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School (NCERT)SEM-1Mathematics

Class 6 Mathematics — Complete Notes with NCERT Solutions & Practice

✍️ WohoTech Team📅 Last Updated: 2026-03-11📄 36 pages · 1.2 MB

Class 6 Mathematics — Complete Study Notes

Class 6 Mathematics builds the foundation for all higher mathematics. This chapter covers all NCERT topics with solved examples, key formulas, and practice questions to help you score full marks.

Chapter 1: Knowing Our Numbers

1.1 Large Numbers — Indian Number System

| Period | Crores | Lakhs | Thousands | Ones | |--------|--------|-------|-----------|------| | Places | Crore, Ten Crore | Lakh, Ten Lakh | Thousand, Ten Thousand | Ones, Tens, Hundreds |

Reading Large Numbers (Indian System):

7,32,45,610 = Seven crore thirty-two lakh forty-five thousand six hundred ten

International System:

Millions, Billions instead of Lakhs, Crores
1 Lakh = 100,000 (1 hundred thousand)
1 Crore = 10,000,000 (10 million)

1.2 Estimation and Rounding

| Rounding Rule | Example | |---------------|---------| | Nearest 10 | 47 → 50; 43 → 40 | | Nearest 100 | 274 → 300; 234 → 200 | | Nearest 1000 | 7654 → 8000; 7234 → 7000 |

Rule: Look at digit to the right of rounding place. If ≥ 5, round up. If < 5, round down.

Chapter 2: Whole Numbers

Natural Numbers: 1, 2, 3, 4, ... (counting numbers)
Whole Numbers: 0, 1, 2, 3, 4, ... (natural numbers + zero)

Properties of Whole Numbers

| Property | Addition | Multiplication | |----------|----------|---------------| | Closure | a + b is always a whole number | a × b is always a whole number | | Commutative | a + b = b + a | a × b = b × a | | Associative | (a+b)+c = a+(b+c) | (a×b)×c = a×(b×c) | | Identity | a + 0 = a (0 is additive identity) | a × 1 = a (1 is multiplicative identity) | | Distributive | a × (b+c) = a×b + a×c | |

Number Line: Numbers increase moving right. Zero is between positive and negative numbers.

Chapter 3: Playing with Numbers

3.1 Factors and Multiples

Factor: A number that divides another exactly
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Multiple: A number obtained by multiplying
- Multiples of 7: 7, 14, 21, 28, 35...

Divisibility Rules:

| Divisible by | Rule | Example | |-------------|------|---------| | 2 | Last digit is 0, 2, 4, 6, 8 | 348 ✓ | | 3 | Sum of digits divisible by 3 | 123 → 1+2+3=6 ✓ | | 4 | Last two digits divisible by 4 | 316 → 16÷4=4 ✓ | | 5 | Last digit is 0 or 5 | 75 ✓ | | 6 | Divisible by both 2 and 3 | 132 ✓ | | 8 | Last three digits divisible by 8 | 1,024 → 024÷8=3 ✓ | | 9 | Sum of digits divisible by 9 | 729 → 7+2+9=18 ✓ | | 10 | Last digit is 0 | 150 ✓ | | 11 | Difference of alternating digit sums divisible by 11 | 121 → 1+1−2=0 ✓ |

3.2 Prime and Composite Numbers

Prime: Has exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...
Composite: Has more than 2 factors. Examples: 4, 6, 8, 9, 10, 12...
1 is neither prime nor composite
2 is the only even prime number

Sieve of Eratosthenes — Method to find all primes up to a given number by eliminating multiples.

3.3 HCF (Highest Common Factor) and LCM

HCF (Greatest Common Divisor):

Largest factor common to two or more numbers
Method 1 — Prime Factorization: find common prime factors, multiply
Method 2 — Division Method: divide larger by smaller, remainder becomes divisor, repeat

HCF of 36 and 48:
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
HCF = 2 × 2 × 3 = 12

LCM (Least Common Multiple):

Smallest number divisible by both numbers
LCM by prime factorization: take ALL prime factors with highest powers

LCM of 4 and 6:
4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 12

Important Relationship: HCF × LCM = Product of two numbers

Chapter 4: Basic Geometrical Ideas

4.1 Points, Lines, and Angles

Point: A specific location, has no dimension. Represented by a dot (·)
Line: Extends infinitely in both directions, has no endpoints
Line Segment: Has two endpoints, fixed length
Ray: Has one endpoint, extends in one direction

4.2 Types of Angles

| Angle Type | Measure | |-----------|---------| | Acute | Between 0° and 90° | | Right | Exactly 90° | | Obtuse | Between 90° and 180° | | Straight | Exactly 180° | | Reflex | Between 180° and 360° | | Complete | Exactly 360° |

4.3 Polygons and Their Properties

| Polygon | Sides | Sum of Interior Angles | |---------|-------|----------------------| | Triangle | 3 | 180° | | Quadrilateral | 4 | 360° | | Pentagon | 5 | 540° | | Hexagon | 6 | 720° |

Formula: Sum of interior angles = (n − 2) × 180°, where n = number of sides

Chapter 5: Understanding Elementary Shapes

5.1 Triangles Classification

By Sides:

Equilateral: all 3 sides equal
Isosceles: 2 sides equal
Scalene: all 3 sides different

By Angles:

Acute-angled: all angles < 90°
Right-angled: one angle = 90°
Obtuse-angled: one angle > 90°

5.2 Quadrilaterals

| Shape | Properties | |-------|-----------| | Square | All sides equal, all angles 90° | | Rectangle | Opposite sides equal, all angles 90° | | Parallelogram | Opposite sides parallel and equal | | Rhombus | All sides equal, opposite angles equal | | Trapezium | One pair of parallel sides |

Chapter 6: Integers

6.1 Introduction to Integers

Integers = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}

Positive integers: 1, 2, 3, 4, ...
Zero: 0 (neither positive nor negative)
Negative integers: −1, −2, −3, ...

Real-life examples of negative integers:

Temperature below freezing: −5°C
Floors below ground: Basement −1, −2
Bank overdraft: −₹500

6.2 Operations on Integers

Addition Rules:

Same sign: Add and keep the sign (+3) + (+5) = +8
Different signs: Subtract and take sign of larger |+3 + (−7) = −4|

Subtraction:

Change subtraction to addition of opposite
(+5) − (+3) = (+5) + (−3) = +2
(+4) − (−6) = (+4) + (+6) = +10

Multiplication/Division:

Same signs: Positive result (+ × + = +), (− × − = +)
Different signs: Negative result (+ × − = −), (− × + = −)

Chapter 7: Fractions

7.1 Types of Fractions

| Type | Definition | Example | |------|-----------|---------| | Proper | Numerator < Denominator | 3/5, 2/7 | | Improper | Numerator ≥ Denominator | 7/3, 9/4 | | Mixed | Whole number + fraction | 2¼, 3⅓ | | Like | Same denominator | 2/5, 3/5 | | Unlike | Different denominators | 1/3, 1/4 | | Equivalent | Same value, different form | 1/2 = 2/4 = 3/6 |

7.2 Operations on Fractions

Adding/Subtracting Unlike Fractions:

Find LCM of denominators
Convert to equivalent fractions
Add/subtract numerators
Simplify if needed

Example: 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12

Multiplying Fractions: Multiply numerators × numerators, denominators × denominators

2/3 × 3/5 = 6/15 = 2/5

Dividing Fractions: Multiply by reciprocal of divisor

3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8 = 1 1/8

Chapter 8: Decimals

8.1 Place Value in Decimals

3  7  2  .  4  8  5
|  |  |     |  |  |
|  |  Units  |  |  Thousandths
|  Tens      |  Hundredths
Hundreds     Tenths

8.2 Converting Fractions to Decimals

1/4 = 25/100 = 0.25
1/5 = 2/10 = 0.2
3/8 = 375/1000 = 0.375

Adding/Subtracting Decimals: Align decimal points, then add/subtract normally.

