Proportion

Proportion
The equality of two ratio is called Proportion.
If a/b = c/d or a : b = c : d then we say a, b, c and d are proportional. It is denoted by : : symbol. In simple words, a : b : : c : d denotes the four numbers are proportional.
Types of Proportion

Mean Proportion:- Mean proportion between a and b is √ab
Example: - Find the mean proportion of 4 and 9
Solution:- Mean Proportion = √4 x 9 = √36 = 9

Third Proportion: - If a : b : : b : c then c is called the third proportion to a and b.
Formula to find Third Proportion: - c = b^2 / a
Example: - Find the third proportion of 4 : 6
Solution: - Third proportion = 6^2 / 4 = 9

Fourth Proportion:- If a : b : : c : d then d is called the fourth proportion to a, b , c and d. Fourth proportion = b c /a
Example: Find the fourth proportion to 3, 7 and 9
Solution: - Here, 3 : 7 : : 9 : x then the fourth proportion = bc / a = 7 x 9 /3 = 21

Basic Rule involved in Proportion

 Invertendo:- For a : b :: c : d the invertendo = b : a : : d : c

 Alternando : - For a : b :: c : d the alterando = a : c :: b : d

 Componendo :- For a : b :: c : d the componendo = a + b : b : : c +d : d

 Dividendo : For a : b :: c : d the dividendo = a – b : b : : c – d : d

 Componendo and Dividendo :- For a : b :: c : d the componendo and dividendo = a + b : a = b = c + d : c – d

Continue...........


Dr Rajesh Kumar Thakur

Ratio

Ratio and Proportion
Ratio is a relationship between two numbers that indicates how many times the first number contains the second. Suppose I bought 2 orange and 8 apples then the ratio of oranges to apple is two to eight.

This ratio can be therefore written as 2: 8 which is equivalent to 1: 4.
A ratio is a comparison of two numbers. For example, if 25 students -- 10 boys and 15 girls -- are in your class, then the ratio of boys to girls is 10 to 15.
The numbers compared in ratio can be any quantities of a comparable kind such as lengths, persons, any objects etc. The ratio of numbers a and b can be expressed as
1. a to b
2. a is to b
3. a : b
In ratio the numbers a and b are called antecedent and consequent respectively.
Ratios are sometimes used with three or more terms. Have you ever gone to a construction sites? A mason mixes sand, cement and water in a proper ratio. If for every 2 kg of sand and 4 kg of cement the mason mixes 9 kg of water then the ratio in this circumstance is 2: 4: 9.

According to the Oxford English Dictionary online, ratio is 'the relation between two similar magnitudes in respect of quantity, determined by the number of times one contains the other (integrally or fractionally)'.
It means the ratio can be expressed in fraction. The ratio of oranges to that of apple as taken above can be written in fraction as 2/8.

There can be a ratio between `100 aand `500 but there can’t be the ratio between `500 and 1000 apples.
Types of Ratio

 Duplicate Ratio: - If two numbers are in ratio, then the ratio of their squares is called duplicate ratio. The duplicate ratio of two numbers a and b is a^2: b^2.
Example: - Duplicate ratio of 5: 6 is 5^2: 6^2 = 25: 36

 Sub- duplicate Ratio: - If two numbers are in ratio then the ratio of their square roots is called sub- duplicate ratio. The sub-duplicate ratio of a and b is a^1/2 : b^1/2
Example: - Sub duplicate of 25: 36 = √25: √36 = 5 : 6

 Triplicate ratio: - If two numbers are in ratio, then the ratio of their cubes is called triplicate ratio. The triplicate ratio of two numbers a and b is a^3: b^3.
Example: Triplicate ratio of 2: 5 is 2^3: 5^3 = 8: 125

 Sub- triplicate ratio:- If two numbers are in ratio, then the ratio of their cube- root is called sub-triplicate ratio. The sub-triplicate ratio of two numbers a and b is a^1/3: b^1/3.
Example:- Sub- triplicate ratio of 8 : 125 is 2: 3

 Inverse Ratio: - If two numbers are in ratio such that when the antecedent and consequent of ratio is interchanged a new ratio is obtained which is called Inverse ratio. The inverse ratio of a: b is b : a.
Example:- Inverse ratio of 4 : 7 is 7 : 4

 Compound Ratio:- If two or more ratio are given, then the antecedent of one is multiplied with antecedent of other and respective consequents are also multiplied then we get compound ratio. If a : b , c : d, e : f are three ratio then their compound ratio will be ace : bdf
Example: The compound ratio of 2 : 5 , 5 : 9 and 7 : 18 is 2 x 5 x 7 / 5 x 9 x 18 = 7 : 81


Dr Rajesh Kumar Thakur

Why should I read Mathematics???

Why should I read mathematics?

Mathematics is omnipresent. It is in your life, in your body, in your kitchen, in your garden, in vegetables, fruits, music, art, painting, medicine, buildings, etc. etc. in one word mathematics is in every particle of the world. If it is so then why there does so much rumor about mathematics. Mathematics is a fun loving subject based on truth. Yes, you have the choice to hate mathematics but can’t think to disassociate yourself with mathematics.
Mathematics is present in everyday life; it surrounds a person wherever a person goes. It is involved in our day to day life. People sometimes don’t realize the amount of math that surround them and the amount of math that they use every day. In order to live happily and be competent to live in a society you should be able to do simple arithmetic like addition, subtraction, multiplication and division. You need to be able to handle money, prepare food do your job, purchase the necessary thing for household and all this require the basic knowledge of mathematics.
The study of mathematics is extremely important for many reasons. Most importantly, math surrounds us in many aspects of our everyday life. Our monetary system, system of measurement, mechanical objects such as automobiles and many other aspects of life we encounter daily are highly dependent upon math. If an individual is unable to perform simple mathematical processes, he or she finds himself in a highly disadvantaged situation that lacks a certain understanding of how things operate. These individuals lack beneficial abilities such as pattern recognition and logical reasoning that are developed in mathematical instruction.

It is necessary for people to study Mathematics for many reasons. The universal language of the world is math, and people have been using it for thousands of years across the world. Today’s society would not be in existence without the application of mathematics. The application of math can be seen everywhere throughout the world, and without it a majority of things would not be possible. Complicated things such as building a bridge or buildings, flying an airplane, setting up an industrial units or mass producing anything would not be possible. Even simple things such as buying vegetables or groceries, paying a mobile, electric, or water bill, or cooking would not be possible without math.

Even if someone has no interest in how to build a bridge, or how things are mass produced, to live in today’s society it is necessary for someone to be able to support themselves financially, and money involves math. In order to cook numerous things exact measurements and temperatures, as well as exact times are needed, all involving math. As much as the “normal person” hates math, no one can escape the need for mathematics in their everyday life. Galileo said, "The great book of nature can be read only by those who know the language in which it was written. And that language is mathematics.” Math is simply logic and reasoning. Math is present in everyday life and is being used by every layman. It is the universal language which is understands by every person regardless of what spoken language one use. The product of 2 and 4 is 8 in India, US, UK and any other part of the world
Let me figure out few more reasons to boost your moral if you think mathematics reading is not important.

• Learning math involves a different type of thinking that is not addressed in other subjects. It helps us to think logically and rationally.

• It helps us keep score- not just in sports, but in everything that we measure such as time, distance, speed, money, cooking quantities, etc.

• It helps us to make our choice better. Suppose you are in a shop to buy a motorcycle and your budget is between ` 50,000-60,000. After discussing much you freeze out on two bikes that fit your budget but the problem is which one to buy. Now you will certainly look out for other option that is mileage. With the increasing cost of patrol you would love to buy that bike which is fuel efficient. Keeping this on mind your next query would be which one is fuel efficient. The first one shown here gives 54 Kmpl and the second one gives 87 Kmpl then certainly your choice would be the second one. This clearly shows that mathematical knowledge is essential in making a better choice.


• Let us see the ingredient of matar- paneer recipe. Matar Paneer Recipe
Prep & Cooking: 40 mts Serves 4-5 persons
Ingredients:
1/4 kg paneer 1small cup fresh green peas
1 large onion, finely chop ginger-green chili paste (1″ ginger piece+3 green chillis)
2 tomatoes, finely chopped 3/4 tsp red chili powder
1 pinch of turmeric powder 1 1/4 tsps coriander powder
1/2 tsp kasuri methi (dry fenugreek leaves) 1pinch of garam masala pwd
1 tsp masala powder 1 tsp malai salt to taste 1 tbsp oil

The above ingredient shows that until you are aware of the mathematical terms like half, quarter, etc or understand the basic measurement you can’t be a good cook. Cooking uses fractions and ratios. If you are baking a cake in microwave you need to understand the timing. In order to make cake you have to understand the measurement. If your measurement is incorrect the recipe will not turn out right.

• The most obvious place where mathematics is used in daily life is balancing the household budget. Whether you are a student, daily wage worker or employee, managing house budget, saving, other expense all depends on mathematical calculation. If you are getting a salary of `25000 pm and as per the requirement you need to save `5000 pm to avoid tax deduction. Besides that you pay `2000 for utility bills like electricity, water, gas and `10000 for your household budget such as grocery, vegetables, milk etc. Now you are left with `8000 and you have other expenses to meet such as tuition fee for your children, medical expenses if any. In order to run your house smoothly you have to decide how to expense the remaining amount judiciously so that your daily requirement is met. This all depends upon the mathematical calculation.

• Many of us use credit card. An understanding of mathematical logic helps you determine whether a credit card is really as great a deal as the bank wants you to think. The bank charges some interest and some banks provide you a grace period and after the end of that grace period interest starts to accrue. The interest rate is called APR that stands for Annual Percentage Rate which accrues on unpaid balances. Suppose you have a credit card with an 18% APR with 25 days grace period. If you purchased goods worth `20000 on your credit card and pays back `5000 within the grace period. Now once the grace period is over you will be charged with 1.5% monthly interest on the balance plus the monthly periodic rate. So after a month you will receive the bill of 20000 x 1.5% + 20000 =23000. Since you paid the sum of `5000, so you will be charged the interest of remaining `18000. The best method of using credit card is to pay the balance in full so that you can avoid paying any interest.

• In order to decorate your home you need to understand measurement, area, perimeter, and simple arithmetical calculation. If you want to plant flowers in your garden, you need to calculate the area in order to determine the number of plants that will fit within the space. To calculate how much soil or stone you will need to fill an area or the amount of water you need to water the plant you must have the knowledge of volume. In order to put wire fencing around the garden you need to understand its perimeter.

The above examples are sufficient to understand that mathematics should be read by all and it should be enjoyed because mathematics is the language of truth. It makes us rational in our approach. Mathematics teaches us judgment, truth, honesty and many other virtues which are important for our life. So be ready to embrace mathematics.



Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Multiplication without tears

Multiplication can be made an interesting exercise by learning some tricks. Don’t you think that you should have knowledge of some super-duper trick which could have been saved your time and multiplication could have been done mentally without using pen and paper?

1. When the sum of unit digit is 10 and ten’s digit are same

Rule: - a) The answer will have two parts. Multiply the unit parts. The product should have exactly two digits. If you get the lesser digit put a zero before.
b) Second Part = Ten’s digit x (Ten’s digit + 1)

Example: - 42 × 48 =?
Solution: - Here the sum of unit digit is 10 (2 + 8 = 10) and ten’s digit in both multiplicand and multiplier are same.
Unit Part = 2 × 8 = 16
Second Part = 4 × 5 = 20
Hence, 42 × 48 = 2016

Example: - 74 × 76 =?
Solution: - Sum of unit digit = 10 (4+ 6 = 10) and ten’s digit in both multiplicand and multiplier are same (7).
Unit Part = 4 × 6 = 24
Second Part = 7 × 8 = 56
Hence, 74 × 76 = 5624

Example: - 99 × 91 =?
Solution: - Sum of unit digit = 10 (9+ 1= 10) and ten’s digit in both multiplicand and multiplier are same (9).
Unit Part = 9 × 1 = 09 (In order to fulfill the condition of two digits in the product 0 is put before 9)
Second Part = 9 ×10 = 90
Hence, 99 × 91 = 9009

2. Sum of ten’s digit is 10 and unit digit are same

Rule:-The answer will have two parts
a)LHS = Product of ten’s digit + Unit digit
RHS = Product of Unit digits
Always remember there should be exactly two digits in the RHS. In case there is a single digit in RHS, put a zero before the product you get

Example: - 84 × 24 =?
Solution: - Sum of ten’s digit = 10 (8+2 = 10) and Unit digit in multiplicand and multiplier are same (4)

LHS = Product of ten’s digit + Unit digit = 8×2 + 4 = 20
RHS = Product of Unit digit = 4 × 4 = 16
Hence, 84 × 24 = 2016

Example: - 93 × 13 =?
Solution: - Sum of ten’s digit = 10 (9+1 = 10) and Unit digit in multiplicand and multiplier are same (3)
LHS = Product of ten’s digit + Unit digit = 9 × 1 + 3 = 12
RHS = Product of Unit digit = 3 × 3 = 09 (one Zero is put due to less number of digit in the product)
Hence, 93 × 13 = 1209

Example: - 75 × 35 =?
Solution: - Sum of ten’s digit = 10 (7+3 = 10) and Unit digit in multiplicand and multiplier are same (5)
LHS = Product of ten’s digit + Unit digit = 7× 3 + 5 = 26
RHS = Product of Unit digit = 5 × 5 = 25
Hence, 75 × 35 = 2625

3. When the difference between the digits to be multiplied is 1

When two consecutive numbers are to be multiplied, follow this technique for quicker multiplication.
a) Square the smaller/ greater number.
b) Add / subtract the smaller/ larger number to the previous result

Example: - Multiply 23 by 24
Solution: - a) Square the smaller number = 23 x 23 = 529 b) Add the smaller number to the previous result =529 + 23 = 552
Hence, 23 × 24 = 552

Example: - Multiply 75 by 76
Solution: - a) Square the smaller number = 75 x 75 = 5625
b) Add the smaller number to the previous result =5625 + 75 = 5700
Hence, 75 × 76 = 5700

Example: - Multiply 95 by 94
Solution: - a) Square the Larger number = 95 x 95 = 9025 b) Subtract the larger number to the previous result = 9025 −95 = 8930
Hence, 95 × 94 = 8930

4. When the difference between the digits to be multiplied is 2

a) Find the mean and square it. The mean of two numbers a and b = a + b / 2
b) Subtract 1
Example: - Multiply 25 by 27

Solution: - a) Mean = 25 + 27 / 2 = 26. (Mean can be found by adding 1 to smaller or subtracting 1 from the larger number in this case.)
b) Square the mean = 26 2 = 676
c) Subtract 1 from the last result = 676 – 1 = 675
Hence, 25 x 27 = 675

Example: - Multiply 74 by76
Solution: - a) Mean = (74 + 76)/ 2 = 75. (Mean can be found by adding 1 to smaller or subtracting 1 from the larger number in this case.)

b) Square the mean = 75 ^2 = 5625
c) Subtract 1 from the last result = 5625 – 1 = 5624
Hence, 74 x 76 = 5624

Example: - Multiply 82 by 84
Solution: - a) Mean = (82 + 84)/ 2 = 83. (Mean can be found by adding 1 to smaller or subtracting 1 from the larger number in this case.)
b) Square the mean = 83 2 = 6889
c) Subtract 1 from the last result = 6889 – 1 = 6888

5. When the difference between the digits to be multiplied is 3
Rule: - a) Add 1 to the smaller number and square it
b) Subtract 1 from the smaller number and add to the previous result

Example: - Multiply 28 by 31
Solution: - Here the smaller number is 28.
a) Add 1 to it = 28 + 1 = 29
b) Square it = 29^2 = 841

c) Subtract 1 from the smaller number and add to the previous result = 841 + 28 – 1 = 868
Hence, 28 × 31 = 868

Example: - Multiply 24 by 27
Solution: - Here the smaller number is 24.
a) Add 1 to it = 24 + 1 = 25
b) Square it = 25^2 = 625

c) Subtract 1 from the smaller number and add to the previous result = 625 + 24 – 1 = 648
Hence, 24 × 27 = 648


6. When the difference between the two numbers to be multiplied is 4

Rule: a) Take the mean of numbers. If the numbers are a and b; where a >b then mean is either a – 2 or b + 2 b) Take the square of mean number.
c) Subtract 4 from the result.

