Topics
Number System(Consolidating the Sense of Numberness)
Number System
Estimation
Ratio and Proportion
Algebra
Numbers in India and International System (With Comparison)
Geometry
Place Value
Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Data Handling
Negative Numbers and Integers
Number Line
HCF and LCM
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
Sets
Ratio
Proportion (Including Word Problems)
Unitary Method
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
Decimal Fractions
Percent (Percentage)
Idea of Speed, Distance and Time
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Substitution (Including Use of Brackets as Grouping Symbols)
Framing Algebraic Expressions (Including Evaluation)
Simple (Linear) Equations (Including Word Problems)
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.
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They are “closed” under addition or multiplication.
-
Order and grouping do not change the sum or product.
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0 and 1 are the identities for addition and multiplication, respectively.
-
Subtraction or division might not give whole numbers.
