Clocks

CLOCKS

The face or dial of a watch is a circle whose circumference is divided into 60 equal parts called minute spaces.

A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand.

ü In 60 minutes, the hand gains 55 minutes on the hour hand.

ü In every hour, both hands coincide once.

ü The hands are in the same straight line when they are coincident or opposite to each other.

ü When the two hands are at right angles, they are 15 minutes spaces apart.

ü When the hands are in opposite directions, they are 30 minutes spaces apart.

ü Angles traced by hour hand in 12 hrs=360⁰.

ü Angles traced by minute hand in 60 min. =360⁰.

Stock and shares

STOCK AND SHARES

·       The total amount of money needed to run the company is called the stock-capital.

·       The whole capital is divided into small units, called shares or stock.

·       The annual profit distributed among share holders is called dividend. Dividend is paid annually as per share or as a percentage.

·       The value of a share or stock printed on the share-certificate is called its face values or nominal value or par value.

·       The stock of different companies are sold and bought in the open market through brokers at stock-exchanges

·       The broker’s charge is called brokerage.

·       The face value of a share always remains the same.

·       The market value of a share changes from time to time.

·       Dividend is always paid on the face value of a share.

Divisibility

TESTS OF DIVISIBILITY

Divisibility by 2:

A number is divisible by 2, if its unit digit is any of 0, 2, 4, 6, 8.

Divisibility by 3:

A number is divisible by 3 only when the sum of its digits is divisible by 3.

Divisible by 4:

A number is divisible by 4 is the sum of its last two digits is divisible by 4.

Divisible by 5:

A number is divisible by 5 only when its unit digit is 0 or 5.

Divisible by 8:

A number is divisible by 8 if the number formed by hundred’s ten’s and unit’s digit of the given number is divisible by 8.

Divisible by 9:

A number is divisible by 9 only when the sum of its digits is divisible by 9.

Divisible by 10:

A number is divisible by 10 only when its unit digit is 0.

Divisible by 11:

A number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

Polynomial

POLYNOMIALS

A polynomial is an algebraic expression, in which no variables appear in denominators or under radical signs and all variables that do appear are power of positive integers.   For example, the trinomial is not a polynomial;  however, the trinomial is a polynomial in the variable x and y.  The numerical coefficients of the terms in a polynomial are called the coefficients of the polynomial.

Monomial: Polynomials which have only one term are known as monomials.

Binomial: Polynomial which have only two terms are called binomials.

Trinomial: Polynomial which have only three terms are called trinomials.

Constant Polynomial: A polynomial of degree zero is called a constant polynomial.

Linear Polynomial: A polynomial of degree one is called a linear polynomial.

Quadratic Polynomial: A polynomial of degree two is called a quadratic polynomial.

Cubic Polynomial: A polynomial of degree three is called a cubic polynomial.

Quadrilaterals

QUADRILATERAL

A closed geometric figure with four sides and four vertices is called a quadrilateral. The sum of all the four angles of a quadrilateral is 360⁰.

Properties of Trapezium:

1.    Sides – One pair of opposite sides is parallel.

2.    Angles – The angles at the ends of each non-parallel side are supplementary.

3.    Diagonals – Diagonals need not be equal.

Properties of Isosceles Trapezium:

1.    Sides – One pair of opposite sides is parallel, the other pair of sides is equal in length.

2.    Angles – The angles at the ends of each parallel side are equal.

3.    Diagonals – Diagonals are equal in length.

Properties of Parallelogram:

1.    Sides – Opposite sides are parallel and equal.

2.    Angles – Opposite angles are equal and sum of any two adjacent angles is 180⁰.

3.    Diagonals – Diagonals bisect each other.

Properties of Rhombus:

1.    Sides – All sides are equal and opposite sides are parallel.

2.    Angles – Opposite angles are equal and sum of any two adjacent angles is 180⁰.

3.    Diagonals – Diagonals bisect each other at right angles.

Real number system

REAL NUMBER SYSTEM

In real life, we use Hindu Arabic numerals – a system which consists of the symbols 0 to 9. This system of reading and writing numerals is called, “Base ten system” or “Decimal number system”.

Natural numbers:

Counting numbers are called natural numbers. These numbers start with smallest number 1 and go on without end. The set of all natural number is denoted by the symbol ‘N’.

N={1,2,3,4,5,…..} is the set of all natural numbers.

Whole Numbers:

Natural numbers together with zero(0) are called whole numbers. These numbers start with smallest number 0 and go on without end. The set of all whole numbers is denoted by the symbol ‘W’.

W={0,1,2,3,4,5,……} is the set of all whole numbers.

Integers:

The whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z={……-2,-1,0,1,2,……} is the set of all integers.

Rational Numbers:

A rational number is defined as a number that can be expressed in the form , where p and q are integers and q≠0. Here p is the numerator and q is the denominator.

