Student Work from Shapes and Designs, Problem 2.2

Angle Sums of Any Polygon

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Shapes and Designs Problem 2.2 asks students to make an important generalization about the angle sum property for any polygon. In the previous Problem 2.1, students have discovered the generalization for regular polygons. Problem 2.2 offers three different strategies from fictitious students Devon, Trevor, and Casey for making the generalization.

Students are introduced to the strategies of three students and asked to make sense of the strategies for finding the angle sums of polygons.


A) Devon began by drawing irregular triangles and quadrilaterals. Then he tore the corners off of those polygones and 'added' the angles by arranging them like this:

Devon triangles


Trevor examined Devon's results from his study of irregular triangles. This gave him a new idea to study polygons with more sides. He divided some polygons into smaller triangles by drawing diagonals from one vertex.

Trevor divides polygons


Casey used Devon's discovery about triangles in a different way. She divided polygons into triangles by drawing line segments from a point within the polygon.

Casey draws line segments from a point within a polygon

Each approach is an illustration of a very important practice in geometry—cutting a given figure into smaller pieces and showing how the properties of those pieces can be used to get results about the whole figure.


A CMP teacher shared a new strategy from a student who continued thinking about the problem at home.

Here is the description of the student work from the classroom teacher.

Yesterday we completed Shapes and Designs Problem 2.2. We had some productive discussions about the different methods posed for finding interior angles sums of polygons by creating triangles inside the polygons.... just as we have many times through the years. Today when the students came to class, a student showed me a picture she drew to show new ways to find the angle sums of polygons. I was excited and asked her to share with the class. At first everyone agreed her idea would work with both shapes. Then another student spoke-up, she didn't think the second shape (the octagon) had the correct interior angle sum. This led us into a wonderful 10-minute exploration about what was going on in the drawings. The second student showed why there was a need to subtract another 360 degrees from the angle measures. The original student did a great presentation of her thinking and the resulting classroom discussion helped to refine the strategy for finding the angle sum for an octagon using the new strategy.

I was so excited that a student went home, explored something on her own, and had the desire to let me know as she came into class. It was really exciting to build off the ideas we had seen in Problem 2.2, where we had to subtract angles "not needed" when placing a vertex inside the shape. I will be using this young student’s work for years to come to extend student thinking during the school year and teacher thinking during professional development in the summer. The ideas from Problem 2.2 can be used in both of the examples; the student just needed a little nurturing to adjust the strategy in the second drawing.


This student work can serve as a guide for planning, teaching, assessing, and reflecting on the mathematics of this Problem. The work can be downloaded for teachers to examine during planning time, for use in a collaborative meeting, or for use in professional development activities.

Also, this student work can be used with students as an extension of the thinking already presented in Problem 2.2. This strategy can deepen students understanding of polygons and their shapes. It is interesting to note that the different examples of student thinking presented in Problem 2.2 may have prompted one student to explore the possibilities of other strategies.