Lesson Notes By Weeks and Term v5 - Grade 6

Lesson Video

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Performance objectives

• Identify four types of transformations.
• Perform translations, reflections, and rotations on a grid.
• Find lines of symmetry in shapes.
• Create your own symmetrical patterns.

Lesson summary

Transformations and symmetry are fundamental concepts in mathematics that help us understand how shapes can be moved and changed while maintaining certain properties. This understanding is crucial not only in mathematics but also in art, architecture, design, and many everyday situations. Imagine designing a beautiful traditional Ndebele pattern, building a stable hut, or even understanding how a soccer ball is constructed – all these involve transformations and symmetry. Understanding these concepts helps us appreciate the beauty and order in the world around us, foster spatial reasoning, and develop problem-solving skills vital for success in various fields.

Lesson notes

Transformations A transformation is a way of changing the position, size, or orientation of a shape. There are four main types of transformations we'll explore: Translation (Slide): A translation moves every point of a shape the same distance in the same direction. Think of it as sliding the shape without changing its orientation.

Example:* Imagine a taxi traveling down a straight road in Johannesburg. The taxi is undergoing a translation.

Explanation:* A translation is defined by a direction and a distance. We can describe a translation using arrows or coordinates.

Reflection (Flip): A reflection flips a shape over a line, called the line of reflection. The reflected image is a mirror image of the original.

Example:* Imagine looking at your reflection in a still dam on a farm. The water acts as the line of reflection.

Explanation:* Each point on the original shape is the same distance from the line of reflection as its corresponding point on the reflected image.

Rotation (Turn): A rotation turns a shape around a fixed point, called the center of rotation. A rotation is defined by the angle of rotation (how much the shape turns) and the direction of rotation (clockwise or counter-clockwise).

Example:* Think about the hands of a clock. They rotate around the center of the clock face.

Explanation:* We describe a rotation by its center, angle, and direction (clockwise or counter-clockwise). Common rotation angles are 90°, 180°, and 270°.

Enlargement/Reduction (Scaling): An enlargement (or reduction) changes the size of a shape. The shape remains similar to the original, but its dimensions are multiplied by a scale factor.

Example:* Think about a photo being enlarged on a computer screen.

Explanation:* If the scale factor is greater than 1, it's an enlargement. If it's between 0 and 1, it's a reduction. The corresponding sides of the original and transformed shapes are proportional. Symmetry Symmetry refers to a balanced and proportionate similarity found in two halves of an object, shape, or image.

Line of Symmetry: A line of symmetry divides a shape into two identical halves that are mirror images of each other. We can fold the shape along the line of symmetry, and the two halves will perfectly match.

Example:* Think about a butterfly. It has one line of symmetry down the middle of its body.

Explanation:* A shape can have one line of symmetry, more than one, or no lines of symmetry at all. Symmetrical vs.

Asymmetrical: A shape is symmetrical if it has at least one line of symmetry. A shape is asymmetrical if it has no lines of symmetry.

Example:* A square is symmetrical (it has four lines of symmetry). A randomly drawn squiggle is likely asymmetrical.

Worked example

Example 1: Translation

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

Solution:*

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

A'(1+4, 1+2) = A'(5, 3)

B'(3+4, 1+2) = B'(7, 3)

C'(2+4, 3+2) = C'(6, 5)

The translated triangle A'B'C' has vertices A'(5,3), B'(7,3), and C'(6,5).

Example 2: Reflection

Reflect square DEFG, with vertices D(1,1), E(4,1), F(4,4), and G(1,4), over the y-axis.

Solution:*

Reflecting over the y-axis changes the sign of the x-coordinate, but the y-coordinate remains the same.

D'(-1, 1)

E'(-4, 1)

F'(-4, 4)

G'(-1, 4)

The reflected square D'E'F'G' has vertices D'(-1,1), E'(-4,1), F'(-4,4), and G'(-1,4).