1. Tessellations By Kiri Bekkers, Jenna Elliott & 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 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

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: Tessellations that do not use regular polygons.

5. Transformational Geometry Flip, Slide & Turn Axis of symmetry Shape Polygons 2D & 3D 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 4 1 2 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 3 4 1 2 3 4 5 6 1 2 3

10. Where the vertices meet...

11. 90* + 90* + 90* + 90* = 360* 120* + 120* + 120* = 360* 60* + 60* + 60* + 60* + 60* + 60* = 360* Where the vertices meet...

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... 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

14. 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*

15. Calculating exterior angles For a hexagon: 6 sides If a full circle is 360* And we know the interior angle is 120* on one vertex then the exterior angle must be the difference between 360* - 120* = 240* CHECK: 120* + 240* = 360* 120* + 120* + 120* + 120* + 120* = 720*

16. Calculating interior angles formula: (180(n-2)/n) where n = number of sides Using this formulae calculate the interior angles for the following... PolygonNo. of sides Interior Angle Triangle360* Square4 Pentagon5 Hexagon6120* Heptagon7 Octagon8 Nonagon9 Decagon10

17. 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)

18. Tiling a foyer... If we want to know how many tiles we will need to cover the foyer, use the information below to calculate... Foyer 600cm (6m) 300cm (3m) Tile 5cm Pythagoras’ Theorem to find height of the equilateral triangle = a² + b² = c² Area of YELLOW equilateral triangle = ½ BH We know that there are 6 equilateral triangles within a hexagon *real life application for a tiler would use mm measurement however for working with this equation, we are using cm to 2 decimal places. A² B²

19. Tiling a foyer... If we want to know how many tiles we will need to cover the foyer, use the information below to calculate... Foyer 600cm (6m) 300cm (3m) Tile Pythagoras’ Theorem to find height of the equilateral triangle = a² + b² = c² 2.5²cm + 5²cm = c²6.25cm + 25cm = 31.25cm² 31.25²cm square root = 5.59cm Area of YELLOW equilateral triangle = ½ BH 5cm x 5.59cm27.95/2 = 13.975cm² (13.98cm²) We know that there are 6 equilateral triangles within a hexagon 13.98cm² x 6 = 83.85cm² = area of hexagon *real life application for a tiler would use mm measurement however for working with this equation, we are using cm to 2 decimal places. 5cm A² B²

20. Tiling a foyer... If we want to know how many tiles we will need to cover the foyer, use the information below to calculate... Foyer 600cm (6m) 300cm (3m) Tile Finding the number of tiles that will be required to tile the Foyer... A = LW or L x W 600cm x 300cm = 180,000cm² 6m x 3m = 18m² So.... Area of ÷ Area of = number of tiles required for foyer 180,000cm² ÷ 83.85cm² = 2146.69 tiles to cover the foyer area Exploring this calculation... 5cm

21. Tiling a foyer... Exploring this calculation... Are we able to calculate this problem In a different way? Foyer 600cm (6m) 300cm (3m) Tile 5cm

22. Tessellating with equilateral triangles… The internal angles of any equilateral triangle always = 180* Therefore each of the 3 corners must be 60* Draw a “funk-if-ied” line on one edge… and cut out

23. Draw a new equilateral triangle on another piece of paper and use your cut out like a template. Align the two shapes by putting your template onto the new triangle and rotate it so that the pattern is sitting against a new side. Trace around the shape against the new side and cut this out… Tessellating with equilateral triangles…

24. Now our shape looks like this… We have one side left that isn’t funky and we need to change that. Find the half way point along that side… Only draw a design on one half of that side… Cut that out and we’re going to again, rotate it to mirror the same half way pattern on the other half of the blank side… Tessellating with equilateral triangles…

25. Now our shape looks like this… We have one side left that isn’t funky and we need to change that. Find the half way point along that side… Only draw a design on one half of that side… Cut that out and we’re going to again, rotate it to mirror the same half way pattern on the other half of the blank side… Tessellating with equilateral triangles…

26. Now it’s time to decorate and tessellate! Tessellating with equilateral triangles…

27. Tessellations around us...

28. 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 Extension Hyperbolic Planes… Example by M.C. Escher – “Circle Limit III” Extension - Working with 3D shapes…