SWBAT describe polygons in terms of their structure. Students will understand how the structure of a polygon can be used in a modeling situation.

Students analyze the structure of polygons and learn how this structure can be used to represent a real-world phenomenon. Will Facebook make our brains grow larger?

10 minutes

Using the slideshow, I display the warm-up prompt for the lesson as the bell rings. The warmup is a somewhat embellished re-telling of the classic "handshake problem": how many handshakes are possible between 3 people? 4 people? I expect most students to solve the riddle by acting out the process of handshakes with their team-mates.

After reviewing the team answers and making sure that everyone understands the correct solutions, I ask: "Now suppose that 10 members get together in public. How many handshakes are there?"

I watch the confusion that follows. Too hard? Can we combine forces with other teams? Who will think of using a geometric model to answer the problem? I look for students who are making a diagram and ask them to share with the class. I ask the class: How can we use geometry to model handshakes? I take suggestions and guide the class if necessary: we can use points to represent people and draw segments between them to represent handshakes.

The warm-up follows our Team Warm-up routine, with students writing their answers in their Learning Journals.

**Goal-Setting**

Displaying the agenda and learning targets, I remind the class that we have been learning to describe geometric objects in terms of their structure. Today we will use the structure of polygons to solve the handshake problem, and to understand a theory that explains why our brains are so large.

30 minutes

As I explain in the Video, the goal of this activity is for students to analyze the structure of a polygon.

**Challenge and Strategy Discussion (10 minutes)**

To launch the activity, I display a challenge problem and give students a minute to read it. I ask how this problem is related to our handshake problem (**MP4**, **MP7**). Does the geometric model make it easier to think about the handshake problem? (**MP1**) I check to be sure students know the meaning of the terms that may be new to some: diagonal, adjacent, non-adjacent. I then give the students 5 minutes to work on the problem.

As students work, I circulate. I am looking to see what sort of approaches students are taking. Most will probably try a brute-force approach: drawing a 10-sided figure with diagonals between each pair of vertices. I am also looking to see whether students are working together or independently. (I encourage those who are having trouble getting started.)

At the end of the time limit, I call the attention of the class to the front board, where I display student work using a document camera. I start with the most common approach (probably the brute-force approach described above). I also include any work that shows promise or is creative and original.

However, if any student finds a method of solution that makes the next part of the activity unnecessary, I praise them and ask them to be prepared to share with the class at the end of the lesson. For example, reasoning abstractly, a student may see that every vertex of an n-gon is connected by either a side or a diagonal to n-1 other vertices, and that the number of line segments obtained this way must be divided by two to correct for double-counting (**MP2**, **MP7**).

Keep in mind that a student may show their understanding by explaining in words or by drawing a picture. Most of my students would not think of using algebraic expressions to represent the numbers of vertices and segments, but they may be reasoning abstractly and quantitatively all the same.

I expect most students to discover that it is really hard to draw and count all the diagonals of a decagon. I suggest a little bit of strategy: *solve a simpler problem *(**MP5**).

**Investigation (20 minutes)**

I display the instructions and distribute the half-sheets for the Diagonal Investigation Activity. This activity follows a Team Jigsaw format, with each student taking a different approach to the problem.

- At the end of the 15 minute time limit, I recap with the class. What problem-solving strategies were useful in solving the problem? What was the advantage of using a list? A table? What did the graph show? What insights occured to students as they drew line segments between each pair of vertices (the approach suggested by the "divergent thinker" handout) (
**MP1,****MP2, MP5**)?

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10 minutes

The goal of this activity is to give students a chance to show that they understood the solution to the previous problem. The context of the new problem is just a little bit different than the context of the handshake problem, but its not a big leap between handshakes and relationships in a social network.

Displaying the instructions, I distribute the handout for the activity, normally 1 per every two students. I ask for students to take turns reading parts of the article aloud to their teams. Students should answer the questions in pairs.

4 minutes

Displaying the Lesson Close prompt, I ask students to summarize what they learned from the lesson with their team-mates, then select the best answer to write on the board. This activity follows our Team Size-Up routine.

**Homework**

I assign problems #21-23 from Homework Set 1. Problem #21 asks students to use polygons in a modeling application. Problem #22 practices vocabulary which was introduced in this lesson. Problem #23 provides practice in spacial visualization, as well as in vocabulary and geometric notation. This problem may reveal a point of confusion for students: the sides of a quadrilateral are line segments, but each of those segments is a part of a line (or ray). Later, it will be important for students to realize that axioms about lines often can be used to reason about segments. For example: there is one and only one line *through* any two points in space, so there is one and only one segment *between* those points.