#### Topics

##### Knowing Our Numbers

##### Comparing Number

- Comparing Multiple Digit of Numbers
- Compare Numbers in Ascending and Descending Order
- Compare Number by Forming Numbers from a Given Digits
- Compare Numbers by Shifting Digits
- Introducing a 5 Digit Number - 10,000
- Revisiting Place Value of Numbers
- Expansion Form of Numbers
- Introducing the Six Digit Number - 1,00,000
- Larger Number of Digits 7 and Above
- An Aid in Reading and Writing Large Numbers
- Using Commas in Indian and International Number System

##### Large Numbers in Practice

##### Whole Numbers

- Concept for Natural Numbers
- Concept for Whole Numbers
- Successor and Predecessor of Whole Number
- Operation on of Whole Number on Number Line
- Properties of Whole Numbers
- Closure Property of Whole Number
- Associativity Property of Whole Numbers
- Division by Zero
- Commutativity Property of Whole Number
- Distributivity Property of Whole Numbers
- Identity of Addition and Multiplication
- Patterns in Whole Numbers

##### Playing with Numbers

- Introduction of Playing with Numbers
- Factors and Multiples
- Concept of Perfect Number
- Concept of Prime Numbers
- Concept of Co-prime Number
- Concept of Twin Prime Numbers
- Concept of Even and Odd Number
- Concept of Composite Number
- Concept of Sieve of Eratosthenes
- Tests for Divisibility of Numbers
- Divisibility by 10
- Divisibility by 5
- Divisibility by 2
- Divisibility by 3
- Divisibility by 6
- Divisibility by 4
- Divisibility by 8
- Divisibility by 9
- Divisibility by 11
- Common Factor
- Common Multiples
- Some More Divisibility Rules
- Prime Factorisation
- Highest Common Factor
- Lowest Common Multiple

##### Basic Geometrical Ideas

- Concept for Basic Geometrical Ideas (2 -d)
- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Ray
- Concept of Intersecting Lines
- Concept of Parallel Lines
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Concept of Polygons - Side, Vertex, Adjacent Sides, Adjacent Vertices and Diagonal
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles

##### Understanding Elementary Shapes

- Introduction to Understanding Elementary Shapes
- Measuring Line Segments
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Right, Straight, and Complete Angle by Direction and Clock
- Acute, Right, Obtuse, and Reflex angles
- Measuring Angles
- Perpendicular Line and Perpendicular Bisector
- 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
- Types of Quadrilaterals
- Properties of a Square
- Properties of Rectangle
- Properties of a Parallelogram
- Properties of Rhombus
- Properties of Trapezium
- Three Dimensional Shapes
- Concept of Prism
- Concept of Pyramid

##### Integers

##### Fractions

##### Decimals

- Concept of Decimal Numbers
- Place Value in the Context of Decimal Fraction.
- Concept of Tenths, Hundredths and Thousandths in Decimal
- Representing Decimals on the Number Line
- Interconversion of Fraction and Decimal
- Comparing Decimal Numbers
- Using Decimal Number as Units
- Addition of Decimal Numbers
- Subtraction of Decimals Fraction

##### Data Handling

##### Mensuration

##### Algebra

##### Ratio and Proportion

##### Symmetry

##### Practical Geometry

- Introduction to Practical Geometry
- Construction of a Circle When Its Radius is Known
- Construction of a Line Segment of a Given Length
- Constructing a Copy of a Given Line Segment
- Drawing a Perpendicular to a Line at a Point on the Line
- Drawing a Perpendicular to a Line Through a Point Not on It
- Drawing the Perpendicular Bisector of a Line Segment
- Constructing an Angle of a Given Measure
- Constructing a Copy of an Angle of Unknown Measure
- Constructing a Bisector of an Angle
- Angles of Special Measures - 30°, 45°, 60°, 90°, and 120°

#### notes

**Lowest Common Multiple:**

The Least Common Multiple of the given numbers is the smallest number that is divisible by each of the given numbers. To find the LCM of the given numbers, we write down the multiples of each of the given numbers and find the lowest of their common multiples.

**a) LCM by Prime Factorization Method:**

Find the LCM of 40, 48, and 45.

The prime factorisations of 40, 48 and 45 are;

40= 2 × 2 × 2 × 5

48= 2 × 2 × 2 × 2 × 3

45= 3 × 3 × 5

The prime factor 2 appears the maximum number of four times in the prime factorisation of 48, the prime factor 3 occurs the maximum number of two times

in the prime factorisation of 45, The prime factor 5 appears one time in the prime factorisations of 40 and 45, we take it only once.

Therefore, required LCM = (2 × 2 × 2 × 2) × (3 × 3) × 5 = 720.

**b) LCM by Division Method:**

Find the LCM of 20, 25 and 30.

We write the numbers as follows in a row:

So, LCM = 2 × 2 × 3 × 5 × 5.

(A) Divide by the least prime number which divides at least one of the given numbers. Here, it is 2. The numbers like 25 are not divisible by 2 so they are written as such in the next row.

(B) Again divide by 2. Continue this till we have no multiples of 2.

(C) Divide by next prime number which is 3.

(D) Divide by next prime number which is 5.

(E) Again divide by 5.

#### Example

In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm, and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?

The distance covered by each one of them is required to be the same as well as a minimum. The required minimum distance each should walk would be the lowest common multiple of the measures of their steps.

Thus, we find the LCM of 80, 85, and 90.

The LCM of 80, 85, and 90 is 12240.

The required minimum distance is 12240 cm.

#### Example

Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.

We first find the LCM of 12, 16, 24, and 36 as follows:

Thus, LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144

144 is the least number which when divided by the given numbers will leave remainder 0 in each case. But we need the least number that leaves remainder 7in each case.

Therefore, the required number is 7 more than 144.

The required least number = 144 + 7 = 151.

#### Example

Add: `17/28 + 11/35`

Let us find the LCM of 28 and 35 in order to add the fractions.

LCM = 7 × 4 × 5 = 140

`17/28 + 11/35 = (17 xx 5)/(28 xx 5) + (11 xx 4)/(35 xx 4) = (85 + 44)/140 = 129/140`.

#### Example

On dividing a certain number by 8, 10, 12, 14 the remainder is always 3. Which is the smallest such number?

2 | 8 | 10 | 12 | 14 |

2 | 4 | 5 | 6 | 7 |

2 | 5 | 3 | 7 |

Let us find the LCM of the given divisors.

LCM = 2 × 2 × 2 × 5 × 3 × 7 = 840.

To the LCM we add the remainder obtained every time.

Hence, that number = LCM + remainder = 840 + 3 = 843

#### Example

Shreyas, Shalaka, and Snehal start running from the same point on a circular track at the same time and complete one lap of the track in 16 minutes, 24 minutes, and 18 minutes respectively. What is the shortest period of time in which they will all reach the starting point together?

The number of minutes they will take to reach together will be a multiple of 16, 24, and 18.

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 3

18 = 2 × 3 × 3

LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144.

They will come together in 144 minutes or 2 hours 24 minutes.