It Stands to Reason
Graphs provide one of the most powerful and revealing ways of representing the relationships between quantities that are the essence of many mathematical and ‘real’ situations. Pupils need to acquire a deep understanding of the qualitative meanings of graphical features, such as maxima, minima, gradients, discontinuities, and so on, over and above the technical skills of choosing scales, plotting points, drawing curves … . They can then use that understanding to reason about particular related quantities; they can solve problems by reasoning qualitatively about the features of appropriate graphs.
What kinds of task will help pupils develop this mastery of the qualitative aspects of graphs? Research (Shell Centre for Mathematical Education, 1985) has shown that the best challenges include those in which pupils:
 try to pairup given sketchgraphs with given written or pictorial descriptions of how particular quantities are related
 try to sketch graphs to represent given relationships (described in writing or using pictures)
 are given sketchgraphs and try to describe, in words or pictures, the relationships that they show.
Here we look at just one (quite hard) context, before we provide the link to an excellent free source of detailed advice about ways of working with these ideas, and that includes a great variety of interesting and relevant tasks.
Example
Suppose that a small ‘thing’, such as a beetle, moves along a track that is a closed loop. The two related quantities that we are interested in are:
 the distance that the beetle has travelled along the track,
 the distance between the beetle and a fixed point, which might be a ‘home’ point on the track, and the point from which it started.
The track may be any shape; for example, it might be rectangular …
Part 1
Each of the pictures, α, β and γ, shows a track of a different shape. In each case the beetle starts from a ‘Home’ point at the bottomleft corner of the track, and does one circuit of the track in an anticlockwise direction.
Each one of the following sketchgraphs, A, B and C, shows the relationship between ‘distancetravelled’ and ‘distancefrom‘home’’ for one of the diagrams above. The pupils’ task is to pairup the diagrams and sketchgraphs, and to explain why their pairing is correct.
Part 2
Using your thinking about Example 1, draw a track that would produce sketchgraph D:
Click to see the track which would fit sketchgraph D
Part 3
Again using your previous observations, sketch a graph to show the relationship between ‘distance from ‘home’’ and ‘distance travelled’ when the beetle does one circuit of this track:
Part 4
The track is a circle of radius 1 unit. The beetle starts from ‘home’, which is a point on the circular track, as shown …
Sketch a graph to show the relationship between ‘distance from ‘home’’ and ‘distance travelled’ when the beetle does one circuit of this track.
Pupils might find this template useful …
An idea of the shape of the graph can be deduced from reasoning about images such as these …
Part 5
For each of the following sketchgraphs, draw a diagram showing the shape of the track and the position of P that would produce a graph with the given features:
Ways of working
Allow pupils time to attempt these tasks in small groups. The atmosphere should be one of conjecture and discussion so that pupils have plenty of opportunities to explain their thinking, perhaps at first tentatively, and obtain feedback from each other.
For each taskpart described above, each group should be encouraged to discuss the ideas of everyone in the group (which might at first be conflicting) until they reach a conclusion with which they all agree. Then invite a representative of each group to explain their conclusions, and how they arrived at them.
Avoid voicing immediate judgements because this may prevent other groups from contributing slightly different or even conflicting explanations.
Further material
This article is about just a few of the many ideas, advice and classroom materials that are beautifully presented in the freetodownload resource The Language of Functions and Graphs, available at the Shell Centre for Mathematical Education Publications Ltd.
Image credit
Page header by Brian Hillegas (adapted), some rights reserved
