aiMOOC

Bearings scale diagrams and direction - Mathematics Year 7

QR-Code

Introduction

Where do you stand when you measure a bearing, and what does a map centimetre represent? Combine three-figure direction conventions with consistent scale units, then check a complete multi-leg route.

You will draw and explain a four-leg model route, convert map lengths to kilometres and distinguish travel distance from displacement. Begin with 065° clockwise from north and a 1:50000 map where 3 cm represents 1.5 km. The model is a mathematical plan, not live ground-navigation guidance.

Declare north and read three-figure bearings

A three-figure bearing describes a clockwise angle from the stated north ray at the starting point. Write leading zeroes where needed: 065° means 65° clockwise from north. It lies between north and east. Turning 65° anticlockwise instead gives a different direction, 295° clockwise; the leading zero does not change the angle’s magnitude.

The main compass directions are north 000°, east 090°, south 180° and west 270°. North-east is 045°, south-east 135°, south-west 225° and north-west 315°. We normally use 000° for north in a 0° through 359° bearing convention; 360° describes the same ray after a complete turn. Label the chosen convention rather than treating north as an unmarked edge of the page.

Measure at the correct point. “The bearing of B from A” begins at A, with its north ray and the ray pointing towards B. Repeating the angle at B without considering the changed starting point describes a different instruction. On a plane diagram with parallel north lines, the reverse bearing differs by 180°: $b_{r e v e r s e} = (b + 18 0^{\circ}) mod 3 6 0^{\circ}$ .

For 065°, the reverse is 245°. For 270°, adding 180 gives 450°; subtract 360 to express the same direction as 090°. A short sketch with two parallel north rays can check which of the two opposite directions you mean. Do not simply negate the three-digit label or always add 180 without reducing a complete turn.

Build a labelled angular model. In a copy of the full-circle source, choose an upward ray as north, mark clockwise east at 090°, south 180° and west 270°. The source colours alone do not declare those cardinal directions. A protractor needs its centre at the actual starting point and its zero reference aligned with the stated north ray, not automatically with the horizontal edge of a screen.

For a school scale diagram, we assume parallel grid-north rays on a flat plan. Magnetic north, grid north and true north are not automatically identical in real navigation; our given reference is the one used by the mathematical task. Keep the model’s stated reference and precision with every bearing rather than claiming a real compass reading that was never taken.

Convert scales and compare map representations

A scale ratio compares lengths in the same unit. On a 1:50000 map, 1 cm represents 50000 cm, which is 500 m or 0.5 km. Therefore $3 c m \times 50000 = 150000 c m = 1.5 k m$ . Multiplying 3 by 50000 and then labelling the answer kilometres would miss the centimetre-to-kilometre conversion.

Two methods for the same distance. One method converts the large centimetre result at the end. Another first establishes 1 map cm=0.5 ground km, then multiplies the map length by 0.5. For 3 cm, both give 1.5 km. The second method is efficient for repeated distances at this same scale; the first keeps the unit-ratio structure explicit.

To work backwards, divide the real distance by the amount represented by one map centimetre. At 1:50000, a real 2 km distance needs 4 map cm. An inverse check multiplies 4 by 0.5 to recover 2. Ratio direction matters: the map is the smaller representation, so a short paper length should not become a tiny fraction of a ground centimetre at this scale.

A different scale is a different relationship. At 1:25000, 1 map cm represents 0.25 km. A map length 4.8 cm therefore represents 1.2 km, and a real 2 km distance needs 8 map cm. Compare with the 4 cm needed at 1:50000: the larger-scale drawing uses twice the paper length for the same real distance.

A properly drawn scale bar represents known real distances by segments on the same map. If the map and its bar are enlarged together, the bar still provides a relative distance reference. The original printed ratio alone no longer describes the new physical centimetre lengths after resizing. A screenshot displayed at an unknown zoom cannot be measured as if its pixels were centimetres.

