#### 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 Circle-graph)
- 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 Circle-graph)
- Graphical Representation of Data as Histograms
- Concept of a Line Graph
- Linear Graphs
- Linear Graphs
- Some Application of Linear Graphs

##### Playing with Numbers

## Notes

**Some Interesting Patterns of Cube Numbers:**

**1. Adding consecutive odd numbers:**

1 = 1 = 1^{3}^{}3 + 5 = 8 = 2^{3}^{}7 + 9 + 11 = 27 = 3^{3}^{}13 + 15 + 17 + 19 = 64 = 4^{3}^{}21 + 23 + 25 + 27 + 29 = 125 = 5^{3}

We can see from the above pattern,

If we want n^{3}, then we needed n consecutive odd numbers such that the sum of consecutive odd numbers is equal to n^{3}.

**2. Cubes and their prime factors:**

Consider the following prime factorisation of the numbers and their cubes.** **

Prime factorisation of a number |
Prime factorisation of its cube |

4 = 2 × 2 | 4^{3 }= 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^{3} × 2^{3}. |

6 = 2 × 3 | 6^{3} = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2^{3} × 3^{3}. |

15 = 3 × 5 | 15^{3} = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3^{3} × 5^{3}. |

12 = 2 × 2 × 3 | 12^{3} = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2^{3} × 2^{3} × 3^{3}. |

Observe that each prime factor of a number appears three times in the prime factorisation of its cube.

If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube.

**1) Is 216 a perfect cube?**

By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3....(factors can be grouped in triples)

Each factor appears 3 times.

216 = 2^{3} × 3^{3} = (2 × 3)^{3} = 6^{3} which is a perfect cube.

**3) Smallest multiple that is a perfect cube**

**Solution:**

## Example

Is 243 a perfect cube?

243 = 3 × 3 × 3 × 3 × 3

In the above factorisation 3 × 3 remains after grouping the 3’s in triplets.

Therefore, 243 is not a perfect cube.

## Example

## Example

^{3}).