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Do quadrilaterals tessellate?

A year 8 teacher wrote on the board, "Any quadrilateral will tessellate".

The teacher said, "If you consider the statement to be always true or never true, explain how you know this. If you think it is sometimes true, explain when and why it is true at some times but not at others."

In groups, the students worked out answers by investigating particular cases. Then the teacher encouraged them to generalise the answer for all cases. She also said, "Try to explain your answer in more than one way."

The diagram below shows an irregular quadrilateral which has been flipped and rotated to cover the space around a point. This group of four identical quadrilaterals can be then be replicated to cover the whole surface.

A shape made up of an irregular quadrilateral in four different orientations around a point, next to a tessellation using the same shape.

Quadrilaterals around a point and a tessellation of quadrilaterals.

Many groups were able to reason well about the statement. You can download a Sample Response to Quadrilateral Tessellation which features a well-explained answer.

This activity is suitable for years 6–10. It could be presented at other levels, using other propositions. You can download a list of other Reasoning Statements to use with students.

Curriculum links

Year 8: Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning