Research Articles

A NEW HORIZON IN MATHEMATICS

Mathematics incorporates arithmetic, algebra and geometry. Each of its constituents took thousands of years to develop into a scientific branch of knowledge as we find it today. Originally it started as a technique for surveying and estimation in agriculture, business and industry in ancient Egypt and Iraq. India, China and Greece brought rapid developments into the system to satisfy the ever-growing needs of the civilized world.

Pythagoras, Plato, Aristotle, Euclid, Archimedes and Ptolemy are only a few Greek names who contributed enormously to advance the scope and application of all related constituents of Mathematics. China developed geometry and India invented the numerals used in the world of mathematics. The Muslims, inspired by the Quran contributed for centuries to all branches of Knowledge. They, not only preserved the Greek works but also translated them into Arabic and other contemporary languages and wrote great commentaries on them. They made valuable additions with original contributions. Mathematics was no exception. Al Khowarizmi (9 th century) may be called the father of Algebra for his advanced contribution. He was the main source of introducing the Indian numerals for the first time to the West, Al-Battani and others worked on trigonometry. Ibnal-Haytham was the founder of geometrical optics. The Persian poet, Omar Khayyam wrote systematic discourses on algebra and engineering.

It was not until late 13 th century when the first European mathematician leonardo da Pisa wrote on arithmetic, algebra and geometry. The calculus is the greatest invention of the 17 th century made by Isaac Newton. The later centuries until the beginning of the 3 rd millennium have seen further developments in this area which consist of general interpretations and increased abstractions. This is a never ending process which motivates mathematicians to contribute consistently and purposefully. The following thesis involving a systematic research on ‘Tables’ has been compiled as a result of passionate and positive efforts of a Pakistani enthusiast, Mr. Mumtaz Khan. Its dogmatic value has been established by the collection and analysis of data obtained through research in school on various mixed ability and age groups of children.

UNIQUE TABLES

Main Features:

Saves time in writing.
Easy to memorise .
Previous style of counting on finger tips to write tables while multiplying or dividing is rendered obsolete.
Speed in table writing enhances the speed of solving sums.
Saved energy can be utilized for other activities.
Easy solution of the sums motivates the child’s interest.
Unique tables introduce such techniques that enable the learners learn to by heart.
Help to make square tables without multiplying.

It’s hoped that this unique technique sooner or later will achieve a global recognition and Pakistan will earn another pride. May Allah provide us with a probing mind ( Amin ) It is observed that if the integers are used to convert the digits shapes into numbers then it becomes easier to work on the tables without multiplication which was always a complicated process for the children.

I. Remember the digit which is changeable by integers.

1, 2, 3, 4 and 5 are not changeable
__
6 = 1 4 changeable

__
7 = 1 3 changeable

__
8 = 1 2 changeable

__
9 = 1 1 changeable

II. way of making numbers with the combined integers. When all the digits of the numbers are changeable. First we write 1 above the given number at the next left side place. Consider place value of 1 after writing and then subtract the actual number from the value and write it at right hand side of 1 with negative sign number. That is the new number equal to the given number.

Example: Finding the equal number Of 77 and 88 with combined integers.

1 0 0 (value of 1)

- 7 7 (actual number)
____
2 3 (difference)

_ _ _ _
Now 77 = 1 2 3 (1 = 100 because at place of hundre And 2 3 means with difference)

______________________________________

10 0

- 8 8
_____
1 2

_ _
1 1 2 = 8 8

If you are able to make equal number then you can easily make tables. Way of making tables:

Write the newly made number above the digit which you want to make table.

Example of table for 6:

_
1 4

0 6

1 2

1 8

2 4

3 0

3 6

4 2

4 8

5 4

6 0

We write the numbers

We write the new form of the number above the number box. First deducted 4 from the 6 because negative integer means subtract and write the obtained number at the next multiple place.
(we got the first multiple of 6 that is 12)

To obtain next multiple further repeat the same process. As 4 cannot be reduced from 2 so we borrow 1 from tens place, now it is 12 so when 4 is deducted we get 8 and write it at unit place of the next multiple row.

