Topics
Theme 1 : Numbers
Number System
Number System (Including Estimation and Approximation)
Number System(Consolidating the Sense of Numberness)
Ratio and Proportion
Estimation
Numbers in Indian and International Systems (With Comparison)
Theme 2 : Ratio, Proportion and Arithmetic Problems
Numbers in India and International System (With Comparison)
Algebra
Natural Numbers and Whole Numbers (Including Number Line and Patterns)
Theme 3 : Algebra
Place Value
Geometry
Theme 4 : Geometry
Negative Numbers and Integers
Sets
Theme 5 : Mensuration
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Negative Numbers and Integers
Fractions
Data Handling
Theme 6 : Statistics
Number Line
Decimal Fractions
HCF and LCM
Playing with Numbers (Including H.C.F. and L.C.M.)
Playing with Numbers
- Simplification of Brackets
- Finding Factors Using Rectangular Arrangements and Division
- Factors and Common Factors
- Multiples and Common Multiples
- Concept of Even and Odd Number
- Tests for Divisibility of Numbers
- Divisibility by 2
- Divisibility by 4
- Divisibility by 8
- Divisibility by 3
- Divisibility by 6
- Divisibility by 9
- Divisibility by 5
- Divisibility by 11
Ratio and Proportion (Including Unitary Method)
Percent (Percentage)
Sets
Ratio
Idea of Speed, Distance and Time
Proportion (Including Word Problems)
Fundamental Concepts (With operations on Algebraic Expressions)
Unitary Method
Framing Algebraic Expressions (Including Substitution and Linear Equations)
Fractions
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Like and Unlike Fraction
- Concept of Equivalent Fractions
- Conversion between Improper and Mixed fraction
- Conversion between Unlike and Like Fractions
- Simplest Form of a Fractions
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of Fraction
- Division of Fractions
- Using Operator 'Of' with Multiplication and Division
- BODMAS Rule
- Problems Based on Fraction
Fundamental Concepts (Including Angles and their Properties)
Triangles (Including Types and Properties of Triangles)
Decimal Fractions
Percent (Percentage)
Quadrilaterals
Idea of Speed, Distance and Time
Circles
Symmetry
Constructions
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Recognition of Solids
Substitution (Including Use of Brackets as Grouping Symbols)
Perimeter and Area of Plane Figures
Framing Algebraic Expressions (Including Evaluation)
Simple (Linear) Equations (Including Word Problems)
Data Handling (Including Mean and Median)
Fundamental Concepts
Angles (With Their Types)
Properties of Angles and Lines (Including Parallel Lines)
Triangles (Including Types, Properties and Constructions)
Quadrilateral
Polygons
The Circle
Symmetry (Including Constructions on Symmetry)
Recognition of Solids
Perimeter and Area of Plane Figures
Data Handling (Including Pictograph and Bar Graph)
Mean and Median
- For Addition
- For Subtraction
- For Multiplication
- For Division
- Key Points Summary
CISCE: Class 6
For Addition
| Property | Definition | Example |
|---|---|---|
| Closure Property | Adding two whole numbers always gives a whole number. | 5 + 8 = 13, 4 + 0 = 4 |
| Commutative Property | Changing the order of addition doesn't change the sum. | 4 + 3 = 7, 3 + 4 = 7 |
| Associative Property | The grouping of numbers doesn’t affect the sum. x + (y + z) = (x + y) + z. |
3 + (5 + 6) = 14, (3 + 5) + 6 = 14 |
| Existence of Identity | Adding 0 to any number leaves it unchanged. | 8 + 0 = 8, 15 + 0 = 15 |
| Additive Inverse | A number and its opposite add to give 0. | 5 + (-5) = 0 |
| Cancellation Law | Adding and subtracting the same number cancels out the result. x + y = x + z ⇒ `\cancel(x)` + y = `\cancel(x)` + z |