Example: - Multiply 65 by 69?
Solution: Here the difference between the numbers to be multiplied is 4. Take its mean.
a) Mean = 65 + 2 = 67
b) Square it = 672 = 4489
c) Subtract 4 from the previous result = 4489 – 4 = 4485 Hence, 65 × 69 = 4485

Example: - Multiply 84 by 88?
Solution: Here the difference between the numbers to be multiplied is 4. Take its mean.
a) Mean = 84 + 2 = 86 b) Square it = 862 = 7396
c) Subtract 4 from the previous result = 7396 – 4 = 7392
Hence, 84 × 88 = 7392



For More on Multiplication without Tears you can buy MATHS MADE EASY published by Rupa Publication

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Mathematics and Industries

3. Mathematics and Industries


Mathematics plays an important role in shaping the industries. It is not merely the investment, selling shares, bonds etc. and finding the profit and loss after each quarter that require the use of mathematics but it will be wise to say that companies decide its every steps on the basis of mathematical calculation.

Here are the two computers on sale in two different shops. The first shop gives 50% +40% discounts where as the second offers 80% discount, the question arises in the mind of a customer which shop to go to buy computer? A customer with no knowledge of mathematics will certainly go to the first shop but a customer who knows that the successive discount given by the first shopkeeper though looks luring but purchasing computer from the second shop will be better.

Suppose the cost of the computer is Rs. 40000
50% discount = 40000 x 50% = 20000
Selling price after the discount = 40000-20000 = 20000
40% discount = 20000 x 40% = 8000
Selling price after the discount = 20000-8000 = 12000
i.e. the customer has to pay Rs12000 for purchasing the computer from the first shop. Now calculate the price one has to pay for purchasing the computer from the second shop.
Cost of computer = Rs40000
80%discount = 40000 x 80% = 32000
Selling price after the discount = 40000-32000 = 8000

I think most of you might be thinking what is the use of differentiation, integration, algebra, and other branches of mathematics in daily life? I was thinking the same until I joined Master degree in Operation Research. During the second year of the course each one of us had to do a project for a company. I, too, had taken a project in IFFCO on Demand Forecasting and Supply Plan for Urea. Working hundreds of hour on computer and applying different statistical forecasting technique I prepared the report. Later I joined an MNC which was developing the Science software mostly on the subjects like Mathematics and Operation Research and there I learnt the use of mathematics in big- big organizations. This part of the chapter will deal with few areas where mathematics is widely used by the companies.

The use of PERT (Programme Evaluation and Review Technique) and CPM (Critical Path Method) technique for the effective management and control of their construction project is widely used by the construction companies.

Use of Assignment models for the allocation of their salesmen to different areas so as to maximize profit is also used by most companies deal in direct sales.

Many companies which manufacture clothes take the help of Linear Programming for blending of cotton. Transportation Method is also used by the companies for minimizing the transportation cost. Manufacturers also make use of calculas in order to be able to utilize raw materials most effectively. Every companies work on the principle: Maximum Profit and Minimum Cost: and that they need the help of LPP. Another problem the companies face is the distribution of materials and products from the plants or warehouses to the shop of retailers or stockists. There are various ways to transport the material from warehousing to shops and cost of each mode is different. Mathematics help the industries to chose the best possible options.

The technique of LPP is also used in business to work out the allocation of resources and to fix levels of production that will create the highest possible profits at the lowest possible cost.

Surveying the market, collection of data from each house before launching the new product in the market is very popular these days; but have you ever thought what the companies do with these data. They take the help of statisticians who apply the statistical method to tell the manufacturer how will the market behave with their products, how much inventory they should keep so that they can meet the market demand etc,etc..

You must have heard of some special brand of products which your family members buy because they trust the quality of that brand. There is a government agencies which test the product and give some mark such as ISI, Agmark etc; this mark on a product guarantee a customer that certain item is a quality product. But you will be surprised to know that whether the product is standard or sub-standard is determined by SQC (statistical quality control) technique.
Companies also use mathematics for Inventory related problem, for minimizing the cost of the product, minimizing the loss and maximizing the profit.

The use of probability is not limited. It has been used to calculate the positions and velocities of electrons orbiting around the nuclei of atom in electron physics. With the help of probability theory it is possible to calculate the percentage of individuals with like and unlike traits in the successive generations.

Recent Cola controversy which put the popularity of Cola companies at stake after the CSE published its report that the amount of pesticides in cola sample is more than the desired level; was based on the statistical method.
The design of an airplane or aircraft requires the use of computational fluid dynamics, partial differential equation and many more mathematical components.

Telecommunication industries, computer industries, pharmaceutical industries are the name of the few industries where mathematics is widely used.

Apart from the above mentioned technique many more techniques are widely used in the industries which are based on mathematics. When you shall grow you will find that mathematics is the most trusted subject and no one can walk a single step without mathematics.

Role of Mathematics Teacher

You and Your MATH Teacher


Teachers are the backbone of an educational system. Children are like clay in a potter’s hands. Just as a potter gives a desired shape to clay in his hands, so do children become what their teachers want them to become. These days, more and more children live in a nuclear family where both parents are working and often, such children rely heavily on their teachers for help, advice and guidance. Teachers have the ability to create a strong like or dislike for the subject they teach their students. Mathematics is considered the most monotonous and insipid subject, but damn sure a true teacher can make it interesting and enjoyable. A good teacher can dispel the darkness from our lives. A good teacher in fact becomes a role model for students. Students tend to follow their teacher in almost every way like manners, style etc. Students tend to get affected by the teacher’s affection as well as love for them.

Dr A P J Abdul Kalam in his address to nation in 2003 on the occasion of Teacher Day said- A student during his school life up to 10+2 spends 25,000 hours in the school campus. His life is, more influenced by the teachers and the school environment. Therefore, the school must have the best of teachers with ability to, teach and love teaching and build moral qualities. Teachers should become role models. Similarly, the student must be alert to build himself with best of qualities and to get ignited with a vision for his or her future life.

Teachers work in close co-ordination with students to help them in building up their future. They mould the students to bring out their skills or improvise them, teaching good habits/attitudes and helping them to become good citizens of the nation.

Picture the scene: - You are in a cave. It is dark, damp, musty, and cold and you are surrounded by solid rock, with nothing but a dim bulb on your helmet and a pick-ax in your hand. But in the face of this bleakness, you keep chipping away and chipping away; because you know that somewhere deep inside that solid rock are some incredible uncut gems and some well disguised nuggets of high grade ores. That is why you become a teacher and you have a duty towards society to give an edge to the knowledge of students by means of your knowledge. There might be some unpleasant moment in your life when you fail to achieve the goal set for yourself but remember problems will crop in when you are doing some innovative work but you should not allow problems to be your masters but you should defeat the problem then only success will sparkle and the taste of this success will be immense. Now let us have a look on the points what a teacher can do to reduce maths anxiety among students.

1. Maths is not merely a test: - Most of teachers base students’ grades on more than just test scores. Many math teachers value neatness, interest in the subject, and punctuality in arriving in class. Let me put you few questions in this regard.

A) Do you hand in homework with your numbers and letters neatly formed?
B) Are your papers unwrinkled and free of stains?
C) Do you appear interested in class?
D) Are you punctual in class?
E) Do you ask questions in class?

Teachers think of these things when composing your grade. So next time you come to class, prepare well in advance the chapter/ topic that is running in class. If you have any doubt, don’t keep silence and raise your hand. If your math teacher asks a question in class, raise your hand and give the answer correctly if possible. Teachers generally make a pre-image of the students and base the grade accordingly. If you are active and paying attention to your teacher then you will certainly not getting an E grade.

2. The Knowledge of Teachers matter most: - Research suggests that greater content knowledge is a contributing factor to increased student learning. A successful mathematics teacher needs a deep understanding of mathematics content to create mathematically rich structures and to adapt to meet the needs of individual students. In addition to content knowledge, a teacher also needs pedagogical content knowledge, or specialized mathematics knowledge for teaching that includes representing the subject to make it understandable to others; knowing what topics will be easy or not; and insight about what pre-conceptions students of different ages and backgrounds have. Teacher can start a new chapter with an interesting story to instill the interest of students in that chapter, tell the story of great mathematicians , let student play some mathematical games etc and can make a class room live with his effort.

