Units of measurement

Units are defined for physical quantities and units of measurements are nothing but definite magnitude of these physical quantities. These units of measurement are derived and adopted after some conventions or laws and now we are using these units as a standard for measurement of the physical quantity.
There are various systems for defining units of measurement:

1.       Traditional System: It was used in ancient time.

2.       Metric Systems: It is the system which is being used universally. Many metric systems have been evolved after the adoption of original metric system in 1791. But the standard metric system which is globally used is the International Systems Of Units (SI), it is also known as the modern form of metric system.

3.       Natural Systems: This system consists of unit values which occur naturally in science. They include atomic units (au), solar mass, the megatons etc.

Here are seven units of measurement which are evolved or derived as a basic set from which all other SI units are derived and their corresponding physical quantities:

1.       Length                            meter (which represents a definite predefined length , i.e., if we say 5                                                         Metre or 5m, we mean 5 times the predefined length called “meter”)

2.       Time                                   seconds

3.       Electric Current                 Ampere

4.       Mass                                    Kilogram

5.       Temperature                      Kelvin

6.       Luminous Intensity           candela (cd)

7.       Amount of substance           mole (mol)

We can derive any other physical quantity by simply multiplying units of measurement.

Units of measurement have been playing a very significant role in various tasks since 1791: like fashioning clothing, construction of dwellings of definite size and shape, physics, medicine, metrology etc.

 Here is one example of how the derivation of other units takes place from the above seven units of measurement:
Area – the SI unit is meter square (m²)
A=Length * breadth (area of rectangle)
A= m * m = m² (meter square)
As SI unit of length and breadth is meter (m).
Like that we derive many more units.

In the next session we are going to discuss Error and magnitude.

different properties to solve problems in Grade IX

In the mathematical field there are a number of properties available to solve the problem related to math subject. We can use different properties to solve problems in math. In the general sense, the properties to solve problems include distributive property, identity of addition, identity of multiplication, inverse of multiplication, associative property, commutative property etc. Here we are going to discuss about the different properties to solve problems.
There are various types of properties, which are elucidated with their explanation in this session. This part helps the students of Grade IX. These properties help to solve maths problems in an efficient manner.
1) Commutative property:  a + b + c = c + b + a
2) Distributive property:   a(b + c) = ab + ac
3) Associative property:  (a + b) + c = a + (b +c)
Let’s show you how to use these properties with solving quadratic equations.
Quadratic equations are those equations which are polynomial equations of the second degree. Here we show you the form of quadratic equation:
   ax² + bx + c = 0
Here x variable is unknown variable, with the constants as a, b and c with a not equal to 0.
Now we show you how to solve the inequalities of quadratic equation. In these we use the different properties of maths to solve problems.  ( to know more about central board of secondary education, here)

Example: Solve for x in given expression 2x² + 4x = x² – x – 6?
Solution: y = 2x² + 4x
               Y = x² – x – 6
       In that part we perform the operation of simplification by properties to solve problems. Let’s show you below:
              2x² + 4x = x² – x – 6
              2x² + 4x- x² + x + 6=0
              x² +5x + 6 =0
Now we perform the factorization of equation to solve it.
                x² +5x + 6 = 0
              (x + 2)(x+3) =0
So we can say that:
          x = -2 and x = -3
In the next session we will discuss about Units of measurement. 

Arc Lengths

Arc lengths:
Let us talk about arc length for grade IX students. An arc length is the length of the part of the circumference of any circle. For measuring arc length, we will first understand the concept of radian.
The equation of radian is:
l=rθ
here, l represents the length of the arc.
Every arc will make an angle θ at the center of the circle whether it is large or very small.
The ratio of the circumference to the diameter of the same circle represents the radian.
In mathematical form,
∏=circumference/diameter      ------------- (1)
Now, as we know that
diameter=2*radius
putting In equation 1, we get
∏=circumference/2*radius.
Or, 2∏=circumference/radius
It is one revolution of a circle. One revolution of a circle can be divided into four right angles. The circumference covers these four right angles. (know more about cbse board, here)
Here,
∏ stands for radian measure for 1 revolution.
The arc length formula is:
θ =l/r,
where, l=arc length
r=radius of circle.
radian measure(θ)=l/r
The ratio l/r is a real number and it is calculated as an angle (θ) at the center. The angle (θ) is measured in radians.
Let us discuss a theorem below,
Theorem: if the ratio of the arc length to the radius of circle is same for the two circles, then they must possess same angle θ at the center.
In mathematical terms,
if l1/r1=l2/r2
then,
θ1=θ2
Let us take an example to calculate arc length.
Example 1: calculate the arc length of a circle if it is given that the radius is 10 cm and central angle is 2.35 radians.
Solution:
As we have discussed earlier,
θ=l/r
l is the arc length of the circle
hence,
l=θ r
θ=2.35 radians,
θ is already in radians.
If it is given in degrees, we will have to first change it into radians.
L=2.35*10=23.5 cm.
Let us discuss another example based on the theorem above.
Example 2: Two circles have the same angle θ at the center created by their arcs.
It is given that r1=20 cm
r2=15 cm
l1=25 cm
l2=?
Solution:
we have to calculate the length of the arc of the second circle.
As given θ1=θ2=θ,
using the theorem,
l1/r1=l2/r2
it implies,
l2=(l1/r1)*r2
l2=(25/20)*15
l2=18.75 cm
In the next session we will discuss about different properties to solve problems in Grade IX