Chapter 9: Data Handling

9.1 Collecting and Organizing Data

Raw Data: Unorganized, collected data
Tally Marks: Groups of 5 (IIII with diagonal = 5)
Frequency: Number of times a value occurs
Frequency Distribution Table: Organizes data into classes

9.2 Types of Graphs

| Graph Type | Best For | Example | |-----------|---------|---------| | Bar Graph | Comparing categories | Marks of 5 students | | Pictograph | Simple data with symbols | Number of books read | | Pie Chart | Showing proportions/percentages | Budget distribution | | Line Graph | Showing change over time | Temperature over a week |

9.3 Measures of Central Tendency

Mean (Average): Sum of all values ÷ Number of values

Marks: 70, 80, 85, 90, 75 → Mean = (70+80+85+90+75)/5 = 400/5 = 80

Median: Middle value when arranged in order

70, 75, 80, 85, 90 → Median = 80 (3rd value)
For even count: average of two middle values

Mode: Most frequently occurring value

Data: 3, 5, 5, 7, 8, 5 → Mode = 5

Chapter 10: Mensuration

10.1 Perimeter Formulas

| Shape | Perimeter Formula | |-------|------------------| | Square | P = 4 × side | | Rectangle | P = 2(length + breadth) | | Triangle | P = a + b + c | | Circle | Circumference = 2πr |

10.2 Area Formulas

| Shape | Area Formula | |-------|-------------| | Square | A = side² | | Rectangle | A = length × breadth | | Triangle | A = ½ × base × height | | Circle | A = πr² |

Note: π ≈ 22/7 ≈ 3.14

Quick Revision Checklist — Class 6 Maths

[ ] Indian vs International number system
[ ] Divisibility rules (2, 3, 4, 5, 6, 8, 9, 10, 11)
[ ] Prime and composite numbers (1 is neither)
[ ] HCF × LCM = Product of two numbers
[ ] Integer operations (sign rules for multiplication)
[ ] Fraction types and operations
[ ] Mean, median, mode formulas
[ ] Perimeter and area formulas
[ ] Types of angles (acute, right, obtuse, reflex)
[ ] Triangle classification (by sides and angles)

📄 Download Complete PDF Notes

Class 6 Maths complete notes covering number systems, fractions, decimals, algebra, geometry, and data handling. NCERT-based with solved examples and practice questions.

36 pages · 1.2 MB · Updated 2026-03-11

Free Download ↓

❓ Frequently Asked Questions

What is an integer?▾

Integers are whole numbers including negative numbers, zero, and positive numbers. Examples: ...−3, −2, −1, 0, 1, 2, 3... Unlike natural numbers, integers include negative values.

What is the difference between a factor and a multiple?▾

A factor of a number divides it exactly (no remainder). Example: factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is obtained by multiplying a number. Multiples of 3: 3, 6, 9, 12, 15...

How do you add fractions with different denominators?▾

Find the LCM of denominators, convert both fractions to equivalent fractions with LCM as denominator, then add numerators. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

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School (NCERT)SEM-1Mathematics

Class 6 Mathematics — Complete Notes with NCERT Solutions & Practice

✍️ WohoTech Team📅 Last Updated: 2026-03-11📄 36 pages · 1.2 MB

Class 6 Mathematics — Complete Study Notes

Class 6 Mathematics builds the foundation for all higher mathematics. This chapter covers all NCERT topics with solved examples, key formulas, and practice questions to help you score full marks.