Angles

ANGLE

Two rays starting from a common point form an angle.

       i.            Acute angle:

An angle whose measure is greater than 0⁰ but less than 90⁰ is called an actue angle.

    ii.            Right angle:

An angle whose measure is 90⁰ is called a right angle.

 iii.            Obtuse angle:

An angle whose measure is greater than 90⁰ and less than 180⁰ is called an obtuse angle.

 iv.            Straight angle:

When the ray of an angle are opposite rays forming a straight line, the angle thus formed is a straight angle and its measure is 180⁰.

    v.            Reflex angle:

An angle whose measure is more than 180⁰ but less than 360⁰ is called a reflex angle.

 vi.            Complete angle:

The angle formed one complete circle that is 360⁰. Such an angle is called a complete angle.

Data handling

DATA HANDLING

Data handling is a part of statistics. The word statistics is derived from the Latin word “Status”. Like Mathematics, Statistics is also a science of numbers. The number referred to here are data expressed in numerical form like,

       i.            Marks of students in a class

    ii.            Weight of children of particular age in a village

 iii.            The amount of rainfall in a region over a period of years

Statistics deals with the methods of collection, classification, analysis and interpretation of such data.

Any collection of information in the form of numerical figures giving the required information is called data.

Raw data: The information which is collected initially and presented randomly is called a raw data. The raw data is an unprocessed and unclassified data.

Grouped data: The data which is arranged in groups or classes is called a grouped data.

Collection of data: The initial step of investigation is the collection of data. The collected data must be relevant to the need.

Symmetry

SYMMETRY

Symmetry is an important geometrical concept commonly seen in nature and is used in every field of our life. Artists, manufactures, designers, architects and others make use of the idea of symmetry. The beehives, flowers, tree leaves, hand kerchief, utensils have symmetrical design.

Symmetry refers to the exact match in shape and size between two halves of an object. If we fold a picture in half and both the halves-left half and right half-match exactly then we say that the picture is symmetrical.

For example, if we cut an apple into two equal halves,  we observe that two parts are in symmetry.

A butterfly is also an example of a symmetrical form. If a line is drawn down the centre of the butterfly’s body, each of the butterfly looks the same.

Symmetry is of different types.

       i.            Line of symmetry or axis of symmetry.

    ii.            Mirror symmetry.

 iii.            Rotational symmetry

Terms

TERMS

A term is a constant or a variable or a product of a constant and one or more variables.

Example: 3x², 6x and -5 are called the terms of the expression 3x²+6x-5.

A term could be

       i.            A constant

    ii.            A variable

 iii.            A product of constant and a variable(or variables)

 iv.            A product of two or more variables

In the expression 4a²+7a+3, the terms are 4a², 7a and 3. The number of terms is 3.

Like Terms: Terms having the same variable or product of variables with same powers are called like terms.  Example: x, -5x, 9x are like terms as they have the same variable x.

Unlike Terms: Terms having different variable or product of variable with different powers are called unlike terms. Example: 6x, 6y are unlike terms.

Integers

INTEGERS

Positive integers, zero and negative integers altogether constitute the integers.

In the number lines, the numbers on the right of 0 are increasing and the numbers on the left of 0 are decreasing.

If the sum of two numbers is zero, then they are additive inverse of each other.

Sum of two positive numbers is positive. The sum of two negative numbers is negative.

The sum of positive number and a negative number is either positive or negative or zero.

Subtracting an integer from another integer is same as adding the additive inverse of the second to the first number.

Point, line, line segment and plane

POINT, LINE, LINE SEGMENT AND PLANE

Points indicate a definite position.

A line is a set of points closely arranged.

A straight line extends in both the direction.

A ray is a line with a starting point.

A line segment is a part of a line between two points.

A plane is flat surface which extends indefinitely in all directions.

Two non-parallel lines intersect at a point.

Lines which do not intersect at any point are called parallel lines.

Two lines which intersect each other at right angles are called perpendicular lines.

Two or more points which lie on the same straight line are called collinear points.

Three or more lines passing through a point are called concurrent lines.

fraction and decimal number

FRACTION AND DECIMAL NUMBER

When a whole number is divided into a number of equal parts, we get fraction.

When we multiply numerators and denominators of fraction by the same number, we get equivalent fraction.

To compare, add or subtract like fraction, we can take only the numerators and perform the operation.

To compare, add or subtract the unlike fraction, convert them into equivalent fraction.

We can find a fraction between any two fractions on the number line.

Decimal fractions are fraction having ten or power of ten as denominators.

A decimal fraction has two parts namely

1.    Integral part

2.    Decimal part

They are separated by the decimal point.

All non-negative integers can be considered as decimal numbers.

In a decimal number, the zeros of the extreme right of decimal point has no value.

While adding or subtracting decimals, arrange the decimal numbers according to the place values as we do in whole numbers.