Topics
Rational Numbers
 Rational Numbers
 Closure Property of Rational Numbers
 Commutative Property of Rational Numbers
 Associative Property of Rational Numbers
 Distributive Property of Multiplication Over Addition for Rational Numbers
 Identity of Addition and Multiplication of Rational Numbers
 Negative Or Additive Inverse of Rational Numbers
 Reciprocal Or Multiplicative Inverse of Rational Numbers
 Rational Numbers on a Number Line
 Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
 The Idea of a Variable
 Expressions with Variables
 Concept of Equation
 Balancing an Equation
 The Solution of an Equation
 Linear Equation in One Variable
 Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
 Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
 Solving Equations Having the Variable on Both Sides
 Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
 Reducing Equations to Simpler Form
 Equations Reducible to the Linear Form
Understanding Quadrilaterals
 Concept of Curves
 Different Types of Curves  Closed Curve, Open Curve, Simple Curve.
 Concept of Polygons  Side, Vertex, Adjacent Sides, Adjacent Vertices and Diagonal
 Classification of Polygons
 Angle Sum Property of a Quadrilateral
 Interior Angles of a Polygon
 Exterior Angles of a Polygon and Its Property
 Concept of Quadrilaterals  Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
 Properties of Trapezium
 Properties of Kite
 Properties of a Parallelogram
 Properties of Rhombus
 Property: The Opposite Sides of a Parallelogram Are of Equal Length.
 Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
 Property: The adjacent angles in a parallelogram are supplementary.
 Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
 Property: The diagonals of a rhombus are perpendicular bisectors of one another.
 Property: The Diagonals of a Rectangle Are of Equal Length.
 Properties of Rectangle
 Properties of a Square
 Property: The diagonals of a square are perpendicular bisectors of each other.
Practical Geometry
 Introduction to Practical Geometry
 Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
 Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
 Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
 Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
 Some Special Cases
Data Handling
 Concept of Data Handling
 Interpretation of a Pictograph
 Interpretation of Bar Graphs
 Drawing a Bar Graph
 Interpretation of a Double Bar Graph
 Drawing a Double Bar Graph
 Organisation of Data
 Frequency Distribution Table
 Graphical Representation of Data as Histograms
 Concept of Pie Graph (Or a Circlegraph)
 Interpretation of Pie Diagram
 Chance and Probability  Chance
 Basic Ideas of Probability
Squares and Square Roots
 Concept of Square Number
 Properties of Square Numbers
 Some More Interesting Patterns of Square Number
 Finding the Square of a Number
 Concept of Square Roots
 Finding Square Root Through Repeated Subtraction
 Finding Square Root Through Prime Factorisation
 Finding Square Root by Division Method
 Square Root of Decimal Numbers
 Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
 Concept of Ratio
 Concept of Percent and Percentage
 Increase Or Decrease as Percent
 Concept of Discount
 Estimation in Percentages
 Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
 Sales Tax, Value Added Tax, and Good and Services Tax
 Concept of Principal, Interest, Amount, and Simple Interest
 Concept of Compound Interest
 Deducing a Formula for Compound Interest
 Rate Compounded Annually Or Half Yearly (Semi Annually)
 Applications of Compound Interest Formula
Algebraic Expressions and Identities
 Algebraic Expressions
 Terms, Factors and Coefficients of Expression
 Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
 Like and Unlike Terms
 Addition of Algebraic Expressions
 Subtraction of Algebraic Expressions
 Multiplication of Algebraic Expressions
 Multiplying Monomial by Monomials
 Multiplying a Monomial by a Binomial
 Multiplying a Monomial by a Trinomial
 Multiplying a Binomial by a Binomial
 Multiplying a Binomial by a Trinomial
 Concept of Identity
 Expansion of (a + b)2 = a2 + 2ab + b2
 Expansion of (a  b)2 = a2  2ab + b2
 Expansion of (a + b)(a  b)
 Expansion of (x + a)(x + b)
Visualizing Solid Shapes
Mensuration
Exponents and Powers
Direct and Inverse Proportions
Factorization
 Factors and Multiples
 Factorising Algebraic Expressions
 Factorisation by Taking Out Common Factors
 Factorisation by Regrouping Terms
 Factorisation Using Identities
 Factors of the Form (x + a)(x + b)
 Dividing a Monomial by a Monomial
 Dividing a Polynomial by a Monomial
 Dividing a Polynomial by a Polynomial
 Concept of Find the Error
Introduction to Graphs
 Concept of Bar Graph
 Interpretation of Bar Graphs
 Drawing a Bar Graph
 Concept of Double Bar Graph
 Interpretation of a Double Bar Graph
 Drawing a Double Bar Graph
 Concept of Pie Graph (Or a Circlegraph)
 Graphical Representation of Data as Histograms
 Concept of a Line Graph
 Linear Graphs
 Linear Graphs
 Some Application of Linear Graphs
Playing with Numbers
 Ungrouped Frequency Distribution Table:
 Grouped Frequency Distribution Table:
 Inclusive Frequency Distribution
 Exclusive Frequency Distribution
Definition
Frequency Distribution Table: When the number of observations in an experiment is large then we can convert it into the tabular form which is called a Frequency Distribution Table.
Ungrouped Frequency Distribution Table: When the frequency of each class interval is not arranged or organized in any manner.
Grouped Frequency Distribution Table: The frequencies of the corresponding class intervals are organised or arranged in a particular manner, either ascending or descending.
Inclusive or discontinuous Frequency Distribution: A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution.
Exclusive or continuous Frequency Distribution: A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution.
Notes
Frequency Distribution table:

