Topics
Integers
- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive Numbers
- Concept of Integers
- Representation of Integers on the Number Line
- Concept for Ordering of Integers
- Addition of Integers
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers
Fractions and Decimals
- Concept of Fractions
- Fraction and its Types
- Concept of Proper Fractions
- Improper Fraction and Mixed Fraction
- Concept of Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Fraction as an Operator 'Of'
- Multiplication of Fraction
- Division of Fractions
- Concept of Reciprocal or Multiplicative Inverse
- Problems Based on Fraction
- The Decimal Number System
- Comparing Decimal Numbers
- Addition of Decimal Fraction
- Subtraction of Decimal Numbers
- Multiplication of Decimal Fractions
- Multiplication of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Fractions
- Division of a Decimal Number by Another Decimal Number
- Problems Based on Decimal Numbers
Data Handling
Simple Equations
Lines and Angles
- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Angle
- Complementary Angles
- Supplementary Angles
- Concept of Angle
- Concept of Linear Pair
- Concept of Vertically Opposite Angles
- Concept of Intersecting Lines
- Introduction to Parallel Lines
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
The Triangle and Its Properties
- Concept of Triangles
- Classification of Triangles (On the Basis of Sides, and of Angles)
- Classification of Triangles based on Sides- Equilateral, Isosceles, Scalene
- Classification of Triangles based on Sides- Equilateral, Isosceles, Scalene
- 3. Classification of Triangles based on Angles: Acute-Angled, Right-Angled, Obtuse-Angled
- 3. Classification of Triangles based on Angles: Acute-Angled, Right-Angled, Obtuse-Angled
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Angle Sum Property of a Triangle
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Sum of the Lengths of Two Sides of a Triangle
- Right-angled Triangles and Pythagoras Property
Comparing Quantities
- Concept of Ratio
- Concept of Equivalent Ratios
- Concept of Proportion
- Concept of Unitary Method
- Basic Concept of Percentage
- Conversion between Percentage and Fraction
- Converting Decimals to Percentage
- Conversion between Percentage and Fraction
- Converting Percentages to Decimals
- Estimation in Percentages
- Interpreting Percentages
- Converting Percentages to “How Many”
- Ratios to Percents
- Increase Or Decrease as Percent
- Basic Concepts of Profit and Loss
- Profit or Loss as a Percentage
- Calculation of Interest
Congruence of Triangles
Rational Numbers
- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
Perimeter and Area
- Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangles
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter and Area
Algebraic Expressions
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Like and Unlike Terms
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
- Use of Variables in Common Rules
Practical Geometry
- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)
Exponents and Powers
- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Crores
Symmetry
Visualizing Solid Shapes
- Introduction
- Interconversion between Improper and Mixed
Introduction
1. Improper Fraction: The fractions whose numerator is greater than or equal to its denominator are called improper fractions, i.e., numerator > denominator. `3/2, 12/7, 18/5` are all examples of improper fractions.
2. Mixed Fraction: A fraction that contains a whole number and a proper fraction is called a mixed fraction. `3 2/3, 4 1/4, 3 7/8` are all examples of mixed fractions.
Mixed fractions will be written as `"Quotient" ("Reminder"/"Divisor")`.
Example:
In 1 `"1"/"2"`, 1 is the integer part, and `"1"/"2"` is the fractional part.
If half of each of the three circles is coloured, the total coloured portion is:
`"1"/"2"`+`"1"/"2"`+`"1"/"2"`= `"3"/"2"`
Since `"1"/"2"`= 1+`"1"/"2"`it can be written as 1`"1"/"2"` which is a mixed fraction.
Interconversion between Improper and Mixed fraction
i) Improper to mixed fraction
1. `17/4 = (16 + 1)/4 = 16/4 + 1/4 = 4 + 1/4 = 4 1/4`.
i.e., 4 whole and `1/4 "more, or" 4 1/4`.
2. `11/3 = (9 + 2)/3 = 9/3 + 2/3 = 3 2/3`.
i.e., 3 whole and `2/3 "more, or" 3 2/3`.
3. `28/5 = (25 + 3)/5 = 25/5 + 3/5 = 5 + 3/5 = 5 3/5`
4. `19/6 = (18 + 1)/6 = 18/6 + 1/6 = 3 + 1/6 = 3 1/6`.
Thus, we can express an improper fraction as a mixed fraction by dividing the numerator by the denominator to obtain the quotient and the remainder.
ii) Mixed to Improper Fraction
1. `10 3/5 = ((5 × 10) + 3)/5 = 53/5`.
2. `9 3/7 = ((7 × 9) + 3)/7 = 66/7`.
3. `8 4/9 = ((8 × 9) + 4)/9 = 76/9`.
Thus, we can express a mixed fraction as an improper fraction as
`(("Whole" × "Denominator") + "Numerator")/"Denominator"`.
or
`"Quotient × Divisor + Reminder"/"Divisor"`.
