Class 6 – Mathematics
Learn Class 6 Mathematics through 30 carefully organised topics covering integers, algebra, equations, ratios, proportions, sets, geometry, mensuration, coordinates, data handling, probability, patterns, logical reasoning and real-life mathematical applications.
🔢 Integers and Number Operations
Understanding Integers and the Number Line
Integers are whole numbers that include positive numbers, negative numbers and zero. They help us represent quantities above and below a reference point.
…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …
On a number line, numbers increase as we move to the right and decrease as we move to the left.
Draw a number line and mark −5, −2, 0, 3 and 6.
Zero is neither positive nor negative.
Comparing and Ordering Positive and Negative Integers
Integers can be compared by looking at their positions on the number line. A number located farther to the right is greater than a number located to its left.
5 > 2
−2 > −6
0 > −4
Arrange −7, 3, 0, −2 and 5 in ascending order.
Among negative integers, the number closer to zero is greater.
Performing Addition of Integers
Integer addition involves combining positive and negative quantities. A number line can help us understand the direction of movement.
5 + 3 = 8
5 + (−3) = 2
−4 + (−3) = −7
Calculate: −6 + 9.
When integers have the same sign, add their magnitudes and keep the common sign.
Performing Subtraction of Integers
Subtracting an integer can be understood as adding its opposite. This rule makes integer subtraction easier to calculate.
7 − 3 = 4
5 − (−2) = 5 + 2 = 7
−3 − 4 = −7
Calculate: −8 − (−5).
Subtracting a negative integer is equivalent to adding the corresponding positive integer.
🔠 Algebra and Equations
Introduction to Algebraic Variables and Constants
A variable is a symbol, usually a letter, that can represent different numbers. A constant has a fixed numerical value.
In x + 5:
x = variable
5 = constant
Identify the variable and constant in y + 12.
Variables allow mathematics to describe general relationships.
Forming Algebraic Expressions
An algebraic expression combines numbers, variables and mathematical operations such as addition, subtraction, multiplication and division.
Three more than x → x + 3
Five times y → 5y
Seven less than n → n − 7
Write an expression for “four times a number plus two”.
Read verbal statements carefully before translating them into algebraic expressions.
Evaluating Simple Algebraic Expressions
Evaluating an expression means finding its numerical value after replacing its variable with a given number.
If x = 4, find 3x + 2.
3(4) + 2 = 12 + 2 = 14
Find 2a + 5 when a = 6.
Substitute the given value first and then perform the operations.
Understanding Simple Mathematical Equations
An equation states that two mathematical expressions have equal values. The equals sign separates the two sides.
x + 4 = 9
The value x = 5 makes this statement true.
Which value makes y + 3 = 10 true?
An equation contains an equality sign, while an expression does not necessarily contain one.
Solving One-Step Equations
A one-step equation can usually be solved by applying one inverse operation to isolate the unknown variable.
x + 6 = 15
x = 15 − 6
x = 9
Solve: y − 7 = 12.
Whatever operation you perform on one side of an equation must preserve the equality.
⚖️ Ratio, Proportion and Sets
Understanding Ratios
A ratio compares two quantities of the same kind. It shows how much of one quantity exists compared with another.
If a basket contains 6 apples and 3 oranges:
Apples : Oranges = 6 : 3 = 2 : 1
Simplify the ratio 12 : 8.
Compare quantities using compatible units.
Understanding Proportions
A proportion states that two ratios are equivalent. Proportions are useful when quantities change at the same relative rate.
2 : 3 = 4 : 6
Both ratios simplify to the same value.
Determine whether 3 : 5 and 9 : 15 form a proportion.
Equivalent ratios represent the same comparison.
Applying the Unitary Method
The unitary method first finds the value of one unit and then uses that value to determine the required number of units.
5 notebooks cost ₹100.
Cost of 1 notebook = ₹100 ÷ 5 = ₹20
Cost of 8 notebooks = 8 × ₹20 = ₹160
If 4 pens cost ₹60, find the cost of 10 pens.
Find the value of one unit before calculating the value of several units.
Introduction to Sets and Set Representation
A set is a well-defined collection of distinct objects. Objects belonging to a set are called its elements.
A = {2, 4, 6, 8}
The set A contains four even numbers.
Write the set of the first five positive odd numbers.
Curly brackets { } are commonly used when listing the elements of a set.
📐 Geometry and Constructions
Understanding Basic Geometrical Ideas
Geometry studies shapes, sizes, positions and relationships in space. Its basic ideas include points, lines, line segments, rays, curves and angles.
A point represents a position.
A line extends endlessly in both directions.
A line segment has two endpoints.
A ray has one endpoint.
Draw and label a line, ray and line segment.
A line, ray and line segment have different endpoint properties.
Exploring Parallel and Perpendicular Lines
Parallel lines remain the same distance apart and do not intersect. Perpendicular lines intersect to form right angles.
Opposite edges of a rectangular notebook are parallel.
Adjacent edges meet at right angles and are perpendicular.
Find two examples of parallel lines and two examples of perpendicular lines in your classroom.
Perpendicular lines form an angle of 90°.
Understanding Complementary and Supplementary Angles
Two angles are complementary when their measures add to 90°. Two angles are supplementary when their measures add to 180°.
30° + 60° = 90° → Complementary
110° + 70° = 180° → Supplementary
Find the complement of 35° and the supplement of 125°.