Chapter 1: Knowing Our Numbers

1.1 Large Numbers — Indian Number System

| Period | Crores | Lakhs | Thousands | Ones | |--------|--------|-------|-----------|------| | Places | Crore, Ten Crore | Lakh, Ten Lakh | Thousand, Ten Thousand | Ones, Tens, Hundreds |

Reading Large Numbers (Indian System):

7,32,45,610 = Seven crore thirty-two lakh forty-five thousand six hundred ten

International System:

Millions, Billions instead of Lakhs, Crores
1 Lakh = 100,000 (1 hundred thousand)
1 Crore = 10,000,000 (10 million)

1.2 Estimation and Rounding

| Rounding Rule | Example | |---------------|---------| | Nearest 10 | 47 → 50; 43 → 40 | | Nearest 100 | 274 → 300; 234 → 200 | | Nearest 1000 | 7654 → 8000; 7234 → 7000 |

Rule: Look at digit to the right of rounding place. If ≥ 5, round up. If < 5, round down.

Chapter 2: Whole Numbers

Natural Numbers: 1, 2, 3, 4, ... (counting numbers)
Whole Numbers: 0, 1, 2, 3, 4, ... (natural numbers + zero)

Properties of Whole Numbers

Number Line: Numbers increase moving right. Zero is between positive and negative numbers.

Chapter 3: Playing with Numbers

3.1 Factors and Multiples

Factor: A number that divides another exactly
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Multiple: A number obtained by multiplying
- Multiples of 7: 7, 14, 21, 28, 35...

Divisibility Rules:

3.2 Prime and Composite Numbers

Prime: Has exactly 2 factors — 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...
Composite: Has more than 2 factors. Examples: 4, 6, 8, 9, 10, 12...
1 is neither prime nor composite
2 is the only even prime number

Sieve of Eratosthenes — Method to find all primes up to a given number by eliminating multiples.

3.3 HCF (Highest Common Factor) and LCM

HCF (Greatest Common Divisor):

Largest factor common to two or more numbers
Method 1 — Prime Factorization: find common prime factors, multiply
Method 2 — Division Method: divide larger by smaller, remainder becomes divisor, repeat

HCF of 36 and 48:
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
HCF = 2 × 2 × 3 = 12

LCM (Least Common Multiple):

Smallest number divisible by both numbers
LCM by prime factorization: take ALL prime factors with highest powers

LCM of 4 and 6:
4 = 2²
6 = 2 × 3
LCM = 2² × 3 = 12

Important Relationship: HCF × LCM = Product of two numbers

Chapter 4: Basic Geometrical Ideas

4.1 Points, Lines, and Angles

Point: A specific location, has no dimension. Represented by a dot (·)
Line: Extends infinitely in both directions, has no endpoints
Line Segment: Has two endpoints, fixed length
Ray: Has one endpoint, extends in one direction

4.2 Types of Angles

4.3 Polygons and Their Properties

| Polygon | Sides | Sum of Interior Angles | |---------|-------|----------------------| | Triangle | 3 | 180° | | Quadrilateral | 4 | 360° | | Pentagon | 5 | 540° | | Hexagon | 6 | 720° |

Formula: Sum of interior angles = (n − 2) × 180°, where n = number of sides

Chapter 5: Understanding Elementary Shapes

5.1 Triangles Classification

By Sides:

Equilateral: all 3 sides equal
Isosceles: 2 sides equal
Scalene: all 3 sides different

By Angles:

Acute-angled: all angles < 90°
Right-angled: one angle = 90°
Obtuse-angled: one angle > 90°

5.2 Quadrilaterals

Chapter 6: Integers

6.1 Introduction to Integers

Integers = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}

Positive integers: 1, 2, 3, 4, ...
Zero: 0 (neither positive nor negative)
Negative integers: −1, −2, −3, ...

