It Stands to Reason
The thoughts about tasks and problems below are a continuation of the discussion in last month's It Stands to Reason about ways of developing pupils’ abilities to imagine and recognise rotations. Such abilities help pupils to reason geometrically, which will be tested explicitly in the new GCSE papers.
When, in any mathematical context, pupils work regularly at visualising features, and describing or drawing what they visualise, their powers of imagination (that they already possess naturally) grow. In particular they become more adept when they look at a diagram of a geometrical problem at visualising features that are not yet shown, at perceiving alternative ways of seeing an image, and also at imagining what can be changed without changing essential relationships.
For example, consider the following (rather tricky) problem from the very useful bank of problems provided by the United Kingdom Mathematics Trust (UKMT) (click to enlarge):
A way of reasoning to the answer becomes clear when the problem-solver visualises the rotation of triangle PSR anti-clockwise about R through 61˚ to see that triangle PSR is similar to triangle QPR, and smaller, and remembers that the longest side of a triangle is opposite its largest angle.
How then can a teacher provide frequent opportunities for pupils to practise visualising (rotations in particular), and describing or drawing what they visualise? A teaching strategy closely related to Say What You See, described in last month's issue, is to challenge pupils to create and describe their own examples of something. As pupils begin tentatively to think of, and to describe to their peers, their examples, the goal is to extend the range of the class of objects that the pupils regard as permissible examples, and to stretch their imaginations as far as possible by asking for another example and another one and another and another ….
Another and Another
Very often when you show pupils your own examples, or examples from a textbook, the pupils do not all experience the examples in the same way as you do; that is, they may not focus on what it is that the author intended the examples to exemplify. By constructing, explaining to others, and discussing, their own examples, pupils can move towards the deep conceptual understanding you are guiding them to acquire.
Here are some examples to try.
- Ask your pupils to split a square into two pieces, and then rotate one piece to form a new shape, as in these four examples.
Ask for another, then another, then another...way of doing this.
What if there are more than two pieces?
- Split a regular hexagon into n pieces of which m pieces are rotated to form a new shape. Vary the values of n and m.
Ask for another, then another, then another…example.
- Challenge your pupils to show how shapes-drawn-on-grids, such as the two grey triangles and two pink squares shown at the top of the diagram, can be split into parts that can be rotated onto each other. The two images on the right show one way (there are many possible ways!) of doing this to each of the four shapes at the bottom.
For each shape ask for another, then another, then another … .
Same and Different
When it is possible to rotate pieces of a shape, such as six equilateral triangles composing a regular hexagon, in different ways to form the same new shape, you can ask “what is the same and what is different?” about the two methods and results. For example, in each case, about which points, and through which angles, are particular triangles in this hexagon rotated to form the parallelogram shown?
Pupils should come to see that labelling points and shapes helps them to describe which triangles are rotated, and how they are rotated. Diagrams showing how each parallelogram can be obtained by rotating parts of the hexagon are here.
Deciding whether two objects are congruent by testing whether each can be mapped on to the other by a rotation is a powerful strategy in geometric reasoning. Struggling to describe, and explain how they know, what is the same and what is different about components of an image such as the one below will help pupils identify congruent shapes.
Some facts that pupils know can be proved visually using rotation. For example, there are many different ways that pupils can use valid reasoning to prove the formula giving the area of a trapezium, each of which corresponds to a different way of seeing the structure of a general trapezium. Looking for justifications for the trapezium area formula would make a good focus for an Another and Another challenge (click to enlarge):
Pupils are sometimes completely stuck on a problem UNTIL they visualise a rotation. For example, consider this question from the UKMT website (click to enlarge):
The solution can be seen, and explained, using only rotation and translation. First, visualise the red trapezium split into an isosceles right-angled triangle and a right-angled trapezium:
Now imagine rotating the green triangle 45˚ anti-clockwise about its bottom right-hand corner, and then translating it:
Mentally re-combine the triangle and small trapezium, and you can see that the shape that is formed can be repeatedly rotated through 90˚ to cover exactly the whole regular octagon:
So you literally see that the area of the whole octagon is four times the given area (3 cm2) of the red trapezium. Therefore the area of the whole octagon is 3 × 4 = 12 cm2.
You might like to offer your pupils this challenging problem from gogeometry.com. It can be proved using only rotations and one translation, and knowledge of the converse of Pythagoras’ Theorem. Enjoy!
Let us know (firstname.lastname@example.org or via Twitter @NCETMsecondary) if you try any of these strategies: the examples you use, the responses your pupils create, and the impact and benefits you notice.
Utah State University has a useful interactive application for exploring rotations with which users can design ‘objects’ (shapes and combinations of shapes), choose and change the rotation angle, move the centre of rotation, move/group/delete their ‘objects’, and turn the background square lattice into a coordinate grid with the origin wherever they want it to be.
The animation Notes sur un triangle from the National Film Board of Canada is well worth watching and studying. Pupils could be challenged to describe, discuss, and possibly try to reproduce in drawings, small parts of it.
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