Students will understand the relationships between the corresponding parts of similar figures.

Students create and investigate their own concave similar polygons.

20 minutes

During the previous lesson students were given an Algebra Review worksheet to finish for homework. I quickly survey the class to see if there were any issues with these problems and to answer any questions.

I then reintroduce one of the similar polygon pairs that the students encountered during the previous day’s lesson, the pair of similar arrow-shaped polygons. I pose the question, “Why are these two polygons different from all the others we looked at yesterday?” Students will, hopefully, eventually observe that two of the vertices have been “pushed in,” and I lead a brief discussion on **convex **and **concave polygons**.

I next give the Arrow Grid handout to each student. I ask my students to use their tools to:

- Find the lengths of all the segments
- Determine the scale factor
- Find the measures of all the angles
- Calculate the sum of the interior angles

I also ask them to classify the polygon, based on the number of sides.

When finding the angle measures, some of the students will find that the angles in triangular portions of their arrows do not add up 180 degrees, which, happily, troubles them. This leads to a good discussion of the limitations of the tool that they are using, and we talk about making adjustments so that the correct degree sum is achieved.

When everyone thinks they have finished the task, we discuss the results. Typically, most of the students do not consider the reflex angles contained in the polygons. This is probably the first time they have seen an angle that measures more than 180 degrees. I lead a discussion in which we calculate the reflex angles using several different approaches (such as subtracting 90 degrees from 360, and adding a straight angle and a right angle). I then have the students recalculate the sum of the interior angles, and it is at this point usually that at least one of the students observes that the formula for the sum of the interior angles of a polygon, *(n – 2)180*, seems to work on this concave polygon. I respond that this is an interesting observation and explain that we are going to do an activity in which they will explore this.

21 minutes

I explain to the students that each of them is going to design his or her own pair of **concave polygons**. I hand out and read with them a written explanation of the project and answer any questions (see Concave Polygon Investigation). I then hand out the Rubric that I will use to grade the projects, and, I take questions. I try to be clear on the minimum each student must include in his/her project:

- Name
- Lengths of sides
- Scale Factor
- Measures of angles
- Sum of interior angles)

I emphasize that I expect students to *explor*e as well. For example:

- What are the identifiable features of your polygon?
- What mathematical concepts can be used to describe or explain your polygons?

The rubric clearly states that their final copy should be neat, readable, colorful, and accurate. It also promotes the idea that creativity is valued (see Examples of Final Versions).

I typically give my students a week to complete the project and I announce the due date before letting kids begin their work. Once everyone seems clear on the goals and requirements of the project, I distribute graph paper and let students begin to create and investigate the “scrap” copy of their design. I circulate around the room, answering questions as they arise.

4 minutes

As class nears the end, I hand out the paper entitled Names of Polygons and I allow the students to look at and discuss the names. I remind everyone of the due date of the task and show them where in the classroom they can find the “good” graph paper to use for the final version of their project. I also mention that they can borrow protractors, straight edges, and colored pencils, and invite them to use my classroom during study halls when my room is not in use or after school.