Visualisation and the use of imagery are important aspects of mathematical learning. Here are some examples.

Imagine a square with one of its diagonals drawn. Imagine cutting the square along a line parallel to the diagonal to make two shapes. Write the names of the two shapes on your whiteboard.

Imagine four right-angled triangles, all the same size, on the table in front of you. Slide two triangles together so that their shortest sides meet edge to edge and their right angles touch each other. Do exactly the same to the other two triangles. Now slide the two pairs of triangles together so that all four right angles touch each other. Write the name of the shape that you have made on your whiteboard.

Imagine a paper square folded in half along a diagonal. Now fold it in half again so that the sides of the square lie on top of each other. What shape have you made? Now imagine making one straight cut across the corner that lies at the centre of the original square. Open out the paper. Without saying anything, quickly draw the shape of the hole. Compare with your partner.

Imagine a parallelogram and cutting from one of corners at right angles to one of the sides. Imagine taking the triangle cut off and attaching to the opposite side of the parallelogram. What shape would be created? What would its area be?

What this might look like in the classroom

Task
Find an image of a Mondrian painting or construct a similar image of your own
Pupil's work in pairs sitting back to back. One pupil has a copy of the image which the other cannot see and describes the picture in mathematical language so that the second pupil can draw his/her version from the description.

Task
Choose the shape of a shadow; e.g. a square. What 3D shapes could cast such a shadow?

Taking this mathematics further

Find out about the 17 wallpaper patterns.

Transformations have a range of applications in other fields, particularly in chemistry and crystallography. Optics in physics also relies on reflection in mirrors to produce a focus point. Biological systems often use symmetry to retain balance and efficiency.

At a higher level transformations can be described by matrices and in advanced mathematics symmetries are often used to teach group theory.

Making connections

Learners will have worked with basic transformations from an early point in their mathematical education. They should be able to carry out reflections, rotations and translations, and be able to describe them.

Understanding basic properties and transformation shapes develops into a formal understanding of similarity and congruence.