Scale & Maps (AQA GCSE Maths)

What is scale?

• In mathematics, scale can have many meanings, but in accurate drawings and constructions scale refers to a ratio
• Maps and technical drawings are usually drawn to a scale
• The scale chosen will depend on the area being mapped out
  • A map of a classroom will use a large scale and can show a lot more detail
  • A map of a country or a world map will use a small scale and can show a much bigger area but in less detail
• Scale can be given as a description, for example "1 cm on the diagram represents 2 m in real life"
• Scale can be given as a ratio, for example "the scale on the map is 1 : 10 000"

How do I convert between a scale length and a real length?

• If you are converting from a scale length to a real length, normally you have to multiply
  • For example, if the scale is "1 cm on the diagram represents 2 m in real life"
    • to convert 6 cm on the diagram to the real-life measurement, we need to multiply by 2 and change the units to m
  • So 6 cm = (6 × 2) m = 12 m

• If you are converting from a real length to a scale length, normally you have to divide
  • For example, if the scale is "1 cm on the diagram represents 2m in real life"
    • to convert 15 m in real life to the length on the diagram, we need to divide by 2 and change the units to cm
  • So 15 m = (15 ÷ 2) cm = 7.5 cm

How are map scales written?

• Map scales are normally given as ratios, 1 : n, without any units
• For example, Ordnance Survey maps are often 1 : 25 000
  • This means 1 cm on the map represents 25 000 cm in real life
  • Or more usefully, 1 centimetre on the map represents 250 metres, or 0.25 kilometres, in real life

How can I convert between map scales and real lengths?

• The scale can be used to convert lengths found on a map to lengths in real life
  • STEP 1
    Use a ruler to measure the distance accurately on the map
    • For example, measuring a length from A to B as 5.8 cm
  • STEP 2
    Use the scale to find the actual distance in the same units (usually cm)
    • For example, if the scale is 1 : 150 000, 5.8 × 150 000 = 870 000 cm
  • STEP 3
    Convert the actual distance to a more suitable unit
    • For example, 870 000 cm = 8700 m = 8.7 km

• The scale can also be used to convert lengths found in real life into a length for a map
  • STEP 1
    Convert the given scale into a scale of 1 cm : [the units the real-life distance is in]
    • For example, if the scale is 1 : 500 000, convert to 1 cm : 5 km
  • STEP 2
    Divide the real-life distance by the new map scale to find the distance on the map
    • For example, if the real-life distance is 20 km, 20 ÷ 5 = 4, so the scale length is 4 cm
  • STEP 3 
    Draw this distance on the map if asked to do so

How are bearings used on maps?

• Bearings are a measure of direction
  • They are measured from the North 
  • They are measured clockwise
  • The angle is always written in 3 figures
    • If the angle is less than 100° the first digit will be a zero
• If you need to use a map to find a direction, use a protractor to find the angle
  • Always begin by drawing the North line
  • Measure the angle clockwise from North
  • Write the angle using 3 figures 
• Knowing the compass directions for the common directions is helpful 
  • Due east means on a bearing of 090°
    • Draw the line directly to the right
  • Due south means on a bearing of 180°
    • Draw the line vertically downwards
  • Due west means on a bearing of 270°
    • Draw the line directly to the left
  • Due north means on a bearing of 360° (or 000°)
    • Draw the line vertically upwards
• Other directions can be found using these, for example 
  • Due Northeast means on a bearing of 045°
    
    • This is halfway between North (000°) and East (090°) 
  • Due Southeast means on a bearing of 135°
  • Due Southwest means on a bearing of 225°
  • Due Northwest means on a bearing of 315°

• Using the above bearings for compass directions will help you to estimate angles for other bearings on the map

How can I convert between scale drawings and real lengths?

• The steps for converting between scale lengths and real lengths are similar as for maps
• However with scale drawings, the scale may be more likely to be given in the form of a description with units, rather than a ratio

• The scale can be used to convert lengths found on a drawing to a length in real life
  • STEP 1
    Use a ruler to measure the distance accurately on the scale drawing, if appropriate
    • For example, measuring a length from A to B as 5.8 cm
  • STEP 2
    Use the scale to find the actual distance in the appropriate units
    • For example, using 5.8 cm, if the scale is "1 cm on the drawing represents 1.5 km in real life", the actual distance = (5.8 × 1.5) km = 8.7 km
• The scale can also be used to convert lengths found in real life into a length for a scale drawing
  • STEP 1
    Use the the scale to convert the actual distance to the scale distance
    • For example, if the scale is "1cm on the drawing is 5 m in real life" and the actual distance is 20 km, the scale distance will be (20 ÷ 5) cm = 4 cm
  • STEP 2
    Draw this distance on the map if asked to do so