## Angles in Polygons

20 Nov

So this is one of the topics that I’m hoping we will eventually phase out of our high school geometry course. But for now, we’re keeping it in since our students don’t come to us knowing the sum of the interior and exterior angles of a polygon.

I like the Math Nspired activity Interior Angles of Regular Polygons because it provides students the opportunity to look for regularity in repeated reasoning.

Students start by looking at the central angles of a regular polygon.

Then they think about the triangles formed by the central angles of a regular polygon and generalize the sum of the interior angles as 180*n-360.

Then they look at the polygon by drawing diagonals from one vertex and generalizing the sum of the interior angles as (n-2)*180.

What is the relationship between these two expressions? An opportunity to look for and make use of structure.

When I first saw this activity, I worried about there being two ways to get to the interior angle sum. Will our teachers think we should only give them one way to do it? Won’t it confuse students for there to be more than one option? But who are we to decide which way makes sense to our students? I heard one of our geometry teachers saying that some of her students preferred subtracting out the central angles…that method made sense to them. I am proud of our teachers providing our students with multiple entry points to making sense of this formula instead of choosing one over the other. I’ve heard another teacher remark recently that some of our students can’t handle there being more than one way to do something. I think that is an unfortunate consequence of our students being raised in our education system. We must change the perception of students (and many of their parents) that there is only one way to get to the right answer.

We also use the Sum of Exterior Angles of Polygons activity, which gives students the opportunity to make sense of why that sum is always 360 degrees. I start by having students draw a triangle with angle measures and then one exterior angle at each vertex. What is the sum of the exterior angles? What do you predict for the sum of the exterior angles of a quadrilateral? Try it. Draw a quadrilateral with angle measures and then one exterior angle at each vertex.

Then we move to the technology to confirm our findings.

I am glad to work with teachers who are willing to step outside their comfort zone as the journey continues ….