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Tessellations

Tessellations. By Kiri Bekkers & Katrina Howat. What do my learner’s already know... Yr 9. Declarative Knowledge: Students will know... Procedural Knowledge: Students will be able to. Declarative Knowledge & Procedural Knowledge.

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Tessellations

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  1. Tessellations By KiriBekkers & Katrina Howat

  2. What do my learner’s already know... Yr 9 Declarative Knowledge: Students will know...Procedural Knowledge: Students will be able to...

  3. Declarative Knowledge & Procedural Knowledge Declarative Knowledge: Students will know...How to identify a polygonParts of a polygon; vertices, edges, degreesWhat a tessellation isThe difference between regular and semi-regular tessellationsFunctions 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 categoriesManipulate 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

  4. 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.

  5. Transformational Geometry • Flip, Slide & Turn • Axis of symmetry • Shape • Polygons • 2D & 3D Tessellations Geometric Reasoning Location & Transformation

  6. Regular Tessellations A regular tessellation can be created by repeating a single regular polygon...

  7. 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...

  8. 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 1 2 3 1 2 4 3

  9. 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 1 2 1 2 3 4 3 1 5 2 4 6 3

  10. Where the vertices meet...

  11. Where the vertices meet... Sum of internal angles where the vertices meet must equal 360* 90* + 90* + 90* + 90* = 360* 120* + 120* + 120* = 360* 60* + 60* + 60* + 60* + 60* + 60* = 360*

  12. Semi-Regular Tessellations A semi-regular tessellation is created using a combination of regular polygons... And the pattern at each vertex is the same...

  13. Where the vertices meet... Sum of internal angles where the vertices meet must equal 360* Semi-Regular Tessellations All these 2D tessellations are on an Euclidean Plane – we are tiling the shapes across a plane

  14. Calculating interior anglesformula: (180(n-2)/n)wheren = 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* + 120* = 720* 90* 90* 180*

  15. Where the vertices meet... Sum of internal angles where the vertices meet must equal 360* Semi-Regular Tessellations 120* 120* + 120* = ? 240* What are the angles of the red triangles? 360* - 240* = 80* 80* / 2 = 40* per triangle (both equal degrees)

  16. Creating “Escher” style tessellations... Some images for inspiration...

  17. Tessellations around us...

  18. Tessellations around us...

  19. Extension Hyperbolic Planes… Extension - Working with 3D shapes… The Hyperbolic Plane/Geometry – working larger than 180* & 360* Circular designs like Escher’s uses 450* - a circle and a half... Working with 2D shapes Example by M.C. Escher – “Circle Limit III”

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