Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify and describe translations, reflections, and rotations.
• Draw the image of a shape after a transformation.
• Identify lines of symmetry in 2D shapes.
• Determine the order of rotational symmetry.
• Describe the effect of combined transformations.

Lesson summary

Transformations and symmetry are all around us! From the patterns in traditional Ndebele art to the design of our houses and even the arrangement of vegetables at the local market, these concepts are fundamental to how we perceive and interact with the world. Understanding transformations helps us describe how shapes can be moved, rotated, or flipped without changing their basic form. Symmetry allows us to appreciate balance and harmony in design and nature. In South Africa, recognizing and utilizing these principles can enhance creativity, problem-solving skills, and an appreciation for our diverse cultural heritage.

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 main types of transformations: Translation (Slide): A translation moves a shape in a straight line without changing its size, shape, or orientation. It's like sliding the shape across a surface. We describe a translation by specifying how far to move the shape horizontally (left or right) and vertically (up or down).

Example: Imagine a tile in a mosaic. You can slide it to another position on the floor without rotating or flipping it. That's a translation. We can use arrows or coordinate grids to represent translations precisely.

Reflection (Flip): A reflection flips a shape over a line, creating a mirror image. This line is called the line of reflection or axis of symmetry. The image is the same distance from the line of reflection as the object, but on the opposite side.

Example: Think about your reflection in a still lake. Your image is a reflection of you across the surface of the water. When reflecting across a vertical line, the x-coordinates change, and the y-coordinates stay the same. When reflecting across a horizontal line, the y-coordinates change, and the x-coordinates stay the same.

Rotation (Turn): A rotation turns a shape around a fixed point called the center of rotation. We describe a rotation by specifying the angle of rotation (e.g., 90°, 180°, 270°) and the direction of rotation (clockwise or anticlockwise).

Example: The blades of a windmill are rotating around a central point. A 360° rotation brings the shape back to its original position. 2.2 Symmetry Symmetry refers to a balanced and proportional similarity found in two halves of an object.

Line Symmetry (Reflectional Symmetry): A shape has line symmetry if it can be folded along a line (the line of symmetry) so that the two halves match exactly.

Example: An isosceles triangle has one line of symmetry. A square has four lines of symmetry. A heart shape has one line of symmetry. Many letters of the alphabet have line symmetry (A, H, I, M, O, T, U, V, W, X, Y). You can find lines of symmetry by folding a shape or drawing a line on the shape that divides it into two identical halves.

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

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 (120°, 240°, 360°). A shape with rotational symmetry of order 1 has no rotational symmetry (other than a full 360-degree rotation).

Worked example

Example 1: Translation

A triangle has vertices at A(1, 2), B(3, 4), and C(2, 5). Translate the triangle 4 units to the right and 1 unit down. What are the coordinates of the image vertices?

Solution:

To translate 4 units to the right, we add 4 to the x-coordinate of each vertex.

To translate 1 unit down, we subtract 1 from the y-coordinate of each vertex.

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

B'(3+4, 4-1) = B'(7, 3)

C'(2+4, 5-1) = C'(6, 4)

Example 2: Reflection

Reflect a rectangle with vertices D(1, 1), E(4, 1), F(4, 3), and G(1, 3) across the y-axis. What are the coordinates of the image vertices?

Solution:

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

D'(-1, 1)

E'(-4, 1)

F'(-4, 3)

G'(-1, 3)