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
 Concept of Equation
 Expressions with Variables
 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)
Mensuration
Visualizing Solid Shapes
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
 Some Application of Linear Graphs
Playing with Numbers
 Ungrouped Data
 Grouped Data
Definition
Unorganised/Raw data: Unorganised form of data is called raw data.
Grouped data: Grouped data means observations are classified into groups.
Frequency: Frequency gives the number of times that a particular entry occurs.
Frequency tables: A frequency table shows the list of categories or groups of things, together with the number of times the items occur.
Class Interval: While arranging a large amount of data in statistics, they are grouped into different classes to get an idea of the distribution, and the range of such class of data is called the Class Interval.
Upper Limit: In each class interval, the greatest number is the upperclass limit.
Lower Limit: In each class interval, the smallest number is the lower class limit.
Class Size: This difference between the upperclass limit and lower class limit for each of the class intervals is equal is called the width or size of the class interval.
Class Mark: The midpoint of each class interval is the class mark.
Class Mark = `"Upper Limit + Lower limit"/2`.
Tally Mark: Tally Mark refers to a group of five marks that should be used as a cross, as shown by `cancel(}`. They are tally marks.
Formula
Class Mark = `"Upper Limit + Lower limit"/2`.
Notes
Organisation Data:

To draw meaningful inferences, we need to organize the data systematically.

The data that is collected needs to be organized in a proper table so that it becomes easy to understand and interpret.
A. Ungrouped Data:

Data mostly available to us is in an unorganized form called raw data.

The observations are not classified into groups.
Consider the following example,
1) Ms. Neelam, class teacher wanted to find how children had performed in English. She writes down the marks obtained by the students in the following way:
23, 35, 48, 30, 25, 46, 13, 27, 32, 38
In this form, the data was not easy to understand. She also did not know whether her impression of the students matched their performance.
Neelam’s colleague helped her organise the data in the following way
Roll No. 
Names 
Marks out of 50 
Roll No. 
Names 
Marks out of 50 
1 
Ajay 
23 
6 
Govind 
46 
2 
Armaan 
35 
7 
Jay 
13 
3 
Ashish 
48 
8 
Kavita 
27 
4 
Dipti 
30 
9 
Manisha 
32 
5 
Faizaan 
25 
10 
Neeraj 
38 
In this form, Neelam was able to know which student has got how many marks. But she wanted more. Deepika suggested another way to organise this data.
Roll No. 
Names 
Marks out of 50 
Roll No. 
Names 
Marks out of 50 
3 
Ashish 
48 
4 
Dipti 
30 
6 
Govind 
46 
8 
Kavita 
27 
10 
Neeraj 
38 
5 
Faizaan 
25 
2 
Armaan 
35 
1 
Ajay 
23 
9 
Manisha 
32 
7 
Jay 
13 
Consider another example:
2) A group of students was asked about their favourite subject. The results were as listed below:
Art, Mathematics, Science, English, Mathematics, Art, English, Mathematics, English,
Art, Science, Art, Science, Science, Mathematics, Art, English, Art, Science, Mathematics, Science, Art.
Which is the most liked subject and the one least liked?
⇒ It is not easy to answer the question looking at the choices written haphazardly.
We arrange the data using tally marks.
Subject  Tally Marks  Number of Students 
Art  `cancel()`   7 
Mathematics  `cancel()`  5 
Science  `cancel()`   6 
English    4 
The number of tallies before each subject gives the number of students who like that particular subject. This is known as the frequency of that subject.

Frequency: Frequency gives the number of times that a particular entry occurs.

Frequency tables: A frequency table shows the list of categories or groups of things, together with the number of times the items occur.
B. Grouped Data:

In grouped data, observations are classified into groups.
 For example, a class of students got different marks in a midterm exam. The data is tabulated as follows:
Mark Interval 
020 
2140 
4160 
6180 
81100 
No. of Students 
8 
14 
45 
39 
8 

This shows how many students got a particular mark range. Grouped data is easier to work with when a large number of data is given.
Class interval related concept:
Class Interval: While arranging a large amount of data in statistics, they are grouped into different classes to get an idea of the distribution, and the range of such class of data is called the Class Interval.
Upper Limit: In each class interval, the greatest number is the upperclass limit.
Lower Limit: In each class interval, the smallest number is the lower class limit.
Class Size: This difference between the upperclass limit and lower class limit for each of the class intervals is equal is called the width or size of the class interval.
Class Mark: The midpoint of each class interval is the class mark.
Class Mark = `"Upper Limit + Lower limit"/2`.
Tally Mark: Tally Mark refers to a group of five marks that should be used as a cross, as shown by `cancel(}`. They are tally marks.