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How can you make an angle of 60 degrees by folding a sheet of paper twice?

Arclets Explained

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.

Bow Tie

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Gibraltar Geometry

Age 11 to 14 Challenge Level:

Why do this problem?

In this problem, we present a photograph that has lots of intriguing patterns that we hope will provoke students' curiosity and prompt them to ask mathematical questions.

The main mathematical themes are tiling, tessellation and symmetry, but there is also an opportunity to think about area and estimation by considering how many tiles would be needed to cover a particular area.

Possible approach

Show the first picture: 


Give students a little while to look at it, and then share the closeup showing detail of the tiles:


Give students a bit more time to look at it, and then invite them to share mathematical questions that occur to them.
Here are some that might arise: 
  • What geometric shapes are there in the tiles?
  • What angles can you work out?
  • How many squares are there?
  • How many eight-pointed stars?
  • How big are the tiles?
  • How many rectangular tiles would you need to tile your classroom?
Students could choose a question to explore, or you could guide them towards a particular topic. This worksheet contains a black-and-white closeup of the tile together with the questions above.

The third picture gives some context by positioning Becky, who is 1.7m tall, in front of the tiles. This opens up the possibility of estimating the size of the tiles, and the number that would be needed to cover a given area.


Key questions

What can you see?
What mathematical questions do you want to explore?

Possible extension

There are lots of suitable follow-up questions in the Angles, Polygons and Geometrical Proof Short Problems Collection.

Possible support

Students could use squared paper or origami paper to draw or fold simpler patterns of their own inspired by the designs in the pictures.