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 a grid.
• Identify lines of symmetry in 2D shapes.
• Understand and find the order of rotational symmetry.
• Use transformations to create tessellations.

Lesson summary

This week, we delve into the fascinating world of transformations and symmetry. Understanding these concepts is crucial because they're everywhere! From the patterns in traditional Ndebele art to the designs of buildings in our cities and even the shapes of objects we use daily, transformations and symmetry help us understand and appreciate the world around us. Think about the symmetry in a beautifully woven Zulu basket or how a tile pattern repeats across a wall – these are practical examples of what we'll be learning. Knowing about transformations allows us to describe how shapes can be moved and changed without losing their essential characteristics.

Lesson notes

2.1 Transformations: Transformations describe how we can move or change a shape on a plane. The original shape is called the pre-image, and the shape after the transformation is called the image. We will look at four main types of transformations: translation, reflection, rotation, and enlargement/reduction (also called scaling).

Translation: A translation (or slide) moves every point of a shape the same distance in the same direction. Think of sliding a book across a table. The shape stays the same size and orientation, it just changes position. We can describe a translation using words like "3 units to the right and 2 units up."

Example: Imagine a square ABCD. If we translate it 4 units to the right, each vertex (A, B, C, and D) will move 4 units to the right. The new square, A'B'C'D', will be identical to ABCD but in a new location.

Reflection: A reflection (or flip) creates a mirror image of the shape across a line called the line of reflection. Think of folding a piece of paper in half and drawing on one side. When you unfold it, you'll see the reflection. The distance from each point of the pre-image to the line of reflection is equal to the distance from the corresponding point of the image to the line of reflection.

Example: Consider a triangle PQR. If we reflect it across a vertical line, each point (P, Q, and R) will be reflected to the opposite side of the line, maintaining the same distance. The image, P'Q'R', will be a mirror image of PQ

R. Rotation: A rotation turns a shape around a fixed point called the center of rotation. We describe a rotation using the angle of rotation (e.g., 90 degrees, 180 degrees) and the direction (clockwise or anticlockwise). Think of turning a steering wheel.

Example: Take a rectangle EFGH. If we rotate it 90 degrees clockwise around point E, the rectangle will turn. Point E stays fixed, while the other points (F, G, and H) move to new positions.

Enlargement/Reduction (Scaling): Enlargement increases the size of a shape, and reduction decreases its size. We use a scale factor to determine how much bigger or smaller the shape becomes. A scale factor greater than 1 represents an enlargement, while a scale factor between 0 and 1 represents a reduction. The shape remains similar to the original but is a different size.

Example: Suppose we have a triangle XYZ and we want to enlarge it by a scale factor of

2. Each side of the new triangle X'Y'Z' will be twice as long as the corresponding side of triangle XYZ. 2.2 Symmetry: Symmetry refers to a balanced and proportionate similarity found in two halves of an object. There are two main types of symmetry we'll consider: line symmetry (or reflectional symmetry) and rotational symmetry.

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. Think of cutting out a heart shape – the fold line is the line of symmetry. Some shapes can have multiple lines of symmetry.

Example: A square has four lines of symmetry (horizontal, vertical, and two diagonals). An isosceles triangle has one line of symmetry. A circle has an infinite number of lines of symmetry.

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

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

3. A circle has infinite rotational symmetry. 2.3 Tessellations: A tessellation (or tiling) is a pattern made of one or more shapes that cover a surface completely without any gaps or overlaps. Many real-world surfaces are tessellated, for example, tiled floors and brick walls. Some shapes tessellate on their own, like squares, equilateral triangles, and regular hexagons. Others require a combination of shapes to tessellate. Transformations, especially translations, are often used to create tessellations. Guided Practice (With Solutions)

Question 1: Triangle ABC has vertices A(1,1), B(3,1), and C(2,3). Translate the triangle 4 units to the right and 2 units up. What are the coordinates of the vertices of the image, A'B'C'?

Solution: To translate a point (x, y) 4 units right and 2 units up, we add 4 to the x-coordinate and 2 to the y-coordinate: (x+4, y+2). A(1,1) becomes A'(1+4, 1+2) = A'(5, 3) B(3,1) becomes B'(3+4, 1+2) = B'(7, 3) C(2,3) becomes C'(2+4, 3+2) = C'(6, 5) Therefore, the vertices of the image are A'(5, 3), B'(7, 3), and C'(6, 5).

Question 2: Reflect the shape below across the y-axis (the vertical axis). Draw the reflection. The shape has vertices D(1,2), E(1,4), F(3,4), G(3,2). What are the coordinates of the reflected shape?