During the first section of this unit, students will construct a house plan, find the area of the house plan, and calculate flooring costs. While finding the area is the focus of this unit, the first few lessons (where students explore the meaning of a polygon, construct house plans, and decompose rectangles into smaller rectangles to find the area) lay the foundation for finding the area of their home plans later on. This also provides students with a meaningful and purposeful context to find the area.
During the second section of this unit, students will investigate dog pen designs and will primarily focus on finding the perimeter, or amount of fencing needed for different dog pens. Students will also explore odd-shaped polygons by finding the area and perimeter of odd-shaped dog pens.
I ask students, "What is a polygon?" and give them time to discuss. My goal is to inspire student thinking. I listen closely to student responses, hoping to identify any misconceptions. If I see any students excluded from student conversations, I say, "Look for someone you can include in your conversations!" In no time, students reach out to other students and soon everyone is involved in math talk!
At this point, I can tell them what a polygon is or I can let them come up with their own definitions. One of the best ways to do this is to show examples and non-examples.
Polygons Verses Nonpolygons
I provide students with the handout: Polygons vs Nonpolygons. Without providing any directions, I ask students to turn and talk with a nearby partner: What do you notice?
After a few minutes, I ask students to make a list of mathematical conjectures about polygons with their group members. I explain: A conjecture is a statement that is believed to be true but not yet proven or disproven. Students like using big fancy words like this!
As a side note: I encourage a lot of group work to help support struggling students, but also to help all students develop communication skills. Besides, learning in a group makes learning more enjoyable!
During this time, I listen for any misconceptions. I don't hesitate to interrupt a group of students or the class to guide the conversation. I say: Macaroni Cheese! and students will respond: Everybody Freeze! Then, I ask, Does a nonpolygon always have to have a curved side? Most students respond, "No...." Turn and talk: Does everyone in your group agree?
Next, we discusse conjectures as a class. As each student or group of students share, I write students' conjectures on the board and we discuss who agrees or respectfully disagrees with each one.
Student conjectures include:
We then developed the definition for a polygon as a class: A polygon is a 2D shape with straight sides and it is a closed figure.
When I teach vocabulary, I try to use TPR (Total Physical Response). As a class, we will develop a simple definition for a vocabulary word as well as hand movements. TPR activates multiple parts of the brain and promotes a stronger memory connection. Often, students are able to recall the meaning of vocabulary words by recalling the hand movements.
Today, we discuss and come up with the following definition and hand movements: Polygon: a 2 D shape (number two on your hand), straight sides (straight arm), closed figure (closed triangle with fingers).
Next, we practice the new vocabulary word several times. To review the meaning of polygon, throughout the unit I say, Turn and Talk: What is a polygon? Students will use the hand movements to recall the definition!
Lesson Introduction & Goal
I begin by introducing today's goal: I can compare polygons to nonpolygons. I explain: Today, we are going to make a t-chart in our journals. On one side, we will write “Polygons” and on the other side, we’ll write “Nonpolygons." I model this t-chart on the board as students complete the t-chart in their math journals. Here's what the end result will look like: Anchor Chart.
Turning & Talking
To make sure students are meeting expectations during pair share opportunities, I refer to our Turn & Talk Guidelines and ask students: What do we do and what do we say when we are turning and talking? Continual review of expectations is important to ensure productive mathematical discourse occurs. Also, by incorporating numerous pair shares into lessons, I engage students in Math Practice 3 (Construct viable arguments and critique the reasoning of others) as students discuss and analyze examples and counterexamples of polygons.
I ask students to turn and talk: What are some examples of polygons? By starting this activity with a peer share opportunity, students learn from one another, strengthen their arguments, and develop the confidence to share ideas with the whole class.
A few minutes later, we discuss polygon examples as a class. One student shares, "A square is a polygon." I draw a square under the polygon side of the t-chart. To encourage students to question and think deeply, I ask: Are you sure? How do you know? Why is a square a polygon? Students refer back to the vocabulary posters and say, "It has straight sides!" "It's a 2D-shape." "It's a closed figure!" I label the square: Straight sides. Students follow along by taking similar notes in their journals.
We continue on in the same fashion. Other studnets offer that a triangle and an L-shape figure are polygons. Again, we discuss and label the evidence.
Next, we move on to discussing and listing nonpolygons in the same manner. Again, I encourage a lot of pair sharing. I ask: Turn and talk... Do you agree that this is an example of a nonpolygon? Why?
During turn and talk moments, I always rotate around the room, providing manipulatives (such as a geoboard and/or whiteboard/marker) for students who are struggling. I use the Polygons vs Nonpolygons handout to help with scaffolding as well.
During this time, I also encourage students to create concave polygons (polygons that “cave in”). Go to this link for more information: http://www.mathopenref.com/polygonconcave.html.
Writing an Explanation
At this point, I want to provide students with the opportunity to summarize today's learning by writing a paragraph to compare polygons and nonpolygons. I write the following prompts on the board and direct students to explain the difference between a polygon and nonpolygon by defining each and providing examples.
Topic Sentence: Polygons are very different from nonpolygons.
A polygon is….
A nonpolygon is…
As a side note, when students are asked to “write a paragraph explaining how a polygon is different from a nonpolygon,” many students will become overwhelmed (unless this is a regular routine). However, by providing students with a topic sentence and paragraph structure, students are more likely to successfully explain their thinking.
As students finish their paragraphs, they share with others and discuss particular points they agree with or disagree with.
To bring closure to this lesson, I ask students to share their paragraphs aloud using the classroom microphone. During this time, I encourage students to share one sentence at a time in order to allow time for other students to process what the student is saying and to analyze the wording of the sentence. For example, some students explain, “All non-polygons have a curvy side, instead of saying “If a figure has a curved side, it is a non-polygon.”
Here's an example of a few students sharing: Polygon vs Nonpolygon. Again, after each student shares, I ask: Does anyone disagree?
To ensure a safe and comfortable environment where students are willing to take risks, I encourage students to say, “I respectfully disagree because…”
As a result of this class discussion, many students revised their paragraphs based on peer feedback. For example, this student, Student Journal & Explanation, changed "a polygon is an open figure" to "a polygon can be an open figure." This revision process is an important part of clearing up misconceptions.