New dimension to look at Magic Square - T.R. Jothilingam

Fun with numbers – Magic Squares

The sweeter side of the MATHS…

        Magic squares are a set of numbers, arranged in a set of 3 x 3 or 4 x 4 or 5 x 5 squares, in which all the Vertical, Horizontal and both diagonal totals are same, called as the “Magic Sum”.

Normally in a 3 x 3 magic square natural numbers 1 to 9 are used, and for 4 x 4 numbers 1 to 16 are to be used.  In general, for a “n x n” magic square numbers, “1 to n²” numbers are to be used.  And the “Magic Sum” of the magic square must be          {(1 + n ²)/2}* n. (First number plus last number to be added, and the sum divided by two, and then multiplied by the number of rows). These magic squares can be done from 3 x 3, 4 x 4… up to any highest number in the world.

Magic squares can also be made with using fractions, zeros, negative numbers, with squares and cubes. We can make magic squares for a given numbers, or size, or for a given date or date of birth also.

 Example:

For 3 x 3 magic square, numbers 1 to 9 are to be used.

The row total will be {(1 + n ²)/2}* n ={(1 + 9)/ 2}*3 = 5 x 3 = 15.

3 x 3 Magic square (Total 15)                     3 x 3 (Total 2019)

8	1	6
3	5	7
4	9	2

 

676	669	674
671	673	675
672	677	670

 

                 

 4 x 4 Magic square (Total 34)    

   {(1+ 16) /2} x 4 = 34                                     4 x 4      (Total 2018)

16	2	3	13
5	11	10	8
9	7	6	12
4	14	15	1

 

512	498	499	509
501	507	506	504
505	503	502	508
500	510	511	497

 

 

                                                                                              

 

6 x 6 Magic Square (Total 111)

35	1	6	26	19	24
3	32	7	21	23	25
31	9	2	22	27	20
8	28	33	17	10	15
30	5	34	12	14	16
4	36	29	13	18	11

 

 

 

 

 

There are many interesting magic squares to enjoy:

 

Smallest Prime number magic square: All numbers in the magic square are prime numbers         

                  (Total 177)

71	89	17
5	59	113
101	29	47

 

 

  

                                      

                        Reverse Magic squares: 

All numbers in a magic square are written from the reverse and that also form a magic square and with the same total.

         (Total 1515)                               (Total 1515)

806	107	602
301	505	709
408	903	204

 

608	701	206
103	505	907
804	309	402

                 

 

Upside-down magic square: 

A magic square when turned to 180⁰    also giving the same total 24.                      

                                                (Total 24)

9	1	8	0	6
0	6	9	1	8
1	8	0	6	9
6	9	1	8	0
8	0	6	9	1

 

 

 

 

This article is written by Mr. T.R. Jothilingam, Retd. Rly. Station Supt., Madurai, He is a poly mathematician and magic square wizard with several record in his name. He is also a winner of Ramanujan Award of All India Ramanujan Maths Club, Gujarat

Contact him at :-  +91 94428 10486

email: mailtojothi@gmail.com

 http://jollymaths.com/

How to motivate a Slow learner : Mr K Bhanumoorty

Desire,Determination,Diligent, Devotion,Destination ---in learning mathematics. 

We the teachers have taken up this profession either by choice/chance. Once taken up, should we not set an example, with devotion, commitment, and accountability for the work we are responsible. 

Parents send their wards with lots of expectations that the school shall definitely bring their children fulfilling their aspirations and ambitions. There may be some children who will be even the first generation of learners who come to school.

 Parents may not be able to help in studies rather in academic matters. In general a class is heterogeneous students of different calibers. There may be some students who may have problems, in, communication skills, understanding and reasoning skills, to grasp quickly. We are sure to see, can say, they bloom  and blossom little late, though  slow in learning, in born abilities are there. 

Many factors are there, first and foremost parents have to understand their children, not to compare with others, to accept the children, to understand the actual problems instead of blaming them. Family issues are to be sorted out if any, for maintaining a healthy environment, support, encouragement at home.  

