#### Topics

##### Integers

- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive Numbers
- Concept of Integers
- Representation of Integers on the Number Line
- Concept for Ordering of Integers
- Addition of Integers
- Addition of Integers on Number line
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers

##### Fractions and Decimals

- Concept of Fractions
- Types of Fraction
- Concept of Proper Fractions
- Improper Fraction and Mixed Fraction
- Concept for Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Fraction as an Operator 'Of'
- Multiplication of a Fraction by a Fraction
- Division of Fractions
- Concept for Reciprocal of a Fraction
- Problems Based on Fraction
- Concept of Decimal Numbers
- Comparing Decimal Numbers
- Addition of Decimal Numbers
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Multiplication of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Numbers by 10, 100 and 1000
- Division of a Decimal Number by a Whole Number
- Division of a Decimal Number by Another Decimal Number
- Problems Based on Decimal Numbers

##### Data Handling

##### Simple Equations

##### Lines and Angles

- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Intersecting Lines
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Complementary Angles
- Supplementary Angles
- Adjacent Angles
- Concept of Linear Pair
- Concept of Vertically Opposite Angles
- Concept of Intersecting Lines
- Parallel Lines
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Checking Parallel Lines

##### The Triangle and Its Properties

- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Classification of Triangles (On the Basis of Sides, and of Angles)
- Equilateral Triangle
- Isosceles Triangles
- Scalene Triangle
- Acute Angled Triangle
- Obtuse Angled Triangle
- Right Angled Triangle
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Angle Sum Property of a Triangle
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Sum of the Lengths of Two Sides of a Triangle
- Right-angled Triangles and Pythagoras Property

##### Congruence of Triangles

##### Comparing Quantities

- Concept of Ratio
- Concept of Equivalent Ratios
- Concept of Proportion
- Concept of Unitary Method
- Concept of Percent and Percentage
- Converting Fractional Numbers to Percentage
- Converting Decimals to Percentage
- Converting Percentages to Fractions
- Converting Percentages to Decimals
- Estimation in Percentages
- Interpreting Percentages
- Converting Percentages to “How Many”
- Ratios to Percents
- Increase Or Decrease as Percent
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Profit or Loss as a Percentage
- Concept of Principal, Interest, Amount, and Simple Interest

##### Rational Numbers

- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers

##### Practical Geometry

- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)

##### Perimeter and Area

- Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangles
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Triangle
- Area of a Parallelogram
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter and Area
- Problems based on Perimeter and Area

##### Algebraic Expressions

- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Like and Unlike Terms
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
- Use of Variables in Common Rules

##### Exponents and Powers

- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Expressing Large Numbers in the Standard Form

##### Symmetry

##### Visualizing Solid Shapes

- Plane Figures and Solid Shapes
- Faces, Edges and Vertices
- Nets for Building 3-d Shapes - Cube, Cuboids, Cylinders, Cones, Pyramid, and Prism
- Drawing Solids on a Flat Surface - Oblique Sketches
- Drawing Solids on a Flat Surface - Isometric Sketches
- Visualising Solid Objects
- Viewing Different Sections of a Solid

- Improper Fraction
- Mixed Fraction
- Conversion between Improper and Mixed fraction

## Definition

**Improper Fraction:** The fractions, where the numerator is greater than the denominator are called improper fractions i.e., numerator > denominator. `3/2, 12/7, 18/5` are all examples of improper fractions.

**Mixed Fraction:** A mixed fraction has a combination of a whole and apart. `3 2/3, 4 1/4, 3 7/8` are all examples of mixed fractions.

## Notes

**Improper Fraction and Mixed Fraction:**

**1. Improper Fraction:**

- The fractions, where the numerator is greater than the denominator are called improper fractions i.e., numerator > denominator.
- Thus, fractions like `3/2, 12/7, 18/5` are all improper fractions.

**2. Mixed Fraction: **

- A mixed fraction has a combination of a whole and apart.
- Thus, fractions like `3 2/3, 4 1/4, 3 7/8` are all mixed fractions.
- Mixed fraction will be written as `"Quotient" ("Reminder"/"Divisor")`.

**3. Interconversion between Improper and Mixed fraction:**

**i) Improper to mixed fraction:**

**1.** `17/4 = (16 + 1)/4 = 16/4 + 1/4 = 4 + 1/4 = 4 1/4`.

i.e., 4 whole and `1/4 "more, or" 4 1/4`.

**2.** `11/3 = (9 + 2)/3 = 9/3 + 2/3 = 3 2/3`.

i.e., 3 whole and `2/3 "more, or" 3 2/3`.

**3.** `28/5 = (25 + 3)/5 = 25/5 + 3/5 = 5 + 3/5 = 5 3/5`

**4.** `19/6 = (18 + 1)/6 = 18/6 + 1/6 = 3 + 1/6 = 3 1/6`.

Thus, we can express an improper fraction as a mixed fraction by dividing the numerator by denominator to obtain the quotient and the remainder. Then the mixed fraction will be written as `"Quotient" ("Reminder"/"Divisor")`

**ii) Mixed to Improper Fraction:**

**1.** `10 3/5 = ((5 × 10) + 3)/5 = 53/5`.

**2. **`9 3/7 = ((7 × 9) + 3)/7 = 66/7`.

**3.** `8 4/9 = ((8 × 9) + 4)/9 = 76/9`.

Thus, we can express a mixed fraction as an improper fraction as

`(("Whole" × "Denominator") + "Numerator")/"Denominator"`.

or

`"Quotient × Divisor + Reminder"/"Divisor"`.