Functions and graphs

You can adopt these techniques and tasks when planning lessons on functions and graphs.

Unless attention is focused on mental processes involved in work on functions and graphs, there is a real risk that pupils will be expected to move rather too quickly from plotting coordinates to tackling challenging generalisations that link algebraic and graphical forms.

Pupils require a higher level of thinking to make connections between real-life contexts and the features of a graph. Development of such skills is best supported by collaborative endeavour, allowing pupils the opportunity to share their emerging understanding and to learn from one another.

Pupils are better able to tackle challenging problems independently if they have first experienced some success in those areas through interactive group work.

Activities relating to functions and graphs can take two main forms: interpreting graphs or generating graphs. These should be developed alongside each other.

Interpreting graphs of functions

Interpreting pre-drawn graphs provides pupils with opportunities to recognise and generalise the relationship between elements in the function and features of the graph.

ICT applications are an ideal medium, both for teacher and pupils, because they provide the means for quickly and accurately testing hypotheses about these links.

Progression

$y=mx$ $$y=c+x$$, $$y=x+c$$ $y=mx+c$	Recognise these graphs for integer values of m and c Note the relationship between families of graphs as values of m and/or c increase or decrease Note which functions represent proportional relationships
$y=a{x}^2$ $y=x^2+c$ $$y=a{x}^2+c$$ $$y=(x+b{)}^2$$ $$y=(x+b{)}^2+c$$ $$y=(x+b)(x+a)$$	Recognise these graphs for integer values of a, b and c Note the relationship between families of graphs as values of a and/or b and/or c increase or decrease

Generating graphs of functions

An important skill is the ability to summarise the key features of a graph through a sketch.

This can be developed alongside skills involving graphical calculators or graph-plotting software. In all cases it is crucial to explore problems, discuss results and explain the relationship between the features of a function and the consequent features of the graph.

Progression

$y=mx$ $$y=c+x$$, $$y=x+c$$ $y=mx+c$	Sketch these graphs for integer values of m and c Explain the relationship between families of graphs as values of m and/or c increase or decrease Explain which functions represent proportional relationships
$y=a{x}^2$ $y=x^2+c$ $$y=a{x}^2+c$$ $$y=(x+b{)}^2$$ $$y=(x+b{)}^2+c$$ $$y=(x+b)(x+a)$$	Sketch these graphs for integer values of a, b and c Explain the relationship between families of graphs as values of a and/or b and/or c increase or decrease

Interpreting graphs arising from real-life problems

Consider using graphs from other subject areas, such as science or geography, or those that appear in newspapers, other published material or on the internet.

Ask pupils to explain what they think the graph might be about.

Discussion about the shape of a graph and how it is related to the variables and the context represented, supports pupils’ understanding.

Progression

linear conversion

a single straight line, interpreting the meaning of points and sections

distance–time

linear sections, interpreting the meaning of points and sections

temperature change

curved sections, interpreting the meaning of points and sections

Generating graphs arising from real-life problems

Use ICT to generate graphs of real data, including application data from other subject areas.

Focus on the degree to which the graph is an accurate interpretation of a real situation (recording temperature change) or part of a mathematical model (distance–time for a cycle journey).

Hypothesising about graphs without scales and headings can draw attention to the way in which different scales and starting points can lead to different interpretations.

linear conversion

a single straight line, interpreting the meaning of points and sections

distance–time

linear sections, interpreting the meaning of points and sections

temperature change

curved sections, interpreting the meaning of points and sections