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
Notes
Division of a Decimal Number by a Whole Number:
-
We use division or long division of decimals by whole numbers as if the decimals were whole numbers ignoring the decimal points.
- In the quotient that we get, we place the decimal point in the same spot as in the number that is being divided.
- For example, 8.4 ÷ 4 = 2.1
1) 6.4 ÷ 2
`= 64/10 ÷ 2`
`= 64/10 xx 1/2`
`= (64 xx 1)/(10 xx 2)`
`= (1 xx 64)/(10 xx 2)`
`= 1/10 xx 64/2`
`= 1/10 xx 32`
`= 32/10`
= 3.2
2) 12.96 ÷ 4
`= 1296/100 ÷ 4`
`= 1296/100 xx 1/4`
`= 1/100 xx 1296/4`
`= 1/100 xx 324`
= 3.24.
Example
Find: 33.73 ÷ 2
33.73 ÷ 2
`= (33.73)/2 xx 1/1`
`= (33.73)/2 xx (1 xx 100)/(1 xx 100)`
`= (33.73)/2 xx 100/100`
`= (33.73 xx 100)/(2 xx 100)`
`= (3373)/(2 xx 100)`
`= (3373 xx 1)/(2 xx 100)`
`= (3373)/2 xx 1/100`
`= 1686.5 xx 1/100`
`= (1686.5)/100`
= 16.865
Example
Find: 4.449 ÷ 4
4.449 ÷ 4
`= (4.449)/4 xx 1/1`
`= (4.499)/4 xx (1 xx 1000)/(1 xx 1000)`
`= (4.449)/4 xx (1000)/(1000)`
`= (4.449 xx 1000)/(4 xx 1000)`
`= (4449 xx 1)/(4 xx 1000)`
`= (4449)/4 xx 1/1000`
`= 1112.25 xx 1/1000`
`= (1112.25)/(1000)`
= 1.11225
Example
Find: 233 ÷ 0.5
`= 233/0.5 xx 1/1`
`= 233/0.5 xx 10/10`
`= (233 xx 10)/(0.5 xx 10)`
`= (233 xx 10)/5`
`= (233)/5 xx 10`
= 46.6 x 10
= 466
Example
Find: 525 ÷ 0.015
525 ÷ 0.015
`= 525/0.015 xx 1/1`
`= 525/0.015 xx (1 xx 1000)/(1 xx 1000)`
`= 525/0.015 xx (1000)/(1000)`
`= (525 xx 1000)/(0.015 xx 1000)`
`= (525 xx 1000)/15`
`= 525/15 xx 1000`
`= 525/15 xx 1000`
= 35 × 1000
= 35000
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