In this activity, students examine examples and non-examples of polygons and identify the defining characteristics of a polygon. 

Developing a useful and complete definition of a polygon is harder than it seems. A formal definition is often very wordy or hard to parse. Polygons are often referred to as “closed” figures, but if used, this term needs to be defined, as the everyday meaning of “closed” is different from its meaning in a geometric context.

This activity prompts students to develop a working definition of polygon that makes sense to them, but that also captures all of the necessary aspects that makes a figure a polygon. Here are some important characteristics of a polygon.

• It is composed of line segments. Line segments are always straight.
• Each line segment meets one and only one other line segment at each end.
• The line segments never cross each other except at the end points.
• It is two-dimensional.

One consequence of the definition of a polygon is that there are always as many vertices as edges. Students may observe this and want to include it in their definition, although technically it is a result of the definition rather than a defining feature.

As students work, monitor for both correct and incorrect definitions of a polygon. Listen for clear and correct descriptions as well as common but inaccurate descriptions (so they can be discussed and refined later).

Display the figures in the first question for all to see. For each figure, ask at least one student to explain why they think it is or is not a polygon. (It is fine if students' explanations are not precise at this point.) Then circle the figures that are polygons on the visual display.

Next, ask students to share their ideas about the characteristics of polygons. Record them for all to see. For each one, ask the class if they agree or disagree. If they generally agree, ask if there is anything they would add or elaborate on to make the description clearer or more precise. If they disagree, ask for an explanation.

Make the key characteristics of a polygon explicit: It is a two-dimensional figure, it is composed of line segments, the line segments meet only one other line segment at each end, and the line segments cross another line segment only at the endpoints. If one of these key characteristics is not mentioned by students, bring it up and revisit it at the end of the lesson.

Tell students we call the line segments in a polygon the "edges" or "sides," and we call the points where the edges meet the "vertices." Point to the sides and vertices in a few of the identified polygons.

If time permits, point out that polygons always enclose a region, but the region is not technically part of the polygon. When we talk about finding the area of a polygon, we are in fact finding the area of the region it encloses. So “the area of a triangle,” for example, is really shorthand for “area of the region enclosed by the triangle.”

This activity has several aims. It prompts students to apply what they learned to find the area of quadrilaterals that are not parallelograms, encourages them to plan before jumping into a problem, and urges them to reflect on the merits of different methods.

Students begin by thinking about the moves they would make to find the area of a quadrilateral and by explaining their preference to their partners. They then consider and discuss the different strategies taken by other students. Along the way, they may notice that some strategies are more direct or efficient than others. Students reflect on these strategies and use their insights to plan the work of finding the area of polygons in this activity and beyond. 

Note that it is unnecessary for students to take the most efficient path. It is more important that they choose an approach that makes sense to them but have the chance to see the pros and cons of various approaches.

To conclude the activity, ask students to choose one quadrilateral they worked on (other than D) and tell their group the first couple of moves they made for finding its area and why. Encourage other group members to listen carefully, check that the reasoning is valid, and offer feedback. 

Students may have noticed that all the approaches involved decomposing one or more regions into triangles, rectangles, or both. If not mentioned by students, point this out. Emphasize that we can decompose any polygon into triangles and rectangles to find its area.

In this activity, students determine the area of an unfamiliar polygon and think about various ways for doing so. The task prepares students to find the areas of other unfamiliar shapes in real-world contexts. It also reinforces the practice of sense-making, planning, and persevering when solving a problem (MP1). Students reason independently before discussing and recording their strategies in groups. 

Because the shape of the polygon is more complex than what students may have seen so far, expect students to experiment with one or more strategies. Consider preparing extra copies of the blackline master for students to use, if needed. 

As students work, monitor for those who:

• Decompose the pinwheel into triangles and find the areas.
• Decompose the pinwheel into rectangles and triangles, rearrange the pieces into parallelograms (right or non-right), and find the areas.
• Enclose the pinwheel with one large square or several smaller rectangles, decompose the extra regions into triangles and rectangles, find the areas of the extra pieces, and subtract them from the area(s) of the enclosing rectangle(s).

Make note of the variations and complexities in how students obtain shapes whose areas can be found. If there is limited variation in strategies, look for different ways of recording the same strategy. 

Students have opportunities to notice and make use of the structure of the pinwheel in their reasoning (MP7). For instance, the pinwheel can be decomposed into four identical pieces (or sets of pieces). Also, enclosing the pinwheel with a square creates four extra regions that are identical.

Use Compare and Connect to help students compare, contrast, and connect the different approaches. Give students time to visit one another’s visual display. Consider displaying the following questions for students to discuss as they investigate others' work:

• “Did this group find the same area as our group? If not, why?”
• “How is their strategy like our strategy?”
• “How is their strategy different from ours?”

During the whole-class discussion, ask students:

• “What did the strategies have in common? How were they different?”
• ​​“Are there any benefits or drawbacks to one representation compared to another?”

Highlight similarities in students’ work in broader terms, as outlined in the activity narrative. Reinforce that all approaches involve decomposing a polygon into triangles and rectangles to find area.