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Every week I learn something new at work. Recently I’ve been struck by the number of things I have taught without fully and completely recognising how they are connected, even what I considered relatively ‘simple’ concepts.
Recently I spent a wonderful couple of hours with Tom Button from MEI, considering the use of dynamic geometry and KS4 geometry content. Several things became apparent. Firstly, it highlighted how little geometry content there is within the KS4 English national curriculum. Secondly, it showed how the nature of our curriculum really does compartmentalise content, potentially leading to disconnected schemes of work. Obviously this has serious consequences for our learners.
One reason I am here working at Cambridge Mathematics is I passionately feel that the Framework can support teachers in developing a more coherent, joined-up sense of mathematics, both for themselves and their students. Developing coherent learning sequences can be more time efficient and allow for a much greater depth of study. It is further supported by considering the tasks and activities we ask pupils to carry out.
Take, for example, transformations.
I don’t know about you but I’ve spent many years teaching rotations using plenty of practical equipment: large shapes to rotate on the board; shapes on a bamboo cane to show how they rotate around a point; jigsaw puzzles; tracing paper and dynamic geometry packages; and I even wrote a chapter on transformations for the CUP GCSE text book. So I felt pretty confident that I knew how to present and work through the topic. Yet now, approaching the content from a much more connected methodology initiated by my work here, I’m questioning the emphasis I put on practising carrying out and describing transformations. I realise now the importance of thoroughly and utterly studying a single rotation, not only as a whole but also in minute detail. In the case of transformations, inspecting the relationship between corresponding points really closely reveals a huge amount.
Much of what I write below may seem obvious, but often what we believe is obvious needs explicitly discussing in class. In doing so we equip students with a deeper resonance with the subject matter.
Many of these tasks could, and I believe should, be investigated before the transformation is formalised. They offer an opportunity really to get to grips with the effects of rotating an object.
Start by tracing the locus of one point (not necessarily a vertex) as it is transformed on the object to its corresponding position on the image. This can be easily accomplished using a program such as Geogebra, but I would encourage some mental visualisation first and reasoning as to why this happens once established.
Why is the locus an arc of a circle? What is significant about the centre of the rotation compared to this arc? Will all points produce an arc? Will they all have the same centre? Radius? What can be said about the proportion of a circle that has been drawn in each case?
I like the idea here of rotating with wet paint to show rotational movement:
Compared with the typical fold-and-paint butterflies it offers a really eye-catching visual.
Now consider what happens when you join corresponding points (not necessary vertices) to the centre of the rotation.
What is significant about the angle between the starting position and rotated position that these line segments make?
As with other transformations joining corresponding points in the object and image also offers an insight. Draw in the arcs and consider a circle theorem involving a chord and the centre of the circle. Constructing the centre of the rotation suddenly becomes easily achievable.
Although I’ve not come across any research, yet, to support the idea explicitly with circle theorems, I do believe that investigating all of the properties and results above on plain paper, potentially with a dynamic geometry environment, and without specific angles, allows learners to concentrate on the rotation itself. We become so concerned with practising set routines again and again that sometimes students can answer questions but don’t fully recognise their implications and construction, and hence struggle with non-standard problems.
How do you teach transformations, particularly rotations? How do you see the links with circles and circle theorems? You can comment below, visit our LinkedIn page or tweet us @CambridgeMaths.
SOMETHING TO TRY:
KS1: After reflecting painted pictures, by folding and letting paint transfer, repeat the task asking pupils to predict (and sketch) where paint will transfer to.
KS2: Ask pupils for the similarities and difference between the three paint pictures. Set them the challenge of recreating them, describing the actions needed to do so and writing instructions to someone in another room to do so.
KS3: What happens when you join corresponding points (not necessary vertices) to the centre of the rotation? How can this help you make sure that your rotation is correct?
KS4: Looking at the diagram below how can you use a circle theorem to carry out a construction to identify the position of an unknown centre of rotation.
KS5: Ask students to think about reflecting a coordinate grid in the y-axis and then to consider the tasks below.
- Give examples of co-ordinates that would be corresponding pairs
- What’s the generalised rule?
- Sketch some examples of graphs that would have the y-axis as a line of symmetry
- Suggest the equations for some of these graphs.
- How can the co-ordinates be used to show/justify this?
Ask students to think about rotating a coordinate grid 180° around the origin.