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
- Concept of Reciprocals or Multiplicative Inverses
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
- Multiples and Common Multiples
Linear Equations in One Variable
- Constants and Variables in Mathematics
- Equation in Mathematics
- Expressions with Variables
- Word Problems on Linear Equations
- 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 Linear Equations
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Basic Concept of Polygons
- Classification of Polygons
- Properties of Quadrilateral
- Sum of Interior Angles of a Polygon
- Sum of Exterior Angles of a Polygon
- Quadrilaterals
- 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.
Data Handling
Practical Geometry
- Geometric Tool
- 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
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
- Ratio
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Calculation of 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
- Classification of Terms in Algebra
- 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) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Exponents and Powers
Visualizing Solid Shapes
Direct and Inverse Proportions
Factorization
- Factors and Common Factors
- 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
Playing with Numbers
- Definition: Equation
- Balancing an Equation
Definition
A balanced equation is one in which the value of the left-hand side (LHS) is equal to the value of the right-hand side (RHS).
Concept of Equation
An equation is a mathematical statement with an 'equal to' (=) symbol between two expressions with equal values.
Example: 5 × 3 = 17 - 2.
There are various types of equations, including linear, quadratic, cubic, and others.
Example:
| Equations | Is it an equation? | |
| 1. | y = 8x - 9 | Yes |
| 2. | y + x2 - 7 | There is no 'equal to' symbol. |
| 3. | 7 + 2 = 10 - 1 | Yes |
Parts of an Equation: There are different parts of an equation, which include coefficients, variables, operators, constants, terms, expressions, and an equal sign.
Example
Write the following statement in the form of the equation:
The sum of three times x and 11 is 32.
Three times x is 3x.
Sum of 3x and 11 is 3x + 11. The sum is 32.
The equation is 3x + 11 = 32.
Rules to Keep an Equation Balanced
An equation is like a balanced weighing scale — both sides are equal.
- Add the same number to both sides.
(e.g., if x = 5, then x + 2 = 5 + 2) - Subtract the same number from both sides.
(e.g., if x = 10, then x − 3 = 10 − 3) - Multiply both sides by the same number.
(e.g., if x = 4, then 2 × x = 2 × 4) - Divide both sides by the same non-zero number.
(e.g., if x = 8, then x ÷ 2 = 8 ÷ 2) - Exchange the two sides of the equation.
(e.g., if x = 7, we can also write 7 = x)
Example
Write the following statement in the form of the equation:
If you subtract 5 from 6 times a number, you get 7.
Let us say the number is z; z multiplied by 6 is 6z.
Subtracting 5 from 6z, one gets 6z – 5. The result is 7.
The equation is 6z – 5 = 7.
Example
Write the following statement in the form of the equation:
One-fourth of m is 3 more than 7.
One-fourth of m is `m/4`.
It is greater than 7 by 3. This means the difference `(m/4–7)` is 3.
The equation is `m/4` – 7 = 3.
Example
Raju’s father’s age is 5 years more than three times Raju’s age. Raju’s father is 44 years old. Set up an equation to find Raju’s age.
Let us take it to be y years. Three times Raju’s age is 3y years.
Raju’s father’s age is 5 years more than 3y; that is, Raju’s father is (3y + 5) years old. It is also given that Raju’s father is 44 years old.
Therefore, 3y + 5 =44.
This is an equation in y. It will give Raju’s age when solved.
Example
A shopkeeper sells mangoes in two types of boxes, one small and one large. A large box contains as many as 8 small boxes plus 4 loose mangoes. Set up an equation which gives the number of mangoes in each small box. The number of mangoes in a large box is given to be 100.
Let a small box contain m mangoes.
A large box contains 4 more than 8 times m, that is, 8m + 4 mangoes.
But this is given to be 100.
Thus 8m + 4 =100.
You can get the number of mangoes in a small box by solving this equation.