Contributions by teachers in abundance are very much essential. A kind of drive is a must, by the parents and teachers, for the progress and attain a set of defined objectives and goals by students.

 

A parent need not know the subjects but regular interaction with son or daughter, on daily basis, what has been taught, shall definitely be fruitful. No doubt parents have to have faith in them, at the same time watch and ward duty is a must, thereby their wards are motivated, shown the right path. Parents to visit the school and have interaction with the teachers to know the performance (to have rapport with the school.) Then comes  the  important  facilitator  teacher  who  has  all concern for progress of  students..

 

In the beginning of an academic year itself, the teachers have bio data / profile of all the parents. There may be even single parent, children living with grandparents, father works at out station etc. Some students lag behind in academic achievements compared to their counterparts in the normal school. Within a matter of 2 months a teacher shall be able to identify who are slow learners. Slow, I would say problems in comprehension ability in understanding or grasping power. In some cases, could be even health problems, family in adequacies. Teachers have to understand what the actual problem is and be diagnosed for early action. No doubt a teacher has limitations to interfere in family matters but as far as the academic matters are concerned he or she should definitely put efforts sincerely to ease out. I have  even  noticed  some  students  who have  more  abilities  but do not use their full potential, there lies the duty and role of a teacher. Interest in the subject to be created and motivated how mathematics is applied in our daily life.

 Let us discuss further, an action Plan, needs to be taken by the teachers.

1) Class may be divided into smaller group, consist of 4-5 students. Care to be taken group is mixed one, student of different abilities. A group leader may be nominated which shall be changed under rotation after a month or so. Students feel comfortable, happy, when they sit with friends, nowadays peer group learning is under practice. 

2) Once a unit is comfpleted, a symposium may be arranged in one period to discuss what are the concepts learnt.

3) Class assignment to conduct, to know, who actually have problems, in learning / working out problems. Questions one  after  another  with  time  frame  to answer, so as to be  aware of time management in answering a question. This will facilitate a teacher for remedial measure. To conduct remedial classes after the school hours, or during weekend with the permission of school authority and intimation to the parents. In remedial classes, to repeat the teaching, in learning new concepts. No doubt lots of patience are required from the teachers. 

4) To prepare graded questions simple to complex, worksheets, to give extra materials, and give them for practice at home, to refer reference books for self-practice. Home works to be corrected then and there, follow up to be done immediately so that the same mistakes are not repeated again. Progress of the student is to be monitored subsequently to inform school authority and parents.

5) To encourage, students to ask doubts, involving in other projects / activity oriented topics. Students have to have respect for teachers but not to have fear to clarify or  ask doubts .

6) To appreciate work done, to give confident, encouragement with positive note, award and reward shall go a long way.  

7) Apart from the above aspects the child has to feel that the teacher has all concern in his or her development, academic career, treats  everyone  equally. ( impartial)

8) Counselling the student at the right time is essential part by the teacher, especially when the students are adolescents.  Interaction more often with students to know the areas of difficult will be a guiding force. Care is to be taken to address individually and separately, never in open class.

9)   A teacher has  to  encourage  students  to participate  in other  school  activities , be impartial, make them realize  school is a place for all round development, not merely academic knowledge.

10) Once a month teacher could plan for Quiz in Mathematics, fun in mathematics (Magic squares, puzzles), to speak on contributions by great mathematicians (Indians as well as other countries.)  This would facilitate a sort of motivation and curiosity among students.

11) If a teacher could contribute, bring a student for self-attainment and made progress in life, he or she has done an immense service. You have done not because they will remember you but because you have done, what you have to.

It is not an exaggeration to say that teachers can influence a student life. Students who are inspired by their teachers can accomplish amazing results and that motivation always stay with them lifelong.

Let me conclude with a quotation by CLAY P. Bedford

“ You can  teach a student a lesson for a day; bur if you can teach him to learn by creating  curiosity , he will continue the learning process as long as he lives.”

