Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe translations, reflections, and rotations.
• Perform these transformations on a grid.
• Find lines of symmetry in shapes.
• Understand and find rotational symmetry.
• Use these skills to solve problems.

Lesson summary

Transformations and symmetry are fundamental concepts in mathematics that help us understand the world around us. From the patterns in traditional Ndebele art to the design of buildings and the arrangement of tiles, these ideas are present everywhere. Understanding transformations and symmetry enhances our spatial reasoning skills and allows us to appreciate the beauty and order in both natural and man-made environments. In South Africa, many cultural patterns, crafts, and architectural designs rely on these principles.

Lesson notes

2.1 Transformations A transformation is a way of changing the position or orientation of a shape. The original shape is called the object, and the new shape after the transformation is called the image. We will focus on three types of transformations: Translation: A translation is a "slide" or "shift" of a shape. The shape moves in a straight line without changing its size or orientation. We describe translations by how far the shape moves horizontally (left or right) and vertically (up or down).

Example: Imagine moving a school desk across the classroom floor without turning it. That's a translation!

Reflection: A reflection is a "flip" of a shape over a line. This line is called the line of reflection. The image is a mirror image of the object.

Example: When you look in a mirror, your reflection is a mirror image of you. The mirror acts as the line of reflection. Think of the reflection of Table Mountain in the Atlantic Ocean.

Rotation: A rotation is a "turn" of a shape around a point. This point is called the centre of rotation. We describe rotations by the angle of the turn (e.g., 90°, 180°, 270°) and the direction of the turn (clockwise or anticlockwise).

Example: Think of the hands of a clock moving around the centre. That's a rotation. 2.2 Symmetry Symmetry refers to a balanced and proportionate similarity that is found in two halves of an object.

We will focus on two types of symmetry: Line Symmetry (Reflectional Symmetry): A shape has line symmetry if it can be folded along a line so that the two halves match perfectly. The line along which the shape is folded is called the line of symmetry (also sometimes called the axis of symmetry).

Example: A square has four lines of symmetry, while a rectangle has two. Many leaves in nature exhibit line symmetry.

Rotational Symmetry: A shape has rotational symmetry if it can be rotated around a central point by less than a full turn (360°) and still look the same as the original. The order of rotational symmetry is the number of times the shape looks the same during one full turn.

Example: A square has rotational symmetry of order 4 because it looks the same after rotations of 90°, 180°, 270°, and 360°. An equilateral triangle has rotational symmetry of order 3.

Worked example

Example 1: Translation

Question: Translate the triangle with vertices A(1, 2), B(3, 2), and C(2, 4) by 4 units to the right and 2 units down.

Solution:

To translate a point, we add the horizontal movement to the x-coordinate and the vertical movement to the y-coordinate.

A'(1 + 4, 2 - 2) = A'(5, 0)

B'(3 + 4, 2 - 2) = B'(7, 0)

C'(2 + 4, 4 - 2) = C'(6, 2)

Therefore, the translated triangle has vertices A'(5, 0), B'(7, 0), and C'(6, 2). We moved each point of the original triangle the same distance in the same direction.

Example 2: Reflection

Question: Reflect the shape with vertices D(2, 1), E(4, 1), F(4, 3), and G(2, 3) across the y-axis.

Solution:

When reflecting across the y-axis, the y-coordinate stays the same, and the x-coordinate changes sign.

D'(-2, 1)

E'(-4, 1)

F'(-4, 3)

G'(-2, 3)

Therefore, the reflected shape has vertices D'(-2, 1), E'(-4, 1), F'(-4, 3), and G'(-2, 3). Notice how the image is a mirror image of the original.

Example 3: Rotation

Question: Rotate the point P(1, 1) 90° clockwise around the origin (0, 0).

Solution:

Rotating a point (x, y) 90° clockwise around the origin results in the point (y, -x).

Therefore, P'(1, -1). Imagine a line segment from (0,0) to (1,1). Rotating this segment 90 degrees clockwise will place the new point at (1,-1).

Example 4: Line Symmetry

Question: How many lines of symmetry does a regular hexagon have? Draw a regular hexagon and show the lines of symmetry.

Solution: A regular hexagon has 6 lines of symmetry. These lines can be drawn from each vertex to the opposite vertex and from the midpoint of each side to the midpoint of the opposite side.