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Secondary Magazine - Issue 133: It Stands to Reason

Created on 17 May 2016 by ncetm_administrator
Updated on 24 May 2016 by ncetm_administrator


Secondary Magazine Issue 133'Close up of The Thinker' by Brian Hillegas
 (adapted), some rights reserved

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 pair-up given sketch-graphs 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 sketch-graphs 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.


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 …

beetle and 'track'

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 bottom-left corner of the track, and does one circuit of the track in an anti-clockwise direction.

alpha picture   beta picture   gamma picture

Each one of the following sketch-graphs, A, B and C, shows the relationship between ‘distance-travelled’ and ‘distance-from-‘home’’ for one of the diagrams above. The pupils’ task is to pair-up the diagrams and sketch-graphs, and to explain why their pairing is correct.

graph A

graph B

graph C

Part 2

Using your thinking about Example 1, draw a track that would produce sketch-graph D:

graph D

Click to see the track which would fit sketch-graph D

track to fit sketch-graph 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:

altered 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 …

track with radius unit 1

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 …

template for graph

An idea of the shape of the graph can be deduced from reasoning about images such as these …

image showing diameter   image showing two radii
image showing two radii and cord   image showing two radii and cord

Part 5

For each of the following sketch-graphs, 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 task-part 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 free-to-download resource The Language of Functions and Graphsavailable at the Shell Centre for Mathematical Education Publications Ltd.

Image credit
Page header by Brian Hillegas (adapted), some rights reserved



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