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 single transformations on 2D shapes on a grid.
• Identify lines of symmetry in 2D shapes.
• Create symmetrical patterns.
• Distinguish between symmetrical and asymmetrical shapes.

Lesson summary

Transformations and symmetry are fundamental concepts in mathematics and are all around us in our daily lives. From the patterns in the vibrant Ndebele art to the symmetrical designs of traditional Zulu beadwork, understanding these concepts helps us appreciate the beauty and structure in our environment. This topic builds on your prior knowledge of shapes and spatial reasoning, developing critical skills in problem-solving and visual analysis. Learning about transformations and symmetry will not only help you understand mathematical concepts but also improve your ability to observe, analyze, and appreciate the world around you.

Lesson notes

2. 1.

Transformations: A transformation is a way of changing the position or size of a shape. We will focus on four types of transformations: Translation (Slide): A translation moves a shape in a straight line without rotating or flipping it. Think of sliding a desk across the floor. The shape stays the same; only its position changes. We describe a translation by how many units it moves horizontally (left or right) and vertically (up or down).

Example: Imagine a triangle ABC with vertices A(1, 1), B(3, 1), and C(2, 3). If we translate it 4 units to the right and 2 units up, the new coordinates of the vertices, A'B'C', will be A'(1+4, 1+2) = A'(5, 3), B'(3+4, 1+2) = B'(7, 3), and C'(2+4, 3+2) = C'(6, 5). The shape and size of the triangle remain unchanged.

Reflection (Flip): A reflection creates a mirror image of a shape across a line called the line of reflection. Think of looking at yourself in a mirror. The image is flipped.

Example: Consider a rectangle with vertices P(1, 2), Q(4, 2), R(4, 4), and S(1, 4). If we reflect it across the y-axis (the vertical line where x=0), the x-coordinates change sign while the y-coordinates remain the same. The new vertices, P'Q'R'S', become P'(-1, 2), Q'(-4, 2), R'(-4, 4), and S'(-1, 4).

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

Example: Imagine a square with vertices X(1, 1), Y(2, 1), Z(2, 2), and W(1, 2). If we rotate it 90° clockwise around the origin (0, 0), the new coordinates will change following the rule (x, y) -> (y, -x). The new vertices X'Y'Z'W' become X'(1, -1), Y'(1, -2), Z'(2, -2), and W'(2, -1).

Enlargement (Scaling): An enlargement changes the size of a shape. We describe an enlargement by its scale factor. A scale factor greater than 1 makes the shape bigger, while a scale factor between 0 and 1 makes it smaller. The shape remains similar.

Example: Let's say we have a triangle DEF with vertices D(1, 1), E(2, 1), and F(1, 2). If we enlarge it by a scale factor of 2 with the origin (0, 0) as the center of enlargement, we multiply each coordinate by

2. The new vertices, D'E'F', become D'(2, 2), E'(4, 2), and F'(2, 4). The triangle is now twice as big, but it has the same shape. 2.

2. Symmetry: Symmetry is when a shape looks exactly the same after being transformed in some way. We will focus on line symmetry.

Line Symmetry (Reflection 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. Think of cutting out a paper heart – the fold line is the line of symmetry. Shapes can have one, several, or no lines of symmetry.

Examples: A square has four lines of symmetry (one vertical, one horizontal, and two diagonal). A rectangle has two lines of symmetry (one vertical and one horizontal). An equilateral triangle has three lines of symmetry. A circle has infinitely many lines of symmetry. An irregular shape like a shoe typically has no lines of symmetry. 2.3 Distinguishing Between Symmetrical and Asymmetrical Shapes: A symmetrical shape can be divided into two identical halves by a line of symmetry. An asymmetrical shape cannot. For example, a perfectly round plate is symmetrical, but a plate with a chipped edge is asymmetrical. Many objects we encounter daily are asymmetrical, such as a hand or a leaf, because they are not perfectly identical on both sides. Guided Practice (With Solutions)

Question 1: Triangle PQR has coordinates P(2, 1), Q(4, 1), and R(3, 3). Translate the triangle 3 units to the left and 1 unit down. What are the new coordinates of the vertices?

Solution: Translation: (x, y) -> (x - 3, y - 1) P'(2 - 3, 1 - 1) = P'(-1, 0) Q'(4 - 3, 1 - 1) = Q'(1, 0) R'(3 - 3, 3 - 1) = R'(0, 2) The new coordinates are P'(-1, 0), Q'(1, 0), and R'(0, 2).

Commentary:* We applied the translation rule by subtracting 3 from each x-coordinate and 1 from each y-coordinate. This gives the new position of the triangle after the slide.

Question 2: Reflect the following shape (a square with vertices at (1,1), (2,1), (2,2), (1,2)) across the x-axis. What are the coordinates of the reflected image?

Solution: Reflection across the x-axis: (x, y) -> (x, -y) (1, 1) -> (1, -1) (2, 1) -> (2, -1) (2, 2) -> (2, -2) (1, 2) -> (1, -2) The coordinates of the reflected image are (1, -1), (2, -1), (2, -2), (1, -2).

Commentary:* Reflecting across the x-axis changes the sign of the y-coordinate while the x-coordinate remains the same.

Question 3: Rotate a point A(2, 2) 90 degrees anticlockwise about the origin. What are the coordinates of the new point A'?

Solution: Rotation 90 degrees anticlockwise: (x, y) -> (-y, x) A'(2,2) -> A'(-2, 2) The coordinates of the rotated point are (-2, 2).

Commentary:* A 90-degree anticlockwise rotation about the origin swaps the x and y coordinates, and changes the sign of the new x-coordinate.