Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Geometry ↓


Symmetry and Transformations


Geometry is a fascinating area of mathematics that deals with the properties of shape, size, and space. In this lengthy exploration, we'll dive deep into one of the exciting topics of geometry: symmetry and transformation. This topic helps students understand how shapes look and change and plays an important role in both mathematics and the world around us.

Understanding symmetry

Symmetry occurs when one part of an object is a mirror image or exact replica of another part. Symmetry exists all around us, from butterfly wings to the structure of snowflakes. Let's explore the different types of symmetry.

Line symmetry

Line symmetry, also known as reflection symmetry, occurs when an object can be divided into two equal parts by a line. The line is called the line of symmetry. For example, human faces often have line symmetry. Let's explore this concept through an example:

Consider a square. If we draw a line through the middle, whether horizontally or vertically, each half is a mirror image of the other.

The red line is the line of symmetry. The left and right sides of the square are equal.

Rotational symmetry

Rotational symmetry is when a shape can be rotated around a central point (less than a perfect circle) and still look the same. A common example of this is a playing card. If you rotate it halfway (180 degrees), it still looks the same card.

The arrows show that no matter how you rotate this shape 90 degrees around the center, it looks the same.

Point symmetry

Point symmetry occurs when every part of an object has a corresponding part the same distance from a central point but on opposite sides. An example of this is the letter "S." The center creates point symmetry where the upper and lower parts mirror each other.

Imagine placing a pencil at a point and rotating it 180 degrees. If the shape or object looks the same, it has point symmetry.

Transformations

Transformation means to change the position, size, or shape of an object. In geometry, understanding transformation is important because it helps to manipulate shapes and analyze their properties. There are several types of transformation:

Translation

Translation means moving every point of an object the same distance in the same direction. Think of it as sliding without rotating or flipping. Imagine sliding a piece of paper across a table; every part of the paper moves the same amount.

The blue square has been moved or shifted to the new position indicated by the red dashed square.

Rotation

Rotation means turning a figure around a fixed point, called the center of rotation. The amount of rotation is measured in degrees. For example, rotating a figure 90 degrees clockwise is called a rotation.

The purple triangle is rotated 90 degrees around the black point to reach the position of the orange dashed triangle.

Reflection

Reflection is the flipping of a figure along a line, producing a mirror image of the original figure. The line is called the reflection line. It is like looking at your own reflection in a mirror.

The red dashed rectangle is created by placing the green rectangle over the blue line.

Scaling

Scaling, or stretching, changes the size of a shape. This can be either enlargement or reduction. The shape will retain its proportions, but its size will change.

The navy circle has been enlarged to become a larger teal dashed circle.

Examples and exercises

Example 1: Line symmetry

Find the line of symmetry for the letter "A".

In the capital letter "A", the line symmetry is vertical. If we draw a vertical line in the center, both the left and right sides mirror each other.

Example 2: Rotational symmetry

Identify whether the letter "E" has rotational symmetry.

The capital letter "E" does not have rotational symmetry. It will not look the same if you rotate it in any way other than a full 360-degree rotation.

Example 3: Transformation

Move triangle ABC 5 units to the right and 3 units up. If the coordinates of triangle ABC are as follows:

A(1, 2), B(3, 4), C(2, 6)

The new coordinates after translation will be:

A'(1+5, 2+3), B'(3+5, 4+3), C'(2+5, 6+3)
A'(6, 5), B'(8, 7), C'(7, 9)

Closing thoughts

Understanding symmetry and transformations allows you to see mathematical relationships and patterns in the world. Whether in art, nature or everyday objects, recognizing symmetry and transformations helps you appreciate the beauty of geometry. Practice by observing the world around you, solving problems and discovering new shapes and transformations. This foundation will enhance your spatial reasoning and analytical skills.


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