This lesson was inspired by a brief post from Hands-on math.  The author mentioned that they talked about shapes in nature that were efficient, specifically hexagons used by bees.

I started to think about how polygons fit together and how we could play with this concept.  This was not our first encounter with polygons, but this could be a fun introductory lesson to the many types of polygons.  First, the students and I reviewed what polygons are.  Poly means many so there are "many" sides.  The shape has to be closed.  Polygons have sides i.e. they have no round parts.  We looked at polygons from triangles through dodecahedrons.

Then, we brainstormed about places one sees polygons in nature.  When the students came upon beehives, we talked about why we thought bees use that shape and how it can be both strong and efficient.  One student pointed out that we could fit more units in an apartment building if the apartment were hexagonally shaped.

Now, to the hands on part.  I had 15-20 of each polygon per student cut out in different scales.  The challenge to the students was to design a nature inspired design using the polygons.  They played quite a bit with different pieces and discovered that polygons with more than 7 sides can't fit together without forming other polygons in between.  I told them they could use more than one shape but that there had to be no waste i.e. as efficient as possible.  Some of the finished pieces:

We didn't have time to color them, but you could have students color if they wanted.  The designs were creative.  Students discovered through the work concepts about how pieces fit together in both nature and real life.  The discussion was lively, particularly as they thought about how their design could be used in nature.  A full lesson plan is available including polygon classification, design, and symmetry detective.