Fives Supplement Arithmetic

Abstract: Abacus arithmetic is the basis for understanding a math problem while doing the problem. Studies prove that learning the abacus improves the speed and accuracy of computation. The abacus does the arithmetic on a fives numbering basis, but by following the process in the beads of the abacus, the answer is shown in the beads based on a tens numbering basis (decimal). You do the problem in fives and get the answer in tens.

I am proposing a method of teaching arithmetic that is similar to abacus arithmetic and does enhance the speed and accuracy of computation. I call this fives supplement arithmetic. By representing numbers using combinations of five or less, the arithmetic becomes more understandable, easier to do, and more accurate. I personally use this methodology for doing arithmetic and find it much easier than conventional methodologies.

A New Approach to Arithmetic based on the Abacus

This is an article on the idea of using new arithmetic methods to make mathematics easier and more understandable. It relates a personal experience with mathematics that ended happily. This article attempts to relate this experience to remedial learning using an abacus and technology, and to the standard processes used in calculating with an abacus.

The goal is to suggest a new way of teaching mathematics based on a different number base that can make numbers more understandable. That is, to break numbers down to values of five or less and combine them from this basis in a method that provides easy understanding of the numbers and the processes.

The Abacus in Education

The abacus has been used for thousands of years (5. Samoly). It provides a way of calculating that does not need paper or other media. Cultures in Europe and Asia developed various forms of the abacus. The version I am discussing is the Chinese version. In this version the top bead area has two five value beads in each column (even though the second five bead is never used). The lower section has five single valued beads in each column (even though you never use the fifth bead). The Japanese abacus, the soroban, only uses the needed beads (the Chinese use has the extra beads to remind you that the value is included in totals, but the Japanese abacus is more efficient).

There have been various experiments, trying the abacus as a special tool in education. Nolan and Morris (1) studied the use of the abacus as an aid in teaching mathematics to the blind. Regular instruction and practice led to significant improvement is speed and accuracy of computation.

Shen (3) reported numerous researchers found that abacus learning has a positive effect on children's learning. It improves understanding of number concepts, and ”increases efficiency in mathematical calculations, enhances concentration skill, improves memory power, and boosts self-confidence”. His research was into working with mentally retarded. He taught them to use a mental abacus for arithmetic. This had the same effect as the use of the abacus in other studies. The participants were able to improve their ability to do mathematics in “real life situations”. The use of the abacus image improved the mathematics education experience.

Rosenblum and Smith (2) studied the use of braille, abacus, and tactile learning by universities with visually impaired students. The study found that most universities combine these methodologies in their training. The study did not indicate success of the training, just that such training was quite common.

Alternate Number Bases

Becker (4) discusses the issues related to teaching new number bases. The point is that, ”students are likely to learn ‘better’ when content is presented in more than one format.” Here the number bases are looked at from different perspectives, to enhance understanding of that number base.

The computer and its unique use of numbers have led to the need to study new number bases. The computer actually uses binary numbers. That is the base two, so we should just do binary. Only binary is not easy to understand.

In fact there are a few people who can read binary. But these are people who work in computer test and repair.

In order to represent the data so a few more people can read it and get some meaning out of it, other bases can be used to represent the computer data. One such base is octal. Octal has the advantage that it is based on eight which is an even power of two (23). This makes it is more useful in representing the data in computers. Because it is a power of two it represents the values in the computer in a symmetric and meaningful way.

Octal as compared with decimal and binary (which octal is representing) is shown in this table:

With octal we run out of numbers in a level at 8. We use 0, 1, 2, 3, 4, 5, 6, and 7. Then we have used the eight numbers that complete the series. Then we need to go to a higher level (or power) so we write 10. 10 does not mean ten here, it means eight. As we count on, 11 means nine.

Octal was used to represent data in computers that had 6 bit bytes. Such a byte could be represented by two octal numbers. Each half byte could hold up to the value 8. Two half bytes form a byte that could hold 64 values (8X8). Presently this is not a popular way to design a computer.

