Research-based guidance and classroom activities for teachers of mathematics

# Functional relations between variables

 We have identified six themes within the key idea of functional relations between variables: function machines, generalise relations, solving equations, polynomial coefficients, functions as mathematical entities, and modelling. Each theme page has links to relevant online activities and resources. A full list of all activities for functional relations between variables is on the right hand side of this page.
##### Equations as intersecting functions

The idea of function is used as the basic concept behind the algebraic and graphical representation of relations. Equations can therefore be seen as particular cases of intersecting functions.

School students are taught how to solve equations in a variety of ways, each with their strengths, limitations and confusions. Graphing has its own problems, some of which are conceptual but may appear to be technical.

##### Pathway to understanding

A full understanding of functions goes beyond connecting equations and graphs, and can take many years and a variety of experiences to develop. To develop this full understanding, learners have to encounter and use a wide range of functions: continuous and discrete; with and without time on the x-axis; smooth and not smooth; calculable and non-calculable. There are several distinct shifts of perspective that need to be made:

• Interpreting graphs
• Tabulation and constructing graphs
• Calculating function values for particular independent variables
• Shifting between pictorial, algebraic, physical and graphical representations
• Identifying functions and their properties and definitions
• Transforming and operating with functions as objects
##### Multiple representational software

While there is no research that indicates a 'best' curriculum ordering, most success has been achieved using multiple representational software to depict situations that learners understand or can understand via a modelling cycle. Such approaches have been shown to avoid conceptual problems with graphing and to improve interpretation of graphs and functions