Introducing polygons and quadrilaterals

Our next unit focuses on quadrilaterals in the coordinate plane. There is some introductory work with polygons that also needs to be covered.

To start things off, I made use of Unexpected Riches from a Geoboard Quadrilateral Activity described in Rich and Engaging Mathematical Tasks Grades 5-9. Students were asked to construct as many quadrilaterals as possible on a geoboard and record the results. As students worked through their constructions, I encouraged them to be creative and think beyond just squares and rectangles.

Excluding rigid transformations, there are 16 shapes that can be created. As students worked, I called out how many different groups had found and told them to keep pushing. Students got to 16 shapes but some were either rotations or reflections of other shapes. Across the room, I did see all 16 shapes.

I displayed the shapes and gave students a chance to compare what they had completed. I numbered the 16 figures and wrote out the numbers for four of the figures on the board. I told the class these four figures all had something in common. I asked the class to describe what the common trait was.

The four figures were the only concave quadrilaterals. Students described how they looked like arrow heads. I pushed them for more clarification. They struggled some with how to describe what was going on. There were incorrect statements about all angles being acute, and the like, which I threw back at the class for verification. The discussion centered around the vertex which defined the concavity of the quadrilateral.

Finally, a student noted that the interior angle of the vertex was obtuse. I wrote down a definition:
A quadrilateral is concave if it has at least one interior angle that is obtuse.

I then noted that the other quadrilaterals were convex quadrilaterals. I wrote down a definition:
A quadrilateral is convex if all its interior angles are less than 180^o.

I then asked the class if a concave triangle can exist. There was some good discussion around this idea. One student noted that he could draw a triangle with an obtuse angle and so it should be called a concave triangle. Other students objected since there wasn't any indentation in the figure and this would require a fourth side to accomplish that result.

I turned back to the definition of a concave quadrilateral and said that we might need a more precise definition. That having an obtuse angle was not enough to define concavity. I also note a couple of the convex quadrilaterals that had obtuse angles. At this, a student suggested that is should state the angle is greater than 180^o and not just an obtuse angle. I re-wrote the definition:
A polygon is concave if it has at least one interior angle greater than 180^o.

This provided a precise mathematical definition for concavity that works for more than just quadrilaterals. The discussion also built upon students' prior knowledge of interior angles of triangles and a natural transference of this idea to quadrilaterals.

With these ideas established, I moved on to polygon names. Most of the standard names through hexagons were known, as was octagon. Students did not know a 7-sided figure was called a heptagon, nor that a 9-sided figure was called an enneagon. With some connections to common words, such as 10 years is called a decade, students came up with a 10-sided figure is called a decagon.

I went through the naming of 11-sided through 19-sided figures as well, only to show how these names were built upon from earlier names, especially figures 13-19.

I then asked students to construct a convex and a concave pentagon on their geoboards. Next, I assigned table groups different figures, hexagon, heptagon, octagon, decagon and asked them to construct a concave figure on their geoboards. It was fun to see their creativity. For example, one group constructed what looked like a crown for the concave decagon.

I concluded with having students record their thoughts and capture definitions in their notebooks.

I still need to cover the ideas of exterior angles and regular polygons. Once these are done, I will move on to look at the sums of exterior angles, the sums of interior angles, and the number of diagonals in polygons.