Integers
Addition of Integers
Same sign: Add absolute values, keep the sign.
Different signs: Subtract absolute values, keep the sign of larger absolute value.
Examples:
1. 5 + 3 = 8
2. -5 + (-3) = -8
3. 5 + (-3) = 2
4. -5 + 3 = -2
Subtraction of Integers
Change subtraction to addition, change sign of the second number.
Examples:
1. 5 – 3 = 2
2. -5 – (-3) = -2
3. 5 – (-3) = 8
4. -5 – 3 = -8
Fractions and Decimals
Addition and Subtraction of Fractions
Make denominators same, then add or subtract numerators.
Examples:
1. 1/4 + 1/4 = 1/2
2. 1/3 – 1/6 = 1/6
Multiplication of Fractions
Multiply numerators, multiply denominators.
Example:
1. 2/3 × 3/4 = 1/2
Division of Fractions
Multiply by the reciprocal of the divisor.
Example:
1. 2/3 ÷ 3/4 = 8/9
Addition and Subtraction of Decimals
Align decimal points and add or subtract.
Example:
1. 3.25 + 1.75 = 5.00
2. 5.50 – 2.25 = 3.25
Data Handling
Mean (Average)
Mean = Sum of observations / Number of observations
Example:
Data set: 4, 8, 6, 5, 3
1. Sum = 26
2. Count = 5
3. Mean = 26 / 5 = 5.2
Median
Middle value in an ordered data set.
Example:
Data set: 3, 5, 6, 8, 9
Median = 6
Mode
Value that occurs most frequently.
Example:
Data set: 4, 4, 5, 6, 6, 6, 7
Mode = 6
Range
Range = Maximum value – Minimum value
Example:
Data set: 3, 5, 6, 8, 9
Range = 9 – 3 = 6
Simple Equations
Solving Simple Equations
Isolate the variable.
Example:
1. x + 5 = 12
x = 12 – 5
x = 7
2. 3x = 15
x = 15 / 3
x = 5
Lines and Angles
Complementary Angles
Sum is 90 degrees.
Supplementary Angles
Sum is 180 degrees.
Vertically Opposite Angles
Opposite angles are equal when two lines intersect.
Adjacent Angles
Share a common arm and vertex.
The Triangle and its Properties
Sum of Angles in a Triangle
Sum is always 180 degrees.
Pythagorean Theorem
In a right-angled triangle, (Hypotenuse)² = (Base)² + (Height)²
Congruence of Triangles
Criteria for Congruence of Triangles
1. SSS (Side-Side-Side)
2. SAS (Side-Angle-Side)
3. ASA (Angle-Side-Angle)
4. RHS (Right angle-Hypotenuse-Side)
Comparing Quantities
Ratio
Ratio of a to b = a/b
Proportion
a/b = c/d implies a × d = b × c
Percentage
Percentage = (Part / Whole) × 100
Profit and Loss
Profit = Selling Price – Cost Price
Loss = Cost Price – Selling Price
Simple Interest
Simple Interest = (Principal × Rate × Time) / 100
Rational Numbers
Standard Form of a Rational Number
Numerator and denominator have no common factors other than 1.
Addition and Subtraction of Rational Numbers
Make denominators the same, then add or subtract numerators.
Multiplication of Rational Numbers
Multiply numerators, multiply denominators.
Division of Rational Numbers
Multiply by the reciprocal of the divisor.
Practical Geometry
Construction of Parallel Lines
Using a ruler and a set square to draw a line parallel to a given line through a point not on the line.
Construction of Triangles
Using a ruler and compass to construct a triangle given its sides or angles.
Perimeter and Area
Perimeter of a Rectangle
2 × (Length + Breadth)
Perimeter of a Square
4 × Side
Area of a Rectangle
Length × Breadth
Area of a Square
Side × Side
Area of a Triangle
(1/2) × Base × Height
Area of a Parallelogram
Base × Height
Area of a Circle
π × Radius²
Circumference of a Circle
2 × π × Radius
Algebraic Expressions
Terms, Factors, and Coefficients
An algebraic expression is a combination of terms separated by + or – signs. Each term is a product of factors, and each factor has a coefficient.
Like and Unlike Terms
Like terms have the same variables raised to the same power. Unlike terms do not.
Monomials, Binomials, and Polynomials
Monomial: An expression with one term.
Binomial: An expression with two terms.
Polynomial: An expression with more than two terms.
Addition and Subtraction of Algebraic Expressions
Combine like terms to add or subtract algebraic expressions.
Exponents and Powers
Multiplying Powers with the Same Base
aᵐ × aⁿ = aᵐ⁺ⁿ
Dividing Powers with the Same Base
aᵐ / aⁿ = aᵐ⁻ⁿ
Power of a Power
(aᵐ)ⁿ = aᵐ×ⁿ
Multiplying Powers with the Same Exponent
aᵐ × bᵐ = (a × b)ᵐ
Dividing Powers with the Same Exponent
aᵐ / bᵐ = (a / b)ᵐ
Negative Exponents
a⁻ᵐ = 1 / aᵐ
Symmetry
Lines of Symmetry
A line that divides a shape into two identical parts.
Rotational Symmetry
A shape has rotational symmetry if it looks the same after a certain amount of rotation (less than 360 degrees).
Visualising Solid Shapes
Faces, Edges, and Vertices
Faces: The flat surfaces of a 3D shape.
Edges: The line segments where faces meet.
Vertices: The points where edges meet.
Nets of Solid Shapes
A net is a two-dimensional shape that can be folded to form a three-dimensional shape.
