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.

11-17-2013 Image008  Screen Shot 2013-11-17 at 5.51.54 PM

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.

11-17-2013 Image009  Screen Shot 2013-11-17 at 5.52.30 PM

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

11-17-2013 Image010 Screen Shot 2013-11-17 at 5.54.42 PM

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.

11-17-2013 Image011 11-17-2013 Image012 11-17-2013 Image013 11-17-2013 Image014 11-17-2013 Image015 11-17-2013 Image016 11-17-2013 Image017

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


Posted by on November 20, 2013 in Geometry, Polygons


Tags: , , , ,

3 responses to “Angles in Polygons

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: