Quadrilateral during Vedic Time

Quadrilateral during Vedic Time:-
Vedas are the rich source of our knowledge. During this period the rituals to please gods by means of Yagna were high. The rituals was an extremely important part of the ancient Hindu religion. Fire altars were constructed to perform yagna and they were made with precision in different shapes and size. Geometrical rules found in the Sulvasutras, therefore, refers to the construction of squares and rectangles, the relation of the diagonal to the sides, equivalent rectangles and squares, equivalent circles and squares, conversion, of oblongs into squares and vice versa, and the construction of squares equal to the sum or difference of two squares. In such relations a prior knowledge of the Pythagorean Theorem, that the square of the hypotenuse of a right-angled triangle is equal to the sum of squares of the other two sides, is disclosed.
Sulba means pieces of Chord or string and Sutra means formula. The Sulba sutras are the mathematical discoveries by famous Indian rishis turned mathematicians at about 1000 BC to 200 BC, using a piece of chord for constructions of various fire sacrifice altars.

Fire altar

The purpose of the rituals was to build an immortal body that would transcend suffering and death, both hallmarks of mortal existence. According to Plofker:-
Many of the altar shapes involved simple symmetrical figures such as squares and rectangles, triangles, trapezia, rhomboids, and circles. Frequently, one such shape was required to be transformed into a different one of the same size. Hence the Śulba-sūtra rules often involve what we would call area–preserving transformations of plane figures, and thus include the earliest known Indian versions of certain geometric formulas and constants. . In an article published in Indian Journal of Histroy of Science “Ritual Geometry in India and its Parallelism in other Cultural areas” – by Mr. A. K. Bag where he writes that the construction of altars having drawn on a base of different figures such as square, circle, semi- circle, isosceles trapezium, triangle, rhombus, falcon or tortoise shape and other led to the development of various geometrical figures, their transformations and calculation of areas involving many Pythagorean relations with rational and irrational numbers leading to its general statement, approximation of the value of √2 and others. Here is the shape of Mahavedi that is in shape of Isosceles Trapezoid trapezium of area 972 sq. units with 24 and 30 units of length for parallel sides and 36 units of length for altitude with the shape of Square and Rectangle within.

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

Quadrilateral -1

Can you guess which one is the largest highway project in India? Don’t worry, I am giving you a clue. This is the project started in 2001 by the PM Atal Bihari Vajpayee. See the map below and concentrate on a mathematical term emerging out from this map. One more thing about this project is that in this project four major metropolitan cities – Delhi, Mumbai, Kolkatta and Chennai will be linked with each other.

Source: - National Highway Authority in India

Don’t worry, Let me tell you the name of this project so that you can guess the topic we will discuss now in details. The name of this project is Golden Quadrilateral.
The above examples are enough to show that quadrilateral is a geometrical shape that consists of four edges (Sides) and four vertices (corner).Quadrilaterals, like triangles are the second most common shape used in geometry. It is a geometric shape that consists of four vertices sequentially joined by straight line segments.
The word ‘quadrilateral is derived from the two Latin word – quadri and latus. The first word quadri means four whereas the word latus refers for side. In other words, quadrilateral is a figure having four sides.
According to the Merriam- Webster Dictionary a quadrilateral is a geometrical shape having four sides. The Collins Dictionary defines quadrilateral as a polygon having four sides and a complete quadrilateral consists of four lines and their six points of intersections. The other words sometimes used for quadrilateral are quadrangle and tetragon.
A quadrilateral is generally of three types – Convex quadrilateral, Concave quadrilateral and Simple quadrilateral.
A quadrilateral that has an angle more than 1800 is called Concave Quadrilateral. A quadrilateral is called simple if the four sides of a quadrilateral meet at vertices. Here we shall only focus on convex quadrilateral where each angle is less than 180.

Dr Rajesh Kr Thakur
rkthakur1974@gmail.com

Fundamental of Mathematics for Secondary and Senior Secondary level

Secondary and Senior Secondary level

Secondary level is the stage where mathematics comes to the students as an academic discipline. It is a stage where students begin to perceive the structure of mathematics. Mathematical terminology gets sophisticated and the concept of proof becomes the central to curriculum. The new branch of mathematics Trigonometry and Co-ordinate geometry get introduced. The working of secondary level is somehow based on the learning the concept of primary and upper primary level. Mathematics at this stage is totally based on concept and reasoning. The theoretical part of mathematics begins and students find it irritating. A bigger question now arises: - What does mathematics really consist of? Mathematics in secondary and senior secondary level consists of Axioms, Theorems, Definitions, Theories, Formulas and Methods. Mathematics could surely not exist without these ingredients; they are all essential. Now the question is what a secondary student should do to enhance his mathematical skill. Polya says – To a mathematician, who is active in research, mathematics may appear sometimes as a guessing game; you have to guess a mathematical theorem before you prove it, you have to guess the idea of the proof before you carry through all the details. Mathematical facts are first guessed and then proven. If the learning of mathematics has anything to do with the discovery to do problems in which he first guesses and then proves some mathematical facts on an appropriate level.

• The student needs to integrate the techniques and basics learnt at the previous level.

• Students at this stage should have practical knowledge of why this and why not this. Reasoning helps to retain the learning for long and you not only understand the principle of mathematics but also understand its practical application.

• Students should have the knowledge of finding square, square root, cube and cube root of any number. It is also expected from the student in secondary level to have memorized the square up to 50 and cube up to 30 so that calculation can be done with ease without error. If possible learn the fourth power and fifth power up to 5 of first 10 numbers.

• They should learn some technique that can help them to calculate speedy. There are ample books available in market which will prove beneficial in order to calculate faster. Vedic Mathematics is now a popular book used by students for fast calculation. The best part of learning some trick to do arithmetical calculation faster is that you save ample time in your examination and the chance of your doing right calculation increase. Besides that student should also learn method to counter check calculation so that the chance of error is avoided. There is one simple and sophisticated technique to counter check the calculation in seconds. The method CASTING OUT NINES method will be beneficial to students in checking the calculation.

• In geometry, first learn about different shapes and its properties. Since the geometry of secondary level is highly based on some theorems so it is advised to students to learn the important statement of theorems so that they can write the statement or name of the theorem in proving a geometrical proof.

• Word problem is another area that needs to be addressed. Understanding mathematical language is equally important to do word problem. So before going through the exercise learn the most frequently used mathematical terms.

• Solving algebraic problem is considered a tough task but I have noticed students performing well in algebra once they understand the language of the word problem. Confusion creates when you don’t remember the formula to be applied at the appropriate problem and look for another way of solving the problem. Algebraic problem may be turned into a joyful game once you understand how to proceed with the problem. So before doing the algebra, learn algebraic identities, get mastery over understanding mathematical language of the problem.

• Trigonometry is introduced in Class 10th in India and students feel uneasy when they solve the problem or prove the identities. You must understand the fact that Trigonometry is the backbone of mathematics in senior secondary level. If you are not good at Trigonometry in secondary level, you will have baskets of problem as Differential and Integral Calculus taught in class 12 can’t be done without clear understanding of Trigonometry. In trigonometry, you must learn the formulas based on trigonometric ratios and understand the principle on which the finding of different angles such as 0, 30, 45, 60 and 90 are based so that when you go in class 11, you can extend these principle in finding the value of different angles such as-120, 135,150,180,---360 and more.

• Last but not the least, at the secondary stage; a special emphasis should be done on experimentation and exploration. Make model and visit mathematics laboratory whenever possible. Mathematics can’t learn alone by merely learning formulas until you master mathematics by practicing. This practical aspect of learning can be achieved only if you make mathematical model, play mathematical brain twister and visit mathematical laboratory to understand the mathematical concept with the help of diagram and mathematical shapes kept in laboratory. Activities in practical mathematics help students immensely in visualization.

Dr Rajesh Thakur
rkthakur1974@gmail.com

Fundamental of Mathematics for Primary and Middle School Students

Fundamentals of Mathematics

Mathematics is like a pyramid. Every new skill requires an understanding of pre requisites to do well. Mathematics is also an analytical subject. It opens the closed gate of our mind. If you are good at math your ways of critical thinking will certainly be far better than those who are not so well in math. Now the BIG question is--- Why someone is good at math and others are not?

The first one is a Kuchha House made by mud and the second one is a Pucca House constructed with bricks, iron rod, cement etc with proper design. Please answer me a simple question--- Which house is stronger and you would love to live in?
Of course, the second one, because it is strong and well constructed. Its foundation is strong as good quality of bricks, cement and iron rods are used while constructing the house. A well planned building last long, so as if the initial preparation is done for mathematics, it will make the mathematical foundation of a student strong enough to cope up with the syllabus of higher class. Now the big question arises --- What one should know in order to make his/ her mathematics good?
I have divided the fundamental principles into two parts:-

a) For Primary and Upper Primary Level
b) For Secondary and Senior Secondary Level

For Primary and Upper Primary Level

The Annual Status of Education Report 2011 has shown an alarming decline in mathematics skill in 6-14 age group. It states that less than a third of class students in rural Indian school can solve simple two digit subtraction problem.

I see many students saying—I hate math. It is not that they hate math, it was that they hated the fact that they didn’t understand math and that they didn’t understand math because they were missing basic building block. Mathematics has a distinction of being most unpopular subject because it requires the learner to think correctly. Most people love to speak about any issue but hate to accept that they are wrong. Mathematics tells right is right and wrong is wrong. The best art to learn mathematics is that you should have the pre-requisite knowledge of the subject. You can’t do addition until you have the understanding of numbers.

The ASER report indicates the same. Two years ago I read an article describing that more than 50% of students passing out class 5 don’t have the adequate knowledge of summing up two numbers. It shows that the fundamental of such students are weak enough to pursue math as carrier. Now the bigger problem is – if students are not aware of eight fundamentals till upper primary level, you can’t think of their doing well in secondary level math.

What is the minimum competency a primary student should possess?

a) Learn Table:- Student should learn table up to 20.

b) Command over Four Fundamental operations:- In primary level students are expected to be well off in four fundamentals of mathematics (Add, Subtract, Multiply, Divide). Not only this student should have a clear understanding of the facts- when to add, subtract, multiply or divide.

c) Number and their properties:- They should be aware of numbers and their properties. Till primary level one should know about Natural Numbers, Whole Numbers, Prime and Composite Numbers, Even and Odd Numbers. On the other hand a student passing class 8 should be aware of Rational Number, Irrational Numbers, Real Numbers, and Perfect Numbers etc. Besides that, student should know the properties of these numbers. Moreover, it is expected from a bright student to be familiar with Triangular numbers, Square Numbers, Cube Numbers, Pythagorean Numbers, and Ramanujan Numbers etc.

d) Place Value System:- The discovery of ZERO made our life easy and we are now in a position to write big- big numbers. In primary level (up to class 5), students should learn to write numbers up to 100 and they should also have sound knowledge of Place Value System. Students should also learn how to write numbers. I have seen many students of class 6 in government school in Delhi fail to differentiate between 40023 and 423. I would advise students that they should develop the ability to recognize the pattern of writing numbers with the help of place value system. They should have enough practice of writing big numbers at home, at school so that they never get confused in writing big numbers either in words or figures. This can only be achieved if students are taught the place value system in an effective manner.

Before decimal After Decimal
Lac Ten-thousand thousand hundred tens one Tenth Hundredth Thousandths
100000 10000 1000 100 10 1 1/10 1/100 1/1000

Clear understanding of the diagram will not only help them understand how the number is written but also it will help student understand the facts why 22-48 is read as Twenty two decimal four eight not forty eight.

e) Conversion of Units:- Students till primary level should also be taught the addition, subtraction, multiplication etc of Decimals, Hours –Minute- Seconds, Rs –Paisa, etc.. They should also know the conversion of one unit into another. 1 inch = 2.54 cms I foot = 12 inch 1 Kg = 1000 gram 1 meter = 100 Centimeter 1 Hour = 3600 seconds 1 Hectare =10000 Sq metre etc.

f) Divisibility Test of Numbers:- It makes me sad when I see students of upper primary level (Class 6 – 8) struggling in converting a fraction into simplest form. This is because they do not remember the divisibility test taught at primary level. Divisibility test pays important role in solving the numerical problem whether it is from simple arithmetic or mensuration. Every student should learn the divisibility of 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Divisibility rule plays a vital role when a fractional number is simplified. It also helps us in deciding the factor of any particular number. In upper primary level or secondary level you will have problem on Factorization of a number, finding square root, cube root by factor method and there you will find the divisibility rule handy.

