Topics
Number System
Number System (Including Estimation and Approximation)
Number System(Consolidating the Sense of Numberness)
Theme 1 : Numbers
Estimation
Theme 2 : Ratio, Proportion and Arithmetic Problems
Numbers in Indian and International Systems (With Comparison)
Ratio and Proportion
Algebra
Natural Numbers and Whole Numbers (Including Number Line and Patterns)
Numbers in India and International System (With Comparison)
Theme 3 : Algebra
Theme 4 : Geometry
Place Value
Negative Numbers and Integers
Geometry
Mensuration
Theme 5 : Mensuration
Natural Numbers and Whole Numbers (Including Patterns)
Sets
Data Handling
Fractions
Theme 6 : Statistics
Negative Numbers and Integers
Decimal Fractions
Number Line
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
Fundamental Concepts (With operations on Algebraic Expressions)
Proportion (Including Word Problems)
Unitary Method
Framing Algebraic Expressions (Including Substitution and Linear Equations)
Fundamental Concepts (Including Angles and their Properties)
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
Triangles (Including Types and Properties of Triangles)
Decimal Fractions
Quadrilaterals
Percent (Percentage)
Circles
Idea of Speed, Distance and Time
Constructions
Symmetry
Fundamental Concepts
Fundamental Operations (Related to Algebraic Expressions)
Substitution (Including Use of Brackets as Grouping Symbols)
Recognition of Solids
Framing Algebraic Expressions (Including Evaluation)
Perimeter and Area of Plane Figures
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
- Introduction
- Types of Brackets
- Order of Removing Brackets
- Rules of Removing Brackets
- Inserting Brackets
- Example 1
- Example 2
- Key Points Summary
Introduction
What Are Brackets?
Think of brackets as mathematical "containers" that help us group numbers and operations together. Just like how we use boxes to organize items, brackets help us organize mathematical expressions!
Why Do We Need Brackets?
Imagine you want to buy 3 pens that cost ₹5 each, plus 2 notebooks that cost ₹10 each. Without brackets, writing this as 3 × 5 + 2 × 10 might be confusing. With brackets, we can write it clearly as (3 × 5) + (2 × 10) = ₹35.
Key Rule: When an expression is enclosed within brackets, treat it as a single unit, even if it contains multiple terms.
Types of Brackets
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( ) - Parentheses (also called small brackets)
-
{ } - Curly brackets (also called middle brackets or braces)
-
[ ] - Square brackets (also called big brackets)
-
‾ - Vinculum or bar bracket (a line drawn over terms)
Order of Removing Brackets
Rules of Removing Brackets
Rule 1: Positive Sign Before Brackets
When there's a + sign before brackets, simply remove the brackets without changing any signs inside.
Example:
-
10 + (7 - 3) = 10 + 7 - 3 = 14
-
a + (b - c + d) = a + b - c + d
Rule 2: Negative Sign Before Brackets
When there's a - sign before brackets, remove the brackets AND flip all the signs inside.
Example:
-
12 - (8 - 5) = 12 - 8 + 5 = 9
-
a - (b - c + d) = a - b + c - d
Memory Tip: Think "negative flips everything!"
Inserting Brackets
Insertion with Positive Sign
When inserting brackets preceded by a positive sign, maintain original signs of all terms within.
Example: a - b + c - d can be written as:
-
a + (-b + c - d)
-
a - b + (c - d)
Insertion with Negative Sign
When inserting brackets preceded by a negative sign, reverse all signs of terms within.
Example: a - b + c - d can be written as:
-
a - (b - c + d)
-
a - b - (-c + d)
Example 1
6a − {a + (2a − `bar" 4 − a "`)}
= 6a − {a + (2a − 4 + a)}
= 6a − {a + (3a − 4)}
= 6a − {a + 3a − 4}
= 6a − { 4a − 4}
= 6a − 4a + 4
= 2a + 4
Example 2
a − [b − {c − (a − `bar" b − c "`)}]
= a − [b − {c − (a − b + c)}] [On removing the bar brackets]
= a − [b − { c − a + b − c}] [On removing the small brackets]
= a − [b − c + a − b + c] [On removing the middle brackets]
= a − b + c − a + b − c [On removing the square brackets]
= 0
Key Points Summary
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Four bracket types: Vinculum (‾), Parentheses ( ), Curly { }, Square [ ]
-
Removal hierarchy: BODMAS order must be strictly followed
-
Sign rules: Positive preserves, negative flips all signs
-
Multiplication principle: Outside coefficient multiplies every inside term