At tens place there is 0 so when we add 0 and 1 we get 1 which we write at tens place. In this way the 2 nd multiple we receive is 18. Continuation of same process will complete the table. We are hundred percent sure that this process, children will learn within two to three days and will get rid of the burden of memorizing tables.

iii. Way of making tables, when the digits of given numbers are mixed, say some are changeable into new format while others can not be changed.

Look at the example: carefully.

a). 72, In this number 7 is only changeable.

b). 8 1 3 7, In this number only 8 and 7 are changeable.

To make tables of such numbers, understand carefully the steps.

i. write the number in rough column.

ii. First write the unchangeable digits above the digit.
                           _
iii. Now as 7 = 1 3 so write 3 above the 7 and 1 above the 3 on the left.
                           _              _
iv. Now as 8 = 1 2 so write 2 above the 8 and 1 on the left of 2
                                           _
v. As a result we received 1 S 2 1 4 3 which is equal to

Table of 8 1 3 7
After writing the equal number above the number Calculate as in the example. So after addition or subtraction you will get the multiples. The new number is in the box.

Steps of process

i.

8 1 3 7

ii.

1 3
8 1 3 7

iii.

1
      _
1 3 3
8 1 3 7

iv

         1
   _       _
1 2 1 3 3
   8 1 3 7

v.

           _   _

1 2 1 4 3 = 8 1 3 7

   _       _
1 2 1 4 3

8 1 3 7

1 6 2 7 4

2 4 4 1 1

3 2 5 4 8

4 0 6 8 5

4 8 8 2 2

Note When we find the new shape of the number or digit, made by both integers (positive and negative), negative sign with the negative integer indicates a subtraction meanwhile positive integers indicate addition, so we follow it with the number written in the series up and down. With the actual digit number and the newly made digit or number we write and find the result after calculation. We find the multiple by addition or subtraction and can write the whole table immediately with multiplication.

Important to remember

1

2

5

6

unchangeable

unchangeable
                                             _
the shape with both integers = 1 4   = (10-4)
                                             _
the shape with both integers = 1 3   = (10-3)
                                             _
the shape with both integers = 1 2   = (10-2)
                                             _
the shape with both integers = 1 1   = (10-1)

Note: As 1 is at tens place so value of 1=10

Keep in mind that when a digit shape is changed then two integers appear which show that the digit now has split into two. Now the place value also becomes 2.

For example 6 is one digit so place value is only ‘unit’.
_ _
6 = 1 4 now 1 has place value of ‘tens” an 4 has the place value of ‘unit’.

While we write the newly made number above the actual number, we write ‘unit’ above the ‘unit’ and ‘tens’ above the tens. Also note that when negative integer appears it increases value of left Side, by 1. (Value increased in negative numbers shows smaller numeral with negative sign.)

Example of the tables consisting of one digit.

Table of 6
_
New shape of 6 =1 4

To make table we write the new shape above the actual and add it to obtain multiples.

_
1 4

0 6

1 2

1 8

2 4

3 0

3 6

4 2

4 8

5 4

6 0

    _
1 4

+  6
_____
1 2

    _
    1 4

+  1 2
______
    1 8

Note: We cannot subtract the positive integer from negative integer.

Table of 7
_
When 7 = 1 3

_
1 3

0 7

1 4

2 1

2 8

3 5

4 2

4 9

5 6

6 3

70

i)

ii)

iv)

    _
1   3

+   7
_____
    14

        _
    1 3

+  1 4
______
     2 1

        _
    1 3

+  2 1
______
     2 8

Table of 8
_
When 8 = 1 2

_
1 2

0 8

1 6

2 4

3 2

4 0

4 8

5 6

6 4

7 2

8 0

    _
1   2

+   8
_____
1 6

        _
    1 2

+  1 6
______
    2 4

        _
    1 2

+  2 4
______
    3 2

Table of 9
_
When 9 = 1 1

_
1 1

0 9

1 8

2 7

3 6

4 5

5 4

6 3

7 2

8 1

9 0

    _
1   1

+   9
_____
1 8

        _
    1 1

+  1 8
______
     2 7

        _
    1 1

+  2 7
______
    3 6

Table of the numbers consisting of two digits.