The use of 8 bit bytes and two 4 bit half bytes has become very popular because of the symmetry of the bytes and the ease of design of 8 bit processes in the computer. The 8 bit bytes can represent 256 different values. The numbering system for representing half byte data is based on 16. Note 16 is 24 an even power of 2. So it provides the symmetrical, and meaningful, representation of the bits in the byte. Hexadecimal (16) is shown in the following table as compared with binary and decimal.

Decimal numbering changes to 10 (the new number needs a higher level - literally higher power) after 9; since we run out of numbers at 9. The point is that using decimal means there are ten numbers in a level of counting. Then we have 0,1,2,3,4,5,6,7,8, and 9 as numbers (ten of them) . So then we need to go to the higher level. In decimal we understand this new level is referring to 10, 20, 30, 40, 50, 60, 70, 80, and 90. Then we get to 100 and are at an even higher level.

Hexadecimal needs to have 16 numbers in each level. These numbers were chosen to be 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. I was hoping for something better. We could make up some neat new numbers. That actually would add more problems and A, B, C, D E, F was much more workable. But here the number written 10 is not ten but sixteen. Number 11 means seventeen. The number written 100 in hexadecimal is equivalent to 256 in decimal.

The pairs of hexadecimal numbers represent what is in each byte of the computer (but held there in binary). Each number represents the value in four bits. Two numbers together represent the eight bits of one byte:

Perhaps these numbers are still mysterious, but many do find useful meaning from such a hexadecimal dump.

My Own Number Base

When I was about 9 my parents sent me to live with my aunt for a summer. She was a school teacher. The point was that I had not memorized the arithmetic tables. This meant I would be doomed to the list of students aimed at becoming factory workers. My parents had this dream of their children going to college.

In spite of my aunts efforts, I did not learn my arithmetic tables while with her for the summer. So I remained with the difficulty of not being able to do the mathematics problems in the book. I was being defined as dumb.

Well somewhere in this period of time I decided that my problem was that we were doing everything in tens, as the teachers said we should. After all we are on the decimal system. But tens are not do-able in our brain. We cannot handle tens issues because there are too many numbers and the mind does not keep track of them or their meaning.

I decided it would be better to do arithmetic in fives. We can keep all the numbers in our brain and not lose track of any of them and what they mean. So when you add 8 and 7; both numbers need to be changed to the fives supplement system. The 7 is 5 +2 and the 8 is 5 +3. So, considering this example, adding the 5s together is simple; giving 10. Then we need to add the fives supplements together; 2 + 3 = 5. So we add 10 and 5 and get 15. That is 8 + 7 = 5 + 3 + 5 + 2 = 5 + 5 + 3 + 2 = 10 + 5 = 15.

This is not exactly abacus arithmetic but very similar. The abacus keeps track of singles until there are four. Then the four singles are pushed and the five is popped. When you get to nine, you push the bunch, four singles and one five, and pop the ten (or more correctly the next digit).

I am doing the same kind of number transposing as abacus users do. Only I never had an abacus and never used one. When I bought an abacus, later in life, and got the book on how it worked, I saw the similarity with my mathematics (which I have used all my life). My method of doing it did not have the beads to help keep track of changes. Having the beads or even being able to visualize the beads would have helped me do it even better. The abacus does the arithmetic in fives. The thinking process in fives is followed by moving the beads. When done with a process the beads show the answer in tens (decimal).

So I was not so dumb. The development of this mathematical trick required some kind of intelligence.

This is a little story that brings back fond memories. After I invented this arithmetic I was asked to go to the board and do one of the problems in the book and then describe how I did it. So I did the arithmetic on the chalk board (I am showing my age) and then it was my turn to explain. So I explained my arithmetic by 5s. After I finished the teacher said “We can’t have you go up and do problems. You will just confuse the other students.” She didn’t say I was wrong.

