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
Subtraction of Algebraic Expressions:
1. Subtraction of Like Terms:
 If we have to subtract like terms then we can simply subtract their numerical coefficients.
 The difference between two like terms is a like term with a numerical coefficient equal to the difference between the numerical coefficients of the two like terms.

Example: 11mn – 5mn = (11 – 5)mn = 6mn.
2. Subtraction of Unlike Terms:
 If we have to subtract the unlike terms then we just have to put a minus sign between the terms.
 Unlike terms cannot be subtracted the way like terms are subtracted means unlike term left as they are.
 Example: 3xy – 7.
Subtraction of General Algebraic Expression:
Horizontal Steps:
 To subtract the general algebraic expressions, we have to arrange them so that the like terms come together, then simplify the terms, and the unlike terms will remain the same in the resultant expression.

Subtract them as we subtract other numbers.

If any term of the same variable is not there in another expression then write is as it is in the solution.

Subtract 24ab – 10b – 18a from 30ab + 12b + 14a
= 30ab + 12b + 14a – (24ab – 10b – 18a)
= 30ab + 12b + 14a – 24ab + 10b + 18a
= 30ab – 24ab + 12b + 10b + 14a + 18a
= 6ab + 22b + 32a.
Vertical Steps:
 We write each expression to be subtracted in a separate row. While doing so we write like terms one below the other.

Subtract them as we subtract other numbers.

If any term of the same variable is not there in another expression then write is as it is in the solution.

The signs in the third row written below each term in the second row help us in knowing which operation has to be performed.

Subtraction of a number is the same as the addition of its additive inverse. Thus subtracting – 6 is the same as adding + 6.
 Subtract 24ab – 10b – 18a from 30ab + 12b + 14a.
30ab  +  12b  +  14a  
  24ab    10b    18a 
  +  +  
6ab  +  22b  +  32a 
Example
7x^{2}    4xy  +  8y^{2}  +  5x    3y  
  5x^{2}    4y^{2}  +  6y    3  
()  (+)  ()  (+)  
2x^{2}    4xy  +  12y^{2}  +  5x    9y  +  3 