Role of Teacher - K Bhanumoorthy

                                      Role of the teacher 

Teachers continue to shoulder a big responsibility to nurture children entrusted to their care, even in the present day world of IT revolution and virtual class rooms. A transformation that a teacher’s role has undergone to develop proper perspectives. Globalization, liberalization, computerization have made a stance impact on every aspect of Human life. Education is one of the areas of human endeavor greatly influenced by these changes. Students have greater access to information, thanks to IT evolution.

 

Multi - Tasking Role of a teacher
(Photo Credit:- http://e-portfolio-53867572.weebly.com/uploads/4/0/0/7/40072649/4878167_orig.jpg)

A teacher has to determine his/her role against dynamically evolving society, to understand social ethos of the present day times and reconcile them with ideals of education. Sincere attempt is to be made to perceive the reality and a teacher has to equip with courage and conviction, make the children, to have faith in eternal values of Truth, Beauty and Goodness.

We are fully aware a teacher is a facilitator who is able to provide the right kind of learning experience, through active interaction thereby basic skills are developed such as, observation, collection of information, drawing inferences and end up with conclusion. Another important aspect is character building which is the need of the hour. If it is firmly laid, definitely learning shall be easier and rest of other things shall fall on line to equip themselves.

 

Sri  Aurobindo  has mentioned 3 principles of learning.

 

(Sri Aurobindo)

“The first principle of true teaching is that nothing can be taught”.

A student is a soul in evolution and contains all possibilities within, a child definitely comes out with success. With due apology, I would say let us not think that students do not know anything. The child is not a blank slate where a teacher could write. The teacher’s work is to show the student how to acquire knowledge, for self. The teacher has to drive the child to understand his instruments of knowledge to, powers of concentration, observation, memory, reasoning, imagination and intuition. The most important thing is to create interest, enthusiasm to learn.

 

A teacher has to perform so many role 

To quote: ”Education is not for sophistication, but it is the manifestation of perfection already in man.”  -----Swami Vivekananda

 

“The second principle is that the mind has to be consulted in its own growth”.

We do believe every individual is unique, has a purpose for which he is born. Sri Aurobindo   says “Everyone has in him something his own, a chance of perfection and strength however small it could be”. The main task is to find out, develop it and use it.

Creativity among the students is widely present. Here comes the role of teacher, making the students think for themselves. Every child has the ability and quest for excellence. In short, “Teaching is a revealing.” So let us make the students learn under a conducive environment.

 

“The third principle of education is take the mind (work  from ) what is near to what is far, from that which is so that which shall be…”.

To be specific, the first step to establish a context which is familiar, relevant to a student.

You have to remember as a teacher everything cannot be forced on the mind. If you wish to have genuine development you have to have a free and natural growth, so to say controlled freedom. At the same time we have to remember a child has to be physically strong emotionally stable, psychologically befitting. There has to be gradual progress in order to sustain .Teaching may be conceived as both a science and an art. The science of teaching is esoteric (mysterious) aspects of life and learning, while teaching as an art, accumulating all the best that is within an individual. Students do have latent capacities in the mind, we teachers have to facilitate, make them realize /understand that they do have enormous potential.

 

Conclusion

A true teacher is a stimulator, does not impose on the other hand, reads the child's mind, motivate ,guides, help, encourage, counselling at the right time, to unlock his full potential and attains one’s personal excellence.

 

To quote Swami Vivekananda-

“What we are now has been the result of our own past action, it certainly follows that whatever we wish to be in future can be produced by our present, so we have to know how to act”.

K Bhanumoorthy 

K Bhanumoorthy is a retired Principal of KVS with more than 30 years of teaching experience. He is the Director of Southern region of All India Ramanujan Maths Club.

 

IMPORTANT PROPERTY OF SQUARE NUMBERS

IMPORTANT PROPERTY OF SQUARE NUMBERS : useful for finding square, square root , last 2 digits and many more.

 

Same property is very useful in many other problems which would be shared in next blog.