Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe translations, reflections, rotations, and enlargements.
• Perform these transformations on 2D shapes.
• Identify lines of symmetry in shapes.
• Distinguish between symmetrical and asymmetrical shapes.
• Describe the effect of transformations on shapes.

Lesson summary

This week, we delve into the fascinating world of transformations and symmetry. These mathematical concepts are not just abstract ideas; they are all around us, from the patterns in traditional Ndebele art to the way buildings are designed. Understanding transformations and symmetry allows us to describe and analyze shapes and patterns, helping us to appreciate the beauty and order in the world around us. Imagine designing a vibrant shweshwe fabric pattern – transformations and symmetry are key! Or consider the careful tiling of a stoep – same principles apply.

Lesson notes

2.1 What are Transformations? A transformation is a way of changing the position, size, or orientation of a shape. The original shape is called the pre-image, and the new shape after the transformation is called the image. We will be focusing on four main types of transformations: Translation (Slide): Moving a shape without changing its size, shape, or orientation. Imagine sliding a R5 coin across a table.

Reflection (Flip): Creating a mirror image of a shape across a line of reflection. Think of looking at yourself in a mirror.

Rotation (Turn): Turning a shape around a fixed point called the centre of rotation. Imagine the hands of a clock moving around the centre.

Enlargement (Scaling): Changing the size of a shape by a scale factor. This makes the shape bigger or smaller. 2.2 Translation (Slide) in Detail: A translation moves a shape a certain distance in a specific direction. We describe a translation by how many units it moves horizontally (left or right) and vertically (up or down).

Example: Translate triangle ABC 3 units to the right and 2 units up. (Imagine this on a grid with A=(1,1), B=(3,1), C=(2,3))

Solution: Each point of the triangle (A, B, and C) moves 3 units to the right and 2 units up. A(1,1) becomes A'(4,3) (1+3, 1+2) B(3,1) becomes B'(6,3) (3+3, 1+2) C(2,3) becomes C'(5,5) (2+3, 3+2) The new triangle A'B'C' is the same size and shape as ABC, but its position has changed. 2.3 Reflection (Flip) in Detail: A reflection creates a mirror image of a shape across a line of reflection. The line of reflection acts like a mirror. The distance from each point on the pre-image to the line of reflection is the same as the distance from the corresponding point on the image to the line of reflection.

Example: Reflect square PQRS across the y-axis. (Imagine a square with P=(-1,1), Q=(-1,3), R=(-3,3), S=(-3,1))

Solution: Each point of the square is reflected across the y-axis (the vertical line where x=0). This means the x-coordinate changes sign, while the y-coordinate stays the same. P(-1,1) becomes P'(1,1) Q(-1,3) becomes Q'(1,3) R(-3,3) becomes R'(3,3) S(-3,1) becomes S'(3,1) The new square P'Q'R'S' is the same size and shape as PQRS, but it is flipped. 2.4 Rotation (Turn) in Detail: A rotation turns a shape around a fixed point called the centre of rotation. We describe a rotation by the angle of rotation (e.g., 90 degrees, 180 degrees, 270 degrees) and the direction of rotation (clockwise or anticlockwise). We often use the origin (0,0) as the centre of rotation.

Example: Rotate triangle XYZ 90 degrees clockwise around the origin. (Imagine a triangle with X=(1,1), Y=(3,1), Z=(1,3))

Solution: For a 90-degree clockwise rotation around the origin, we swap the x and y coordinates and change the sign of the new y-coordinate. X(1,1) becomes X'(1, -1) (Swap and change sign of new y) Y(3,1) becomes Y'(1, -3) (Swap and change sign of new y) Z(1,3) becomes Z'(3, -1) (Swap and change sign of new y) The new triangle X'Y'Z' is the same size and shape as XYZ, but it has been turned. 2.5 Enlargement (Scaling) in Detail: An enlargement changes the size of a shape. We describe an enlargement by its scale factor. If the scale factor is greater than 1, the shape gets bigger. If the scale factor is less than 1 (but greater than 0), the shape gets smaller.

Example: Enlarge rectangle ABCD by a scale factor of 2, using the origin as the centre of enlargement. (Imagine a rectangle with A=(1,1), B=(3,1), C=(3,2), D=(1,2))

Solution: Multiply each coordinate of each point by the scale factor (2). A(1,1) becomes A'(2,2) (1x2, 1x2) B(3,1) becomes B'(6,2) (3x2, 1x2) C(3,2) becomes C'(6,4) (3x2, 2x2) D(1,2) becomes D'(2,4) (1x2, 2x2) The new rectangle A'B'C'D' is twice as big as ABCD. 2.6 Symmetry: A shape has symmetry if it can be folded along a line (called the line of symmetry) so that the two halves match exactly. A shape can have one line of symmetry, multiple lines of symmetry, or no lines of symmetry. Shapes with at least one line of symmetry are called symmetrical. Shapes with no lines of symmetry are called asymmetrical.

Examples: A square has 4 lines of symmetry. A rectangle has 2 lines of symmetry. A circle has infinite lines of symmetry. A scalene triangle has no lines of symmetry. Consider a traditional Zulu shield pattern - many have rotational symmetry. Guided Practice (With Solutions)

Question 1: Translate triangle DEF 2 units to the left and 4 units down. D(2,5), E(4,5), F(3,2).

Solution: D(2,5) becomes D'(0,1) (2-2, 5-4) E(4,5) becomes E'(2,1) (4-2, 5-4) F(3,2) becomes F'(1,-2) (3-2, 2-4)

Commentary: Remember to subtract from the x-coordinate for leftward movement and from the y-coordinate for downward movement.

Question 2: Reflect quadrilateral GHIJ across the x-axis. G(1,2), H(3,4), I(5,2), J(3,0).

Solution: G(1,2) becomes G'(1,-2) H(3,4) becomes H'(3,-4) I(5,2) becomes I'(5,-2) J(3,0) becomes J'(3,0)

Commentary: Reflecting across the x-axis changes the sign of the y-coordinate.