Polygons: Introduction & Investigation

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SWBAT compare attributes of regular and irregular pentagons, hexagon, heptagons and octagons. They will construct viable arguments about what is/isn't a polygon and critique the reasons of other students who are doing the same.

Big Idea

Interacting with polygons and testing the definitions can be engaging, fun and build a powerful, individualized understanding. (Lesson updated: January 6, 2015)

What is a Polygon?

20 minutes

In this section of the lesson, students are working on Mathematical Practice 3 as they construct viable arguments and critique the reasoning of others.  In order to effectively debate, they need to also employ Mathematical Practice 6 and use precise language in their justifications and critiques.

Students read the definition of a simple polygon with examples. 

As they are doing that, I draw some more examples up on the whiteboard.  I ask them to think about (and talk quietly with a neighbor) which of the forms on the board are polygons, which are not, and why.  I include concave polygon, an open figure, an angle, a circle, a curvy figure and a line.  Then I call them up to explain each one.  

Example student responses:

I know that this is not a simple polygon because even though it has straight sides it is not a closed figure.

This figure is closed and has 3 or more straight sides but it isn't a simple polygon because the lines intersect.

This figure is a simple polygon because it has straight sides, they don't cross each other, and they all meet.  (When they use general language draw out more specifics - vertex, intersection, closed figure, angle).  

There are several points of debate during this segment of the lesson.  When we come to the example of the line as something that's not a polygon, one child argues it is a 2-sided polygon. Another student points out that it has 4 sides (I had drawn the line with the thick edge of a marker).  

This leads to a rich conversation about lines and the possibility of a 2-sided polygon. When I ask my students if they can draw a 2-sided polygon, more than 1/2 of them say yes.  I have some examples of what they did and where the conversation went from there.

Also, I review the word arbitrary with them.  Just as the rounding up from the midpoint was an arbitrary decision someone made long ago, so is the point at which a line is no longer considered a really, really thin rectangle.  



Draw Some Basic Simple Polygons

20 minutes

Students divide two pieces of blank paper into eighths.  We go through the steps... "One divided in half = 1/2.  One half divided in half = 1/4  One quarter divided in half = 1/8."

I guide students in drawing a square (a special kind of rectangle with 4 equal sides), a rectangle, a regular (isosceles) trapezoid, an irregular trapezoid, a parallelogram or rhombus and a kite. 

On the second sheet of paper, students are guided in drawing the following. 

equilateral triangle (measure the sides),

a right triangle (briefly discuss 90 degree angles, 180 degrees = a line and 360 degrees = a circle). 


  • scalene and isosceles triangles  (measure the sides)
  • regular pentagon, irregular pentagon
  • Heptagons are very difficult to draw and have the sides meet up.  I would skip them for now for this reason!
  • regular octagon, irregular octagon

Students write the name of each polygon with careful attention to their handwriting (tall letters tall, short letters 1/2 the size of a tall letter), chunking the syllables, and 2 letter phonograms. 

For students with less developed fine motors skills, you may want to have them do this on a whiteboard so they don't end up with a messy, unclear paper.  They can be provided with a reference sheet later to keep in their math folder/binder! 

Polygons - Initial Exploration

15 minutes

For this part of the lesson, students work with partners at computers. They use an interactive polygon generator from Math is Fun to create polygons with 20 sides or less.  In teaching this lesson a second and third time, I have limited them to polygons with 10 or fewer sides so that they focus on more common polygons and attributes instead of becoming sidetracked with the larger quantities.  They pick several that they find interesting, take screen shots of them, paste the screen shots into a Google document or Word document, and write several observations about their work.

Here is a brief video explaining this student guided exploration of regular and irregular polygons.

This part of the lesson is an open-ended exploration.  There is a place for this in the classroom, if it has a purpose.  The purpose here is for children to familiarize themselves with the interactive polygon program, to create several polygons that they will use in the next section, and to generate authentic, individual observations.  An open-ended activity allows the teacher an opportunity to interact with students and ask them individualized questions to prompt deeper thinking and to gently nudge them into clearing up their own misconceptions.  Also, the comments/observations they write about the polygons they created help me gauge their ability to employ precise language, which then informs me about their developing understanding of the characteristics of polygons and the relationships between vertices, sides, numbers and size of angles, and the differences and similarities between a regular and irregular polygon. 


5 minutes

Students use one of these words (pentagon, hexagon, octagon) in a written or oral sentence that shows correct word use AND provides enough information to demonstrate understanding of the term.

  • I drew an octagon.  (falls below)
  • An octagon has eight straight sides.  (approaching)
  • An octagon is a simple polygon with eight straight sides and 8 vertices. (meets)
  • An octagon is a simple polygon with eight straight sides, 8 vertices, and it is closed.  If the sides are all the same length and the figure has a line of symmetry it is a regular octagon.  (exceeds)

I usually ask students to add rich adjectives and descriptive clauses so they can make a mind movie but that would be a tricky task with some of these math terms.