x + 8 = 5 + 8 ⇒ x + `\cancel(8)` = 5 + `\cancel(8)` ⇒ x = 5 |
CISCE: Class 6
For Subtraction
| Property | Definition | Example |
|---|---|---|
| Closure Property | Subtracting two whole numbers doesn't always result in a whole number. | 8 - 3 = 5 (Whole number), but 15 - 18 = -3 (Not a whole number). |
| Commutative Property | Subtraction doesn't work the same as addition; order matters. x − y `\cancel(=)` y − x. |
15 - 8 = 7, but 8 - 15 = -7. |
| Associative Property | Subtraction does not satisfy the associative property. x − (y − z) `\cancel(=)` (x − y) − z |
15 - (10 - 7) = 12, but (15 - 10) - 7 = -2. |
| Distributive Property | Subtraction distributes over multiplication. x × (y − z) = x × y − x × z and (y - z) × x = y × x − z × x |
x = 3, y = 5, z = 2, 3 × (5 − 2) = 9 and 3 × 5 − 3 × 2 = 9 |
| Existence of Identity | No identity number exists for subtraction, but 0 is its own identity for subtraction. | 5 - 0 = 5, but 0 − 5 ≠ 5 |
| Existence of Inverse | Subtraction doesn't have an inverse for non-zero whole numbers. | No inverse exists for subtraction of non-zero whole numbers. |
CISCE: Class 6
For Multiplication
| Property | Definition | Example |
|---|---|---|
| Closure Property | Multiplying two whole numbers always results in a whole number. | 5 × 4 = 20 12 × 0 = 0 |
| Commutative Property | The order of multiplication does not affect the result. | 4 × 5 = 20, 5 × 4 = 20 3 × 0 = 0, 0 × 3 = 0 |
| Associative Property | The grouping of numbers does not change the result. | 4 × (8 × 10) = 320, (4 × 8) × 10 = 320 |
| Distributive Property | Multiplication distributes over addition. | 5 × (3 + 4) = 35, 5 × 3 + 5 × 4 = 35 |
| Existence of Identity | The identity for multiplication is 1. Multiplying any number by 1 gives the same number. | 9 × 1 = 99 , 15 × 1 = 15 |
| Multiplicative Inverse | The inverse of a number is the number that, when multiplied by the original number, gives 1. | The inverse of 1 is 1: 1 × 1 = 1 No inverse exists for numbers like 2 in whole numbers. |
| Cancellation Law | If both sides of an equation are multiplied by the same non-zero number, that number can be "cancelled out". | 3 × a = 3 × b Cancel 3 → a = b m × 7 = n × 7 cancel 7 → m = n |
CISCE: Class 6
For Division
| Property | Definition | Example |
|---|---|---|
| Closure Property | Division of whole numbers does not always result in a whole number. | 5 ÷ 8 is not a whole number. So, closure property does not exist for division. |
| Commutative Property | Division of whole numbers is not commutative; changing the order changes the result. x ÷ y `\cancel(=)` y ÷ x. |
3 ÷ 5 ≠ 5 ÷3 8 ÷ 13 ≠ 13 ÷ 8 |
| Associative Property | Division of whole numbers is not associative; grouping changes the result. x ÷ (y ÷ z) `\cancel(=)` (x ÷ y) ÷ z |
(10 ÷ 5) ÷ 2 = 1, but 10 ÷ (5 ÷ 2) = 4 |
| Existence of Identity | There is no identity element for division in whole numbers. | No identity exists for division of whole numbers. |
| Existence of Inverse | There is no inverse for division of whole numbers. | Inverses do not exist for division. |
Note:
- a ÷ a = 1, i.e., 5 ÷ 5 = 1, 12 ÷ 12 = 1, 28 ÷ 28 = 1, etc.
- a ÷ 1 = a, i.e., 5 ÷ 1 = 5, 16 ÷ 1 = 16, 28 ÷ 1 = 28, etc.
- 0 ÷ a = 0, i.e., 0 ÷ 8 = 0, 0 ÷ 23 = 0, 0 ÷ 47 = 0, etc.
- a ÷ 0 is not defined, i.e., 8 ÷ 0 is not defined, 24 ÷ 0 is not defined, etc.
CISCE: Class 6
Key Points Summary
-
Whole numbers begin from 0 and go on forever.
-
They are “closed” under addition or multiplication.
-
Order and grouping do not change the sum or product.
-
0 and 1 are the identities for addition and multiplication, respectively.
-
Subtraction or division might not give whole numbers.
