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The Guides for Effective Instruction in Mathematics Revised

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E1.1 Identify geometric properties of tessellating shapes and identify the transformations that occur in the tessellations.

Activity 1: The Tessellations Around Us

Planks for hardwood flooring. People planning a kitchen renovation. A brick wall.

In this activity students analyze various tessellations and recognize a practical and aesthetic use of the properties of two-dimensional shapes and transformations.

A few days before the activity, ask the students to reproduce, on a piece of paper, a tiling that they see at home (for example, bathroom or kitchen tiling, brick wall, driveway or sidewalk pavement). On another sheet, they draw one of the patterns of the life-size tiling. On the day of the activity, ask the students to get into teams of four and compare the tessellations by discussing the two-dimensional shapes that make them up, the patterns present, the size of the patterns (using the second sheet prepared at home) and transformations used to create these patterns. In order to encourage the exchange between the team members, invite the students to discuss the reasons that may have motivated the creation of these tessellations (for example, shapes, colours, transformations, materials, aesthetics, originality). Then ask each team to share certain findings with the class using the mode of presentation of their choice (for example, poster, oral presentation, discussion).

Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 39-40.

Activity 2: Turn, Turn!

Ask students to check if it is possible to create a tessellation from a triangle in Appendix 6.1 (Pairs of Triangles) using only rotations whose centre is located on one of the vertices of the triangle.

Note: This is sometimes possible, either with the C, D or E triangles.

Image A hexagon formed by a triangular paving. "A paving with the triangle "C". A square formed by a triangular paving. "A paving with the triangle "D".A triangle formed by a triangular paving. "Paving with triangle "E".

Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 90.

Activity 3: Dear Escher

Note: Maurits Cornelis Escher (1898-1972) was a Dutch artist who became famous for, among other things, his mosaics that give the illusion of metamorphosis (for example, Reptiles, 1943) and his depictions of impossible worlds (for example, Ascent and Descent, 1960). These works, based on the properties of geometric transformations and the laws of perspective, have particularly fascinated and inspired mathematicians.

Show students some of Escher's work and ask them to observe how the artist was able to cover the plane. Point out that they will have the opportunity to create a similar work using the same principles. Have students draw a large triangle of some sort on a piece of cardboard approximately 10 cm x 10 cm, cut it out and determine the midpoint of each side. Show students how to modify their triangle into an original shape from which they can create a tessellation:

  • draw any shape starting from a vertex of the triangle and ending at the midpoint of a side adjacent to that vertex;
Image A figure is plotted by starting from a vertex of a triangle.
  • cut out the shape;
A figure of some kind is plotted by starting from a vertex of a triangle. This figure is then cut out.
  • rotate this shape, using the midpoint of the side as the centre of rotation, until the two halves of the side of the triangle are overlapping, and tape the two pieces together.
A figure is drawn by starting from a vertex of a triangle. This figure is then cut out and rotated.

If students wish, they can repeat this process starting at each of the other two vertices. Then have students create a tile from the resulting shape using rotations from a centre of rotation on the midpoint of one of the sides. Suggest that they add lines inside each of the shapes to further highlight what they look like (for example, bird, fish, leaf) and that they colour their tile to create a better effect.

Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Fascicule 2, p. 91.

Activity to Do at Home: Fantastic Tessellation!

Create a tessellation on which we can see all the following shapes:

  • congruent right-angle triangles;
  • congruent rectangles;
  • congruent trapezoids;
  • rhombuses;
  • hexagons.

Colour your tessellation so that your creation is truly unique.

Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).