3. Involvement in Class activity:- During the class, the teacher should keep circulating , listening carefully to student’s thought and discussion; observing the dynamics of group work and questioning individuals or groups about the strategy they are using and their findings. Teacher should ask students appropriate questions to help them clarify the direction they are taking in solving problem and provide hints to students who need then to solve the problem in a different way. Moreover, it is important for teachers to encourage students to make extension or generalizations of the solutions. Teacher should also encourage students to reflect on their solutions and on the processes they used. Effective teachers engage the classes in productive discourse by letting students communicate their ideas. The classroom activities influence on students’ achievement is two or three times more than the school level. Classroom teaching is nearly a universal activity designed to help students to learn. It is the process that brings the curriculum into contact with students and through which educational goals are to be contacted with students and through which educational goals are to be achieved.

4. Expectation from students: - Expectation from students also affects the learning outcomes of students. Teachers with low expectations of their students will most likely to teach only remedial skills and not move on to teach more advanced concepts or stress conceptual understanding. In common practice, teacher judge the students on the basis of marks/ grade obtained in previous class and make a pre- decision about the students ability in mind. This pre- concept in the mind of students divide the whole class into three sections—students with high IQ, students with medium IQ and students with low IQ and teacher handle the class accordingly. In order to achieve the 100% pass result in mathematics, teacher generally give some suggestion like--- List of Important Questions, list of chapters that require less effort, list of important theorems etc so that even a mediocre student or student with low IQ get at least the minimum passing marks. Teacher too often believe that attempting to teach High Order Thinking is inappropriate when students are struggling. If teachers consistently set high expectations by engaging all students in challenging mathematics, I am sure all students will deliver a positive response. Good teachers set an example and motivate students to take the challenge. Have you heard the story of Isaac Barrow, the teacher of Isaac Newton who resigned from the chair of professor and recommended the University to appoint Isaac Newton instead? Ramanujan, Gauss were the great mathematician and their success was due to their respective teachers.

5. Involvement of Parents: - Teacher should be polite with parents who show up for Parent Teacher meeting held once in a month in every school to know the progress of their children. Teacher should discuss the ways to help the grades of poor performer in a particular test. A unit test or class test can’t decide that students are backward in learning mathematics. The collective efforts of teacher and active involvement of parents can work like a magic for the students. It is common in our society that we assume in advance that mathematics is not meant for this particular student. Mathematics is for all. Euclid whom we regard as FATHER OF GEOMETRY used to draw lines and shapes in his copy and hardly used to care what his teacher was teaching in class. The teacher got annoyed with his behaviour and finally wrote to his father that your son had got mad and you should better consult the doctor for his treatment. Euclid’s father took him to doctor but his madness of drawing lines, geometrical shapes didn’t stop. But in the whole episode Euclid’s mother was highly affirmative and was sure that his son was not mad and he would himself prove this. One day Euclid began to jump with and came yelling to his mother that he had discovered an interesting branch in mathematics. The main aim to write this story was that a parent should have confidence in ability of a student. If parents start losing hope in their children then the frustration of students would certainly increase resulting to dislike of the subject.

6. Never discourage the students: - If you are teaching too fast during a lesson and a student raise his hand and politely ask you to speak more slowly then never scold the students. It is not so that rest of class are following you, there might be many students who aren’t understanding whatever you are teaching but still keeping mum. Listen to his request and teach slowly. Disheartening a student for a mistake that he /she has not committed will take him/her away from the subject. Give your students chance to interact with you. Encourage them to put up question if they are having problem in understanding. Let them be inquisitive, give them space to explore their knowledge. It is easy to learn a subject from a person you respect and students have great respect for his teachers. Your little effort might bring a change in your student’s life. So come forward and listen to the individual problem, though this practice will hamper the class functioning for sometime but imagine the amount of love and affection you will get at the end when 100% of your students will say sir, learning mathematics is fun.

Let me put one example from the life of our former president Dr A P J Abdul Kalam in his own words.
“When I think of my second teacher, I am reminded of my childhood days when I was studying in 8th class at the age of 13. I had a teacher, Shri Siva Subramania Iyer. He was one of the very good teachers in our school. All of us loved to attend his class and hear him. One day he was teaching about a bird's flight. He drew a diagram of a bird on the blackboard depicting the wings, tail and the body structure with the head. He explained how birds create the lift and fly. He also explained to us how they change direction while flying. For nearly 25 minutes he gave the lecture with various information such as lift, drag, how the birds fly in a formation of 10, 20 or 30. At the end of the class, he wanted to know whether we understood how birds fly. I said, I did not understand. When I said this, the teacher asked the other students whether they understood or not. Many students said that they also did not understand. He did not get upset by our response since he was a committed teacher.

Our teacher said that he would take all of us to the sea shore. That evening the whole class was at the sea shore of Rameswaram. We enjoyed the roaring sea waves knocking at the sandy hills in the pleasant evening. Birds were flying with sweet chirping voice. He showed the sea birds in formations of 10 to 20 numbers. We saw the marvelous formations of birds with a purpose and we were all amazed. He showed us the birds and asked us to see that when the birds fly, what they looked like. We saw the wings flapping. He asked us to look at the tail portion with the combination of flapping wings and twisting tail. We noticed closely and found that the birds in that condition flew in the direction they desired. Then he asked us a question, "Where the engine is and how it is powered"?

The bird is powered by its own life and the motivation of what it wants. All these things were explained to us within fifteen minutes. We all understood the dynamics from this practical example. How nice it was. Our teacher was a great teacher; he could give us a theoretical lesson coupled with a live practical example available in nature. This is real teaching. ”


7. Discuss the problem of students: - Furner and Berman (2002) write about the ways to prevent and reduce math anxiety. A great point they made is that teachers need to get an understanding of what their students are going through in maths by journaling and talking with their students. If there are some students in your class who have special needs, such as hearing problem or poor eyesight let them sit in the front row so that they can catch every word and see the chalkboard more clearly. Let your students have a good attitude and perception towards learning mathematics. If a student feels good about learning as a person then learning maths will be less challenging for them. If the student understands the math concepts then he or she is more likely to truly understand the concepts and develop confidence when learning math. It is truly said- A teacher’s purpose is not to create students in his own image but to develop students who can create their own image.

8. Change your teaching style occasionally: If despite your more than 100% effort, your students are not coming up to your expectation, your teaching style is sure to blame. Research suggested that teaching style exerted effects on student achievement that were independent of students’ characteristics. The promise that – one teaching style fits all is attributed to a teacher centered teaching style and it is not fit for a growing number of diverse student populations. Problems occur when teaching styles conflicts with students learning styles resulting to limited or no learning. Leaner centered approach will meet the diverse need. In teacher centered approach the control is in hand of teacher and students have a minimum say in this approach. The general approach of teaching is – I teach, you listen. This is the wrong way of communication. This style of teaching will do no good to students so teacher should adopt another teaching style i.e. I learn, you learn and together we learn. An open ended discussion and change in teaching style with a focus on student learning will give greater opportunity to students to interact with teachers and inquire their doubts with ease. Cooper (1998) found that student learn more in classes where they spend most of their time being taught or supervised by teachers, rather than working on their own. The extent of students’ opportunity to learn in a free and fearless environment directly and decisively on students’ achievement. Jacques Barzun said: - Teaching is not a lost art, but the regard for it is a lost tradition.

9. Provide students opportunity to learn: - The opportunity to learn refers to the amount of time students are given to learn the curriculum in and outside the class. Since the class time is limited say a maths class of 30-40 minutes runs every day so the teacher is bound to give some additional work to students to be completed at home known as Homework. Homework is seen as a contribution towards students’ learning, extending the curriculum beyond the classroom and it can be conceived as one facet of opportunity to learn in the sense that home assignments offer students the opportunity to continue school work after regular school hours. The homework provides students chance to solve the problem independently and if mistakes occur during this process then that will be clearly examined and instructed by the teacher in the very next class. This gives the students chance to explore the knowledge and improve the mistake if any.

