#### Topics

##### Rational Numbers

- Concept of 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 - 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 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

**Rational Numbers between two rational numbers:**

If m is an integer, then m + 1 is the next bigger integer. There is no integer between m and m + 1. The integers that lie between any two non-consecutive integers can be counted. However, there are infinitely many rational numbers between any two rational numbers.

There are unlimited numbers of rational numbers between two rational numbers.

**List rational numbers between – 2 and – 1.**

Let us write – 1 and – 2 as rational numbers with denominator 5.

We have, `- 1 = -5/5 and -2 = - 10/5`

`"So", -10/5 < -9/5 < -8/5 < -7/5 < - 6/5 < -5/5 or - 2 < -9/5 < -8/5 < -7/5 < - 6/5 < -1.`

The rational numbers between –2 and –1 would be, `(- 9)/5, (- 8)/5, (- 7)/5, (- 6)/5`.

**Taking another example, let us find numbers between the rational numbers `1/2 and 3/5`.**

Let us first convert each of the numbers into their equivalent fractions.

For example, `1/2 = 5/10, and 3/5 = 6/10`

On the number line, there are points showing the numbers `5/10 and 6/10`.

Let us find the mid-point of the line segment joining these two points and see what number it shows.

`1/2 (5/10 + 6/10) = 11/20`

This is the mid-point of the line segment.

Because `6/10 - 11/20 = (12 - 11)/20 = 1/20` and

also `11/20 - 5/10 = (11 - 10)/20 = 1/20`

Thus, `11/20 "is the number that lies exactly in the middle of" 5/10 and 6/10`.

It means that, `11/20 "is a number that lies between"1/2 and 35`.

Similarly, we can find numbers that lie between `1/2 and 11/20 "and between" 11/20 and 3/5`.

#### Example

– 2 can be written as `(-20)/10 "and" 0 "as" 0/10`.

Thus we have `(-19)/10, (-18)/10, (-17)/10, (-16)/10, (-15)/10, ..........., (-1)/10` between - 2 and 0.

#### Example

Find any ten rational numbers between `(-5)/6 "and" 5/8`.

#### Example

Find a rational number between `1/4 "and" 1/2`.

`(1/4 + 1/2) ÷ 2 = ((1 + 2)/4) ÷ 2 = 3/4 xx 1/2 = 3/8.`

`3/8 "lies between" 1/4 "and" 1/2`.