It Stands to Reason
Both the use and study of rotations provide opportunities for our pupils (and we teachers) to develop their (our!) natural powers of imagining and of expressing to others what they (and we) are imagining. By working over time to refine the skills of ‘picturing in the mind’, all pupils can acquire a personal problem-solving resource to use widely.
Recently, in trials of sample assessment materials, many Higher Level pupils were unable to answer the following GCSE question:
It is possible that, had they been given more opportunities throughout their whole mathematics learning to practice visualising adaptations to, and movements in, images, they might have considered rotating triangle CEF and then been able to construct a valid argument (click to enlarge):
The diagram on the right shows the position of triangle CEF (coloured green) after a clockwise rotation of 90˚ about C. Triangle CE1F1 (the image of triangle CEF after the rotation) is isosceles with equal angles, of size y, at E1 and F1. In the parallelogram ABCD, angle BCD is opposite (and so equal) to the angle of size 2x. As a result of the rotation, we can see that the two equal angles in triangle CE1F1 (each of size y) are opposite to the exterior angle of size 2x. Because any exterior angle of any triangle equals the sum of the opposite interior angles, 2x = y + y = 2y. Therefore y = x.
What are some effective tasks that will provide opportunities for pupils to picture rotations in their minds? What teaching strategies will help them visualize rotations, and think hard about what they are visualizing? What kinds of questioning will develop and extend their ability to imagine rotations so that they can use that skill to reason with confidence and fluency in geometrical situations?
Say What You See
Prompting pupils simply to look at images and to imagine parts moving makes use of pupils’ natural powers to imagine and express what they are imagining, and helps those powers to develop. For example, you could ask pupils to say what rotations of one shape onto another they see in an image such as:
Different pupils will usually focus on different aspects – they will see different things, or see things differently. When a pupil describes something that she sees, the attention of other pupils may be directed to features or ideas that they would not otherwise have noticed. So it is valuable for pupils to hear what others have to say.
Suppose pupil A visualizes…
…and says “the red triangle at the top can be rotated through 180˚ onto the blue triangle on the right.” Then pupil B might respond (correctly) “I think the angle of rotation is 60 degrees”, because that pupil is visualizing this…
It may be tempting to tell the class that both pupils are right. But, in order to encourage reasoning, a better strategy would be to ask an open question such as “Why don’t pupils A and B agree about the angle of rotation?”. This is likely to draw other pupils into the discussion while motivating pupils A and B to refine (re-state or add to) their initial statements, in particular by explaining where in their particular mental images the centre of rotation is.
Some pupils may point out that after pupil B’s rotation the orientation of the triangle is different to its orientation after pupil A’s rotation...
If no pupils show this awareness it is well worth prompting discussion in that direction.
It is important that pupils understand that you are not asking for ‘right answers that you already know’. Also, pupils respond best in an atmosphere in which tentative statements are valued and may be offered as starting points for further modification by the same pupil or by other pupils.
Pupils ‘saying what they see’, perhaps just to themselves, is sometimes an important aspect of pupils’ activity (even) when the task is not specifically to ‘say what you see.’ For example the task might be to create a given image by repeatedly rotating a polygon. Suppose the given image is an ‘outline-only’ version of the previous image...
These are examples of polygons and sequences of rotations that create the whole image...
The different polygons and sequences of rotations shown here correspond to particular ways of seeing the whole image. That is, it can be seen as two overlapping triangles, as three overlapping rhombuses, as six touching triangles, or as three crossed polygons. Pupils need to learn to look flexibly at images, to be able to switch between alternative ways of seeing. This is an important skill that they can make good use of in problem-solving, particularly when they are stuck!
The same task on this modification of the whole image...
...provides more opportunities to see an image in different ways...
You and your pupils may see other possibilities!
Other images of which pupils can say what rotations they see, or can describe ways of making them by repeatedly rotating a single polygon, are not difficult to create. Here are some more examples...
The excellent tasks presented in Attractive Rotations from NRICH are likely to interest and involve pupils. The page “provides a simple starting point for creating attractive patterns using rotations, with the potential to go much further”. The images given, and pupils’ own creations, would make good subjects for ‘Say what you see’ sessions.
Pupils’ mental and physical activity when they tackle, reflect on and discuss this kind of task will help them:
- visualize rotations;
- understand that two rotations with different centres and angles can put an unmarked symmetrical shape into the same position (so that it looks the same) although it is actually in a different orientation;
- use their developing facility with rotation to recognize relationships between parts of geometrical structures;
- use their developing facility with rotation to modify images in order to simplify the construction of valid chains of reasoning.
Not only do pupils gain insight by learning to look for different ways of seeing images, but they also become more skilful reasoners by exploring alternative problem-solving methods. Therefore it is useful that any problem-solution or proof by congruence can be translated into a proof by transformation, and vice versa. For example, how would you and your pupils construct a proof of the following fact?
This is a very simple proof by one rotation and use of the properties of an isosceles triangle:
Many other mathematical facts can be proved visually using only rotation; thinking of such proofs is usually very satisfying, as is this one:
In the next issue we will look at some more problems in which thinking about rotation enables reasoning. In the meantime, do share similar activities that you know, or have used with your own pupils.
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