10. Brush up your knowledge regularly: - Teaching is a lifelong learning process. Even if you are completely knowledgeable about teaching standards based mathematics, using current technology and understanding new fields of mathematics, within six months or a year there would be a new need for professional development. You can’t be static when the world is in a state of dynamic flux and we daily witness changes in technology, mathematics, schools, students and society. You have to be innovative and adapt the technology in the changing world. You must know that the Fermat Last Theorem proposed in year 1670 was solved in 1995 after a span of more than 350 years. It means you have to be patient and more vigilant towards your goal and keep updating your knowledge so that you can make aware about the changing mathematical scenario to your students. For long pi (π) was considered as the standard notation to refer the ratio of circumference to diameter but in 2012 mathematicians have proposed to change pi with new symbol tau as the circle represent the 360 degree that is equal to 2π , hence the value of pi should be changed from 3.14 to 6.28. Such information should be passed to students and that require that you update your knowledge with recent development in mathematics. In the year 2012 itself Prof Ken Ono has solved the 110 years old Ramanujan Puzzle.

I hope if teacher become friend to students and the fear of learning mathematics disappears from the mind of students, the gap of learning and teaching will be minimum and mathematics learning will be an enjoyable activity. The idea of doing something new will come in the front path and the idea of rote learning that too in mathematics will take a back seat. Once teacher ignite the candle of why and how in mathematics, the learning approach will be a smooth drive.
In order to make teaching learning process more effective I am putting here one table with a hope that it will lighten the stress of teacher and make classroom teaching enjoyable.

Teaching Action Purpose

BEFORE

1. Read the problem... Discuss words or Illustrate the importance of reading
phrases students may not understand carefully; focus on special vocabulary
2. Use whole-class discussion to focus on Focus on important data, clarification
importance of understanding the problem process
3. (Optional) Whole-class discussion of Elicit ideas for possible ways to solve
possible strategies to solve a problem the problem

DURING

4. Observe and question students to Diagnose strengths and weaknesses
determine where they are
5. Provide hints as needed Help students past blockages
6. Provide problem extensions as needed Challenge early finishers to generalize
7. Require students who obtain a solution Require students to look over their work
to "answer the question" and make sure it makes sense

AFTER

8. Show and discuss solutions Show and name different strategies
9. Relate to previously solved problems Demonstrate general applicability of
or have students solve extensions problem solving strategies
10. Discuss special features, e.g. pictures Show how features may influence approach
__________________________________________________________
Table 1 (Adapted from Lester, Garofalo, & Kroll, 1989, P. 26)

The above table will help the teacher to decide which action plan should be implemented for better result in class.



DR RAJESH KUMAR THAKUR
(THIS ARTICLE IS TAKEN FROM MY BOOK ENRICH YOUR MATHS SKILL -- PUBLISHED BY PRABHAT PRAKASHAN)

rkthakur1974@gmail.com

TIPS TO HELP IMPROVE MATH SKILLS TO YOUR KIDS

Tips to Help Improve Math Skills to Your Kids



• Help your child master the basic facts.

It is saddening that the ASER reports shows that more than 70% of students of class 8 cannot do two digit multiplication. The onus falls on all of us as the RTE 2009 has taken the toll and students are doing drastic in Mathematics. If you ask a class 10 students to tell you the table of 15 or square of 34, cube of 14 etc. in maximum number of time you will get a wrong answer. Mastery of the basic facts means that your child can give an answer to a math fact in three seconds or less. Drills are recommended to be able to give quick and accurate responses. My teacher used to say that Drill Mathematics until you get mastery over it.

• Make sure your child understands mathematical concepts.



Mathematics need to visualize. If you fail to make mathematics a lovable subject it is obviously your fault. You need to co-relate mathematics with daily needs or daily uses. You have to put the notion in the head of students that life without mathematics is nothing and they should have a proper attention to master the basic facts, then only they can be master in Mathematics. Explain and review the concept with your child and check for understanding.

• Teach your child to write numbers neatly.


Mathematical errors may occur because of messy number writing or not lining numbers up correctly. Let your child understand the types of numbers with its property as well. Let them differentiate between prime and rational, even and odd, natural and whole. They should understand the place value system and face value of a number and while noting numbers in their copy, they should be aware of the facts which number should be written where and at what place. Improve your child's number writing skills by having them trace over numbers you have written, or by using graph paper to write the numbers neatly and more organized.

• Help correct skill difficulty before your child moves to the next skill level.



Math is a subject in which everything builds upon what has been previously learned. If the foundation stone is missing you can’t expect a building to last long. If your child has a problem in doing addition and subtraction he/she will have difficulty in multiplication and division too. If one is not good in fraction he will not be able to arrange them properly and will fail in doing percentage problem.

• Review skill technique with their math homework.


Homework pays an important role in building the math concept in a child. Let your child be in a habit of doing homework regularly. Check your kids’ copy and give suggestion if a mistake is done. Let your child practice with worksheet or sample papers. Doing math homework reinforces the skills your child is learning in class.

• Encourage your child to do more than the assigned work.


Only the text book is not sufficient. Let your child practice with some other books and sample papers so that he/she can encounter a varieties of problems to solve. It is well said -Practice makes perfect and this is true with your lovely subject mathematics too. Practice is necessary for your child to become comfortable with their math skills.

• Explain how to solve word problems.


Solving word problems is a tough job even for a teenagers. I have seen the students of Higher class in trouble. Let your child be prepared to face such problem boldly and for that you need to Teach your child to read a word problem at least two times, if not more, in order to fully comprehend what is being asked of them. Relate the problem with a real life situation to understand it fully. You can make a chart in order to describe it. Ask them what is being asked, what type of operation is necessary, and discuss the steps necessary to solve the problem. Operation dealing is an area of concern and let your child understand the facts the sign changes when you change the side.

• Help your child learn the vocabulary of mathematics.



Understanding Mathematics words and learning it is important. Let your child understand the word with proper definition. Buy a small dictionary for your child and help them to learn it. You can teach the meaning with examples and proper definition.

• Teach them how to do math in their head- "Mental Math".



This is an era of doing fast calculation. You can read book such as – VEDIC MATHEMATICS, Trenchberg system of fast mathematics and let your child do faster calculation mentally. ABACUS also plays handy in doing fast calculation. Help them to understand that if they know 2+3=5, then they can solve 20+30=50. Similarly if 2 x 3 = 6, so as 200 x 300 = 60000.

• Make math a part of your child's daily life.



The sad parts of Mathematics is that it has the stigma of being unpopular or insipid. The reason behind it that teacher hardly relate mathematics with the daily life. Math will become more meaningful when your child sees how important it is in real life situations. Encourage them to use math in practical ways. I remember that when my son Nilabh was 3 years I used to teach them 3 digits numbers or 4 digits numbers while driving with the help of car or bike number plate. I also use to give him a 10 rupee note and tell him to buy chips or something and teach him the concept of minus, plus or addition.


DR. RAJESH KUMAR THAKUR

युद्ध और गणित (war and mathematics)

आप सोच रहे होंगे की गणित को युद्ध में कैसे प्रयोग किया जा सकता है पर चौंकिए नहीं युद्ध में गणित का प्रयोग प्राचीन समय से होता आ रहा है . ईसा पूर्व 18 वी शदी के बेबीलोनियन पट्टी में युद्ध में पकडे सैनिको के बारे में जानकारी मिलती है साथ ही युद्ध की योजना जैसे- बंकर बनाने में कितने ईंटो की आवश्यकता होगी , छिपने के लिए जमीन को कितना गहरा कर खोदना पड़ेगा और युद्ध बंदियों को रखने के लिए बनाये जाने वाले जेल में कमरे की संख्या का आकलन करने के लिए प्राचीन समय में भी गणना के लिए गणित का प्रयोग किया जाता रहा है .

ईसा पूर्व 18 वी शदी के बेबीलोनियन पट्टी त्रेता युग में जब सीता का अपहरण कर रावण लंका ले गए और श्री राम के लाख प्रयासों के बाबजूद युद्ध अवश्यम्भावी हो गया तो श्री राम की सेना ने लंका की ओर कूच किया और इसके लिए समुद्र पर 100 योजन लम्बा और 10 योजन चौड़ा पुल का निर्माण किया गया. श्री राम की विशाल सेना देख रावण आशंकित हो गया और अपने गुप्तचर शुक को श्री राम की सेन्य बल का पता लगाने के लिए भेजा . शुक बानर का रूप धारण कर श्री राम के सेना में शामिल हो गया और वहां के युद्ध तैयारी का पता लगाया और फिर लंका पहुंचकर श्री राम की सेना के वारे में जो सुचना दी उसका विवरण श्री वाल्मीकि रामायण में इस प्रकार दिया है – इस श्लोक में 1060 तक की संख्या जिसे महौघ कहा जाता था का उल्लेख दिखता है .

अतः श्री राम की विशाल सैन्यबल की कल्पना करना अपने आप में एक डर पैदा करता है . क्या आप जानते है की पृथ्वी का क्षेत्रफल 1016 वर्ग फीट से कम है तो इतनी बड़ी सेना के लिए धरती पर खड़ा होना और युद्ध करना भी असंभव प्रतीत होता है और शायद इसीलिए गोस्वामी तुलसीदास ने रामचरित मानस में लिखा – राम ते अधिक राम कर दासा. इसी प्रकार महाभारत की युद्ध में भी सेना के प्रकार और उसमे सेना की संख्या के बारे में गणना का विवरण दिखता है जी गणित के सार्थक ज्ञान के विना संभव नहीं था . इस श्लोक के अनुसार प्रत्येक रथ में 4 घोड़े को बांधा जाता था और 100 धनुष रखने की व्यवस्था थी . एक रथ के पीछे 10 हाथी और एक हाथी के पीछे 10 घोड़े और एक घोड़े के पीछे 10 पैदल सेना चलती थी . इसका मतलब 1 रथ के पीछे 10 हाथी , 100 घोड़े और 1000 सैन्यबल चलता था और यह सभी संख्या गुणोत्तर क्रम में दिखते हैं. यही नहीं 1 सेना में 500 हाथी और 500 रथ होते थे जिसका सीधा मतलब ये है की एक सेना में पैदल सैन्यबल = 500 x 1000 = 500000. महाभारत में अक्षोनी सेना के बारे में भी वर्णन मिलता है. अक्षोनी सेना का प्रयोग चतुरंग सेना के सन्दर्भ में होता है. एक चतुरंग सेना के चार अंग जिसमे – 1. रथ 2. हाथी 3. घोड़े और 4. पैदल सेना होते थे . एक अक्षोनी सेना के पास 21870 रथ , 21870 हाथी , 65610 घोड़े और 109350 पैदल सेना होते थे और मजे की बात ये है की इनका अनुपात 1 : 1 : 3 : 5 है . यदि एक रथ में सारथी और योद्धा मिलाकर 2 लोग मान लिया जाये तो इनके पीछे चलने वाले हाथी पर 2 लोग सवार हो तो 1 अक्षोनी में 183680 योद्धा होंगे. 2 × 21870 + 2× 21870 + 65610 + 109350 = 262440 उपरोक्त श्लोक के अनुसार पांडवो के पास 7 अक्षोनी सेना और कौरवों के पास 11 अक्षोनी सेना थे अर्थात दोनों के हिस्से में कुल 18 अक्षोनी सेना थे या यूँ कहा जाये तो 18 अक्षोनी में 4723920 सेना उनके तरफ से लड़ रहे थे . इस तरह के सेना किसी चिर परिचित राजा द्वारा उनके समर्थन में लड़ रहे हो सकते हैं पर पूरी महाभारत पढने के बाद आपको लगेगा की ये सेना की संख्या इससे कही अधिक थी . मसलन युद्ध समाप्ति के बाद जब धृतराष्ट्र ने युधिष्ठिर से पूछा की युद्ध के उपरांत कितने सैनिक जीवित बचे और कितने वीर गति को प्राप्त हुए तो युधिष्ठिर का जबाब कुछ और ही बाते बयां करते हैं . युधिष्ठिर के अनुसार युद्ध में मरने बाले सैनिको की कुल संख्या 1 अरब 66 करोड़ 20 हज़ार तथा जिन्दा बचे सैनिको की संख्या 2 लाख 40 हज़ार एक सौ पेशा पैसठ है . महाभारत की लड़ाई की सबसे खास बात चक्रब्यूह की है . चक्रव्यूह या पद्मव्यूह सेनाओं का ऐसा समूह है जिसमे सैनिक 7 सकेंद्रिय वृत के रूप में एक चक्र की रचना करते थे और इसमें घुसने के एक ही रास्ते थे और योद्धा के इस व्यूह में घुसते ही व्यूह दायीं और घूम जाता और निकलने का रास्ता बदल जाता था और व्यूह के योद्धा अन्दर प्रवेश करने वाले योद्धा को मार देते थे . विकिपीडिया के अनुसार चक्रव्यूह की संरचना सैनिकों की एक वहू परतीय रक्षात्मक गठन को दिखता है जो ऊपर से देखने से एक कमल फूल जैसा दिखता था . इसमें हर परत के साथ सैनिकों की संख्या और योग्यता का स्तर बढ़ता जाता था और कृष्ण, अर्जुन , प्रद्युम्न और अभिमन्यु को छोड़ कर किसी पांडव योद्धा को चक्रव्यूह को तोड़ना नहीं आता था,

चक्रव्यूह सैनिकों के युद्ध में वृताकार , स्तम्भ , सारिणी , वर्गाकार और कील के आकार के रचना का उल्लेख पुस्तकों में मिलता है जहाँ सेनापति अपने सेना को अलग अलग आकृति के व्यूह रचना में युद्ध के लिए तैयार करते थे. ग्रीक लोगों के द्वारा वर्गाकार रचना में युद्ध लड़ने का उल्लेख मिलता है. क्या आप जानते है की युद्ध के लिए बनाये जाने वाले अधिकांश उपकरण – जहाज, पनडुब्बी , हथियार, टेलिस्कोप, गोला –बारूद, कैमरा , दूरबीन बिना गणितीय ज्ञान के बनाना संभव ही नहीं है . सेराक्युस शहर की रक्षा के लिए आर्कमिडीज ने एक पंजा डिजाईन किया था जिसमे एक क्रेन जैसे भुजा थी जिससे एक बड़ा धातु का हुक लटका हुआ था .जब इस पंजे को एक आक्रमण करते हुए जहाज पर डाला जाता था तो इसकी भुजा ऊपर की ओर उठती थी और जहाज को उठाकर पानी में डूबा देती थी. यही नहीं सेराक्युस की घेराबंदी के दौरान आर्कमिडीज ने अपने द्वारा बनाये बड़े- बड़े दर्पण का उपयोग कर समुद्र में ही सेरिक्युस की ओर आने वाले जहाज को राजमहल से ही दर्पण द्वारा सूर्य के प्रकाश को फोकस कर आग के हवाले कर देते थे.

Archimedes Burning Mirror

15 वी शदी में तरताग्लिया (? 1500-1557) ने युद्ध प्रक्षेपकी के लिए एक नया सिधांत विकसित किया और इसका उपयोग गोली और गोला दागने में किया जाने लगा और सम्भबतः गणित का युद्ध के तकनीक में यह पहला प्रयोग हो.


राजेश कुमार ठाकुर

HOW TO READ FOR MATHEMATICS EXAMINATION

How to Study For Examinations?

Examination!! Isn’t it a word that makes you tense?

Let me quote a Sanskrit text first and then we shall start discussing how to prepare you for the examination?
तावद भयस्य भेतव्यं यवाद भयमनागतम
आगतं तु भयं विक्ष्यम प्रतिकुर्याधथोचितम
The meaning of the text is ---- Get afraid of fear as long as it doesn’t comes to you. The moment fear approaches you, come forward and embrace it.

Hey guys , don’t get tensed over the examination. The tension at little amount is essential as it motivates you to work harder. Prepare well in advance and keep the tension away. Here are few tips and strategies that will work well if you follow it as advised.

1. Exams are terrible and stressful things to study for, especially knowing that they can make or break your final mark. In the age of Technology, we have different types of books available in the market or e-book available on internet. You can buy a ten years question paper or sample paper to see the pattern of question paper. If you are net savvy, you can download the previous year question paper or sample papers from the CBSE /ICSE or respective Boards’ websites. Besides that, you can download the blue- print of the syllabus which will guide you to decide which chapter will fetch you maximum or minimum marks. The blue print will also guide you about the number of questions being asked in the examination and marking scheme. The most interesting part of the question paper available on the internet on the CBSE website (www.cbse.nic.in) is it has also a solved question paper with detail description of the step-wise marking scheme. If you go through this solved paper, you will get the knowledge about the marks being awarded in the Board Examination. Even this pattern is being followed in the school.


2. You can buy sample papers available in the market. The publishing company has now designed the sample paper on the CCE pattern for Secondary classes. For the senior secondary classes you can get books on sample papers. This book contains solved and unsolved question paper on the Board pattern. Now look for a peaceful space in your home where you can sit 3 hours continuously to solve the sample paper. Mind one thing, you won’t be able to study while there are distraction like ----- A) your little sister or brother is running screaming around the house. B) Your favourite program or film is running on TV. C) There is dim light in your room. D) You are in a mess. Statistics say that 75 % of guys that study in a better light will focus better on study. For girls it is indicated that 90% of time, they study and focus better in a brighter room with little noise, their chance of grasping the matter is better. So chose a peaceful place where you don’t get any distraction. Close the room and sit there for three hours. Don’t practice in installment. Though you can take 1-2 minute break in between but that too in emergency call.
3. After practicing 2-3 papers, reduce the time by 15-30 minutes so that you can keep these 15 minutes for revising the papers. 4. Once you complete the paper, get it examined by your class teacher or tutor so that you get to know the area where you need to put extra labour. 5. The ideal time to start sample paper is 2- month before the final examination takes place. A sample paper book of 20 odd papers should be finished in 30 days so that before the examination is scheduled to happen, you get to revise all the sample papers at least 2-3 times. 6. Solving sample paper will not only control the mistake you make in examination but it will help you to learn the effective way of writing in examination. Besides that, you will learn how to complete your paper within the stipulated time. Solving sample paper before examination will serve three purpose: - a) It will control the mistake b) manages your time effectively c) you learn how to write the paper effectively. What to do a night before examination?



If you are studying one night before examination, you must have some hours left to prepare. So don’t get frightened, you need time management and you can manage to get pass or get good marks depending upon your earlier preparation.


1. Eat some brain food:- A night before examination is a crucial moment for your life. The tension grows if you haven’t had studied much in past or you have examination phobia. Get some brain food. Brain food not only gives us energy and helps us to live longer but it can also help you to get good marks. Green tea is a best brain food as it contents polyphenols. The bitter tasting substance protects the brain from wear and tear plus it helps in mood enhancement. You can eat walnuts, eggs, dark chocolate etc. Brain food boosts your brain’s workability by giving it what it needs to function properly. Moreover, by eating something before study you will be less tempted to get hungry and quit study early.


2. Dress comfortably: - Hey, I am not joking. Go to the bathroom and dress comfortably so you don’t fall asleep. Prepare your body as much as you can so that when you start the study season you have no excuses to get up and go somewhere.

3. Organize your study materials:- Get all the materials such as class-notes, text-book, reference book out on your desk so you can see what you have to work with. This preparation should be done in advance so that you don’t have to break your study every now and then.


4. Don't cram. Cramming the night before is proven to be useless, because you're taking in so much information at once that it's impossible to memorize it at all -- in fact, you'll hardly retain anything. I know it's been preached to you many times before, but it's true: Studying before and going over it multiple times really is the best way to learn the material. This is especially true with things like history and subjects dealing with theory. Most of the math is not memory work but recognizing patterns and applying useful techniques to problems. The only way to learn mathematical skills is to practice.

5. Plan: Always create a plan before you start studying, not to mention that this plan has to be achievable. If out of 5 lessons 3 are easy and can be finished fast, finish them first, that way you can spend quality time on the difficult lessons without fretting. Small tricks like these will help you complete your portions quickly. Go through the previous year question paper to make a list of important questions so that you don’t have to run through all pages. Check whether you have gone through the solved examples or not.


6. Don’t read whole night: - Though Examination is a do or die situation but you can do well in exam if you are a regular reader. Remember one thing that mathematics can’t be read in overnight. There is nothing you can gain just by cramming few hours/days before examination. So make a habit of regular reading. Our brain works like a muscle. It must be exercised regularly to be strong. Don’t put much pressure a night before examination. Let your mind take a rest. Watch film, do some recreational work instead so that your mind is relaxed and can perform better. Study less, with more concentration, if possible. If you want to study for a long time then you can take breaks in between to regain your concentration after you return from the break. It would be a better choice to study in 1 hour slot followed by a 5 minute break. If you try to study indefinitely for hours and hours, your brain will overload and you will have to work to regain your focus on studying.


7. Sleep well: - Nothing will make you do worse on a test than pulling an all- nighter. You may be tempted to study whole night and cram in as much as is possible, but this practice will do no good to you. Sleep well before exams, so your mind will be alert before, after, and during the exam. Reading whole night and sleeping less than the usual hours of sleeping will get you feel tired. It's better to study for two hours in one day than to try and cram in that daily hour of studying at two in the morning. Excess amount of reading one night before exam will make you feel sleepy in the exam hall also. Remember this analogy: 'Study without ambition is a bird without wings'.


8. Don’t call your friend: - Many students keep calling his/ her friend for help or to inquire important questions. Avoid making any call especially one night before examination, because you may get confused. Don't depend heavily on the help of others. Avoid stress from other people, if at all possible. Don't hang around friends that worry and stress. This will rub off on you. Remember don’t call your competitor, although he may look like your good friend. I have seen my competitor calling me to tell the list of important questions one night before examination only to distract my study. Your competitor may make some plan to disturb you so that you get less mark and he gets chance to prove his excellence in class/ board. So better consult your friend a month ago not a night before examination to avoid any last minute mistake.



9. Overcome test anxiety: - Examination and anxiety are supplementary words. You will hardly find any persons on the earth who are not afraid of examination. To a certain extent it is normal to be nervous about taking test, even if you are a very good student. While mild anxiety can be a motivating factor, high anxiety triggers your flight response and makes you avoid the subject that is a dangerous issue. You can go through the chapter how to overcome the math anxiety in the book itself.


10. Get up early and REVISE: - This is one of the last steps I shall advise you before you go for test. Get up early and go through the important formula, HOTS, etc. STOP reading 3 hours before examination. Have breakfast so that your mind gets proper nutrients it needs to work at full capacity. What to do in Examination Just as it is important to think about how you spend your study time, it is important to think about what strategies you will use when you take a test. Good test taking strategies can make a big difference to your score. Strategies to write a math test • Read the instruction first. The instruction is provided to help you and make strategies accordingly.

In many tests there are provision of negative marks so if you haven’t gone through the instruction, you may lose your marks.

• CBSE provides extra 15 minutes to read question paper so look over the entire paper to get a sense of its length. Try to identify those problems you are sure about it and you are having some doubt.

• Write important formula in the margin you think you will need for the exam. Write them all out on your text the moment it gets handed to you. The answer sheet should be folded into three parts. The middle part should have more space and the first and last fold should have less space. Once you finish your paper, cross the rough area to avoid any confusion.

• Go through the question paper properly. Do the easiest one first and save the hardest for last. It is not a wiser idea to try with the hardest. Writing the easiest question first will build your confidence and help you focus your time and energy on the tough ones later. This is a good way to manage your time as well.


• Don’t get panic after reading the question paper. Give yourself time to adjust. Take a long breathe and let it out slowly. Don’t watch other students working as it may put pressure on mind. Cool down and see how many question you can solve. Attempt those question first you find easy. Once you start solving you will find you know some more problems to solve.

• Don’t lose point through careless errors. Check each step as you go, and make sure that it really follows correctly from the previous step. Watch for things like dropped minus signs or missed exponent or mistake you generally make.

• Work properly, writing all steps. Discipline yourself to sow all the steps in your solution and show them one after the other, not little bits of math written here and there. Jumping steps should be avoided. Give reasoning whereas necessary. Underline the important formula. Put your answer in a box. This gives a good look to your answer sheet and makes a better impression in the mind of examiner.


SEND YOUR VALUABLE COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com


Question: - Sum of angle of a triangle is 180 degree. Given: - A triangle ABC To prove :-

Proof: - Since BC // DE and AB is a transversal line. <2 = <4 ------------- (Alternate Angle)

(1) Again, BC // DE and AC is a transversal line. <3 = <5 ----------- (Alternate Angle)


(2) Adding equation 1 and 2 we get, <2 + <3 = <4 + <5 -----------


(3) Add <1 both side in equation (3), we get, <1 + <2 + <3 = <1+<4+<5 Since, <1+<4+<5 = 1800 (Angle on straight line) Hence, <1 + <2 + <3 = 1800 Q.E.D.
• In order to answer the geometric proof the order should be ---

a) Given b) Required to Proof c) Construction d) Proof The proof should have proper reasoning. If you are applying the theorem in order to proof a question, don’t forget to mention it.


• In graph, write the scale and also write the equation of the line. Let us now plot a velocity-time (v - t) graph for the following data. Velocity in m/s 0 10 20 30 40 50 Time in Seconds 0 2 4 6 8 10

• Leave at least two- three lines between two consecutive questions so that it makes a good impression to the examiner.


• Time is very crucial factor in exam. If there are 20 questions to be solved in 3 hours test, then one question will have on an average 9 minutes. Don’t spend much time in one question. If you are working on objective question give less time to it so that you have some extra time to work out in long type question. Keep checking the time so that you can manage the time well.

• Never waste time in erasing. If you got some steps wrong just put it in a box and draw a line to show you have cut that part. Not only does erasing waste precious time, but you may erase something useful and for that you need to repent later.

• Write the dimension in the answer. If you are asked to find the volume of a solid objects the answer will contain the cubic unit (m3 ,cm3 ,--). If you are asked to find the area of any objects never forget to write the unit in square unit such as m2 , cm2 etc. Does the problem ask how old the woman was? Make sure that your x really did stand for the woman’s age and not her daughter’s age. Students lose lots of points on every test because they didn’t answer the question that was actually asked, but answered some other question instead. Make sure you have worked the problem to the end and that your answer is in the right form, and that you’ve answered all parts of the question. • Especially in problem of Trigonometry, Geometry, Co-ordinate geometry, Mensuration etc, make clean diagram. The diagram is essential in such question and the visual representation will clear your mind and help you figure out a way to solve it.


• Don’t copy or use unfair means in exam. Never match your answer with your friend Students sometimes ask their friend for help and sometimes cross check the answer. This is a dangerous practice. I still remember one incident, when I was a student of B.Sc. (Math Hons). In a Real analysis paper I answered a question that was wrong but I was confident of my answer. A friend of mine sitting back asked me the answer of that question and I told him the answer I got. Since I was a brilliant student and most of my friend used to trust me blindly so he crossed his answer and solved it again with the method I used. When I went home and checked my answer I found that I had applied wrong concept thus got wrong result. This made me sad because I managed to get 80 marks in that paper but my friend who could have scored 70 got 55 in that paper.


• Don’t leave the paper in hurry. Have you heard the story of Tortoise and Rabbit? Doing your paper fast and submitting it without re-checking it will not bear fruit. Many students like to leave a test paper early to show their fellow friend that how good he is to finish the paper early. Regardless of your motive, leaving the exam room early is a mistake. If you have finished the paper well in advance, re- checks it two to three times. Make sure you have actually answered each question correctly. Write your variables clearly so that you don’t confuse them with other ones. Ask yourself if your answer makes sense. Are the units correct? Is your answer consistent with the parameters of the question? Check your formulas etc before submitting it to the teacher. • The last but the important point is that your writing should not be illegible. Good handwriting makes a good impression in mind of teacher and illegible writing brings a wrong impression and your illegible writing sometime may be the reason of your poor marks in exam. Hope these tips will help you to prepare for your class test or board exam.


Always remember a good strategies and well planned preparation will help you to score good in exam.

Issac Newton

Isaac Newton With the possible exception of Albert Einstein no scientist, so altered the human perception of the Universe as Isaac Newton. His cosmos of absolute space and time drew upon the work of others such as Kepler and Galileo but it was Newton who brought together Galileo’s mechanics and Kapler’s law of planetary motion to fashion a universe which could run without the benefit of continual divine intervention. His successor have pointed Newton as the supreme intellect that the human race has produced—he who in genius surpassed the human kind. The English Physicist, Mathematician Isaac Newton was born on 25th December 1642 in Woolsthrope in England. His father Isaac died at the age of 37, before the birth of his son. Newton was a premature child. His mother Hannah Ayscough married the next door neighbour Barnabas Smith and left the 3 years old Newton to the care of his grandmother. As a child Newton was not robust and was forced to shun the rough games of boys of his age. Instead of playing with them he invented his own diversions in which his genius first showed up. Newton received his early education in the common village school of his vicinity. His maternal uncle William Ayscough was the first able man to recognise the talent of Newton. He persuaded Newton’s mother to send his son to Cambridge instead of keeping at home. On his uncle’s advice he had been sent to the Grantham Grammar School. While preparing for Cambridge Newton lodged with a Mr. Clarke and fell in love with his step daughter Miss Storey to whom he became engaged before leaving for Cambridge. But in Cambridge his growing absorption in work put his romance in the back seat and Newton never married. Newton attributed his success to three greatest mathematicians - Descartes, Kepler, and Galileo. Newton writes- If I have seen a little further than others it is because I have stood on the shoulders of giant. In June 1661, Newton entered Trinity College Cambridge where his maths teacher was Dr. Isaac Barrow. Newton passed his BA degree in January 1664. Till 1664, Newton did nothing great. Though a manuscript dated May 20, 1665 shows that Newton had scientifically developed the principles of Calculus at the age of 23. He called his method Fluxions. He had also discovered the Integration and Binomial Theorem in 1665. The second of his great discovery is the law of universal gravitation in 1666 at Woolsthrope’s apple garden. Newton always preferred to view the Universe as a gigantic watch spring wound up by the hand of God and left to run down on its own. He believed that it was impossible for science to transcend the barrier between our world and the spiritual world and was quite content to leave such speculations to other natural philosophers. He wrote:- We have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power... I have not been able to discover the cause of those properties of gravity from phenomena , and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypothesises---- have no place in experimental philosophy. In 1667, Newton was elected a Fellow of Trinity and in 1669, at the age of 26, he succeeded Barrow as Lucasian Professor of mathematics. It is said the Isaac Barrow resigned from the Lucasian Professorship of mathematics in favour of his incomparable pupil Newton. His first lecture was on Optics. In 1668, he constructed a reflection telescope with his own hand and used it to observe the satellite of Jupiter. His master piece Principia was published at the cost of his friend Edmund Halley in 1687. It contained the Dynamics, Laws of Gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws, by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation. The Principia had no mention of his Calculus. From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multi-coloured spectrum into white light. He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. This is known as Newton's theory of colour. Newton was also a member of the Parliament of England from 1689 to 1690. In 1696, at the age of 54, Newton became warden of the Royal Mint. His job was to reform the coinage. In the same year Johann Bernoulli and Leibniz concocted two challenges to the mathematician of Europe. The problem baffled the mathematician for next six months and none of the great mathematician of that very time could come up with the exact solution. Newton heard of the problem when a friend communicated him. He was at that time returned from his Mint office. Newton took his dinner and sat to solve the problem and finally got the solution of the two challenging problems. He communicated the solution to the Royal Society anonymously. On seeing the solution Bernoulli at once exclaimed, “Ah! I recognize the lion by his paw.” In 1701-02, Newton represented Cambridge University in Parliament and in 1703 was elected President of Royal Society. In 1705, he was knighted by Queen Anne. Newton had done a great service to mankind by his discovery but he was so humble that he once said: - I don’t know what I may appear to the world; but to myself I seem to have only like a boy playing on the seashore and diverting a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her husband, until his death in 1727.Newton died in his sleep in London on 31 March 1727 and was buried in Westminster Abbey. Let’s close the biography of Greatest mathematician of the world with the line of English Poet Alexander Pope:- Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light Rajesh Kumar Thakur rkthakur1974@gmail.com