There are two key learning points when solving problems with angles in polygons. The first is to understand why all the exterior angles of a polygon have a sum of 360°. The second is to understand the interior and exterior angles appear on the same straight line.
Students can be told these two facts and attempt to internalise them by repeated practice. However, this would be an opportunity missed. Exploring the properties of polygons through construction can be a fantastic discovery of patterns and geometric properties.
I begin the lesson by asking students to categorise a selection of 2D shapes into groups. This can be done through discussion or in their books. I don’t use mini-whiteboards this lesson as the student’s desks will be busy with rulers, pairs of compasses, plain paper and protractors.
Through the class discussion I draw out any misconceptions about how to define a regular polygon. This is essential knowledge for the remainder of the lesson.
I explain to the students that the aim of today’s lesson is to discover some (I do not say how many) angle properties of polygons. To make this as simple as possible all the polygons will need to be regular. Later, we will extend these properties to calculate other facts about polygons.
We begin by constructing a regular triangle within a circle. I ask all students to construct their own circles using a pair of compasses. This may seem a simple task, however it often proves to be the most challenging part of the lesson as compasses often slip and students can lack dexterity.
The class and I construct the regular triangle and quadrilateral together. When the polygon has been constructed I ask students to extend the edges to make a windmill shape. Later, this will help to visualise the exterior angles. The windmill analogy proves helpful to visualise the sum of the exterior angles as 360°.
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Having constructed the triangle and square I leave the students to work independently to construct a pentagon and hexagon. They use the constructions to measure the interior and exterior angles.
About 40 minutes into the lesson most students have completed up to a hexagon. At this point I stop the class to remind them of the aim of the lesson.
I explain to the class that to identify a pattern it is helpful to display our results within a table. I use student’s results to partially complete the table as this provides an opportunity to discuss accuracy in the construction or measurement of angles.
I ask the students to discuss any patterns they notice and to use these patterns to predict the likely results for the next four polygons. The predictions can be tested, and if needed rethought by constructing the heptagon and octagon.
In the plenary I ask students to discuss the patterns they have noticed and how they can be explained. All the students identify the sum of the exterior angles as 360°. When I challenged them to explain why the majority refer back to the initial circle we constructed. I ask them to think deeper and with a greater focus on the windmills created by the extended edges. Eventually, word begins to spread that the all exterior edges rotate around a circle which is why they have a sum of 360°.
I ask the class to look for another pattern. A short while later a student shares the exterior and interior angles lie along a straight line which is why they add to 180°.
I challenge the class to check whether these two patterns work with their own results. If they do not can we explain why not? Students are quick to point out any instances where they do not work can be explained through inaccurate measurements or constructions.
Constructing the polygons, measuring the interior and exterior angles, and completing the table of results typically takes a full 60 minute lesson and homework. In the next lessons students use the properties of regular polygons to solve a range of complex problems involving multiple angle properties.
My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.
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