Tessellations By Kiri Bekkers & Katrina Howat What do my learners already know... Yr 9 Declarative Knowledge: Students will know... Procedural Knowledge: Students will be able to... Declarative Knowledge & Procedural Knowledge Declarative Knowledge: Students will know... How to identify a polygon Parts of a polygon; vertices, edges, degrees What a tessellation is The difference between regular and semi-regular tessellations Functions of transformational geometry - Flip (reflections), Slide (translation) & Turn (rotation) How to use functions of transformational geometry to manipulate shapes How to identify interior & exterior angles Angle properties for straight lines, equilateral triangles and other polygons How to identify a 2D shape They are working with an Euclidean Plane Procedural Knowledge: Students will be able to... Separate geometric shapes into categories Manipulate geometric shapes into regular tessellations on an Euclidean Plane Create regular & semi-regular tessellations Calculate interior & exterior angles Calculate the area of a triangle & rectangle Tessellations Tessellation: Has rotational symmetry where the polygons do not have any gaps or overlapping Regular tessellation: A pattern made by repeating a regular polygon. (only 3 polygons will form a regular tessellation) Semi-regular tessellation: Is a combination of two or more regular polygons. Demi-regular tessellation: Is a combination or regular and semi-regular. Non-regular tessellation: (Abstract) Tessellations that do not use regular polygons. Transformational Geometry Flip, Slide & Turn Axis of symmetry Shape Polygons 2D & 3D Geometric Reasoning Location & Transformation Regular Tessellations A regular tessellation can be created by repeating a single regular polygon... Regular Tessellations A regular tessellation can be created by repeating a single regular polygon... These are the only 3 regular polygons which will form a regular tessellation... Axis of Symmetry Axis of Symmetry is a line that divides the figure into two symmetrical parts in such a way that the figure on one side is the mirror image of the figure on the other side Axis of Symmetry Axis of Symmetry is a line that divides the figure into two symmetrical parts in such a way that the figure on one side is the mirror image of the figure on the other side Where the vertices meet... 90* + 90* + 90* + 90* = 360* 120* + 120* + 120* = 360* 60* + 60* + 60* + 60* + 60* + 60* = 360* Where the vertices meet... Semi-Regular Tessellations A semi-regular tessellation is created using a combination of regular polygons... And the pattern at each vertex is the same... Where the vertices meet... Semi-Regular Tessellations All these 2D tessellations are on an All these 2D tessellations are on an Euclidean Plane we are tiling the shapes across a plane Calculating interior angles formula: (180(n-2)/n) where n = number of sides We use 180* in this equation because that is the angle of a straight line For a hexagon: 6 sides (180(n-2)/n) (180(6-2)/6) 180x4/6 180x4 = 720/6 (720* is the sum of all the interior angles) 720/6 = 120 Interior angles = 120* each 120* + 120* + 120* + 120* + 120* = 720* Where the vertices meet... Semi-Regular Tessellations 120* + 120* = ? 240* What are the angles of the red triangles? 360* - 240* = 80* 80* / 2 = 40* per triangle (both equal degrees) Creating Escher style tessellations... Some images for inspiration... Tessellations around us... The Hyperbolic Plane/Geometry working larger than 180* & 360* Circular designs like Eschers uses 450* - a circle and a half... Working with 2D shapes Extension Hyperbolic Planes Example by M.C. Escher Circle Limit III Extension - Working with 3D shapes