Mathematics Class 8: Algebra Concepts

Prime Factor Decomposition

• Finding the prime numbers that multiply together to make a given number
• A prime number can only be divided by 1 and itself.
• Example: 24 = 2 x 2 x 2 x 3

Laws of Indices

• Explain how to multiply and divide numbers with powers
• Example: x^m x x^n = x^(m+n)
• Example: x^m / x^n = x^(m-n)

Powers of 10

• Used to express very large or very small numbers in a more concise way
• Uses powers of 10 to represent the number of digits in a number
• Example: 1000 = 10^3, 0.01 = 10^-2

Rounding and Estimating

• Approximating a number to the nearest whole number or decimal place
• Used for making calculations easier
• Example: 12.34 rounded to the nearest whole number is 12

Simplifying Expressions

• Combining like terms and using laws of indices
• Goal: to make the expression easier to understand
• Example: 2x + 3x - 5 = 5x - 5

Expanding and Factorising

• Expanding brackets: Multiplying each term inside the brackets by the term outside the brackets
• Factorising: Writing an expression as a product of two or more factors
• Example: Expanding (x + 2)(x - 1) = x^2 + x - 2
• Example: Factorising x^2 + 5x + 6 = (x + 2)(x + 3)

Substituting

• Replacing variables in an expression with given values
• Evaluates the expression to find the answer
• Example: If x = 2, then 3x + 1 = 3(2) + 1 = 7

Solving

• Finding the value of a variable that makes an equation true.
• Isolate the variable on one side of the equation by performing the same operation on both sides
• Example: To solve for x in 2x + 3 = 7, we can subtract 3 from both sides to get 2x = 4, then divide both sides by 2 to get x = 2

Solving One-Step Equations

• Equations that require just one arithmetic step
• Example: x + 5 = 8; Subtract 5 from both sides to get x = 3

Solving Two-Step Equations

• Equations that require two arithmetic steps in a row
• Example: 2x + 5 = 11. Subtract 5 from both sides to get 2x = 6, then divide both sides by 2 to get x = 3

More Complex Equations

• Equations involving more than one variable
• Involve rearranging to solve for specific variables.
• Example: To solve for y in 2x + 3y = 12, subtract 2x from both sides to get 3y = 12 - 2x, then divide both sides by 3 to get y = (12 - 2x) / 3

Angles in Polygons

• The sum of the interior angles of a polygon can be found using the formula (n-2) x 180 degrees, where 'n' is the number of sides
• Example: The sum of the interior angles of a hexagon (6 sides) is (6-2) x 180 = 720 degrees

Triangles, Parallelograms and Trapeziums

• Triangles: Have three sides and three angles
• Parallelograms: Have four sides with two sets of parallel sides
• Trapeziums: Have four sides where only one pair of sides is parallel
• Example: The area of a triangle is (1/2) x base x height

Perimeter and Area of Compound Shapes

• Perimeter is the total length of the outside edges of a shape
• Area is the amount of space inside a shape
• Compound shapes can be divided into simpler shapes to calculate their area and perimeter
• Example: A compound shape made up of a rectangle and a semicircle can have its area calculated by adding the area of the rectangle and the area of the semicircle

Surface Area

• The total area of all the faces of a 3-dimensional object
• Example: The surface area of a cube with side length 's' is 6s^2

Volume

• The amount of space a 3-dimensional object takes up.
• Example: The volume of a rectangular prism is length x width x height