Use Corbettmaths’ original bearings examples to identify the starting point, north ray and clockwise turn. Its visible 325° example is separate from our 065° and four-leg route; combine the direction method with the independently stated scale here. Original: corbettmaths, Bearings - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Before reporting a length, state the scale, the map unit and the real unit. Check by reversing the conversion and by considering whether the result is a plausible size for the chosen model.

Construct and audit a complete four-leg route

An original four-leg route on a 1:50000 plan. Choose north as increasing y and east as increasing x. The map coordinates, in declared centimetres, are A(0, 0), B(3, 0), C(3, 4), D(0, 4), then back to A. This makes a closed rectangular route; the diagram is yours to construct with labelled north rays and lengths.

Leg	Three-figure bearing	Map distance	Real distance
A to B	090°	3 cm	1.5 km
B to C	000°	4 cm	2 km
C to D	270°	3 cm	1.5 km
D to A	180°	4 cm	2 km

Plot each next endpoint from the current point, not repeatedly from A. Starting at A, move 3 units east, then 4 north, then 3 west, then 4 south. The final coordinate is again (0, 0). The horizontal moves cancel and so do the vertical moves, which supplies an independent closure check.

The total travelled map length is 3+4+3+4=14 cm, so the model ground length is 7 km. The final displacement is zero because the route returns to A; zero displacement does not mean zero travel. If you stop at C instead, the path A→B→C is 7 map cm or 3.5 km, but the straight-line A→C distance is 5 map cm or 2.5 km in the ideal right-triangle model.

Reverse a leg as a check. The bearing of A from B is 270°, opposite the 090° first leg. The bearing of B from C is 180°, opposite 000°. Keep both the start and end labels in the answer; the phrase “the bearing between A and B” without a direction is incomplete.

Error clinic. A learner starts measuring 065° at the destination rather than the starting point. Another uses 65° anticlockwise, which corresponds to 295° clockwise. A third treats 150000 cm as 150000 km. Correct the first changed reference or unit, then verify with a reverse bearing or inverse scale conversion.

For unfamiliar transfer, use a 1:25000 plan: 4.8 cm means 1.2 km, and 2 km needs 8 cm. Explain why enlarging a diagram changes the physical ratio while a jointly enlarged scale bar can remain useful. Retain your full four-leg diagram, bearing labels, unit conversions and independent checks. Actual field navigation or instrument accuracy requires separate real observations; this course does not invent them. After a genuine break, retry one reverse bearing and one scale conversion and keep your actual response.

Interactive Tasks

Quiz: Test Your Knowledge

How is the stated bearing 065 degrees measured? (Clockwise from north at the starting point) (!Anticlockwise from north at the destination) (!Clockwise from east at any point) (!Anticlockwise from south at the starting point)

What is the three-figure east bearing? (090 degrees) (!90 degrees without a leading zero) (!180 degrees) (!270 degrees)

What is the reverse of 065 degrees on the stated plane model? (245 degrees) (!295 degrees) (!115 degrees) (!065 degrees)

What is the reverse of 270 degrees? (090 degrees) (!450 degrees as a reduced bearing) (!180 degrees) (!000 degrees)

What real distance does 3 cm represent at 1 to 50000? (1.5 km) (!150000 km) (!0.15 km) (!15 km)

How many map centimetres represent 2 km at 1 to 50000? (4 cm) (!8 cm) (!2 cm) (!1 cm)

What bearing takes B to C in the route? (000 degrees) (!090 degrees) (!180 degrees) (!270 degrees)

What is the total real route distance? (7 km) (!0 km) (!3.5 km) (!2.5 km)

What is the displacement after all four legs? (Zero) (!7 km) (!14 km) (!2 km north)

What does 4.8 cm represent at 1 to 25000? (1.2 km) (!2.4 km) (!12 km) (!0.12 km)

Memory Game

Bearing	Clockwise direction from a stated north ray
Scale	Ratio of represented to real lengths in the same unit
Reverse	Opposite direction along a stated leg
Displacement	Change from initial to final position
Route	Ordered path through successive locations
Reference	Declared starting ray or unit for interpretation