Direct indirect measures

Direct and indirect measures in maths for Grade IX:
Direct measure is a measurement of something that we actually want while the indirect measurement is that we are measuring something other than the actual thing of interest.
Let us learn Direct and indirect measures through examples:
Direct measure is looking at actual samples of students’ work produced in the programs organized by the school authority while indirect measurement is gathering information from the sources other than looking at the actual programs those were attained by the students. (know more about cbse sample papers, here)
A direct measurement is counting the number of people those who have entered in a stadium through the gate while the indirect measurement is counting the number of people entering in the stadium with the help of tickets sold from the ticket window. It can be false or not accurate because of all the tickets those that were sold, it is not necessary that they have been used to get entry in the stadium.
If we count the number of students using term papers taken by them by the number of copies then it is the direct measurement. If the number of students is counted on the paper basis then it is indirect measurement because all students may not have taken the exam.
Written surveys and questionnaires are also the examples of indirect measurement.
Let’s have another example. If a person goes to a doctor than he might get a form to fill in which some questions are like:
• Any person in your family affected by cancer.
• Is that person taking medicines like painkiller regularly?
• Is that person is allergenic to anything.
This is like the survey and definitely would not give accurate result. It’s just a measurement or survey that will provide the general information related to the population. A lot people fill these types of forms. So a general percentage can be calculated but it might not be accurate. Accuracy is far away from the result drawn.
In the next session we will discuss about Arc Lengths.

Rate, distance in geometry

In the mathematics we study a lot of concepts which are related to our daily life. These concepts help to solve the problems related to mathematical calculations. In the mathematical field there are many ways to represent the things in the measurable quantity. These measurable quantities provide an easy and convenient way to represent the things, which help in to performing the various tasks related to our daily routine life.
Here we are going to discuss about the rate and distance in geometry. Rate contains the collection of several measurable quantity or measures to specify the quantity of the product or things. In simple terms rates is a ratio, which demonstrates the relationship of a thing to the measurable quantity.

Suppose John makes a trip to Hollywood, in which he covers a distance of 250 miles in 5 hours then what is the speed of his car?
This would be represented by dividing the distance by the time.
            Let’s see                    250/5 = 50 miles per hour
It means john covers the distance at the rate of 50miles/hour. Here ( / )symbol represent ‘ at per’.
Now we show the relationship between rate and distance. As we describe rate termed as ratio, which represents the comparison of two different numbers to display them in measurable quantity.
Now we will see an example to represent the relationship between the rate and distance:
Example: Suppose there is a shop where jack purchases 5 liter milk. Now he pays $15 for the milk. So find the amount of 1 liter milk.
Solution: Here given that purchase milk = 5 liter
                               Paid amount             = $ 15
                         So, amount of 1 liter   =       $15 / 5
                                                                =   $3/liter
                                                 
In the next session we are going to discuss time and angle measures

time and angle measures

Time and angle measures are one of the important types of mathematical problems. We all know that time problems (clock angle measure problem) are all about finding the angles between the hands (Hands are minute hand or hour hand or second hand) of an analog clock. This is very important part of geometry. In geometry the measurement unit for angle is the “Degree”. A circle is divided into 360 equal degrees.  Moreover, degrees are divided into two terms, known as Minutes and Seconds. This division of degree is not universal. (To get help on central board of secondary education books click here)
These problems generally relate to two different measures which are angles and time. For such problems we always consider the change rate of the angle in degree per minute (deg/min) and we may also use arc lengths in this. In an analog clock the hour hand of the clock turns 360 degree in 12 hours which are 720 minutes or we can say 0.5 degree per minute. In the same analogue clock the minute hand turns 6 degree per minute or 360 deg in 1 hour (60 minutes).
Now to solve related problems we need to apply some formulas in form of general equations. These equations are:
1.      Equation for angle of the hour hand of the clock:
Ahr = ½ Me = ½ ( 60 H + M)
Here Ahr is the angle of the hand measured clockwise from the exact time 12’o clock. M is the minute past hour and Me is minute past 12’o clock.
2.      Equation for degree on minute hand of the clock:
Amin  = 6*M
Where Amin is the angle of the hand measured clockwise from the time 12’o clock and its unit is degree.
For example: Time given is 5:30 then the angle of hour hand in degrees:
Ahr  = ½(60*5 + 30) =165 degree
Similarly for the same time the angle of minute hand in degree:
Amin  = 6*30  = 180 degree.
So today we learnt about Times and Angle measures. In the next session I will tell you about arc lengths and In the next session we will discuss about Rate, distance in geometry. 

