Diagonals in regular polygons - a first take

To preface this post, I didn't get as far as I hoped to as our school had a lock-down drill today that took up half the class period.

I asked students to consider how many diagonals were contained within different regular polygons. I drew a triangle, a square, and a pentagon on the board and drew the diagonals within each. As I drew, I defined a diagonal as a line segment that connects two non-adjacent vertices. I wrote the number of diagonals below each figure.

I told the class they were going to investigate the number of diagonals and look for patterns. Specifically,. they were to draw and determine the number of diagonals present in a hexagon, heptagon, and octagon. They were to use the patterns they were seeing to help determine the number of diagonals in a 50-gon. Finally, they were to use what they did to determine a formula for calculating the number of diagonals in any regular polygon.

I asked students to work on this task individually, what I call "engaging in individual think time." I re-visited some of the norms for working on an individual: explore, identify characteristics, connect to other work they may have done, and to check their work. I asked them to work individually for 5 minutes and then they could work with their table groups.

As I walked around the room, I could see that students were struggling with the basic idea of diagonals and that they wanted to jump straight to drawing the octagon and its diagonals.

I wrote the definition of a diagonal on the board and told students they needed to draw all the diagonals for all of the figures. I continued to walk around and help point out where diagonals were missing.

Once students started to discuss what they were seeing there was some progress. We confirmed that a hexagon has 9 diagonals and a heptagon has 14 diagonals. It took a while for students to confirm the heptagon value. Some students had counted 10 diagonals. I asked students to consider if this value made sense given all the information they had. The diagonal pattern developed so far was 0, 2, 5, 9. Does it look reasonable that the next diagonal count would be 10? Students admitted that 10 didn't look right.

Some students had counted 16 diagonals for the octagon. I repeated my question of does that look like a reasonable value? Students are supposed to try correctly counting diagonals in an octagon and to try to use this pattern to determine the number of diagonals in an enneagon and a decagon.

There continues to be a distinct lack of effort when it comes to pushing thinking and true problem solving. Next class may not progress very far as my intention is to push the class until they identify patterns that make sense to them and can then model their patterns using mathematics. If all goes well, students will push through and be able to calculate the number of diagonals in a 50-gon and have a formula they can use for any n-sided regular polygon.