Drag and Drop

090 degrees	East bearing
180 degrees	South bearing
270 degrees	West bearing
245 degrees	Reverse of 065 degrees
1.5 km	Real length of 3 cm at 1 to 50000

Crossword Puzzle

BEARING	Clockwise direction from north
NORTH	Reference direction for the stated bearing
CLOCKWISE	Required turn direction
SCALE	Represented-to-real length ratio
DISTANCE	Length of a specified path
DIRECTION	Orientation of travel from a start

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Label a compass model: Mark 000, 090, 180, 270 and 065 clockwise from declared north.
Name the starting point: Explain the difference between bearing B from A and A from B.
Convert the source example: Show 3 cm to 150000 cm to 1.5 km at 1:50000.
Check reverse bearings: Find reverses of 065, 090 and 270 without retaining a full extra turn.

Standard

Draw the whole route: Plot A, B, C, D and Awith all four labelled bearings and lengths.
Check closure and distance: Show horizontal/vertical cancellation, total 7 km and zero final displacement.
Compare conversion methods: Use real-centimetre conversion and 0.5 km-per-map cm for the same leg.
Interpret a partial route: Distinguish A→B→C path 3.5 kmfrom ideal diagonal 2.5 km.

Advanced

Transfer to another scale: Calculate 4.8 cm→1.2 km and 2 km→8 cm at 1:25000.
Evaluate a resized map: Explain ratio changes and a scale bar enlarged with the map.
Correct reference and unit errors: Diagnose destination-centred angle, anticlockwise reading and centimetre/kilometre confusion.
Retrieve later with assumptions: Keep an actual later bearing and scale task with reference, units and an independent check.

Learning Check

Three-figure convention: Explain 065 and the main cardinal directions.
Scale example and inverse: Convert the original 3 cm at 1:50000 and reverse 2 km.
Complete multi-leg route: Draw and label all four legs.
Independent route check: Explain closure, path and displacement.
Unfamiliar scale and error analysis: Use 1:25000 and diagnose the first errors.
Evidence and later retrieval: Retain the full labelled plan and actual later work.

Worked checks and assessment guidance

065 is 65 clockwise from north at the start. Main bearings 000/090/180/270; 065 reverse 245 and 270 reverse 090. The full-circle source has no native North/numbers, you declare the model’s reference.
3×50000=150000 cm=1.5 km; one map cm=.5 km, so 2 km needs 4 cm. Same units belong in the ratio; do not label a centimetre product kilometres without conversion.
A 0, 0→B 3, 0 bearing 090/3 cm; B→C 3, 4 bearing 000/4 cm; C→D 0, 4 bearing 270/3 cm; D→A 0, 0 bearing 180/4 cm. Ground distances 1.5/2/1.5/2 km, total 7.
Net x 3−3=0, net y 4−4=0; final displacement 0 but travel 7 km. Partial A→B→C path 3.5 km versus ideal 3-4-5 diagonal 2.5 km. Reverse leg labels include their starting point.
One map cm=.25 km; 4.8 cm→1.2 km and 2 km→8 cm. Wrong angle origin/direction and cm-as-km change the problem; inverse/reverse checks expose them. Resizing changes physical scale ratio.
Keep north rays, three-figure bearings, scale, units and independent checks. Jointly resized scale bar can retain relative meaning; uncalibrated pixels are not physical centimetres. No actual field-navigation or tool-accuracy record is invented.

Learning evidence

Keep a full multi-leg scale plan, three-figure bearings, unit conversions and reverse/inverse checks. Actual navigation and later retrieval require actual retained evidence.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, geometry,number,fluency,reasoning and problem solving. Year 7 allocation is editorial within KS3. All numerical tasks,models,diagrams and worked answers are original. Actual Commons sources retain their creator/licence credits below; original-publisher videos supplement the written teaching. Reviewed 12 November 2026.

Protractor (vector image).svg, Frap at English Wikipedia, Public domain. Unchanged original; caption and learning activities are original.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.