Isosceles triangle theorem

Hello friends. In this blog we will learn about the isosceles triangles and the theorems related to such triangles. The isosceles triangles are used in the geometry of grade IX standard of the geometry mathematics. As per the definition of the isosceles triangles, the isosceles triangles are those triangles whose two sides are congruent to each other; this means that a triangle which has its two sides equal is an isosceles triangle. Now if the two opposite sides of any triangles are equal then the both opposite angles will also be equal in the magnitude. The figure of the isosceles triangles can be given as:

Now, let us talk about the isosceles triangle theorems. The theorem for isosceles triangles states that the angles apposite the both equal sides are also equal. The isosceles triangle theorems with their relative proof can now be given as:
Theorem: The first theorem for isosceles triangle states that if the two sides of a triangle are congruent (Congruent stands for similarity in the shape and size of the sides in triangle) then the angles opposite them will also be congruent. To prove the theorem, let us draw a triangle XYZ in which two of the sides (side XY and side YZ) of triangle are congruent to each other. For the purpose of proving let us draw a bisector line YM, which intersects the base line XZ at a mid point M. Now the two triangles XYM and ZYM in the original triangle XYZ are congruent to each other. They are congruent because the line YM is common between them and also congruent sides for both triangles and that’s why the other lines XY and YZ both are also congruent.  The angles between them (angle XYM and angle ZYM) are also congruent to each other and from figure angle YXZ and YZX are both corresponding angles of the triangles XYM and ZYM. So, they are also congruent. Hence it is proved that the angles opposite the both equal sides are also equal. (To get help on ICSE Board Syllabus click here)

The same theorem in some other words can also be explained as, that if two opposite angles of a triangle are equal then the opposite sides will also be congruent and In the next session we will discuss about time and angle measures. 

Rectangular coordinate system in Grade IX

In this section, we will discuss about the topic Rectangular Coordinate System, which you need to study in grade IX.
Mainly there are two types of coordinate System.
1-cartesian coordinate System
2-polar coordinate System

Cartesian coordinate system:
As you can see in above figure, every point on x Axis and y axis is shown uniquely in the plane. Every point either on x axis or on y axis are at equal distance from each other. If you are given 3 Cartesian coordinate in the plane you can determine the exact position of point with the help these coordinates. Using coordinate system, geometrical things like curve can be described by Cartesian equation.
For example, a circle of radius 16 can be described in x and y coordinate can be written as x²+y²=4.
There are 3 types of dimensions
One dimension System-
In one dimension only x coordinate is there rest coordinate are 0
For example-(5, 0, 0) here dimension of x is 5 and dimension of y & z is 0
Two dimension System-
For two dimensions there are two coordinate one is in x direction and other is in y direction and z coordinate is 0.
For example – (5, 4, 0) here dimension of x is 5 ,dimension of y is 4 & dimension of z is 0.
Three dimension System-
For three-dimension all the 3 coordinate are present
(4, 5, 6)
Here dimension of x is 4, dimension of y is 5 and dimension of z is 6.
Now we will move to next coordinate system that is polar coordinate system
In this system the given points are in two dimensions and each point in the plane is determined by a distance and direction is decided by angle. The point which is fixed is called is called pole. And the ray from the pole in the fix direction is called polar axis. The distance from the pole is called radius and angle is called polar angle. The radial coordinate is given by r and the angular coordinate is given by θ or t. Angle in polar System are generally expressed as degree or radian (1 radian=360degree).
Conversion of polar coordinate to Cartesian coordinate:
The two coordinate r and θ can be converted to x and y coordinates by the following formulas X=rcosθ and y=rsinθ. (To get help on cbse sample papers click here)
In the next topic we are going to discuss Rate, distance, time, angle measures, and arc lengths and In the next session we will discuss about Isosceles triangle theorem. 

Math Blog on Planar cross-sections, perpendicular lines and planes

A plane section consists of two-dimensional flat surface. The two dimensional analogues are the points with zero-dimensions, points with one-dimension and also with three-dimensional plane. From the theory of Euclidean geometry planes are seen as the parts of that space with higher dimensional space.
If we take a two-dimensional Euclidean space then we can say that it is a whole space and most of tasks in geometry and trigonometry are performed by using it. Planar cross sections are the important concepts in math. When a plane surface can be crossed by the intersection of a solid plane then it is known as the cross section. Some properties of planar cross section are given below,
Two planes either intersect in a line or they are parallel.
Two lines are parallel to a plane surface, they are contained in the same plane or meet at a single point.
Two lines are said to be parallel to each other if these lines are perpendicular to each other.
Perpendicular lines: If two lines intersect then four angles are formed at the intersecting points of the two lines. The condition of perpendicular line is that two lines are said to be perpendicular if four angles which are formed are equal and also if the slope of one is the negative reciprocal of the other then two lines are perpendicular. Let us suppose that if the slope of one line is n, then slope of the other is equal to the negative reciprocal that is -1/n. The perpendicular makes a right angle or 90° and it shows a relationship between two lines. On other hand parallel Lines are those lines which never intersect each other and so that parallel lines have the same slope. Perpendicular lines are those lines that intersect only at right angles. Because of this reason product of slopes of parallel lines is equal to – 1. (To get help on CBSE Sample papers click here) and In the next session we will discuss about Rectangular coordinate system in Grade IX.