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# Transformation Units of Work

Learning Sequence

Curriculum Achievement Objectives

Specific Learning Outcomes
The students will be able to:

Units of Work

Level 1

Level 1 Transformation AO1

• identify lines of symmetry in objects
• make patterns which have ine symmetry
• describe line symmetry in their own words

Pattern Matching

Level 1 Transformation AO1

• make patterns that involve translations, reflections, and rotations
• identify translations, reflections, or rotations in patterns

Making Patterns

Level 2

Level 2 Transformation AO1

• make shapes with tessellations
• investigate shapes that tessellate
• make geometric patterns by translating, reflecting and rotating

In The Garden

 Level 2 Transformation AO1 Level 2 Shape AO1
• explain in their own language what line symmetry is
• describe the process of making shapes with line symmetry.
• name common two-dimensional mathematical shapes
• describe the differences between common two-dimensional mathematical shapes in relation to number of sides

Fold and Cut

Level 2 Transformation AO1

• fold paper systematically
• cut shapes from folded paper
• find number patterns derived from folding and cutting using a table

Fold and Cut 2

Level 2 Transformation AO1

• create simple tessellations involving squares and dominoes
• identify the repeating element(s) in simple tessellations involving squares and dominoes

Tessellating Tiles

Level 3

 Level 3 Transformation AO1 Level 3 Shape AO1
• find all the lines of reflection symmetry in a given shape
• identify the order of rotational symmetry of a given shape (how many times it "maps" onto itself in a full turn)
• create designs which have reflection symmetry
• rotational symmetry (orders 2, 3, 4, 6) and translational symmetry

Level 3 Transformation AO1

• demonstrate why a given tessellation will cover the plane
• create regular tessellations

Keeping In Shape

Level 4

 Level 4 Transformation AO1 Level 4 Shape AO1
• create regular and semi-regular tessellations of the plane
• demonstrate why a given tessellation will cover the plane

Fitness

Level 4 Transformation AO1

• alter polygons to create unique shapes that tessellate
• describe the reflection or rotational symmetry of a shape or tessellation

Tessellating Art

 Level 4 Transformation AO1 Level 4 Measurement AO2 Level 4 Number strategies and knowledge AO3
• follow instructions, in diagram form, to construct two-dimensional mathematical shapes, e.g. triangles, quadrilaterals, pentagons and hexagons
• enlarge and reduce two-dimensional mathematical shapes by a given scale factor
• identify invariant properties when enlarging and reducing two-dimensional mathematical shapes
• convert between mm and cm measurement
• multiply whole numberss by a decimal

Team Puzzles

Level 5

 Level 5 Transformation AO1 Level 5 Measurement AO4 Level 5 Number strategies and knowledge AO4
• use scale factors to investigate areas being enlarged
• use scale factors to investigate volumes being enlarged
• solve real life context problems involving scale factors

Scale Factors for Areas and Volumes

 Level 5 Transformation AO1
• find the scale factors for length, area and volume
• identify the centre point of an enlargement
• place similar objects to show a negative enlargement

Russian Boxes

 Level 5 Transformation AO2 Level 5 Number strategies and knowledge AO4
• measure the lengths of the sides of sets of similar right angled triangles and find the ratio of sides
• investigate the relationship between these ratios and the angle size
• use calculators or tables to find the sine, cosine and tangent of angles

Introducing Trig

 Level 5 Transformation AO2
• label right angle triangles with respect to a given angle.
• use trigonometric ratios to calculate the length of opposite and adjacent sides in right angled triangles
• use trigonometric ratios to calculate the size of angles in right angled triangles

Using Trigonometry

 Level 5 Transformation AO2
• state and explain Pythagoras' theorem
• use Pythagoras' theorem to find the unknown sides of right angled triangles

Pythagoras' Theorem

 Level 5 Transformation AO2
• find lengths of obejcts using Pythagoras' Theorem
• understand how similar triangles can be used to prove Pythagoras' Theorem
• understand that Pythagoras' Theorem can be thought of in terms of areas on the sides of the triangle

Gougu Rule or Pythagoras' Theorem

 Level 5 Transformation AO2
• describe and demonstrate how trigonometry can be used to find the height of a tall building or tree
• describe and demonstrate how trigonometry can be used to find the height of a high hill, or other high object where one cannot stand directly beneath the highest part
• describe in broad terms how trigonometry might be used to find the distance between the earth and the moon

Trigonometric applications outside the classroom

 Level 5 Transformation AO2
• use cos to solve problems involving right-angled triangles
• solve equations of the form cos(θ) = a, for a between –180 and 360 degrees
• state the value of cos(θ) in special cases
• graph y = cos(θ)

Investigating the Idea of Cos

 Level 5 Transformation AO2
• use sin to solve problems involving right-angled triangles
• solve equations of the form sin(θ) = a, for a between –180º and 360º
• state the value of sin(θ) in special cases
• graph y = sin(θ)
• describe some of the ways in which the sine, cosine and tangent functions are related

Investigating the Idea of Sin

 Level 5 Transformation AO2
• use tan to solve problems involving right-angled triangles
• solve equations of the form tan(θ) = a, for a between –180ºand 360º degrees
• state the value of tan(θ) in special cases
• graph y = tan(θ)

Investigating the Idea of Tan

Level 5 Transformation AO2

• measure lengths and angles accurately
• find the height of objects using trigonometry

Dizzy Heights

### 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 geo-boards to construct shapes that are symmetric about some axis;
• investigate polygonal shapes that will tessellate;
• analyse and construct Escher-type 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 right-angled 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 anti-clockwise turns. They should also be beginning to recognise that two-dimensional 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 columes are transformed by enlarging.

Throughout, angle should be used where it aids communication and understanding in Shape, Position and Symmetry