Creative Problem Solving Can be as Simple as Drawing Triangles

Looking back at my first blog post from this year, I realized that one aspect of problem solving that I have not discussed much yet is creativity. Often creating creative lessons seems like a difficult task for teachers, especially in the area of math. We have all spent hours creating an intricate lesson that we hope will inspire our students through deep thought, play, and creativity. Sometimes these lessons work out, and other times they flop, regardless it is often still a learning experience for all. I want to share one of my favorite creative lessons. I love this lesson because it lets the students create and explore various methods, while being surprisingly simple in set up. Unfortunately, I cannot take any credit for the idea, since it came from our curriculum (CPM Core Connections Geometry). I do, however, want to shed light on how I used this lesson to get students excited about their learning and the justification behind the topic.

The geometry lesson focuses on interior angles of polygons. Students are prompted to break up a pentagon into triangles in as many ways as possible. Then they try to determine the sum of the interior angles based on those triangles, if possible. I let students play with this idea for about 15 minutes. At first students look at you like, “So you just want me to draw some triangles in here?” I just giggle a little to myself and respond, “Yep, split it up however you’d like!” Every student is able to be part of this problem and they all have unique ideas. As they are working, I ask students which of their drawings were helpful for finding interior angles and why. This sparks discussion, and students start realizing that some of their drawings are a bit more complicated than necessary. I help groups that are struggling to find the interior angles sum by guiding their discussion toward the sum of triangle angles and then point out the interior angles that they want.

After letting students play with this idea for about 15 minutes, I bring them back together and ask groups to share all of the drawings they came up with. Then one-by-one we narrow down the methods that are useful. In both classes, the two most effective showed up (see diagrams below). Then together we discuss how students found the sum of the interior angles from these drawings.

I don’t spend long on this, since their next task is to use both methods to find the sum of the interior angles for polygons ranging from triangles to decagons. Students spend quite a while on these shapes and keep track of their work in a table with the number of sides and sum of interior angles. As they work, groups start to notice some patterns and start to find little shortcuts in calculating the sums. I spend a bit more time with groups who are not becoming as efficient. I ask questions such as:

• How are the number of triangles used in your calculation?
• How are the number of sides related to your calculation?
• How are the two methods similar or different?

In one of the groups that often struggles, I spent quite a bit of time working through these questions. By the time I left the group, two of the students who usually sit back and listen were leading the conversations and taking pride in their creations. When we came back together as a class, one of these students even shared and explained his work on the board.

I had students share their drawings and calculation on the board for both methods on two different shapes. I was incredibly impressed with how excited students were to share. They took ownership of the methods they had created and were very well spoken in their explanations. Then I helped the class develop an equation generalization for each method. We had great discussions about “n” (the number of sides) in each equation and where we saw each part of the equation in their drawings. Discussing the algebraic relationship between the equations was also a quick and fun algebra review. I then reminded students that although they should write down and feel free to use the equations they don’t need to memorize them. Since they took the time to create and understand the equations, the process is something they can recreate and use anytime.

I loved how proud students were at the end of this lesson. They felt a sense of accomplishment, but this feeling didn’t necessarily come easily. Many ideas led to roadblocks and once they found useful ones, the ability to understand and generalize required a lot of discussion. They had a much deeper understanding and appreciated for the concept by creating it themselves rather than being told the equations. Breaking a larger shape into smaller ones is also a concept something that shows up often in geometry. The second method of breaking a polygon into triangles that meet at the center is even used a few lessons later when they find the area of regular polygons by first finding the area of the triangles.

It’s incredible that all of this came from an easy setup and a simple prompt. Every single student was able to participate in this lesson, and it even gave students who usually feel disadvantaged a place to shine. It allowed students to think and draw creatively, while still encouraging students to look for efficient methods. The discussions that resulted about the number of sides and/or triangles also brought out many verbal justifications. Most importantly, my students were proud of their work and were excited to share it with their peers.