Four pentagons inquiry
Mathematical inquiry processes: Interpret; explore; find rules; generalise. Conceptual field of inquiry: Measurement of angles; interior and exterior angles of polygons.
The four pentagons inquiry prompt was developed by Colm Sweet (a mathematics teacher in West Sussex, UK). He tried it with classes in years 9 and 10. Students' observations and questions that generated inquiry in his classroom related to:
Exterior and interior angles
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.
After using the prompt, Colm described how students who are normally quiet in class can become enthusiastic to discuss their findings publicly. Indeed, a group of previously reticent girls in year 10 took a leading role in the inquiry.
Galina Zuckerman in her research on Moscow classrooms claimed such 'breakthrough groups', as she labelled them, are capable of changing the dynamic in the classroom. As the group embraces the social and mathematical norms of inquiry, it leads the class into exploration, conjecture and generalisation.
In a structured inquiry students derive the formula for the interior angle of a regular polygon before exploring a series of designs comprised of four regular polygons.
Initially, the teacher directs students to find different angles and explain their reasoning.
As the inquiry develops, the students take the initiative by conjecturing and generalising. One line of inquiry involves the class in trying to find a connection between an n-sided polygon and the shape enclosed by the four polygons.
The structured inquiry ends with the class trying to express one angle in terms of another. In the illustration, for example, angle y is expressed in terms of x.
Mixed attainment inquiry
A year 7 mixed attainment class at Drayton Manor High School in Ealing (London, UK) used the four pentagons prompt to extend their inquiry into the 2-dimensional shapes prompt.
The students followed the structured inquiry to derive the formula for the sum of the interior angles of a polygon. They then applied their knowledge to other diagrams composed of four polygons, recording their calculations below each diagram (see pictures).
The inquiry ended with students explaining how they had deduced their solutions. The teacher used a visualiser to project a diagram and calculations on the board.
In this final phase, the teacher insisted on precise mathematical language related to angle properties of points, lines, and polygons.
In reviewing the reasoning of their peers, students suggested that long chains of calculations (such as, 7 - 2 = 5 x 180 = 900) are inaccurate and should be replaced by separate calculations.
Those students whose calculations contained such chains were keen to make corrections, which they did at home and brought to the next lesson.
Students' responses to the prompt
Question and notice sheets
Colm's year 9 class posed questions about the angles in the diagram and the area of the pentagon. They used properties of polygons to find the angles and started to draw on their knowledge of Pythagoras' Theorem and trigonometry to work out the area.
Presenting the results of inquiry
After presenting the results of inquiry, students exchanged their exercise books to peer review each other's ideas.
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."