The work of M C Escher

20th May 2005 at 01:00
The picture on the front of the poster shows "Relativity", which was produced in 1953 by the Dutch graphic artist Maurits Cornelius Escher. It is one of his "impossible" pictures and contains many contradictions, although at first glance it may seem fairly realistic.

Talking about the picture

Ask the children to look closely at the poster and count how many examples of various objects they can see. Looking for trees and plants will make them realise that the "outdoors" is not in one obvious place; counting the number of stairs will lead them to realise that it is not always clear what counts as a stair, particularly at either end of the flight. For example, one stair doubles as the vertical side of a doorway.

Children can be encouraged to look for, and discuss, shapes in the picture.

This can be as simple as looking for curved shapes or straight lines.

Discussion of curved or round shapes can extend to looking at archways and discussing which shapes count as round and why. For example, how would children classify plant pots, the objects some of the figures are carrying, or the spheres on top of the posts? The picture could lead to discussion of how three-dimensional shapes are represented in two dimensions, or of parallel lines and associated angles.

Look for contradictions. This may be helped by asking children to decide which surfaces are which. Some double for others (for example, floors and walls). And, while some surfaces are clearly "walls", it is not clear which way is up. Encourage children to look for particular contradictions (for example, on two of the staircases, both sides function as top and one staircase contains two figures moving in the same direction, but one appears to be going up and the other down).

Creating a poster

Having looked at the poster, children might be encouraged to create a picture or poster of their own. They could create a picture using all or some of the ideas in the poster, for example, staircases, railings and archways. An alternative is to challenge them to produce a picture using only pencil, ruler and compasses, or using squared paper or other mathematical paper, such as triangular (isometric), dotty or polar.

More able pupils may try to draw a picture incorporating impossible elements in a believable way.

Using the activities

The activities provided build on some of the images introduced in the poster. You can differentiate them for different ability groups by omitting some of the questions or changing the numbers. The notes suggest ways in which the activities can be approached or simplified for younger children.

Extensions are also suggested that extend the number patterns to introduce simple algebraic ideas.

Arches, posts and rails

* Younger children could approach this activity by building linked arches using wooden blocks for the posts and either arch-shaped blocks or cuboids to place on top. Building arches using different numbers of posts should help to introduce the idea that the number of posts needed is one more than the number of arches.

* The activities on the sheet move from the simple relationship outlined above to slightly harder ones. The middle activity requires children to move from the number of posts to the number of rails, whereas the final activity requires them to work backwards from the number of rails to the number of posts.

* All parts of this activity can be extended by asking children to consider general cases. Algebraic notation can be used. For example, a pencil fence with p posts needs 2(p - 1) rails. A fence with p posts and triple rails needs 3(p - 1) rails.

Staircase maths

* One approach to this activity is to build staircases like the ones in the pictures using multi-link cubes. The first staircase offers opportunities for children to count in steps of two and they can predict how many cubes are needed to build the next staircase.

* The first activity on the sheet moves gently to talking about the staircase with 5 steps and then 6 steps. Some children may be helped by continuing in this way though others may be able to move straight to considering 30 steps, hopefully without having to draw or make the staircase. They will certainly need an alternative strategy to drawing if they are to answer the final part of the question dealing with 500 steps.

* The second sequence is based on triangular numbers and it is harder to find a connection between the number of the stairs and the number of squares. The staircase with s steps will have 12s (s+1) squares. This can be shown by drawing or building two such staircases and putting them together to form a rectangle with sides s and s+1.

Banisters and stairs

* The simplest parts of this activity involve naming shapes, which could be accompanied by talking about their properties and drawing them. Children may call the shapes in the first picture quadrilaterals or (more precisely) trapeziums. The shapes in the second picture have six straight sides and are therefore hexagons, though they may not match children's views of what a hexagon looks like.

* The activities on this sheet can be extended by building on the question about octagons and the one following it. This could lead to a sequence of banister and staircase pictures in which every time an extra step is put between two posts, the shape formed has two more sides.

* The third picture starts by looking at shapes but is then extended to include angles. In the picture, the parallelograms have two angles of 60x and two angles of 120deg. Other shapes produced in a similar way should give pairs of equal angles, so that adjacent angles add up to 180deg.

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