Class 8 – Mathematics
Master 30 Class 8 Mathematics topics covering real numbers, squares and cubes, exponents, scientific notation, algebra, factorisation, linear equations, coordinate geometry, polygons, three-dimensional solids, compound interest, growth and depreciation, pie charts, histograms and data interpretation.
🔢 Real Numbers, Squares and Cubes
Understanding Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a ratio p/q of two integers, where q is not zero. Their decimal expansions are non-terminating and non-repeating.
√2, √3 and π are irrational numbers.
√2 ≈ 1.414213…
Identify which are irrational: 1/2, √5, 0.75, π and 4.
Not every square root is irrational. For example, √9 = 3.
Introduction to the Real Number System
Real numbers include both rational and irrational numbers. They can be represented as points on the number line.
−5, 0, 3/4, 2.6, √2 and π are all real numbers.
Classify −3, 2/5, √7 and 6.25 as rational or irrational.
Real Numbers = Rational Numbers + Irrational Numbers.
Understanding Properties of Rational Numbers
Rational numbers follow important arithmetic properties such as closure, commutativity and associativity for suitable operations, along with distributivity.
Commutative property of addition:
2/3 + 1/4 = 1/4 + 2/3
Distributive property:
a(b + c) = ab + ac
Verify the commutative property of multiplication using 2/5 and 3/7.
Subtraction and division are not commutative.
Understanding Squares and Square Numbers
The square of a number is obtained by multiplying the number by itself. Numbers that are squares of integers are called perfect squares.
4² = 16
7² = 49
12² = 144
Find the squares of 13, 15 and 20.
1, 4, 9, 16, 25, 36, 49, 64 and 81 are some perfect squares.
Finding Square Roots
A square root of a number is a value that produces the original number when multiplied by itself.
√64 = 8 because 8 × 8 = 64.
√144 = 12 because 12 × 12 = 144.
Find √81, √121 and √225.
For a positive number, the symbol √ usually denotes its principal, non-negative square root.
Understanding Cubes and Cube Numbers
The cube of a number is obtained by multiplying the number by itself three times.
2³ = 2 × 2 × 2 = 8
4³ = 64
5³ = 125
Find the cubes of 6, 8 and 10.
A perfect cube is the cube of an integer.
Finding Cube Roots
The cube root of a number is the value which, when multiplied by itself three times, gives the original number.
∛27 = 3 because 3³ = 27.
∛125 = 5 because 5³ = 125.
Find ∛64, ∛216 and ∛1000.
Cube roots can also be negative: ∛(−8) = −2.
⚡ Exponents and Scientific Notation
Understanding Laws of Exponents
Laws of exponents make calculations involving powers easier when the bases are the same.
am × an = am+n
am ÷ an = am−n, a ≠ 0
(am)n = amn
Simplify 2³ × 2⁵ and 5⁷ ÷ 5³.
These exponent rules require careful attention to the bases and the conditions under which the expressions are defined.
Working with Negative and Zero Exponents
A non-zero number raised to the power zero equals 1. A negative exponent represents the reciprocal of the corresponding positive power.
5⁰ = 1
2−3 = 1/2³ = 1/8
Simplify 7⁰, 3−2 and 10−3.
For a ≠ 0, a⁰ = 1 and a−n = 1/an.
Understanding Scientific Notation
Scientific notation is a compact way to write very large or very small numbers using powers of 10.
5,600,000 = 5.6 × 10⁶
0.00032 = 3.2 × 10−4
Write 7,500,000 and 0.000045 in scientific notation.
In standard scientific notation, the coefficient is at least 1 and less than 10.
🔠 Algebra, Identities and Factorisation
Multiplying Algebraic Expressions
Algebraic expressions are multiplied by applying multiplication to coefficients and variables while using the distributive property when brackets are involved.
3x × 4x = 12x²
2x(x + 5) = 2x² + 10x
Expand 3a(2a + 4).
Multiply every required term inside the bracket.
Understanding Algebraic Identities
An algebraic identity is an equality that is true for all values of the variables for which both sides are defined.
(a + b)² = a² + 2ab + b²
Expand (x + 4)² using the identity.
An identity is different from an equation that is true only for particular variable values.
Applying Standard Algebraic Identities
Standard identities allow algebraic expressions and numerical calculations to be expanded or simplified efficiently.
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
Use an identity to expand (x − 5)².
Choose the identity that matches the structure of the expression.
Introduction to Factorisation
Factorisation is the process of rewriting an expression as a product of simpler factors. It reverses the process of expansion.
6x + 12
= 6(x + 2)
Factorise 8x + 16.
Look for a common factor before using more advanced methods.
Factorising Algebraic Expressions
Algebraic expressions can be factorised using common factors and, for suitable expressions, standard identities.
x² − 9
= x² − 3²
= (x + 3)(x − 3)
Factorise x² − 25.
a² − b² = (a + b)(a − b).
Solving Linear Equations with Variables on Both Sides
When a variable appears on both sides of an equation, equivalent operations can be used to collect variable terms on one side and constant terms on the other.
5x + 3 = 2x + 18
5x − 2x = 18 − 3
3x = 15
x = 5
Solve 7x − 4 = 3x + 20.
Preserve equality by performing valid equivalent operations.
📈 Linear Relationships and Coordinate Geometry
Understanding Graphs of Linear Relationships
A linear relationship between two variables can be represented graphically by a straight line. Each point on the line represents a pair of related values.
For y = 2x:
x = 1 → y = 2
x = 2 → y = 4
x = 3 → y = 6
The points (1,2), (2,4) and (3,6) lie on the same straight line.
Create a value table for y = 3x when x = 0, 1, 2, 3 and 4.
