Type of Class:
           High School Geometry Class (all levels)

 Related VA SOL: G.9 

Time Frame:
              90 minute block period

Objectives:

  • Students will know and understand why which regular polygons tessellate and which do not.

  • Students will be able to tessellate regular polygons using Geometer’s Sketchpadâ.

  • Students will further understand and be able to give a precise and correct definition of a tessellation.

Materials:

  • Computer per student (or pairs of students) with Geometer’s Sketchpadâ (version 3.01)

  •  Paper

  • Scissors

  • Colored pencils/crayons/markers (coloring tools)

  • Rulers

Procedures:

1.      Have students in pairs and provide them with paper, ruler, scissors and some coloring tools (three different colors).  Have students use their rulers to draw a scalene triangle whose longest side is 3 to 4 inches long.  Have students color their triangles in the following way:

  1. Find the midpoint of each side and pick a point somewhere in the interior of the triangle.

  2. Connect each midpoint to the point inside the triangle so that they can color each vertex a different color.

      Then have students make about six of these triangles (all identical).

2.      Have students figure out how they need to arrange the triangles so they tessellate.  [They should be familiar with the definition of tessellation and how to make a tessellation.]  Ask students the following questions:

  1. Looking at the vertices of your triangles, are all the angles of the same color congruent?

  2. Do all three colors appear at each vertex in your tessellation?

  3. How many times does each color appear at a vertex?

  4. What is the sum of the interior angles of a triangle?  By your tessellation, how do you know this?

3.      Next, have students draw a quadrilateral and color it so that each vertex is a different color.  Students should make at least 6 copies of their quadrilateral and tessellate the shape.  Again, students should answer the following questions:

  1. Looking at the vertices of your quadrilaterals, are all the angles of the same color congruent?

  2. Do all four colors appear at each vertex in your tessellation?

  3. How many times does each color appear at a vertex?

  4. What is the sum of the interior angles of a quadrilateral?  By your tessellation, how do you know this?

4.      Have students get into small groups and discuss regular polygons and regular tessellations. You should guide their discussions by asking them questions that may begin with what makes a polygon regular?  Now that you understand what a regular polygon is, what do you think a regular tessellation is?  After discussing this, direct students individually or in pairs go to a computer with Sketchpad.  Have them open the following scripts: 

  1. 3byedge.gss

  2. 4byedge.gss

  3. 5byedge.gss

  4. 6byedge.gss

Have students construct and select two points to play back each script. Have students measure the interior and exterior angles of each polygon.  Have them then select two points on their polygon and play the script again.  If nothing happens, have them select the points in the opposite order.  Have students continue this process to produce a tessellation.  Ask students the following questions:

  1. Which regular polygons tessellate?

  2. Why don’t all of the regular polygons that you tested tessellate?  What requirements do you think that the regular polygon must meet in order for it to tessellate?

  3. Will a heptagon tessellate?  Why or why not?  What about an octagon? 

  4. For regular polygons, what is the greatest number of sides for which it will tessellate?  Why?

  [Regular polygons will tessellate if and only if the measure of each angle divides 360 evenly.  It is impossible for fewer than three angles to surround a vertex, so no regular polygons with more than six sides will tessellate.]

Assessment:
           This activity can be assessed in different ways.  You could have students create a tessellation using Geometer’s Sketchpadâ by having them design one of the regular polygons and then tessellate that polygon.  The assessment may also wait until another lesson on tessellations is completed so that students understand how to make a tessellation using transformations and then have students create a tessellation on Sketchpad on their own.

Resources:
Bennett, Dan  (1996).  Exploring Geometry with The Geometer’s Sketchpadâ.  Berkeley, CA: Key Curriculum Press.