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.

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.