Tessellation, Transformation and Art

Purpose

In this unit students to learn about transformations and tessellations, then apply this knowledge to create a piece of art work that creatively demonstrates one or more tessellated transformations.

Achievement Objectives
GM4-5: Identify classes of two- and three-dimensional shapes by their geometric properties.
GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).
Specific Learning Outcomes
• Using translations, rotations and reflections to create Escher-type tessellations.
• Apply knowledge of tessellations to the creation of a piece of art.
Description of Mathematics

The idea of this unit is for students to develop their understanding of the mathematical concepts that underlie the creation of Escher type tessellations and apply this to the creation of an art piece.

The unit of work is based on Escher’s work.

Main resource: Creating Escher-Type Drawings. E.R Ranucci and J.L. Teeters, Creative Publishers Inc 1977.

Associated achievement objectives:

The Arts:

Understanding the arts in context

• Investigate the purpose of objects and images from past and present cultures and identify the contexts in which they were or are made, viewed, and valued.

Developing practical knowledge

• Explore and use art-making conventions, applying knowledge of elements and selected principles through the use of materials and processes.
Required Resource Materials
• Scissors
• Glue
• A3 paper
• Packet of note paper
• Tracing paper
• Colour pencils
• Plastic sleeves to store student work
• A4 box lids (one per group) to collect waste paper for recycling.
• A3 box to keep student work for each lesson.
Activity

This unit of work is presented as a series of six sessions, however, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period.

This unit of work has most of the mathematics front-loaded to support students’ ideas for their piece of art. The teacher supports this work with examples of art that are based on the ideas of the mathematics being explored.

Session 1

Introduction to Escher power point. Outline of unit of work by teachers explaining what is required of students in each subject.

SLOs:

• Setting the scene for the unit and making the connection between the mathematics and how it will be used in art.
• Introduction to tessellations.
• Recognising that area and length are invariant in tessellations.

Activity 1

Introduction to Escher’s work, both his wider art and his art based on tessellations through a PowerPoint. An outline of the unit of work is given, explaining what is required of students in each subject.

Activity 2

Beginning tessellations:

Students are introduced to the idea of tessellations, referring to the examples given in the introduction. Discussions can include examples of other places students see tessellations in the real world, for example, paving patterns, ceramic tiles, beehives, wallpaper.

Students complete the activities Stingrays, Hexastars, and Butterflies (Copymaster 1).

Use the activities to draw out the ideas around invariant properties of tessellations – specifically area and length remain the same.

Shape recognition:

While students are working on the tessellation activities, there is a great opportunity to circulate around your classroom and discuss the shapes the students are going to meet during this unit.  You may wish to add additional activities while they are doing the beginning tessellations activities so you can teach shape recognition and the properties of the shapes used in this unit. Shapes are equilateral triangle, square, rectangle, rhombus, parallelogram, kite and hexagon.

Session 2

This session explores learning to create a base pattern for a tessellation that has translation only.

SLOs:

• Learning about translation.
• Using translation to create a tessellation.

Activity 1

In this activity students are learning about translations.

Translations involve a linear shift or slide of a figure in a plane.

In the figure above, quadrilateral ABCD has been translated to a new position in the plane (A’B’C’D’).  Note that lines AA’, BB’, CC’ and DD’ are all parallel.

We would say quadrilateral ABCD maps to quadrilateral A’B’C’D’ under translation to the right of 6.5cm.

The properties of size, shape and orientation remain invariant (unchanged) under the operation of translation.

Translation allows us to repeat patterns.

If a more in-depth look at translation was wanted teachers could explore Session 2, Activity 1 “teacher and student activity using translations” in the Transforming Shapes unit of work.

Activity 2

Making a base template with translation only.

Use the Escher type tessellations instructions card above to show how a tessellating tile can be made using translation.  These instructions use the idea of taking a nibble and translating that across the square.  Other ideas include altering a side and translating the alteration across the square, both ideas are explored in the student activities given later.

Demonstrate this using an example or two, for example:

Ideas around creativity can be mentioned here by the teacher. Birds/owls/spaceships etc. Have a selection of pictures of Escher’s work to leave on your board.

Students can then practice and have a go at making their own using Copymaster 2.  These are best printed A3 size.

Key ideas to reinforce in this include the conservation of area, perimeter and orientation.  If students work with a 5cm square piece of card, create a pattern to repeat on a sheet of A4 paper to create a tessellation (as in the instruction card), what area does their pattern cover… (the new shape still has the same area as the square).

When students are working on the activities Using Translation to Tessellate they should show on their completed tessellation the direction(s) and distances of the translations used in their tessellation.  For example:

Alternatively, they might write ABCD maps to A’B’C’D’ under translation 2.1cm to the right; ABCD maps to A’’B’’C’’D’’ under translation 2.1cm down.

Session 3

This session explores learning to create a base pattern for a tessellation that has rotation (and translation).

SLOs:

• Learning about rotation.
• Using rotation to create a tessellation.

Activity 1

In this activity students are learning about rotations.