g) Mathematical Laws: - When I teach the secondary school students and find them not knowing about Commutative law, Associative law, Distributive law, Identity law, Inverse law, I feel very sad about it. The first three rules are taught in Primary level and the last two rules are taught in upper primary level. If a student is not familiar with these laws then they won’t be able to solve the problem of Binary operations in secondary level. Moreover, in secondary level when they are taught about characteristics of numbers, matrix they will hardly understand about such laws. Hence, it becomes the duty of teachers and parents to teach these rules to students.
A + B = B + A A X B = B X A --------- Commutative law for + and x
A + (B + C) = (A+B) +C A x (B x C )=( A x B) x C – Associative law for + and x
A x (B + C) = A x B + A x C Distributive law
A +0 = A Additive identity (0)
A x 1 = A Multiplicative identity (1)
A + ( - A ) = 0 Additive inverse
A x 1/A = 1 Multiplicative inverse

h) Knowledge about Mathematical Shapes and their Properties:- In Primary level, students should be given the detailed knowledge about all geometrical shapes ( Types of Triangle and Quadrilaterals). The properties of geometrical shapes such as Triangle (Right angle, Equilateral, Isosceles, Scalene, Acute angle, Obtuse angle), Quadrilateral (Parallelogram, Trapezium, Rhombus, Square, and Rectangle) should be understood by students. Mere recognizing the mathematical shapes do not work unless you also know about the properties of the geometrical figures. Moreover, students should be shown the 2-Dshapes like Circle, Triangles, Quadrilaterals etc and 3-D shapes like Cone, Cylinder, Sphere, Hemisphere, Cube and Cuboids so as to use it explicitly when the application of such mathematical figure come. In primary level wherever possible students should be made acquaintance with such 2-D and 3-D shapes with mathematical models, power-point presentation etc so as to understand the concept of areas and volumes clearly. In villages, women draw Rangoli where they use curves, mathematical 2-D shapes, properties of symmetries. Such traditional knowledge should be passed to students. Appreciating the relevance of such mathematical values used by the villagers can enrich the child’s perception of mathematics and they can enjoy mathematics.

i) Fractions and Decimals: - Fractions and decimals constitute another problem area. I have seen students of secondary level sometimes fail to sum up or subtract the algebraic problem involving fractions. There is some evidence that the introduction of operations on fractions coincides with the beginning of fear of mathematics. It is the responsibility of primary teacher to clarify how to do fundamental operations in fractions. I am putting here some examples that need special care while teaching. If teacher takes the responsibility to make the fundamentals of why and how in such types of operations in fraction understandable to students then they will be able to apply the same in algebra.

Addition / Subtraction :- a) 2 ± 3/7 b) ½ + ¾ c) ¾ ± 5
Division a) 5/8 ÷ 10/12 b) 2/3 ÷ 4 c) 5 ÷3/7
In addition to this, students fail in multiply, add, subtract and divide the problem on decimals. Primary students generally face problem of placing decimals after multiplying or how to place different decimal numbers while doing addition or subtraction. The primary level courses designed by different text book publishing company don’t put much emphasis on the application of fractions. Such things have largely disappeared from the text book and that need special attention of teachers because the importance of fraction in conceptual structure of mathematics is undeniable.

j) Understanding of mathematical laws and formulas: - The syllabus of upper primary level is somehow dependent to that of primary level. In upper primary level, most of student encounters problem in algebra and applied geometry. In algebra the most confusing part is why---
i) + 5 – 7 = - 2
ii) – 2 x – 3 = + 6
iii) 6 x – 5 = -30
iv) 8 ÷ (- 2 ) = - 4

The root of the problem is that students are asked to learn the rule without giving any proper explanation. Mathematics is a reasoning based subject and until students are taught why and how in this subject, it can’t be made popular. I, therefore request all mathematical lover to be inquisitive and learn why and how of everything. Moreover, chapter like Percentage, Average, Exponent, Factorization, Ratio and Proportion, Profit and Loss, Mensuration, HCF and LCM, Square and Square root, Cube and Cube root should be dealt with rigor. You must learn the basic rules, definitions and formulas related to the above chapters. When I see students of class 10th unaware of the terms – cyclic factorization, componendo and dividendo, surds, rule of three etc, I feel sad about the level of mathematics being taught in our school. I am not blaming the students only the fault lies in our education system. The syllabus designing bodies should look into the matter. The NTSE, Math Olympiad asks question of high qualities and that require a sound knowledge of the subjects. If you compare the syllabus of the competitive examination with that of school curriculum you will find that there is a huge gap and that gap need to be bridged with standard textbook. Buy qualities text book and go through the chapters minutely. Consult other book for the same chapter and you will notice that your knowledge get updated every time you consult a new book.


Dr Rajesh Thakur
rkthakur1974@gmail.com

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