Complementary → 90°
Supplementary → 180°
Constructing Angles Using a Protractor
A protractor is used to measure and construct angles accurately. Most school protractors display measurements from 0° to 180°.
To construct 60°:
1. Draw a ray.
2. Place the protractor centre on its endpoint.
3. Mark 60°.
4. Draw another ray through the mark.
Construct angles of 45°, 75° and 120°.
Align the baseline of the protractor correctly before measuring.
Constructing Line Segments and Perpendiculars
Geometrical construction uses instruments such as a ruler and compass to create accurate line segments and perpendicular lines.
A line segment AB of length 6 cm can be drawn using a ruler. A perpendicular through a point can be constructed using a compass and straightedge.
Draw AB = 7 cm and construct a perpendicular through its midpoint.
Accuracy is important in geometrical constructions.
Understanding Polygons and Their Classification
A polygon is a closed plane figure made from straight line segments. Polygons can be classified according to their number of sides.
3 sides → Triangle
4 sides → Quadrilateral
5 sides → Pentagon
6 sides → Hexagon
8 sides → Octagon
Name polygons having 7 and 10 sides.
A polygon must be closed and its sides must be straight line segments.
Exploring Three-Dimensional Geometrical Nets
A net is a two-dimensional arrangement of faces that can be folded to form a three-dimensional solid.
A cube has six square faces. Certain arrangements of six connected squares can be folded to form a cube.
Draw one possible net of a cube and identify its six faces.
Not every arrangement of six squares forms a valid cube net.
📏 Mensuration and Coordinates
Introduction to Mensuration
Mensuration deals with measurements of geometrical figures, including length, perimeter, area and, at later stages, volume.
Rectangle of length 8 cm and breadth 5 cm:
Perimeter = 2 × (8 + 5) = 26 cm
Area = 8 × 5 = 40 cm²
Find the perimeter and area of a rectangle measuring 10 cm × 4 cm.
Perimeter measures boundary length, while area measures surface covered.
Calculating Areas of Triangles
The area of a triangle depends on its base and the perpendicular height corresponding to that base.
Base = 8 cm
Height = 5 cm
Area = ½ × 8 × 5 = 20 cm²
Find the area of a triangle with base 12 cm and height 7 cm.
Use the perpendicular height corresponding to the chosen base.
Understanding Coordinate Points Informally
Coordinates describe the position of a point using an ordered pair. The first number tells the horizontal position and the second tells the vertical position.
Point A(3, 2) means move 3 units horizontally and 2 units vertically from the reference origin.
Draw a simple coordinate grid and plot A(2,3), B(4,1) and C(5,4).
Coordinates are written in order: horizontal coordinate first, vertical coordinate second.
📊 Data Handling and Probability
Organising Data Using Frequency Tables
A frequency table organises data by showing how often each value or category occurs.
Scores: 2, 3, 2, 4, 3, 2
2 → 3 times
3 → 2 times
4 → 1 time
Create a frequency table for:
1, 2, 3, 2, 4, 1, 2, 3, 3, 2
Frequency means the number of times a value occurs.
Representing Data Using Pictorial Graphical Methods
Pictorial representations use symbols or pictures to display data visually. A key explains how much each symbol represents.
If ★ represents 5 students:
★★★ represents 15 students.
Create a pictograph showing the number of books read by four students. Choose a suitable symbol and key.
Always read the pictograph key before interpreting the symbols.
Introduction to Double Bar Graphs
A double bar graph displays two related sets of data side by side, making comparison easier.
A double bar graph could compare the number of books read by Class A and Class B during January, February and March.
Collect the marks of two students in four subjects and represent them using a double bar graph.
Use the same numerical scale for both data sets.
Understanding Basic Probability Through Experiments
Probability describes how likely an event is to occur. Simple experiments with coins, dice and coloured objects help us understand chance.
When a fair coin is tossed, the possible outcomes are:
Head or Tail.
Each outcome has an equal chance in the ideal model.
Toss a coin 20 times. Record the number of heads and tails. Compare your experimental results.
Experimental results may not be perfectly equal in a small number of trials.
🧠 Patterns, Reasoning and Applications
Recognising Algebraic Number Patterns
Number patterns follow rules. Algebra helps us express these rules in a general form rather than listing every number individually.
3, 6, 9, 12, 15, …
Each term is 3 times its position.
A simple rule is 3n.
Study 5, 10, 15, 20, … and write a rule for the nth term.
Look for how the terms change before deciding the rule.
Solving Mathematical Problems Using Logical Reasoning
Logical reasoning involves analysing the given information, identifying relationships and following valid steps to reach a conclusion.
A number is greater than 20, less than 30, even and divisible by 3.
Possible even numbers: 22, 24, 26, 28
Only 24 is divisible by 3.
Therefore, the number is 24.
A number is between 30 and 40. It is odd and divisible by 5. What is the number?
Use every condition given in the problem before choosing your answer.
Applying Mathematics to Everyday Situations
Mathematics is used in shopping, budgeting, travel, measurement, cooking, construction, time management and many other everyday activities.
A student buys 3 notebooks costing ₹35 each and 2 pens costing ₹15 each.
Notebook cost = 3 × ₹35 = ₹105
Pen cost = 2 × ₹15 = ₹30
Total = ₹135
You have ₹500. You buy a book for ₹225 and stationery for ₹145. How much money remains?
Real-life mathematical problems often require more than one operation. Identify the information and required result before calculating.