Perfect thirds

What kind of fractions of a given length can we fold?

Key stage: 3
Subjects: mathematics, shapes, angles, fractions


What has mathematics to do with paper folding? What kind of fractions of a given length can we fold?

In this activity we provide a context to explore a kinaesthetic study of number, fraction, proportion, similarity of shapes, properties of angles on straight lines, algebra of straight lines, simultaneous equations as well as ‘knowing for sure' versus ‘having numerical evidence'. The session develops naturally to prove a result.

The lesson has been trialled as a full lesson for Y9, Y10 and as a starter for sixth form students (it can be an example to use when addressing proof in AS-level mathematics).

The context also allows for making a connection between manual actions (folding paper) and abstract concepts such as straight lines and geometrical shapes. Teachers can talk about mathematical constructions as models for reality: models are simple enough to allow us to make predictions about reality.


  • Pupils fold squares of paper into halves, thirds and other parts. They compare fractions by comparing lengths folded.
  • Pupils investigate: how to generalize a method to fold 1/3, 1/5, 1/7, ...
  • Pupils are asked to research: other paper folding constructions, e.g. approximation methods for folding 1/5..., equilateral triangle

Curricular links:

  • Proof, conjectures
  • Mathematical modelling
  • Similar shapes
  • Angles on parallel lines
  • Simultaneous equations
  • Problem solving, conjectures
  • Fractions

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