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Four pentagons inquiry

The 4 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.


You can follow Colm Sweet on twitter @MrCSweet.
An A-level student's response to the prompt
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 here 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:
Polygon of n sides Number of sides of central shape
 0 mod 4 n - 4
 1 mod 4 n - 1
 2 mod 4 n - 2
 3 mod 4 n + 1
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."
Year 9 students' responses to the prompt
(1) Questions and comments sheets
(2) Presenting the results of inquiry (with peer assessment)