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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

Tessellations

By KiriBekkers & Katrina Howat


What do my learner s already know yr 9

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

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


Tessellations1

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.


  • Shape

  • Polygons

  • 2D & 3D

Tessellations

Geometric Reasoning

Location & Transformation


Regular tessellations

Regular Tessellations

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


Regular tessellations1

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

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

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Axis of symmetry1

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

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Where the vertices meet1

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*


Semi regular tessellations

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 meet2

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


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*


Where the vertices meet3

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)


Creating escher style tessellations

Creating “Escher” style tessellations...

Some images for inspiration...




Extension hyperbolic planes

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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