Ratio, proportion and rates of change: use scale factors, scale diagrams and maps

The idea

A scale tells you how many real units one drawing unit stands for. A map scale of 1 : 25 000 means 1 cm on the map is 25 000 cm on the ground, which is 250 m or 0.25 km. When the scale is written as 1 : n with the map length first, you multiply by n to go from the map to real life and divide by n to come back. So a 6 cm line on that map is 150 000 cm, which is 1.5 km. On a scale drawing you also measure with a ruler and a protractor, so work to the nearest millimetre and the nearest degree. Between two similar shapes the scale factor is a length on the new shape divided by the matching length on the old one, and lengths that are not matching pairs give the wrong factor.

What earns the marks

• Stating the scale as a multiplier, for example that 1 cm represents 25 000 cm.
• The method of multiplying or dividing a measured length by that scale, even if the final unit is wrong.
• A length measured within the tolerance allowed on the drawing, a small amount either way.
• The real distance given in the unit the question asks for.

Common mistakes and how to put them right

• Reversing a scale of 1.

  Put it right: 50 000, so a map length is divided by the scale instead of multiplied by it.

• Turning 250 000 cm into kilometres by dividing by 1000 only, giving 250 km rather than 2.5 km.

• Measuring from the end of the ruler instead of the zero mark, so every length is out by the same amount.

• Taking a scale factor from a pair of sides that do not correspond, such as the base of one shape against the height of the other.

Exam tips

• Use a sharp pencil and measure to the nearest millimetre, because scale drawing marks are awarded inside a narrow tolerance.
• Write the unit beside every line of a map calculation; most lost marks here are centimetres reported as metres.
• A scale written as a ratio with no units has both sides in the same unit, so convert only once at the end.