Presentation of data in ascending or descending order can be quite timeconsuming.

When the number of observations in an experiment is large then we can convert it into the tabular form which is called a Frequency Distribution Table.

A frequency table shows the list of categories or groups of things, together with the number of times the items occur.

There are two types of frequency distribution table:
(i) Ungrouped Frequency Distribution Table
(ii) Grouped Frequency Distribution Table
A. Ungrouped frequency distribution table:
When the frequency of each class interval is not arranged or organised in any manner.
Consider the marks obtained (out of 100 marks) by 30 students of Class IX of a school:
10, 20, 36, 92, 95, 40, 50, 56, 60, 70, 92, 88, 80, 70, 72, 70, 36, 40, 36, 40, 92, 40, 50, 50, 56, 60, 70, 60, 60, 88.
Marks  Number of students (i.e., the frequency) 
10  1 
20  1 
36  3 
40  4 
50  3 
56  2 
60  4 
70  4 
72  1 
80  1 
88  2 
92  3 
95  1 
Total  30 
B. Grouped Frequency Distribution Table:

Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.

Presenting data in this form simplifies and condenses data and enables us to observe certain important features at a glance. This is called a grouped frequency distribution table.
Grouped data could be of two types as below:
1. Inclusive or discontinuous Frequency Distribution:
A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution.
While analysing a frequency distribution, if there are inclusive type of class intervals they must be converted into exclusive type. This can be done by extending the class intervals from both the ends.
2. Exclusive or continuous Frequency Distribution:
A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution.
1) Consider the following marks (out of 50) obtained in Mathematics by 60 students of Class VIII:
21, 10, 30, 22, 33, 5, 37, 12, 25, 42, 15, 39, 26, 32, 18, 27, 28, 19, 29, 35, 31, 24,
36, 18, 20, 38, 22, 44, 16, 24, 10, 27, 39, 28, 49, 29, 32, 23, 31, 21, 34, 22, 23, 36, 24,
36, 33, 47, 48, 50, 39, 20, 7, 16, 36, 45, 47, 30, 22, 17.
If we make a frequency distribution table for each observation, then the table would
be too long, so, for convenience, we make groups of observations say, 010, 1020, and so on, and obtain a frequency distribution of the number of observations falling in each group.
Groups  Tally Marks  Frequency 
0  10    2 
10  20  `cancel() cancel()`  10 
20  30  `cancel() cancel() cancel() cancel() `  21 
30  40  `cancel() cancel() cancel()`   19 
40  50  `cancel()`   7 
50  60    1 
Total  60 

Each of the groups 010, 1020, 2030, etc., is called a Class Interval.

In the class interval, 1020, 10 is called the lower class limit and 20 is called the upperclass limit.

This difference between the upperclass limit and lower class limit for each of the class intervals 010, 1020, 2030, etc., is equal, (10 in this case) is called the width or size of the class interval.
2) Let us now consider the following frequency distribution table which gives the weights of 38 students of a class:
Weights (in kg) 
31  35  36  40  41  45  46  50  51  55  56  60  61  65  66  70  71  75 
No. of students  9  5  14  3  1  2  2  1  1 
In this class, we cannot added new students with 35.5 kg and 40.5 kg because there are gaps in between the upper and lower limits of two consecutive classes. So, we need to divide the intervals so that the upper and lower limits of consecutive intervals are the same. For this, we find the difference between the upper limit of a class and the lower limit of its succeeding class. We then add half of this difference to each of the upper limits and subtract the same from each of the lower limits.
So, the new class interval formed from 3135 is (31 – 0.5)  (35 + 0.5), i.e., 30.535.5.
Now it is possible for us to include the weights of the new students in these classes. But, another problem crops up because 35.5 appears in both the classes 30.535.5 and 35.540.5.
By convention, we consider 35.5 in the class 35.540.5 and not in 30.535.5.
Now, with these assumptions, the new frequency distribution table will be as shown below:
Weights (in kg)  Number of students 
30.5  35.5  9 
35.5  40.5  6 
40.5  45.5  15 
45.5  50.5  3 
50.5  55.5  1 
55.5  60.5  2 
60.5  65.5  2 
65.5  70.5  1 
70.5  75.5  1 
Total  40 