Research-based guidance and classroom activities for teachers of mathematics

Generalise relations

Generalise relations is a theme within functional relations between variables. Links to relevant activities and resources are on the right hand side of this page. 

What do we mean by 'relation?'

The word ‘relation’ is often used in school mathematics to try to describe what functions do, i.e. they relate two sets of numbers, the input of independent variables and the output of dependent variables. However, the word ‘relation’ has other meanings in mathematics which are not closely related to functions and some critics like to avoid it in the function context. We use it here because it is in common use when describing the general connection between the input, usually expressed as x, and the output variable, usually expressed as y. Functions are usually expressed as actions on the independent variable, f or f(x).

Maintaining its common school usage, it is useful for learners to get some experience in expressing relations between x and y in relatively simple linear contexts as an introduction to the behaviour of linear functions. This includes being able to distinguish between the term-to-term pattern in the y-values and the function itself connecting the x and y values. This is one of the many places in mathematics where an additive mindset has to be replaced with a multiplicative perspective.

Another value in this kind of task is learning to use conventional notations to encapsulate a relationship which consists of a string of unary operations. Using linear and simple quadratic expressions to describe something that is already understood can be a confidence-building start to understanding algebra at an age where some students might find it too abstract.