Topics
Rational Numbers
 Concept of Rational Numbers
 Closure Property of Rational Numbers
 Commutativity Property of Rational Numbers
 Associativity of Rational Numbers
 Distributivity of Multiplication Over Addition for Rational
 Identity of Addition and Multiplication
 Negative of a Number
 Additive Inverse of Rational Number
 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  Regular Polygon, Irregular Polygon, Convex Polygon, Concave Polygon, Simple Polygon and Complex Polygon
 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 Or 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
notes
Drawing a double Bar Graph:
1) The number of boys and girls in a school is given. Draw a double bar graph to show this information.
Class  5^{th}  6^{th}  7^{th}  8^{th}  9^{th}  10^{th} 
Boys  52  68  67  50  62  60 
Girls  57  63  64  48  62  64 
Solution:
Steps for drawing a Joint Bar Graph:
 On a graph paper, draw the Xaxis and Yaxis and their point of intersection.

Keeping the distance between two sets of joint bars equal shows the classes on Xaxis.

Choose a scale for the Yaxis. For example, 1 unit = 10 girls/boys. Mark the numbers of boys and girls on the Yaxis.

Using the scale, work out the height of columns required to show the numbers of boys and girls in each class. Use different colours to show the different bars in each set.
2) A mathematics teacher wants to see, whether the new technique of teaching she applied after a quarterly test was effective or not. She takes the scores of the 5 weakest children in the quarterly test (out of 25) and in the halfyearly test (out of 25):
Students  Ashish  Arun  Kavish  Maya  Rita 
Quarterly  10  15  12  20  9 
Halfyearly  15  18  16  21  15 
To represent this data in the form of a double bar diagram, here are the steps.
 On a graph paper, draw the Xaxis and Yaxis and their point of intersection.
 Keeping the distance between two sets of joint bars equal shows the students on Xaxis.
 Choose a scale for the Yaxis. For example, 1 unit = 5 marks. Mark the numbers of marks on the Yaxis.
 Using the scale, work out the height of columns required to show the marks of quarterly and halfyearly of each student. Use different colours to show the different bars in each set.
Shaalaa.com  Drawing a Double Bar Graph
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The number of saplings planted by schools on World Tree Day is given in the table below. Draw a joint bar graph to show these figures.
School Name 
Name of sapling  Almond  Karanj  Neem  Ashok  Gulmohar 
Nutan Vidyalaya  40  60  72  15  42  
Bharat Vidyalaya  42  38  60  25  40 
The table below shows the number of people who had different juices at a juice bar on a Saturday and a Sunday. Draw a joint bar graph for this data.
Days  Fruits  Sweet Lim  Orange  Apple  Pineapple 
Saturday  43  30  56  40  
Sunday  59  65  78  67 
The following numbers of votes were cast at 5 polling booths during the Gram Panchayat elections. Draw a joint bar graph for this data.
Persons  Booth No.  1  2  3  4  5 
Men  200  270  560  820  850  
Women  700  240  340  640  470 
The maximum and minimum temperatures of five Indian cities are given in °C. Draw a joint bar graph for this data.
Temperature  City  Delhi  Mumbai  Kolkata  Nagpur  Kapurthala 
Maximum temperature  35  32  37  41  37  
Minimum temperature  26  25  26  29  26 
The numbers of children vaccinated in one day at the government hospitals in Solapur and Pune are given in the table. Draw a joint bar graph for this data.
City  Vaccine  D.P.T. (Booster) 
Polio (Booster) 
Measles  Hepatitis 
Solapur  65  60  65  63  
Pune  89  87  88  86 
The percentage of literate people in the states of Maharashtra and Gujarat are given below. Draw a joint bar graph for this data.
State  Year  1971  1981  1991  2001  2011 
Maharashtra  46  57  65  77  83  
Gujarat  40  45  61  69  79 
There are four polling booths for a certain election. The numbers of men and women who cast their vote at each booth is given in the table below. Draw a joint bar graph for this data.
Polling Booths  Navodaya Vidyalaya 
Vidyaniketan School 
City High School  Eklavya School 
Women  500  520  680  800 
Men  440  640  760  600 