Maths and how to study maths or mathematics

MATHS

I have been teaching maths from quite a lot time and my experience with maths has made me all the more popular with students who want to learn math. I started teaching maths when I was at the age of 15. I was in 9th standard then and I used to teach maths to students from my class as well as from lower classes. I used to frame my own questions on maths by using the things and variable from around our world. Like being from the country side, I would always give example of cows and buffalos in numbers and as variables what the city teachers use as toffees and chocolates. Maths isn't about cramming up maths formula and putting the relevant numbers of the questions in the formulae, but it is about knowing how well you can think with numbers. Maths is actually a play, a game. In this game you try to manage numbers in the same way as you would manage your pocket money. Think what would it be like if you forget maths for a weekend. Nothing ! Just your parents will save a little money because you will be happy with whatever pocket money they give, provided you don't know numbers and maths to count them :)

Maths isn't about scoring and forgetting. It is all the more important in our daily lives too. If you don't know maths you can be cheated at every moment. A layman doesn't know much about loans and interest he pays to the bank. It is not because he is illiterate and cannot read or write, but it is because he is mathematically unaware. He simply cannot calculate and relies on other. If you can learn little maths you can make a wise choice between the offers of two banks.

Maths is same to children as Ocean is to sailor. But ultimately the sailor sails and at one time he crosses the whole big ocean. Similarly the children initially fear maths but when the start learning maths, they will learn it eventually. And yes this maths is helpful to them in their later years of life. Be it professional or even social or personal too. Maths is so important that it is included in everybody's life whether you learn it from school or not. When you did not join school even then you have learnt maths in your life. Example- When a mother offers a chocolate to a small kid. He accepts it. The mother does it for five days. The sixth day mother offers him two chocolates. This continues for next 5 days. Now the small child has learnt his numbers. He knows difference between one and two though he may not know difference between 10 and 20. But maths has started to work in his brain. Next day if he is shown two chocolates and given only one. He will cry or push the whole house on his head until he has his second chocolate.

So Maths is something very inherent in all of us.

So why Run away from maths. Maths is not a nightmare. It is only that you are trying to learn maths in darkness that you are not able to see it clearly. If you learn it with clarity and clear concepts, it is very easy and interesting.

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Video Learning 4 Kids - 5 ( SIMILAR TRIANGLES )

SIMILARITY OF TRIANGLES

What actually is similarity ?
It just means that the angles are same. So two triangles are similar if all three angles of first triangle are equal to the three angles of the second triangle.
Suppose I say that the two angles A & B of first triangle are equal to the two angles say P & Q of the second triangle, then it is obvious that the third angle of both will be same because the sum of angles in triangle is 180.
That is why it is enough to say Angle-Angle theorem (AA) and not AAA.
We here must also note that the ratio of sides of triangles is same corresponding to the equal angles.
We must also be aware that capital letters are used for angles and small letters are used for the sides.
So if A=P, B=Q & C=R, then we will have a/p = b/q = c/r
Suppose this ratio turns out to be one. Then these become congruent triangles.
Another way to show similarity is SAS, in this the two sides of the two triangles are in same ratio and the angle included between them is equal. This becomes the SAS theorem (Side Angle Side).



Source:Khanacademy.org








Source : khanacademy.org

Count on Your Time, your @ssets

Ttime is the most important asset of a student. But this asset does not grow. It only slips out of hand. The wise use of time is very crucial today as there are more candidates to any opportunity than the opportunities itself. The time today most of the students spend on is Facebook. Instead of serving the purpose of networking between people for some or any purpose it has become a network of purposeless activities to kill time and to keep busy for no good reason. The main motive behind such social networking sites my dear friends is not to keep login for whole day and publish your new and indifferent status and feeds, rather it should form a forum for you to serve for a purpose, a means to keep in touch and not keep login. Even the founder of Facebook might not be using his idea too that extent to keep all the day hanging on it. If was it true, he could never have been so capable and developed to develop such a mastermind. The question is not quitting, infact the answer is wisely using.
Talking of time I just recalled another puzzling question about time. Consider this:
Problem :
Mohan lives 24 km from the railway station. He has one hour to catch the train. He leaves home at the speed of  18 km/hr. Travels half the distance and then increases his speed to 30km/hr. He  with this speed for the next quarter of the distance and then plans to speed up again. Now what should be his speed in order to catch the train.
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Counting On Matches

Problem:
 There are 17 teams to play in a Game. Matches are played between two teams and No match ends is draw. The loser is out of the game and the winner stays in the game to play ahead. The matches continue to be played among the teams until one team emerges as the winner. So can you tell me how many matches will be played in all to decide the winner of the games.

Solution:


It’s so simple, only if you think it practically and at the same time becomes much difficult the moment you try making any sort of equations etc.

I can explain the answer to you in a very easy way.

Suppose you and 15 of your friends come and play with me. So you all are 16 and me is the 17^th person. Right.

Since No match ends in a draw, so for me to emerge as the sole Winner, I will have to play with each one of you and defeat you to return you back to pavilion. So tell me how many people I will send back if I win. Sixteen. To defeat sixteen people, then how many matches have I played ? Sixteen.

Agreed!!!

That’s it. So simple.  Bubye have a nice concept !

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You are welcome to leave any comment or post your queries to solveaquestion@gmail.com
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FROM THE FUNCTIONS

Here is another simple but a really good concept building problem from the topic Functions and Graphs
The problem must be approached systematically to arrive at the result. After solving the question you will build some confidence to solve the problems related specially to the Greatest Integer Functions.


Problem :

The number of solutions of | [x] – 3x | = 6, where [x] is the greatest integer < or = x, is

1.   2
2.   4
3.   3
4.   No solution

 Solution:
          First of all we must know what is [x].

[x] is the greatest integer function i.e. if the value of x is 2.5 then the greatest integer less than 2.5 is 2. If the value of x is -2.5 then the greatest integer less than -2.5 is -3 and Not -2.

In this problem the nature of x is not specified, so we have to consider the whole range of real numbers.

First we will consider that x is an integer and later as a non-integer.

When x is an integer, which means
[x] = x,

hence the equation | [x] – 3x | = 6 reduces to | x – 3x | = 6 which can be easily solved as below:

x – 3x = 6    &   -(x – 3x) = 6

-2x = 6        &   -(-2x) = 6

x = -3          &      x = 3

When x is not an integer, we can write x as x = n+ k. Where n is the integer and k is the fraction part such that 0 < k < 1.

so the equation | [x] – 3x | = 6 will now reduce to | [n+k] – 3(n+k) | = 6 i.e.

| n – 3n - 3k | = 6

| – 2n - 3k | = 6

What is important to note is that for the result to be an integer as can bee seen on the R.H.S. the factor 3k on L.H.S. must be an integer which is possible only when 3k=1 i.e. when k = 1/3

With k = 1/3, the equation becomes | -2n - 1 | = 6 which can be solved very easily as

-2n - 1 = 6       &      -( -2n - 1 ) = 6

-2n = 7            &        2n = 5

n = -7/2          &        n = 5/2

Hence we see that four solutions are possible to the question. Hence the option (b)

This was a very simple problem and hardly takes two minutes.

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For support and queries mail to mathsprobe@gmail.com

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IIT JEE EXAM 2011- EXAM DATE 10 April 2011, Last date to Apply = 15 December 2010

PROBABILITY OF SOLVING THIS PROBLEM

Probability can be a real fun to solve and scoring topic too but only when right approach is used. A wrong approach can take you far from the answer. Its Probability is much more a case to be thought practically rather than jumping here and there with equations. Probability of solving questions on this topic is inversely proportional to the probability of your losing concentration. So check your concentration.....



Problem :
From a pack of n cards no 1,2,3,4,5.......... n. A draws a card and then puts it back. Now B draws a card. Find the probability that B has drawn the same card as A.
Solution :
A can draw card in n ways.
Similarly, B can also draw card in n ways.
So the total no ways in which A and B can select a card is = n x n.
This becomes the total no of possible cases.

Now A and B can draw the same card nos like in these cases(1,1) or (2,2) or (3,3) or (4,4)....... or (n,n). Hence the total no of favourable cases are n.

So P = No of favourable cases / Total no of cases = n/(nxn) = 1/n



That was quite a simple problem. But the important thing to consider is that how you start to think. If you start with a negative approach of  " its too difficult ..very hard problem... bla bla bla....", it oughts to become the same.  So be positive and explore the ways to achieve the desired results.

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Your suggestions and querries are welcome.
Mail to mathsprobe@gmail.com
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IIT JEE EXAM 2011- EXAM DATE 10 April 2011, Last date to Apply = 15 December 2010