allowing access to calculators and digital tools, so the investigations are more about spatial reasoning than calculation.The difficulty of tasks can be varied in many ways including: Regular sharing of ideas will encourage students to extend their thinking and will allow them to learn from each other. encouraging students to work collaboratively in partnerships, and to share and justify their ideas. Make the table accessible to students so they can make predictions about sets of shapes that will, and will not, tessellate organising the data about regular polygons in a table, especially the measures of internal angles.directly modelling examples of tessellations and explaining why the combinations of shapes around each vertex will work.explicitly teaching angle as a measure of turn through physical action, at first, then diagram drawing, and the use of a protractor to measure the amount of turn.asking students to justify why they believe patterns occur.Encourage anticipation of results, e.g., “Will regular hexagons tessellate? How do you know?” providing physical (or digital) manipulatives and regular polygons so that students can experiment with shapes.The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include: There is a unit on that at Level 4, Tessellating Art, though some of the concepts there would be accessible to students at Level 3. Related to the idea of tessellations is that of Escher drawings. This unit also investigates the possibility of non-regular tessellations. Semi-regular tessellations involve two or more regular polygons. In the regular case it shows that regular tessellations can be made only with equilateral triangles, squares and regular hexagons. The Problem Solving lesson Copycats, Geometry, Level 3 could also be used as part of the Exploring stage of this unitįitness, Level 4 follows on from this unit and looks at both regular and non-regular tessellations. You might use Measuring Angles, Level 3 for this purpose. Therefore, a necessary precursor to this unit is a lesson or series of lessons that give the class a sound knowledge of angles in degrees. Either the corners of the basic shape all fit together to make 360°, or the corners of some basic shapes fit together along the side of another to again make 360°. These tessellations provide a strong structure for their two different purposes.Ī key features of tessellations is that the vertices of the figure, or figures, must fit together, meaning that there are no gaps or overlaps in the pattern created, and that the pattern completely covers a given two-dimensional space. Bees use a basic hexagonal shape to manufacture their honeycombs (a tessellation of regular hexagons). Brick walls are made of the same shaped brick repeatedly laid in rows (a tessellation of rectangles). Tessellations have other, practical uses. They also demonstrate an application of some of the basic properties of polygons. When assembled according to a simple set of matching rules, an infinite number of distinct tessellations can be formed by Penrose tiles, and none of them are repeating! These tiles have a number of other interesting properties, many of them related to the Golden Ratio.Ĭlick here to browse products related to tessellations.Tessellations can be found in a variety of contexts, including in kitchen and bathroom on tiles, linoleum flooring, patterned carpets, parquet wooden floors, and in cultural patterns and artworks. These are named after their inventor, English mathematician and theoretical physicist Sir Roger Penrose. In fact, there is a famous family of tessellations based on two tiles known as "Penrose" tiles. Tessellations do not have to be repeating, or periodic. Repeating and non-repeating tessellations Basically, anytime a surface needs to be covered with units that neither overlap nor leave gaps, tessellations come into play. Examples include floor tilings, brick walls, wallpaper patterns, textile patterns, and stained glass windows. Tessellations are widely used in human design. The three regular and eight semi-regular tessellations are collectively known as the Archimedean tessellations. There are eight semi-regular tessellations. (A vertex is a point at which three or more tiles meet.) There are only three regular polygons that tessellate in this fashion: equilateral triangles, squares, and regular hexagons.Ī semi-regular tessellation is one made up of two different types of regular polygons, and for which all vertexes are of the same type. A regular tessellation is one made up of regular polygons which are all of the same type, and for which all vertexes are of the same type.
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