A linear graph has a constant rate of change.
Plotting Points on the Cartesian Plane
The Cartesian plane consists of a horizontal x-axis and vertical y-axis that meet at the origin.
To plot A(3,2), move 3 units to the right from the origin and then 2 units upward.
Plot A(2,3), B(−2,4), C(−3,−2) and D(4,−1).
Coordinates are written as (x,y), not (y,x).
Understanding Cartesian Coordinates
The coordinate plane is divided into four quadrants. The signs of the x- and y-coordinates depend on the quadrant containing the point.
Quadrant I → (+,+)
Quadrant II → (−,+)
Quadrant III → (−,−)
Quadrant IV → (+,−)
Identify the quadrant containing each point:
(4,5), (−3,2), (−5,−4), (6,−2).
A point on an axis is not located inside any quadrant.
📐 Polygons, Quadrilaterals and Solid Geometry
Understanding Properties of Polygons
A polygon is a closed two-dimensional figure formed from straight line segments. Polygons are classified according to their number of sides and other properties.
Triangle → 3 sides
Quadrilateral → 4 sides
Pentagon → 5 sides
Hexagon → 6 sides
Name polygons having 7, 8, 9 and 10 sides.
A regular polygon has all sides equal and all interior angles equal.
Understanding Interior and Exterior Angles of Polygons
The interior-angle sum of a polygon depends on its number of sides. For a convex polygon with n sides, the sum is (n − 2) × 180°.
Pentagon:
(5 − 2) × 180° = 540°
The sum of one exterior angle at each vertex of a convex polygon, taken in the same direction, is 360°.
Find the interior-angle sum of a hexagon and an octagon.
Interior sum = (n − 2) × 180°.
Exploring Properties of Special Quadrilaterals
Different quadrilaterals have special properties involving their sides, angles, parallel sides and diagonals.
Square → 4 equal sides and 4 right angles
Rectangle → opposite sides equal and 4 right angles
Rhombus → 4 equal sides
Parallelogram → opposite sides parallel and equal
Compare the properties of a square, rectangle and rhombus.
A square is both a rectangle and a rhombus because it satisfies the defining properties of both.
Understanding Euler’s Formula for Polyhedra
For many familiar convex polyhedra, the numbers of vertices, edges and faces are connected by Euler’s polyhedron formula.
Vertices (V) = 8
Edges (E) = 12
Faces (F) = 6
V − E + F = 8 − 12 + 6 = 2
A tetrahedron has 4 vertices, 6 edges and 4 faces. Verify Euler’s formula.
For convex polyhedra: V − E + F = 2.
Calculating Surface Area of Three-Dimensional Solids
Surface area measures the total area covering the outer surfaces of a three-dimensional solid.
Side = 5 cm
Total surface area = 6 × 5²
= 6 × 25
= 150 cm²
Find the total surface area of a cube with side 8 cm.
Surface area is measured in square units.
Calculating Volume of Cylinders
The volume of a cylinder is found by multiplying the area of its circular base by its perpendicular height.
Radius = 7 cm
Height = 10 cm
Volume = πr²h
= 22/7 × 7 × 7 × 10
= 1540 cm³
Find the volume of a cylinder with radius 3 cm and height 8 cm, using π ≈ 3.14.
Volume is measured in cubic units.
💰 Compound Interest, Growth and Depreciation
Understanding Compound Interest
Compound interest is calculated on the original principal and on interest that has already been added during previous compounding periods.
Principal = ₹10,000
Rate = 10% per year
Time = 2 years
After Year 1 = ₹11,000
After Year 2 = ₹12,100
Compound Interest = ₹12,100 − ₹10,000
= ₹2,100
Find the amount on ₹5,000 at 10% per year compounded annually for 2 years.
For annual compounding, Amount = P(1 + r/100)n.
Solving Problems Involving Growth and Depreciation
Repeated percentage increase can model growth, while repeated percentage decrease can model depreciation. Each new percentage change is applied to the updated value.
A machine worth ₹50,000 depreciates by 10% in one year.
Depreciation = ₹5,000
New value = ₹45,000
A population of 20,000 grows by 5% in one year. Find the population after one year.
For repeated changes, calculate each percentage from the current value rather than always from the original value.
📊 Data Representation and Interpretation
Representing Data Using Pie Charts
A pie chart represents a whole using sectors of a circle. The size of each sector is proportional to the category it represents.
If 25% of students prefer Mathematics:
Sector angle = 25/100 × 360°
= 90°
In a survey, 40% prefer cricket, 30% football, 20% badminton and 10% other sports. Find the angle of each pie-chart sector.
All sectors together represent 100% or 360°.
Understanding Histograms
A histogram represents continuous grouped numerical data using adjacent rectangular bars. For equal class widths, the height of each bar can represent frequency.
Height interval | Frequency
120–130 | 4
130–140 | 7
140–150 | 10
150–160 | 6
Create a histogram from a grouped frequency table containing four equal-width class intervals.
Unlike an ordinary bar graph for separate categories, histogram bars normally touch because the intervals represent continuous data.
Interpreting and Comparing Different Data Representations
The same data can often be represented using tables, bar graphs, double bar graphs, pie charts, line graphs or histograms. The best representation depends on the type of data and the comparison being made.
A pie chart is useful for showing parts of a whole.
A bar graph is useful for comparing categories.
A line graph is useful for showing change over time.
A histogram is useful for grouped continuous numerical data.
Choose a suitable representation for:
1. Monthly temperature over one year
2. Favourite sports of students
3. Percentage distribution of a household budget
4. Grouped heights of 100 students
Do not choose a graph only because it looks attractive. Choose the representation that communicates the data accurately and clearly.