Shape and Space Reflections The aim of this unit is to teach pupils to: Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning Understand congruence and similarity Identify and use the properties of circles Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp 184-197. Reflections

Reflection An object can be reflected in a mirror line or axis of reflection to produce an image of the object. For example, Each point in the image must be the same distance from the mirror line as the corresponding point of the original object.

The image is congruent to the original shape. 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’ D’ Explain that we call the original shape the object and the reflected shape the image. The image of a shape 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: S2 2-D shapes - congruence mirror line or axis of reflection The image is congruent to the original shape.

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’ C’ D’ object image mirror line or axis of reflection Stress 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).

Reflecting shapes Use this activity to dynamically prove the result on the previous slide, regardless of the shape or the position of the mirror line.

Reflecting shapes by folding paper We can make reflections by folding paper. Draw a random polygon at the top of a piece of paper. Fold the piece of paper back on itself so you can still see the shape. Pierce through each vertex of the shape using a compass point. When the paper is unfolded the vertices of the image will be visible. Join the vertices together using a ruler.

Reflecting shapes using tracing paper Suppose we want to reflect this shape in the given mirror line. Use a piece of tracing paper to carefully trace over the shape and the mirror line with a soft pencil. When you turn the tracing paper over you will see the following: Place the tracing paper over the original image making sure the symmetry lines coincide. Draw around the outline on the back of the tracing paper to trace the image onto the original piece of paper.

Reflection on a coordinate grid y The vertices of a triangle lie on the points A(2, 6), B(7, 3) and C(4, –1). A’(–2, 6) 7 A(2, 6) 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 –4 What do you notice about each point and its image? 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 the original point × –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. –5 –6 –7

Reflection on a coordinate grid y The vertices of a quadrilateral lie on the points A(–4, 6), B(4, 5), C(2, –2) and D(–5, 3). A(–4, 6) 7 6 B(4, 5) 5 4 3 D(–5, 3) 2 1 C(2, –2) 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, –2) D’(–5, –3) –2 –3 –4 What do you notice about each point and its image? 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 the original point and the y-coordinate of the image point is the same as the y-coordinate of the original point × –1. In other words, the x-coordinate stays the same and the y-coordinate changes sign. –5 B’(4, –5) –6 A’(–4, –6) –7

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 B(7, –1) x –1 –2 –3 –4 What do you notice about each point and its image? 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 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. –5 –6 –7 C(2, –6)