Four pentagons inquiry

The prompt

Mathematical inquiry processes: Interpret; explore; find rules; generalise. Conceptual field of inquiry: Measurement of angles; angle properties of regular pentagons and rhombuses; interior and exterior angles of polygons.

The four pentagons inquiry prompt has been developed by Colm Sweet (a mathematics teacher in West Sussex, UK). He has tried it with classes in years 9 and 10. Students' comments and questions that have generated inquiry relate to:

  • Exterior and interior angles

  • Tessellations

  • Symmetry

  • Angle properties of parallel lines

  • Area of pentagons

  • Four other regular polygons

  • Angles in polygons

  • The relationship between the central shape and the number of sides of the polygons.

In a recent discussion, Colm described how students who are normally quiet in class can become enthusiastic to discuss their findings publicly in inquiry lessons. Indeed, a group of previously reticent girls took a leading role the last time he used the four pentagons inquiry. Galina Zuckerman in her research on Moscow classrooms claimed such 'breakthrough' groups, as she labelled them, are capable of changing the dynamics within the whole inquiry classroom.

Year 9 students' responses to the prompt

Question and notice sheets

Presenting the results of inquiry and peer assessment

Structured inquiry

Having derived the formula for the interior angle of a regular polygon, students are given a series of designs comprised of four regular polygons. Initially they are directed to find the different angles, but, as the inquiry develops, they are expected to identity the angles themselves.

An A-level student's inquiry

James Thorpe - a maths teacher at John Taylor High School, Staffordshire (UK) - used twitter to set the prompt as a challenge to his A-level class. The pages shown below are the response of Joe Tilley, one of the students in the class. On the first sheet, Joe has deduced the ratio of the area of a pentagon to the area of the central quadrilateral using trigonometry. On the second, he develops the inquiry by looking at the number of sides of the central shape as the number of sides of the polygon increases. Joe notices that the results can be grouped by using modular arithmetic. With a modulus of four, he has determined expressions for the number of sides of the central shape:

This is an impressive piece of independent mathematics, which was commended when it appeared on twitter. The next stage might be to attempt to prove the results. James commented that, "The geometry prompts lend themselves particularly well to an extended level of inquiry."