \(300\,000 \div 25\,000 = 12\) cm.
\(118^\circ + 180^\circ = 298^\circ\).
\(237^\circ - 180^\circ = 057^\circ\).
- The bearing of A from B is the back bearing: \(050^\circ + 180^\circ = 230^\circ\).
- The bearing of C from B is \(140^\circ\).
- Angle ABC is the turn between these directions: \(230^\circ - 140^\circ = 90^\circ\).
- (a) PQ = \(12 \div 2 = 6\) cm east; QR = \(16 \div 2 = 8\) cm north, with a new north line at Q.
- (b) PR measures 10 cm, so \(10 \times 2 = 20\) km. (Check: \(\sqrt{12^2 + 16^2} = 20\).)
- (c) The angle clockwise from north at P is about \(037^\circ\).
- (d) Back bearing: \(037^\circ + 180^\circ = 217^\circ\).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
A map has a scale of 1 : 20 000. Two churches are 3.6 km apart. How far apart are they on the map?
- A18 cm
- B1.8 cm
- C180 cm
- D0.18 cm
The bearing of A from B is \(048^\circ\). Find the bearing of B from A.
- A\(228^\circ\)
- B\(312^\circ\)
- C\(132^\circ\)
- D\(138^\circ\)
The diagram shows the north line at P and the direction of Q from P. Find the bearing of Q from P.
- A\(320^\circ\)
- B\(040^\circ\)
- C\(220^\circ\)
- D\(140^\circ\)
A plan of a room uses a scale of 1 cm to 5 m. On the plan the room is 4.6 cm long and 3 cm wide. Find the real floor area of the room.
- A\(345\text{ m}^2\)
- B\(69\text{ m}^2\)
- C\(13.8\text{ m}^2\)
- D\(38\text{ m}^2\)
A walker goes from A to B on a bearing of \(065^\circ\), then turns and walks from B to C on a bearing of \(200^\circ\). Find angle ABC.
- A\(45^\circ\)
- B\(135^\circ\)
- C\(115^\circ\)
- D\(315^\circ\)
From a lighthouse A, buoy B is on a bearing of \(130^\circ\) and buoy C is on a bearing of \(220^\circ\). AB = AC. Find the bearing of C from B.
- A\(265^\circ\)
- B\(085^\circ\)
- C\(175^\circ\)
- D\(220^\circ\)
Quiz key
\(3.6\text{ km} = 3.6 \times 100\,000 = 360\,000\) cm. Real to map: divide by the scale factor, \(360\,000 \div 20\,000 = 18\) cm. The tempting 1.8 cm converts km to cm with \(\times 10\,000\) instead of \(\times 100\,000\). 180 cm uses \(\times 1\,000\,000\), and 0.18 cm is a further factor of 10 out.
A back bearing differs by \(180^\circ\): \(048^\circ + 180^\circ = 228^\circ\). The tempting \(312^\circ\) is \(360^\circ - 48^\circ\), a reflection in the north line, not the opposite direction. \(132^\circ\) is \(180^\circ - 48^\circ\), and \(138^\circ\) adds only \(90^\circ\).
Turning clockwise from north to the line PQ goes nearly all the way round: \(360^\circ - 40^\circ = 320^\circ\). The tempting \(040^\circ\) measures anticlockwise; \(040^\circ\) would point east of north. \(220^\circ\) and \(140^\circ\) are the back bearings of \(040^\circ\) and \(320^\circ\), pointing towards P rather than away from it.
Real length \(= 4.6 \times 5 = 23\) m and real width \(= 3 \times 5 = 15\) m, so the area is \(23 \times 15 = 345\text{ m}^2\). The tempting 69 finds the plan area \(4.6 \times 3 = 13.8\text{ cm}^2\) and multiplies by 5 only once, but area needs the scale applied to BOTH lengths. 13.8 is the plan area itself, and 38 is the real perimeter \(2(23 + 15)\).
At B, the bearing back to A is \(065^\circ + 180^\circ = 245^\circ\), and the bearing to C is \(200^\circ\). Both are measured from the same north line at B, so angle ABC \(= 245^\circ - 200^\circ = 45^\circ\). The tempting \(135^\circ\) is \(200^\circ - 65^\circ\), the angle of turn, measured as if both bearings were taken at A. \(115^\circ\) is \(180^\circ - 65^\circ\), and \(315^\circ\) is the reflex angle at B.
Angle BAC \(= 220^\circ - 130^\circ = 90^\circ\), so the base angles are \((180^\circ - 90^\circ) \div 2 = 45^\circ\). At B the bearing of A is \(130^\circ + 180^\circ = 310^\circ\). C is \(45^\circ\) anticlockwise of BA (it lies to the south-west of B), so the bearing of C from B is \(310^\circ - 45^\circ = 265^\circ\). The tempting \(085^\circ\) is the bearing of B from C, the back bearing. \(175^\circ\) adds \(45^\circ\) to \(130^\circ\) at the wrong point, and \(220^\circ\) is the bearing of C from A, not from B.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.