Real-life examples of negative integers:

Temperature below freezing: −5°C
Floors below ground: Basement −1, −2
Bank overdraft: −₹500

6.2 Operations on Integers

Addition Rules:

Same sign: Add and keep the sign (+3) + (+5) = +8
Different signs: Subtract and take sign of larger |+3 + (−7) = −4|

Subtraction:

Change subtraction to addition of opposite
(+5) − (+3) = (+5) + (−3) = +2
(+4) − (−6) = (+4) + (+6) = +10

Multiplication/Division:

Same signs: Positive result (+ × + = +), (− × − = +)
Different signs: Negative result (+ × − = −), (− × + = −)

Chapter 7: Fractions

7.1 Types of Fractions

7.2 Operations on Fractions

Adding/Subtracting Unlike Fractions:

Find LCM of denominators
Convert to equivalent fractions
Add/subtract numerators
Simplify if needed

Example: 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12

Multiplying Fractions: Multiply numerators × numerators, denominators × denominators

2/3 × 3/5 = 6/15 = 2/5

Dividing Fractions: Multiply by reciprocal of divisor

3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8 = 1 1/8

Chapter 8: Decimals

8.1 Place Value in Decimals

3  7  2  .  4  8  5
|  |  |     |  |  |
|  |  Units  |  |  Thousandths
|  Tens      |  Hundredths
Hundreds     Tenths

8.2 Converting Fractions to Decimals

1/4 = 25/100 = 0.25
1/5 = 2/10 = 0.2
3/8 = 375/1000 = 0.375

Adding/Subtracting Decimals: Align decimal points, then add/subtract normally.

Chapter 9: Data Handling

9.1 Collecting and Organizing Data

Raw Data: Unorganized, collected data
Tally Marks: Groups of 5 (IIII with diagonal = 5)
Frequency: Number of times a value occurs
Frequency Distribution Table: Organizes data into classes

9.2 Types of Graphs

9.3 Measures of Central Tendency

Mean (Average): Sum of all values ÷ Number of values

Marks: 70, 80, 85, 90, 75 → Mean = (70+80+85+90+75)/5 = 400/5 = 80

Median: Middle value when arranged in order

70, 75, 80, 85, 90 → Median = 80 (3rd value)
For even count: average of two middle values

Mode: Most frequently occurring value

Data: 3, 5, 5, 7, 8, 5 → Mode = 5

Chapter 10: Mensuration

10.1 Perimeter Formulas

10.2 Area Formulas

| Shape | Area Formula | |-------|-------------| | Square | A = side² | | Rectangle | A = length × breadth | | Triangle | A = ½ × base × height | | Circle | A = πr² |

Note: π ≈ 22/7 ≈ 3.14

Quick Revision Checklist — Class 6 Maths

[ ] Indian vs International number system
[ ] Divisibility rules (2, 3, 4, 5, 6, 8, 9, 10, 11)
[ ] Prime and composite numbers (1 is neither)
[ ] HCF × LCM = Product of two numbers
[ ] Integer operations (sign rules for multiplication)
[ ] Fraction types and operations
[ ] Mean, median, mode formulas
[ ] Perimeter and area formulas
[ ] Types of angles (acute, right, obtuse, reflex)
[ ] Triangle classification (by sides and angles)

📄 Download Complete PDF Notes

Class 6 Maths complete notes covering number systems, fractions, decimals, algebra, geometry, and data handling. NCERT-based with solved examples and practice questions.

36 pages · 1.2 MB · Updated 2026-03-11

Free Download ↓

❓ Frequently Asked Questions

What is an integer?▾

Integers are whole numbers including negative numbers, zero, and positive numbers. Examples: ...−3, −2, −1, 0, 1, 2, 3... Unlike natural numbers, integers include negative values.

What is the difference between a factor and a multiple?▾

A factor of a number divides it exactly (no remainder). Example: factors of 12 are 1, 2, 3, 4, 6, 12. A multiple is obtained by multiplying a number. Multiples of 3: 3, 6, 9, 12, 15...

How do you add fractions with different denominators?▾

Find the LCM of denominators, convert both fractions to equivalent fractions with LCM as denominator, then add numerators. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.