Table of 16
_
When 16 = 2 4

_
2 4

1 6

3 2

4 8

6 4

8 0

9 6

1 1 2

1 2 8

1 4 4

1 6 0

       _
   2  4

+  1  6
_____
    3 2

        _
    2 4

+  3 2
______
     4 8

Note: As we cannot reduce Positive number from negative number so from positive the negative number is subtracted.

Table of 19
_
When 19 = 2 1

_
2 1

1 9

3 8

5 7

7 6

9 5

1 1 4

1 3 3

1 5 2

1 7 1

1 9 0

       _
   2  1

+  1  9
_____
    3 8

        _
    2 1

+  3 8
______
     5 7

        _
    2 1

+  5 7
______
     7 6

Table of 27
_
When 27 = 3 3

_
3 3

2 7

5 4

8 1

1 0 8

1 3 5

1 6 2

1 8 9

2 1 6

2 4 3

2 7 0

       _
   3  3

+  2  7
_____
    5 4

        _
    3 3

+  5 4
______
     8 1

Table of 88
_ _
When 88 = 1 1 2
_ _
New shape of the 88 = 1 1 2

_ _
New shape of the 88 = 1 1 2

_ _
1 1 2

8 8

2 6 4

3 5 2

4 4 0

5 2 8

6 1 6

  7 0 4

  7 9 2

  8 8 0

      _ _
   1  1 2

+   8  8
________
   1 7 6


       _ _
   1  1 2

+ 1 7  6
_________
   2 6 4

Table of 79
_ _
When 79 = 1 2 1

_ _
1 2 1

7 9

1 5 8

2 3 7

3 1 6

3 9 5

  4 7 4

  5 5 3

  6 3 2

7 1 1

7 9 0

      _ _
   1  2 1

+   7 9
________
   1 5 8


       _ _
   1 2 1

+ 1 5 8
_________
   2 3 7

Table of 126
_
When 126 = 1 3 4

_
1 3 4

1 2 6

2 5 2

3 7 8

5 0 4

6 3 0

7 5 6

8 8 2

1 0 0 8

1 1 3 4

1 2 6 0

         _
   1  3 4

+ 1 2 6
________
   2 5 2


       _
   1 3 4

+ 2 5 2
_________
  3 7 8

Table of 263
_
When 263 = 3 4 3

_
3 4 3

2 6 3

5 2 6

7 8 9

1 0 5 2

1 3 1 5

1 5 7 8

1 8 4 1

2 1 0 4

2 3 6 7

2 6 3 0

   _
    3 4 3

+ 2 6 3
________
   5 2 6


         _
    3 4 3

+ 5 2 6
________
   7 8 3

Table of 718
_ _
When 718 =1 3 2 2

_ _
1 3 2 2

7 1 8

1 4 3 6

2 1 5 4

2 8 7 2

3 5 9 0

4 3 0 8

5 0 2 6

5 7 4 4

6 4 6 2

7 1 8 0

      _   _
   1 3 2 2

+ 7 1 8
________
  1 4 3 6


   _   _
  1 3 2 2

+1 4 3 6
________
   2 1 5 4

Table of 999
_
When 999 =1 0 0 1

_ _
1 0 0 1

9 9 9

1 9 9 8

2 9 9 7

3 9 9 6

4 9 9 5

5 9 9 5

6 9 9 4

7 9 9 2

8 9 9 1

9 9 9 0

     _   _
   1 0 0 1

+ 9 9 9
________
  1 9 9 8


         _
  1 0 0 1

+1 9 9 8
________
   2 9 9 7

Table of 4 digit numbers:

Table of 1982

Forms of digits changed by integers.