A rotation involves turning a figure in a plane about a given point in the plane, called the centre of rotation.  The properties of size, shape and orientation remain invariant (unchanged) under the operation of rotation. In the picture below, the figure is rotated 90o clockwise about the point P. S has been rotated to a new position in the plane (object S’).  Point P is the centre of rotation.

The centre of rotation may or may not be a point on the figure itself.

Half turn rotations

Half turn rotations are when the amount of the rotation is 180o.  Half turn rotations are common in Escher type tessellations.

If a more in-depth exploration of half turn rotations was wanted teachers could explore Session 2, Activity 3 “teacher and student activity using half turns (and vertical reflections) ” in the Transforming Shapes unit of work.  These ideas could be adapted for other turns, e.g. quarter turns.

Activity 2

Making a base template by altering the sides using rotation.

In this activity we are going to explore the other designs part of the Escher type tessellations instructions card.  As with translation a nibble can be taken and then rotation about a vertex or rotation can happen at the midpoint on a side. Also altering a side and then rotation about a vertex or rotating about a midpoint on a side is also a viable way or creating a base pattern to use in the tessellation.

Use cut outs on board to demonstrate how to alter a shape and then use rotation to create a tessellation.

Ideas around creativity can be mentioned here by the teacher. Have a selection of pictures of Escher’s work to leave on your board that show rotations.

Students can then practice and have a go at making their own using Copymaster 3.  These are best printed A3 size.

Designs using rotations can become quite complicated. Shapes can be left-handed and right-handed. This is a good thinking exercise for students who need some extension.  They could also try shading triangular tiled tessellations so that there are not equal numbers of each colour of tile.

Key ideas to reinforce in this include the conservation of area, length and orientation. As with the previous tessellation example the concept of area can be explored. Students should be describing the rotation(s) in the tessellation including angle of rotation and centre of rotation.  They should also include the translation description as these two transformations combine to make the tessellation.

Session 3

This session explores learning to create a base pattern for a tessellation using reflection.  Rotations and translations are also used.  Using reflections is more challenging than using rotations and translations.

SLOs:

• Learning about reflection.
• Using reflection to create a tessellation.

Activity 1

In this activity students are learning about reflections.

A reflection is the flipping of points of the plane about a line, called a mirror line.  The properties of size and shape remain invariant (unchanged) under the operation of reflection. In the figure below, the figure is reflected through line m. D’ is the reflected image of D, and D is the reflected image of D’.

Mirror lines are usually labelled with small italic letters.

The properties of size and shape remain invariant (unchanged) under the operation of reflection.

If a more in-depth look at reflection was wanted teachers could explore Session 2, Activity 2 “teacher and student activity using horizontal and vertical reflections ” in the Transforming Shapes unit of work.

Activity 2

Making a base template with reflection.

Use cut-outs on board to demonstrate how to alter a shape and then use reflection and rotation to create a tessellation.

Original triangle               Showing the nibble         Cut and reflect the nibble            Rotate the nibble

Note that the “nibble” in the final two is flipped, so the side showing is the blank side.  It is useful to colour one side as stated previously.

Showing how the reflection and rotation work together.

Students can then work on Copymaster 4 to practice and then create their own base templates.

Ideas around creativity can be mentioned here. Have a selection of pictures of Escher’s work to leave on your board that show reflection and rotations.

Key ideas to reinforce in this include the conservation of area and length. As with the previous tessellation example the concept of area can be explored. Students should be describing the reflections in the tessellation though not as straight forward.  They should also include the rotation and translation description as these three transformations combine to make the tessellation.

Session 5

In this session students are pulling on their previous learning experiences to show how tessellating shapes are developed.

SLOs

• Applying knowledge of Escher type tessellations.
• Creating own Escher type tessellation poster.

Activity 1

Students choose a tile from the set in Copymaster 5.

Students should trace or make several copies of the tile to see if the shape tessellates.  If the shape tessellates students are to describe the transformations involved in the tessellation and to describe how the shape was created (e.g. using slides (translations), flips (reflections) and/or turns (rotations)).

Activity 2

Students then choose one of these shapes or one of the previous shapes they have created or design a new shape and produce a tessellation onto an A3 size poster.

To add colour, students could trace their shapes onto square note pad paper and glue together to make the tessellation, or they could transfer their shape onto paper and colour using pencils, crayons or felts.

Examples showing the use of coloured paper:

Session 6

In this final session students can add their imagination to their design through colour, texture and imagination.

SLOs:

Finishing off Escher type tessellation designs.

Introduction to the art component of the unit.

Activity 1

Activity to show use of colour and texture: Hexagon to small rhombuses cut-out demonstrated on the board.

The final part of the challenge is for students to draw something creative onto their base pattern they have used in their tessellation.

Activity 2

Once students have completed their A3 poster from Session 5 they can start the art focused component of this unit.

They begin by researching art-work that is creative and includes transformations and tessellation components in the design.

Students then design and produce their art-work.

Examples of student created work: