Mathematical inquiry processes: Test cases; reason; generalise to other cases; prove. Conceptual field of inquiry: Tessellation; rotation and translation.
The prompt originated in a suggestion by Dan Walker, a secondary school mathematics teacher. He proposed the statement: "All quadrilaterals tessellate." By including two types of shape, the prompt could encourage students to generalise their findings about triangles and quadrilaterals to other polygons.
The prompt is true. In the tessellation of a triangle, each vertex (the point of intersection of three or more tiles) is made up of two sets of each of the three angles in the triangle. For the tessellation of a quadrilateral, each vertex is made up of one each of the four angles in the shape.
Initially students might decide the prompt is true by designing triangles and quadrilaterals and testing whether they tessellate. The teacher could encourage them to deepen their reasoning by pointing out that angles that meet at a vertex must sum to 360o. The final stage of the inquiry might involve students seeing a proof of the prompt.
A year 7 mixed attainment class at Haverstock school (Camden, UK) inquired into the prompt during a 50-minute lesson. Andrew Blair reports on how the inquiry progressed:
The students had been studying angles. The prompt gave them an opportunity to see angle facts in a new context. They had no prior knowledge of tessellations and, unsurprisingly, that was their first question about the prompt:
What does tessellate mean?
Does it mean that triangles fit into quadrilaterals? Do they "perfectly overlap"?
Do triangles and quadrilaterals do it in the same way?
Will it work with all the types of triangles?
Does it work with all 4-sided shapes?
After showing them pictures of tessellations, the students began to construct an understanding of the concept:
"The shapes fit side-by-side."
"They join together."
"They are a copy of each other."
"The shapes are perfectly patterned."
We compared their ideas with a formal definition (below) and agreed that they were consistent.
To tessellate is to cover a surface with a pattern of repeated shapes, especially polygons, that fit together closely without gaps or overlapping.
As our time was limited, I directed the students to cut out a triangle or quadrilateral from card and, after measuring and noting down the interior angles, tessellate their shape on paper. The quadrilaterals presented a challenge even to the students with the highest prior attainment, particularly when the size of the angles were similar. They had to think carefully about how to transform the shape.
I encouraged students to write in the angles that met at a point to verify that they summed to 360o. At the end of the inquiry, I displayed some of the tessellations under a visualiser, which elicited an intriguing question from one of the students who had noticed the angles chosen for the quadrilaterals were all less than 180o: "Would it work if the quadrilateral has a reflex angle?"
The tessellations shown here are from Suad, Alim, Mohamed, Ruhan and Era.
Quadrilaterals and pentagons
The kites (above left) tessellate because the angles at the vertices sum to 360o. Each of the four angles in the kite meet at a vertex. There are two types of vertex in the tessellation of the rhombuses (above centre). Either three or six equal angles form a vertex.
The third pattern (above right) is an example of an irregular pentagon that tessellates. Currently, there are 15 types of convex pentagons that are known to tile the plane using the same shape. Unlike the other two diagrams, the third is not an edge-to-edge tessellation because adjacent tiles do not share one full side. Students could go on to research the pentagonal tiles that tessellate.
Tessellations of regular polygons
The only regular polygons that tessellate are equilateral triangles, squares and hexagons (below) because the size of their interior angles are factors of 360o.
Another line of inquiry involves students researching the eight semi-regular tessellations (right), which are made up of two or more regular polygons. Each tessellation can be described using the number of sides of the polygons that meet at a vertex, starting with the smallest number.
From left to right, the configurations are:
22.214.171.124 126.96.36.199.4 (or 188.8.131.52) 184.108.40.206.4 (or 33.42) 4.8.8 (or 4.82)
220.127.116.11.6 (or 34.6) 18.104.22.168 3.12.12 (or 3.122) 4.6.12
Draw any triangle. Rotate the triangle around the midpoint of one of the sides (Rotation 1). This always creates a quadrilateral that can be rotated again around the midpoint of a side (Rotation 2) or translated. You can tessellate the shape across the plane with further rotations and translations.
Draw any quadrilateral. Rotate the quadrilateral around the midpoint of a side (Rotation 1). You can tessellate the shape across the plane with further rotations around the midpoints of sides or by translating the shape.