Polygon Angle Sums: Develop, Don't Give.

A common way to get students to see that the sum of three angles is 180° is to get them to tear the three angles and rearrange them to create a straight line. Rather than giving the angle sum formula for any polygon, I wondered if I could use this approach would work with other polygons so as to develop the formula instead. I tried this with a couple of grade 9 Applied classes.

I visibly random grouped the students in to threes and gave each group a different paper quadrilateral. It didn't take long for the to rearrange the angle to form a complete turn and for them to tell me that quadrilaterals have an angle sum of 360°.

Moving on to pentagons, I wanted to make sure that the angle would rearrange clearly into one and half complete turns. I figured that the best way to do this was to give pentagons that had two right angles like this one:

This allowed the students to rearrange like so:

...and then tell me that pentagons have an angle sum of 540°

In a similar way, I then gave each group of students a different, hexagon, heptagon,  or octagon whose angles could be torn off and easily rearranged into full and half turns such as these below:

This nudged the students into quickly rearranging the angles:

As the students were finding the angle sums, I recorded their results using Desmos so that we could all see what was happening:

I could now ask the class "What do you notice? What do you wonder?"

They quickly noticed that the angle sum was increasing by 180° each time the number of sides increased by one. Some students wondered if this was something to do with the angle sum of a triangle.

So, I then sent them in their groups to the vertical whiteboards and asked them (one polygon at a time) do choose a single vertex, and from there, draw as many diagonals as possible to any other vertex. 

As this decomposes the polygon into smaller triangles (each of which has an angle sum of 180°) they could then confirm their earlier results.

I now challenged them to predict the angle sum of a dodecagon. As they were now recording their data in a table, it made it easier for them to spot and extend this pattern:

I followed this by asking them to tell me the angle sum of a 102-sided shape and then to generalise for any shape. They were able to see that each polygon with n sides could be split into (n-2) triangles and so the angle sum is (n-2)×180°. 

This was pleasing as we had not only used the concrete-diagrammatic-symbolic continuum but also the next-near-far-any continuum for patterns. The students seemed pretty chuffed that they had 'discovered' this rule themselves. I finished the lesson by giving them these questions to consolidate what they had just learned.