Toothpick Exploration Leading to Angle Sums of Polygons

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SWBAT find the sum of the interior angles of any polygon.

Big Idea

How many toothpicks does it take? Students explore and develop their own formulas - leading to their discovery of the formula for the interior angle sums of polygons.

Do Now

10 minutes

As the students enter the room, a variety of polygons are displayed on the board, and each student has the Do Now on his or her desk. The students are asked to name as many of the polygons as they can.

When all appear to have finished, I ask the class what the term polygon means to them. Typically, they offer a number of different explanations. I am listening for students who are focusing on the prefix, poly-, and how this shapes the meaning.

Once we discuss the poly in polygon, we will discuss the names of all of the polygons displayed on the board. My students are often intrigued by the names, their derivations, and their links to more common words. For example, we discuss the fact that September and October used to be the seventh and eighth months (linking them to septagon and octagon) before Julius Caesar changed the calendar. I also discuss the prefix dec-, representing tens (as in decagon and dodecagon). I also mention that relatively recently mathematicians decided to use Greek, rather than Latin, prefixes, so that heptagon is now more common than septagon.

Toothpick Activity: How Many Do You Need?

40 minutes

In class today, my students will spend about forty minutes building polygon figures using toothpicks. Here is an example of Student Work on the poster that they will create as they work. This VIDEO explains the activity in detail. The activity is fun for students, helps them to focus on the structure of polygons, and gives them practice at interpreting numerical patterns and making an algebraic generalization.

Finding the Formula for the Sum of the Interior Angles

15 minutes

Now that the students have experience in looking at patterns and generalizing a formula (MP8), I hand out small whiteboards and markers to all the students. Still working in their groups, I ask the students to work together to find a formula for the sum of the interior angles of any polygon.

Before setting them loose, I ask the students to think about two questions:

  • For what polygon do you know the angle sum? 
  • How might you apply this knowledge to other polygons?

This activity requires that the students pay close attention to the structure of each polygon; i.e. they must understand that each polygon can be seen as the sum of a series of triangles (MP7).

During this section of the class, I find it enlightening to listen to the students' conversations and their critiques of each other's arguments (MP3). Their discourse often helps me to formulate good discussion questions for future lessons. I have included one such Sample Discussion Question that I use after the students have learned about regular polygons.

Putting Our New Knowledge to Work

15 minutes

We work on Angle Sums of Polygons as a class, going through the assignment one problem at a time. I allow the students time to work together and discuss a problem, before discussing it as a group. 

My intent in this exercise is to give the students time to process and practice using the interior angle sum formula, and also time to recall and reinforce the parallel line concepts that they learned earlier in the year. 

Ticket Out the Door and Homework

10 minutes

I announce that the homework is to complete proofs #4 and #5 in Puzzle Proofs (from a previous lesson in this unit) and to prepare for a writing exercise. Based on their explorations of constructions and triangle congruence in the previous lesson, I also announce that they will, in the next class meeting, be writing an essay for me in which they explain each of the methods of proving triangles congruent (SSS, ASA, SAS, AAS, and HL), and also which are not sufficient (AAA and SSA) and why. 

In order to assess each student's understanding of the angle sum formula, I hand out the Ticket Out the Door in which I ask the students to explain the formula and its derivation. After class, I will look through the Tickets Out the Door with an eye for any students who are not able to articulate the logic behind the interior angle sum formula, and will meet individually with any who don't seem to have a solid grasp of the topic.