Tessellating the Plane USC Upstate Faculty镶嵌平面SC州学院.docVIP

Tessellating the Plane Reference: Discovering Geometry: An Inductive Approach by Michael Serra, pp. 316 - 340. Review of Prerequisites: 1. A polygon is a closed figure that does not intersect itself and is formed by a finite number of line segments. The endpoints of the segments are called the vertices of the polygon; the segments are called the sides of the polygon. Vertices belonging to the same side of a polygon are called adjacent vertices. 2. An interior angle of a polygon is an angle that has as its vertex a vertex of the polygon, sides (rays) determined by the sides of the polygon intersecting at that vertex, and an interior that intersects the interior of the polygon. The sum of the measures of the interior angles in a triangle is 180E. 3. An exterior angle of a polygon is an angle that has as its vertex a vertex of the polygon, sides (rays) determined by the sides of the polygon intersecting at that vertex where one side has been extended, and an interior that does not intersect the interior of the polygon. 3. A diagonal of a polygon is a line segment connecting two nonadjacent vertices. 4. A tessellation, or tiling, of the plane is a drawing in which the plane is covered with non-overlapping, congruent copies of a figure. Objectives: 1. To develop an understanding of those regular polygons that can tessellate a plane by themselves; 2. To investigate tessellations by other figures than regular polygons. Materials: 1. Geometer’s Sketchpad 2. Colored paper and scissors 3. Laminated polygonal shapes Activity 1: Regular Tessellations 1. For each of the polygons in the figure below, choose a vertex, and draw all diagonals of the polygon that have the selected vertex as an endpoint. Complete the table that follows. Notice that each angle of the triangles that are formed by the diagonals is part of an angle of the polygon, and that the measure of each angle of the polygon may be expressed as a sum of measures of angles belonging to t