Research-based guidance and classroom activities for teachers of mathematics

Multiplicative relationships

Multiplicative relationships is a theme within ratio and proportional reasoning. Links to relevant activities and resources are on the right hand side of this page. 

The core idea that underpins ratio and proportion is that any two numbers can be expressed as multiples of each other. Comparing two quantities can therefore be expressed as scaling, and the comparison written as a ratio. If two quantities are compared additively it is usually easy to visualise and represent the difference between them, but their ratio cannot be easily represented. If learners only think of multiplication as to do with repeated addition, arrays, and ‘times tables’ it is hard to apply multiplication to proportional relationships.

Many situations that require an understanding of ratio can be solved by ad hoc methods which are specific for that situation or those particular numbers. Examples:

  • ‘one bucket holds 3 times what another bucket holds’ can be visualised
  • ‘a sweater shrinks to four-fifths of its former size’ can be done by scale drawing
  • ‘the gradient of a linear function is 2’ can be thought of as doubling
  • ‘if it takes me 12 minutes to walk to the shops and it takes you 8 minutes, the duration of your walk is 2/3 the duration of mine’ can be represented with a numberline.

To get beyond this and deal in a general way with ratio takes several years and many experiences. It can also take imagination and the development of mental models to allow learners to make appropriate comparisons. Traditionally, there are classic problems involving proportional and inverse proportional relations that require such comparisons. Typical problems would be ‘if 5 kilos of potatoes cost £3, how much will 8 kilos cost?’ and ‘if it takes 3 days for 5 men to dig a ditch, how many men are needed to finish the job in one day?’. To decide how to tackle these questions the situations have to be imagined.