Level 1 Transformation
Achievement Objectives  Learning Outcomes  Unit title 
GM15 

Pattern Matching 
GM15 

Making Patterns 
Level 2 Transformation
Achievement Objectives  Learning Outcomes  Unit title 
GM27 

In The Garden 
GM27 GM23 

Fold and Cut 
GM27 

Fold and Cut 2 
GM27 

Tessellating Tiles 
Level 3 Transformation
Achievement Objectives  Learning Outcomes  Unit title 
GM36 GM34 

Logo Licenses 
GM36 

Keeping In Shape 
Level 4 Transformation
Achievement Objectives  Learning Outcomes  Unit title 
GM48 

Tessellating Art 
GM48 GM45 

Fitness 
GM48 GM42 NA43 

Team Puzzles 
Level 5 Transformation
Achievement Objectives  Learning Outcomes  Unit title 
GM59 

Russian Boxes 
GM59 GM54 NA54 

Scale Factors for Areas and Volumes 
GM510 NA54 

Introducing Trigonometry 
GM510 

Using Trigonometry 
GM510 

Pythagoras' Theorem 
GM510 

Gougu Rule or Pythagoras' Theorem 
GM510 

Trigonometric applications outside the classroom 
GM510 

Dizzy Heights 
GM510 

Investigating the Idea of Cos 
GM510 

Investigating the Idea of Sin 
GM510 

Investigating the Idea of Tan 
Transformation
Transformation begins at Level 1 with the identification of lines of symmetry in objects and making simple symmetrical patterns and culminates in Level 5 with the exploration of the invariant properties of objects under transformation. Along the way, it involves making geometrical patterns using translation, reflection and rotation (Level 3) it develops to describing patterns using the words translation, reflection and rotation (Level 4) and enlargement with scale factors (Level 5). There is a distinct step here from following instructions with a given set of words to using the words for oneself (van Hiele Stages 1 and 2).
Levels One and Two: Some activities here include:
 finding objects from magazines that have symmetry in them;
 folding paper in half and cutting shapes out of the fold;
 ‘tiling’ paper with cut out squares.
Levels Three and Four: Here students might:
 use geoboards to construct shapes that are symmetric about some axis;
 investigate polygonal shapes that will tessellate;
 analyse and construct Eschertype drawings.
Tessellations are an aspect of geometry that involves an understanding of many key geometric ideas, including: shape, angle, space, and transformations.
Level 1
At this stage the only idea of tessellation students are likely to have is an ability to cover a surface by physically joining shapes together or to draw shapes tessellating. It is unlikely that they will be able to correctly identify any common attributes of shapes that tessellate, but they may be able to use trial and error to identify some shapes (such as circles or octagons) that do not tessellate.
Level 2
By this stage students are starting to develop their own ideas of why shapes tessellate and will be able to explain some kind of reasoning, even if it is not totally correct. For example they might say that shapes with straight sides tessellate because they know that circles do not.
Level 3
By level three students are able to use their knowledge of right angles to show that squares, rectangles and rightangled triangles are able to tessellate. They will probably be able to identify that hexagons tessellate but will be unable to give a good explanation of why.
Level 4
By this stage students recognise that shapes that tessellate fit together round a point. They recognise that the angle at a point is 360 degrees (though may not use the correct term) and can use their knowledge of angle to explain that equilateral triangles and hexagons tessellate.
Level 5
At level five students will investigate the effect of scale factors on the properties of shapes.
Angle
Angle is a fundamental notion in geometry and pervades Shape, Position and Symmetry. At Level 1 the basis for angle should be given through the idea of an amount of turning by the body. At this Level there is only need to consider quarter and half turns. However, students should realise that each of these turns can be made in any direction (to their left or right) and that these turns do not depend on the initial direction that the student is facing. In other words, a quarter turn is still a quarter turn regardless of whether the student is facing along a wall or at some angle to it.
At Level 2, students should be comfortable with clockwise and anticlockwise turns. They should also be beginning to recognise that twodimensional shapes have angles at their ‘corners’. The sizes of these angles can be compared by putting one shape on top of another (but there is no need for exact measurement at this Level). So it is clear that squares and rectangles have the same sized angle; that regular hexagons have larger angles than rectangles; and that triangles can have bigger and smaller angles than rectangles.
Level 2 students should call the angle at the corner of a rectangle a right angle. They should see that a quarter turn is a right angle and that a half turn is two right angles.
In all this it should be emphasised that angle involves an amount of turning and that angles of the same size can be made starting from any given line.
More complicated angles, again defined as the amount of turning from some fixed line to another, should be constructed and compared at Level 3. This knowledge should be used to check that tilings do cover the entire floor on which they are laid. Children at this Level should know that one right angle has a measure of 90° and that a whole turn is 360° .
At Level 4, students should be using protractors to measure and construct almost any angle in degrees. They can practice on polygons and can use their skills in constructing shapes and building objects. At this stage too they should also realise that the (interior) angles in any triangle add to 180° ; that the interior angles of any quadrilateral add to 360° ; and that the interior angles of any pentagon add to 540° .
At Level 5, students are able to use scale factors to investigate how areas and volumes are transformed by enlarging.
Throughout, angle should be used where it aids communication and understanding in Shape, Position and Symmetry