Actual number

Formation of new number:

   _
1 1 _
   1 2
                                 _
1       2               = 2 0 2 2

1 9 8 2

_
Now 1 9 8 2 = 2 0 2 2

Table:

_
2 0 2 2

1 9 8 2
3 9 6 4
5 9 4 6
7 9 2 8
9 9 1 0
1 1 8 9 2
1 3 8 7 4
1 5 8 5 6
1 7 8 3 8
1 9 8 2 0

_
2 0 2 2

+1 9 8 2
________
3 9 6 4

_
2 0 2 2

+3 9 6 4
________
5 9 4 6

Table of 6 9 2 8

Actual number

Formation of new number:

   _
1 4 _
   1 1
           2

               _            _ _    _
           1   2       =1 3 1 3 2

6 9 2 8

Table:

_ _ _
1 3 1 3 2

6 9 2 8
1 3 8 5 6
2 0 7 8 4
2 7 7 1 2
3 4 6 4 0
4 1 5 6 8
4 8 4 9 6
5 5 4 2 4
6 2 3 5 2
6 9 2 8 0

_ _ _
1 3 1 3 2

+6 9 2 8
________
1 3 8 5 6

_ _ _
1 3 1 3 2

+1 3 8 5 6
________
2 0 7 8 4

Table of 9 3 8 6

Actual number

Formation of new number:

   _
1 1
      3
           _           _ _ _
      1   2       =1 1 4 1 4
              _
           1 4

9 3 8 6

_ _ _
Now 9 3 8 6 =1 1 4 1 4

Table:

_ _ _
1 1 4 1 4

9 3 8 6
1 8 7 7 2
2 8 1 5 8
3 7 5 4 4
4 6 9 3 0
5 6 3 1 6
6 5 7 0 2
7 5 0 8 8
8 4 4 7 4
9 3 8 6 0

_ _ _
1 1 4 1 4

+9 3 8 6
________
1 8 7 7 2

_ _ _
1 1 4 1 4

+1 8 7 7 2
________
2 8 1 5 8

Table of 3 6 8 4

Actual number

Formation of new number:

3
   _
1 4
      4
           _           _    _
      1   2       =4 4 5 2

3 6 4 8

_ _
Now 3 6 8 4 = 4 4 5 2

Table:

_ _
4 4 5 2

3 6 8 4
7 2 9 6
1 0 9 4 4
1 4 5 9 2
1 8 2 4 0
2 1 8 8 8
2 5 5 3 6
2 9 1 8 4
3 2 8 3 2
3 6 4 8 0

_ _
4 4 5 2

+3 6 8 4
________
7 2 9 6

_ _
4 4 5 2

+3 6 8 4
________
1 0 9 4 4

Appraisal

“The Unique Tables” technique was offered to the practising teachers in KG and Primary sections of a number of schools in Karachi. It was observed that the children of KG sections failed to respond positively to the method, as their understanding of four basic rules of arithmetic (+ - × ÷) was not very strong yet. A small number of them accepted the challenge and tried to grasp the technique, but ended up in confusion. They lacked clarity of concept and speed of working. At Primary (9+) level, the children absorbed themselves in understanding, learning and retaining the technique and once grasped they started using it enthusiastically to solve all long and short problems. Where use of traditional tables was essential, they took advantage of the ‘Unique Tables’ technique and with rapid practice, they succeeded in solving the long and short division and multiplication sums within remarkably less time. It relieved them of the cumbersome and monotonous typical rote learning of tables.

Before introducing the ‘Unique Tables’ to children, teachers are advised to get conversant with the process themselves and understand the basic concept with repetitive workings. They will find how easy and interesting the technique is, and how effectively its adoption can remove the obstacle of traditional tables during the course of learning and teaching mathematics at all post-primary levels of education. May Allah grant us wisdom and patience.