![]() Three regular polygons, triangles, squares, and hexagons tessellate the plane with a monohedral pattern (Figures 3 and 4). Ask them to explain why certain patterns work. Printing a hard copy of students’ work will preserve interesting patterns they discover. Have students drag the original figure to change its size and show that the pattern remains unchanged. Many more patterns exist than can be shown here. The extent of this activity is limited only by students’ willingness to experiment. Teacher’s Guide: Tessellations and Tile Patterns Construct and Investigate: 1. Geometric Investigations on the Voyage™ 200 with Cabri Show examples that work, and explain why. Find other hexagons that also make a monohedral tiling pattern. A regular hexagon makes a monohedral tiling of the plane. Show examples, and give the conditions under which these figures produce a tessellation. Experiment to see whether any pentagons make a monohedral tiling of the plane. In part 1 above, you discovered that a regular pentagon will not make a monohedral tiling of the plane. Experiment to see whether every quadrilateral tessellates the plane? What about a concave quadrilateral? Explain with examples or counterexamples, or both.Įxplore: 1. Rectangles and trapezoids are quadrilaterals. Does a general trapezoid always tessellate the plane? Explain with examples and/or counterexamples. Some trapezoids tessellate because of their special properties (see Figure 2 and the chapter Investigating Properties of Trapezoids). Experiment with ways to tessellate the plane with a trapezoid. Does any rectangle always tessellate the plane? 5. Are there special cases that work better than others? Give examples and explain. Find at least two ways to tessellate the plane with a rectangle. Does a scalene triangle always tessellate the plane? Explain by showing examples and counter examples. Experiment with ways to tessellate the plane with a scalene triangle. Does an isosceles triangle always tessellate the plane? Explain with examples or counterexamples, or both. Use the transformational geometry tools Translation, Rotation, Reflection, and Symmetry to experiment with ways to tessellate the plane with an isosceles triangle. Classify the regular polygons by the types of tiling patterns they make.Ģ. Repeat this procedure on some or all sides of the polygon and its reflections to show a tiling pattern (Figure 1). Draw a regular polygon, and use the Reflection tool to reflect the polygon across one of its own sides. Using the Regular Polygon tool, determine which regular polygons form monohedral, dihedral, or other tessellations of a plane. (Grunbaum and Shephard, 1987.) Construct and Investigate: 1. Trihedral tiling-tessellation using three different congruent figures. Dihedral tiling-tessellation using two different congruent figures. ![]() Monohedral tiling-tessellation made up of congruent copies of one figure. Tessellation-covering, or tiling, of a plane with a pattern of figures so there are no overlaps or gaps. Tessellations and Tile Patterns Definitions:
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