Hub Cap SymmetryAuthor(s): Lesley JonesSource: Mathematics in School, Vol. 20, No. 2 (Mar., 1991), pp. 10-13Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214769 .

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Lesley Jones, Goldsmiths' College, University of London

In the National Curriculum document the non-statutary guidance declares its commitment to the development of cross curricular work. This has been the traditional way of working in the primary school, where it is relatively simple to implement because of the structure within the school. Mathematics arises naturally from most topics and can be developed in the context of other work, so that children understand where the mathematics arises and what is its purpose. The ideas described in this article were developed in connection with a topic on transport.

A fascination with pattern and symmetry had caused me to notice the variety of patterns contained in the hub caps of car wheels. It was very easy to put together a collection of 36 photographs of wheels, each showing a different pattern. Well, maybe a few raised eyebrows as I knelt beside the kerbside with my Cannon Sureshot, but finding the variety of pattern was not a problem.

Reflectional Symmetry The work was undertaken with a class of fourth year juniors in a local school. These children had used Logo quite extensively and they showed a good awareness of angle. However, it was some time since they had studied symmetry, so this was taken as a starting point.

An initial discussion about line symmetry was followed by a look at shapes with 1, 2, 3 or 4 lines of symmetry. An exploration of the different tetrominoes (see Fig. 1) allowed the children to consider which were symmetrical and to find their lines of symmetry.

There was a familiar ring to the way in which they continued to test the S-shaped tetrominoes, even after they had tested with mirrors and by folding. Children seem to show an awareness of symmetry beyond the concept of line symmetry. This persistence is often expressed in statements such as: "It ought to be symmetrical, but I can't find the line". This provides the perfect cue for introducing the idea of rotational symmetry. We discussed this in terms of the shape "fitting its own outline" and "looking the same when it has been turned through a certain angle". These seem to be the most usual ways of introducing the idea, but they do seem to cause difficulties when children first encounter rotational symmetry.

Fig. 1

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Rotational Symmetry The formal language for rotational symmetry is very awk- ward. "Order of symmetry" doesn't trip lightly off the tongue and seems to make heavy weather of an idea which is not essentially so difficult for children. It is not the concept which is so difficult, but the formalisation and expression of it. Certainly these children coped well with establishing the number of ways in which the patterns could be turned, yet still look the same, before returning to their original position. Theirfacility with angle helped them to translate this into the angle which you would turn through to reach the next part of the pattern.

However, there is no doubt that some children do find symmetry a difficult topic. It is possible that we confuse our explanations in ways which we think are interchange- able, but which learners do not see as being so. In line symmetry, for instance, we use folding as a way of creating patterns with line symmetry (the ubiquitous ink blots) and then use folding as a way of checking for symmetry. Quite logical to us, but does the child see the implied logic of this?

Mirrors are used for checking, but if you place a mirror along a line, the resulting shape has (by definition) reflective symmetry. Now you need to check by looking back at the original to see if it looks the same as the shape in the mirror. Confused? Did the rules change again? The pleth- ora of terminology: mirror symmetry, line symmetry, reflective symmetry, reflection symmetry does not help to present the ideas simply. Rotational symmetry can be equally confusing. Leaving aside the formal language, we talk about turning the shape until it looks the same, or about turning it until it fits its own outline. Yet this latter explanation is not really useful after the initial idea has been grasped. If we consider a complex pattern such as the one shown below (Fig. 2), the rotational symmetry is a source of fascination, but the notion of fitting it into its own outline does not fully explain the symmetry. The "outline" is just that: the perimeter, the edge. The fasci- nation about the symmetry is that every point is repeated in an equivalent place. The playing card displays the same property. The card would certainly fit into its own outline, but the interesting feature of the symmetry is that every part of the picture is repeated in such a way that we cannot tell if the card is upside down or the right way up (Fig. 3).

Fig. 2

Practical Experience The children continued to explore rotational symmetry, using cut-out shapes set in rubber mats which could be turned and fitted into their outline. Then they were asked to produce patterns showing rotational symmetry, using coloured paper and scissors. They each produced 4 ident- ical motifs which had to be placed with great care and precision along the folds of the paper, to make a complete pattern (Fig. 4).

