Throughout key stage 2, pupils have learnt about and used 2 types of mathematical relationship between numbers: additive relationships and multiplicative relationships. In year 6, pupils should learn to represent the relationship between 2 given numbers additively or multiplicatively, as well as use such a representation to calculate a missing number, including in measures and statistics contexts. Consider the following: Holly has cycled 20km. Lola has cycled 60km. We can describe the relationship between the distances either additively (Lola has cycled 40km further than Holly; Holly has cycled 40km fewer than Lola) or multiplicatively (Lola has cycled 3 times the distance that Holly has cycled).
As pupils progress into key stage 3, the ability to relate, recognise and use multiplicative relationships is essential. A pupil who can think multiplicatively would, for example, calculate the cost of 1.2m of ribbon at 75p per metre as 1.2×75p, whereas a pupil who was still thinking only in terms of additive relationships would use the approach of finding the cost of 0.2m (15p) and adding it to the cost of 1m (75p). During key stage 3, pupils will regularly use calculators to solve problems with this type of structure, and the multiplicative approach is more efficient because it involves fewer steps. Given any 2 numbers (related by a whole-number multiplier), pupils must be able to identify the additive relationship (in the example above, +40 and −40) and the multiplicative relationship (in the example above ×3 and ÷3). Though multiplicative relationships should be restricted to whole-number multipliers, pupils should be able to connect division by the whole number to scaling by a unit fraction: in the example above, this corresponds to understanding that because 60÷3=20, is one-third times the size of 60. When given a sequence of numbers, pupils should be able to identify whether the terms are all related additively or multiplicatively, identify the specific difference or multiplier and use this to continue a sequence either forwards or backwards. Pupils will need to use formal written methods to calculate larger numbers in sequences.
Language focus “The relationship between 2 numbers can be expressed additively or multiplicatively.”