G13 Reflection and symmetry Boardworks KS3 Maths 2009 G13 Reflection and symmetry G13 Reflection and symmetry This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry

Boardworks KS3 Maths 2009 G13 Reflection and symmetry An object can be reflected in a mirror line or axis of reflection to produce an image of the object. Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflecting shapes If we reflect the quadrilateral ABCD in a mirror line we label the image quadrilateral A’B’C’D’. A B C D A’ B’ object image C’ Explain that we call the original shape the object and the reflected shape the image. The image of an object can be produced by any transformation including rotations, translations and enlargements, as well as reflections. Define the word congruent to mean the same shape and size. Link: G5 Polygons and congruence D’ mirror line or axis of reflection The image is congruent to the original shape.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflecting shapes If we draw a line from any point on the object to its image the line forms a perpendicular bisector to the mirror line. A B C D A’ B’ object image Emphasize that the mirror line is always perpendicular (at right angles) to any line connecting a point to its image. The mirror line also bisects the line (divides it into two equal parts). C’ D’ mirror line or axis of reflection

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflecting shapes Use this activity to dynamically illustrate the result on the previous slide by changing the shape reflected and the position of the mirror line.

Reflecting shapes by folding paper Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflecting shapes by folding paper This animation demonstrates how to reflect shapes by folding paper. Pupils could be shown the animation then attempt the activity themselves, using a different polygon.

Reflecting shapes using tracing paper Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflecting shapes using tracing paper This activity demonstrates how to draw reflections of shapes using tracing paper. Pupils could attempt the activity themselves, using a different shape.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflect this shape Choose one of the buttons down the right-hand side to change the position of the mirror line. Modify one of the pentagons by dragging its vertices, and choose it as the original. Position the shape by dragging on its centre. Ask a volunteer to come to the board and reflect the shape by modifying the other pentagon. If necessary, use the pen tool set to draw straight lines to connect the points in the object to the mirror line. Instruct the volunteer to reflect these lines to find the positions of the image points.

Reflection on a coordinate grid Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflection on a coordinate grid y A’(–2, 6) A(2, 6) The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). 7 6 B’(–7, 3) 5 B(7, 3) 4 3 2 Reflect the triangle in the y-axis and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 C’(–4, –1) –2 C(4, –1) –3 Pupils should notice that when a shape is reflected in the y-axis, the x-coordinate of each image point is the same as the x-coordinate of the original point multiplied by –1 and the y-coordinate of the image point is the same as the y-coordinate of the original point. In other words, the x-coordinate changes sign and the y-coordinate stays the same. –4 What do you notice about each point and its image? –5 –6 –7

Reflection on a coordinate grid Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflection on a coordinate grid y The vertices of a quadrilateral lie on the points A(–4, 6), B(4, 5), C(2, 0) and D(–5, 3). A(–4, 6) 7 6 B(4, 5) 5 4 3 D(–5, 3) 2 1 C(2, 0) Reflect the quadrilateral in the x-axis and label each point on the image. –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 C’(2, 0) D’(–5, –3) –2 –3 Pupils should notice that when a shape is reflected in the x-axis, the x-coordinate of each image point is the same as the x-coordinate of the original point and the y-coordinate of the image point is the same as the y-coordinate of the original point multiplied by –1. In other words, the x-coordinate stays the same and the y-coordinate changes sign. –4 What do you notice about each point and its image? –5 B’(4, –5) –6 A’(–4, –6) –7

Reflection on a coordinate grid Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflection on a coordinate grid y x = y B’(–1, 7) The vertices of a triangle lie on the points A(4, 4), B(7, –1) and C(2, –6). 7 6 A’(4, 4) 5 4 A(4, 4) 3 2 C’(–6, 2) Reflect the triangle in the line y = x and label each point on the image. 1 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 x –1 B(7, –1) –2 –3 Pupils should notice that when a shape is reflected in the line x = y, the x-coordinate of each image point is the same as the y-coordinate of the original point and the y-coordinate of the image point is the same as the x-coordinate of the original point. In other words, the x- and y-coordinates are swapped around. –4 What do you notice about each point and its image? –5 –6 –7 C(2, –6)

Boardworks KS3 Maths 2009 G13 Reflection and symmetry G13.2 Reflection symmetry

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflection symmetry If you can draw a line through a shape so that one half is a reflection of the other then the shape has reflection or line symmetry. The mirror line is called a line of symmetry. Some pupils may find it easier to visualize a line of symmetry if they visualize it as a fold line rather than a mirror line. Explain that if you fold along a line of symmetry the two sides should match up exactly. Explain that there is no way to fold a parallelogram so that one side fits exactly on top of the other. Demonstrate this if necessary by folding a piece of paper cut in the shape of a (non-rectangular) parallelogram along lines suggested by pupils. one line of symmetry two lines of symmetry no lines of symmetry

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Reflection symmetry How many lines of symmetry do the following designs have? one line of symmetry five lines of symmetry three lines of symmetry

Make this shape symmetrical Boardworks KS3 Maths 2009 G13 Reflection and symmetry Make this shape symmetrical Ask a volunteer to come to the board and drag the vertices to make the shape symmetrical about the given mirror line.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Planes of symmetry Is a cube symmetrical? We can divide the cube into two symmetrical parts here. Challenge pupils to draw all the planes of symmetry for a cube on isometric paper. There are nine altogether. This shaded area is called a plane of symmetry. How many planes of symmetry does a cube have?

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Planes of symmetry We can draw the other eight planes of symmetry for a cube, as follows: Reveal the images on this slide to compare with the pupils solutions. Point out that these images are shown from a different angle to the ones drawn on an isometric grid. The last plane of symmetry on the slide can only be shown as a line because of the angle chosen. Link: G8 3-D shapes and representations

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Planes of symmetry How many planes of symmetry does a cuboid have? A cuboid has three planes of symmetry. Ask pupils to draw the three planes of symmetry of a cuboid on squared or isometric paper. Point out that each corner of the plane of symmetry will touch half-way along the corresponding edge of the cuboid. A square prism will have two further (diagonal) planes.

Boardworks KS3 Maths 2009 G13 Reflection and symmetry Planes of symmetry How many planes of symmetry do the following solids have? An equilateral triangular prism A square-based pyramid A cylinder Establish that an equilateral triangular prism has four planes of symmetry, a square-based pyramid has four planes of symmetry and a cylinder has an infinite number. Explain that a right prism is a prism whose opposite faces are parallel. Explain why any right prism will always have at least one plane of symmetry.

Investigating shapes made from four cubes Boardworks KS3 Maths 2009 G13 Reflection and symmetry Investigating shapes made from four cubes Ask volunteers to come to the board and make three-dimensional shapes with given numbers of planes of symmetry. If other members of the group believe the shape constructed has more (or fewer) planes of symmetry, they should describe these fully and give an alternative shape.