During this work the children had practical experience of rotational symmetry, but also they had used skills of

Fig. 3

estimating, measuring, calculating angles. It was apparent that they were ready to proceed to the next stage.

The hub cap photos were spread on the table and the children were asked to sort them into groups according to any criteria they chose. Each photo was numbered for ease of identification. Not surprisingly, they were somewhat cautious about the open-ended nature of this task, and found it difficult to believe that there wasn't one right answer to be found. However, their comments were very revealing. They noticed the different types of symmetry in the patterns. They noticed the different numbers and shapes which occurred.

The ideas generated here were used to introduce the children to IDELTA on the computer. This is a simple database which sorts items into a binary "tree" form, similar to the "key" task used in secondary science. The questions have to differentiate between two items at a time and are phrased in such a way that the answer must be "YES" or "NO". The questions varied in their mathemat- ical content, but many showed a good awareness of pattern:

* Inside is it shaped like a cross? * Are the shapes in groups of three? * Is the middle a square? * Has it a five point star? * Has it got a line of symmetry? * Are the holes shaped like boots?

(Next time you walk down the road see if you can spot the hub cap with boot-shaped holes!)

Fig. 4

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Fig. 5

12 Mathematics in School, March 1991

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One group of children continued with this task, whilst the others were asked to chose a photo of one wheel and to attempt to represent it as accurately as they could by drawing it (Fig. 5). They were provided with a sheet with a large circle drawn on it, compasses, circle stencils, rulers, protractors, pencils and calculators. This task brought together many skills. The children needed to measure the diameter of their circle and of the photo of their wheel, then to work out an approximate scale. The task differed then, according to the photograph chosen. For most they had to calculate and measure angles, work out the size of the holes and either make a template or choose the appropri- ate circle size. They had to decide on the placing of the shape in relation to the overall diagram. They were given very little guidance at this stage. It was posed as a problem solving exercise, though help was at hand if they asked.

In all cases they had to consider the proportions of their drawing, relative to the original photograph, to use a pair of compasses and a protractor skillfully and to make careful linear measures. They were encouraged to round the measurements up or down to simplify the task as necessary, thus giving experience of approximation. They found the

assignment quite difficult, but were clearly delighted by their successful results and they used a great deal of mathematics on the way.

The direct cross-curricular links of this work would be with art and design, language and science. The indirect links would emerge from the rest of the transport topic. A perusal of the National Curriculum shows that we were working on Attainment Targets 1, 9, 10, 11, 12 and 13 at levels 3 and 4 and Attainment Target 8, levels 4, 5 and 6. Most important of all, the children's confidence in handling reflectional and rotational symmetry in a practical way developed from the simple starting point of car wheels.

Reference

IDELTA is available from ILLECC for RML Nimbus computers.

Acknowledgement

Figure 2 reproduced from The Mathematical Association publication Maths in a Multicultural Society.

WHAT IF*

4. rid patterns by Jim Smith, Maths Education Centre, Sheffield Polytechnic

This starting point is probably best displayed on OHP, and consists of a hundred square on one transparency and a shape on a second overlay. For example:

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

I ask pupils to describe any patterns that they can see

in the numbers in the grid, and uncritically collect their observations on the board. We then consider which of the

observations will remain true if I translate the shape to a

different position on the grid, the idea being to focus on

the invariants of the situation. For example, although the

middle number will vary, it will of course always be ten more than the number beneath it.

An interesting line of development is to consider the

total of all of the numbers in the shape. Move the shape one square to the right, and work out the new total. What

happens? What happens to the total when you move the

shape one square left? What happens to the total when

you move one square down? Try some other movements

of the shape and see if you can tell what will happen to

the total. Can the total ever be 250? 251? 123543?

Pupils can then work on their own 100 squares, (prefer- ably preprinted), and I ask them to find out as much as

they can about the activity, but using a shape of their own choice, drawn on tracing paper.

The task is easy to extend, for example ... What if you tried another shape? If you reflected it? If you enlarged it?

If you numbered the grid in a different order? If you had

a different shape grid? If the grid was in modulo 5? A hexagonal grid?

Puzzle Section:

not all of these are on the same sort of grid!

21 254

?? ?? x-i x

33 ?? 35 263 x+5

You could try making up a puzzle of your owni

Mathematics in School, March 1991 13

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