Teaching TILINGS / TESSELLATIONS with Geo Sphinx Grunbaum and Shephard defined TILINGS /TESSELLATIONS in 1987 as "partitionings of the plane into regions, or tiles". Most students will easily recognise that any regular triangle, quadrilateral, and hexagon can tile a plane surface. However, to determine whether any shape different from the above-mentioned will cover the plane without gaps or overlaps can be a far more challenging task. The SPHINX SHAPE provides a perfect extension activity for tessellation lessons. It is made up of 6 equilateral triangles as shown in the diagram on the left. Geo Sphinx and REP-TILES A rep-tile is a geometric shape which can be fitted together with copies (replicas) of itself to form a larger similar figure. A simple example of rep-tile is the square: 4 congruent squares will build a larger square; with 4 of these large squares an even larger square can be made etc. By continuing this activity many times, the plane will be tiled. Therefore, we can say that the square is a rep-4 tile. So is the Sphinx. In fact, with 4 Sphinx shapes, a bigger Sphinx can be formed. A curious fact is that every rep-4 tile is also a rep-9 tile (see diagrams below). Sphinx as a Rep-4 Rep-tiles help students to acquire spatial sense and to visualise and represent geometric figures as well as to explore their transformations. Creating a tiling pattern requires such mental imagery as visualising the possible rotations and placements of a tile in the tiling pattern, thus further developing spatial awareness. Sphinx as a Rep-9 Activities using Geo Sphinx Sphinx as a Rep-4 (size 2) Ask students to form a Sphinx from 4 pieces. This is called a rep-4 size 1 Sphinx. Ask them to work out what size 2 would be (4x4=16). Ask groups of 4 to build it. Solution: see diagram on the left. Now let them calculate how many Sphinxes are needed for a rep-4 size 3 (4x4x4=64) and have them build it. Encourage them to verbalise and record a rule which allows to determine the number of Geo Sphinx shapes necessary to make ANY SIZE rep-4 sphinx. Solution Rep-4 Sphinx size n = 4 x 4n-1 = 4 n Sphinx as a Rep-9 (size 2) The experiment can be repeated with the Sphinx as a rep-9 tile. Size 2 requires 81 Geo Sphinx pieces (see diagram left). Size 3 requires 729 Geo Sphinx pieces. In this case, the summarising rule to determine the number of Geo Sphinx shapes necessary to make ANY SIZE rep-9 Sphinx is: Rep-9 Sphinx size n = 9 x 9n-1 = 9 n More Sphinx Ideas A different way of looking at the size of a Sphinx is to consider the pieces at its base. Assuming that the longest side of a Geo Sphinx piece has the value 1, the other sides will be worth 2/3 and 1/3. This way, the Sphinx also provides an interesting cue for fractions activities. Here on the left-hand side are some examples of how we can calculate the size of a Sphinx following the above method. Any Sphinx is always made up of as many pieces as the square of its size (base length). Size 2, 4 pieces Size 4,16 pieces Size 3, 9 pieces. Base: 2/3 + 1 + 1 + 1/3 = 3 Size 6, 36 pieces. Base: 1/3+1+1+1/2+1/3+1+1+1/2= 6 We made a size 2 & a size 3 Sphinx. Is it possible to build a Sphinx which has ANY PRIME NUMBER as its base length?? Geo Sphinx as a puzzle game At the end of the lesson, the Sphinx can be made into an entertaining challenge by asking the children to guess how 4 pieces were joined to make outlines such as the ones on our Task Page. You can then ask pupils to think of what outlines THEY could make with 5 or 6 pieces.