Scale factors, scale diagrams and maps

OCR J560 sets scale-drawing and map-reading questions across all calculator papers (J560/02, /03, /05, /06). The skill links to ratio (R-strand) and proportional reasoning.

Scale notation

A scale of 1:n means 1 unit on the diagram represents n units in real life.

• 1 cm : 1 m means 1 cm on the map = 100 cm in real life (because 1 m = 100 cm). Equivalent to 1 : 100.
• 1 : 50 000 means every 1 cm on the map = 50 000 cm = 500 m = 0.5 km in real life.
• Scales use the same unit on both sides — convert before computing.

Map reading

Most OS maps in the UK use 1:25 000 (1 cm = 250 m) or 1:50 000 (1 cm = 500 m).

To find a real distance from a map: measure on the map, multiply by the scale factor, convert units sensibly.

To go the other way: divide the real distance by the scale factor.

Scale factors and similar shapes

A scale factor of k applied to a 2D shape:

• Lengths: × k
• Areas: × k²
• Volumes (3D, if scaling 3D): × k³

Example: a model car is 1/20 the size of the real car (length scale factor 1/20).

• Surface area of model = (1/20)² × real = 1/400.
• Volume of model = (1/20)³ × real = 1/8000.

Bearings (often appears with scale drawings)

A bearing is measured clockwise from North, given as a 3-digit angle.

• North = 000° (or 360°), East = 090°, South = 180°, West = 270°.
• Always 3 digits: 047°, not 47°.
• "Bearing of B from A" — start at A, face north, turn clockwise to face B.

Constructing a scale diagram

1. Decide the scale (e.g. 1 cm represents 5 km).
2. Convert real distances to drawing distances.
3. Use a ruler for lengths, a protractor for angles.
4. Mark all features clearly.

OCR mark scheme conventions

• "Use a scale of 1 cm to N km" — must use that scale; using a different scale loses A1.
• M1 for the correct conversion or correct measurement; A1 for the value.
• Bearings must be 3 digits; "47°" instead of "047°" loses the A1.