E1.1 Identify geometric properties of tessellating shapes and identify the transformations that occur in the tessellations.
Skill: Identifying Geometric Properties of Tessellating Shapes and the Geometric Transformations That Occur
It is important to understand that a tessellation uses at least one polygon combined in a repeating pattern to cover an area without creating overlap or leaving gaps. Due to the fact that they are regular polygons (angles and sides are congruent), only equilateral triangles, squares, and regular hexagons can create regular tessellations on their own.
Image A paving made out of regular polygons, in this case triangles. The paving is made of triangles, placed one upright and the next one upside down, in multiple rows.
It is possible to use irregular polygons to create tessellations,
Image Paving made with non-regular polygons, in this case trapezoids. They are placed one upright and the other upside down, in multiple rows.
Image Paving made of non-regular polygons, in this case non-regular hexagons. They are all laid in the same way and are joined together in multiple rows.
or use at least two polygons to avoid spaces.
A tessellation constructed using regular polygons is a regular tessellation, and a tessellation constructed with two or more types of regular polygons is a semi-regular tessellation.
The angles where tiles meet must add to 360°.
Complex tessellating tiles can be designed by decomposing shapes and rearranging the parts using combinations of translations, reflections, and rotations. If a shape can be transformed through a series of rotations, reflections and translations (for example, by being turned, flipped, or slid), and still look the same, the shape is symmetric.
Tessellations are often used to create artistic designs, including wallpaper, quilts, rugs, and mosaics.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Some tools are useful in helping students better understand transformations and see the results of various transformations. These include dot paper, graph paper, and tracing paper.
The Mira is particularly useful when reflecting geometric shapes.
Image On a gridded surface there is an initial figure, two squares below it, an axis of reflection and two squares below it, an image. It is a vertical reflection. On a squared surface on the left there is an initial figure, one square on the right, a reflection axis and one square on the right, an image of a horizontal reflection. On a squared surface in the left corner is an initial figure. The vertex, of the initial figure, closest to the axis of reflection is placed on the same square as the vertex of the image closest to the axis. The axis of reflection is placed diagonally, it crosses the common cube of the two patterns and divides it into equal parts. It is an oblique reflection.
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. 31.
Dynamic geometry software and applications are particularly useful for performing rotations according to a specified fraction of a turn and they help students better understand this transformation.
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. 34.
Note that students consolidate their understanding of the transformations by comparing them and using them to create friezes or tessallations.
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. 27.
Knowledge: Tessellation
A tessellation uses tiles to cover an area without gaps or overlaps. The angles where tiles meet must add to 360°. Tessellating tiles are composed of one or more shapes and fit together in a repeating pattern.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.
Knowledge: Geometric Properties of Polygons
To master the concept of tessellation, certain vocabulary words and certain concepts must be well understood.
A polygon is a two-dimensional shape formed by a broken and closed line.
A convex polygon has interior angles less than 180° and all vertices point outwards.
Image an irregular pentagon, the measurements of the angles are, 60 degrees, 140 degrees, 110 degrees, 140 degrees. A rectangle with 2 diagonals. A rhombus, the measurements of the angles are, 110 degrees, 65 degrees, 113 degrees, 72 degrees.
A concave polygon has at least one interior angle more than 180°, and at least one vertex pointing inwards.
Regular polygon : a polygon whose sides and interior angles are all congruent. The regular polygon is symmetrical and convex.
Irregular polygon : a polygon whose sides and interior angles are not congruent.
Source: translated from Guide d’enseignement efficace des mathématiques de la 4e à la 6e année, Géométrie et sens de l'espace, Document d'appui, Fascicule 1, p. 24-26.
Knowledge: Geometric Transformation
The transformations used for tessellations are translations, reflections, and rotations, all of which produce congruent images.
- Translations “slide” a shape by a given distance and direction (vector).
- Reflections “flip” a shape across a reflection line to create a mirror image.
- Rotations “turn” a shape around a centre of rotation by a given angle.
Source: The Ontario Curriculum. Mathematics, Grades 1-8 Ontario Ministry of Education, 2020.