Now back to the pictures of ten and five:

With the first set most people have to count to see how many. With the second, most people can tell without counting that there are five. Handling five is intuitive. That is the basis of abacus arithmetic. But by putting the result in beads on the abacus we see the final answer immediately in decimal.

At five we push the four singles and pop the five. At ten we push both the four singles and the five and pop the next digit. Then a single in the original digits is added to the ten and gives eleven.

Addition

Adding and subtracting uses arithmetic complements so 6 + 7 = 13 becomes 6 - 5 + 2 + 10 = 13:

6 + 7 = 13

5 + 1 - 5 + 10 + 2 = 13

5 + 5 + 1 + 2 = 13

10 + 3 = 13

The 5 and 1, and the 10 minus 5 are figured in the head. Then the two fives are added to get 10 thus the two fives are pushed and the next digit (10) is popped. Then the 1 and 2 are added and marked in singles by combining beads. The beads show the answer. The abacus may require more manipulations than straight decimal arithmetic, but doing arithmetic using the abacus can be much faster and easier than on paper. Another side to this is that arithmetic becomes a process rather than the result of memorizing tables. That means understanding arithmetic as number relationships is possible the abacus way.

Subtraction

Abacus subtraction uses complements. I used compliments (of 5 and 10, but more often just 5) in subtraction also.

11 - 7 = 4 becomes 11 - 10 + 3 = 4. The 3 is the 10s compliment of 7. The abacus arithmetic can use both 10s compliments and 5s compliments. We know we need to lose the ten. So we decide how much of the ten we will lose, 10 - 7 or 3. Then we add three and just hold the single, left from the eleven, to get four.

11 - 7 = 4

11 - 10 + 3 = 4

1 + 3 = 4

I tended to always use five’s compliments and supplements, rather than use ten’s compliments and supplements. This is my way of doing the problem:

11 - 7 =

5 + 5 + 1 - 5 - 2 =

5 - 5 + 1 + 5 - 2 =

1 + 3 = 4

Multiplication

Multiplication is slightly more complicated. I needed to base my multiplication on 5. That is the definition of the method. So I start with the 5s. This is not necessarily the way it is done on the abacus. But I needed to supplement my lack of remembering the multiplication tables, so I developed the fives multiplication methods.

First the ones are simple. 1 2 3 4 5 6 7 8 9 10

This is one times each other number, which is just counting.

The fives have a special character, related to the ones, that makes the multiples easier:

1/2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Shift decimal 5 10 15 20 25 30 35 40 45 50

The number 9 is special. Multiply 9 times anything and you get the shifted value minus the value.

Value 10 20 30 40 50 60 70 80 90 100

Subtract 1 2 3 4 5 6 7 8 9 10

9 X 9 18 27 36 45 54 63 72 81 90

Eight can be done similarly:

Value 10 20 30 40 50 60 70 80 90 100

Subtract 2X 2 4 6 8 10 12 14 16 18 20

8 X 8 16 24 32 40 48 56 64 72 80

Six might also be done this way with 5 rather than ten:

5 5 10 15 20 25 30 35 40 45 50

Add 1 2 3 4 5 6 7 8 9 10

6 X 6 12 18 24 30 36 42 48 54 60

I actually prefer the following (5s) method for six through eight. For these Intermediate numbers, you use the fives and fives supplement to make it easier:

6 X 8 = (5 + 1) X (5 + 3) = (5 X 5) + (5 X 3) + (1 X 5) + (1 X 3) = 25 + 15 + 5 + 3 = 45 + 3 = 48

This can be done quickly in the head once you recognize the process.

7 X 8 = (5 + 2) X (5 + 3) = (5 X 5) + (5 X 3) + (2 X 5) + (2 X 3) = 25 + 15 + 10 + 6 = 50 + 6 = 56

8 X 8 = (5 + 3) X (5 + 3) = (5 X 5) + ( 5 X 3) + (3 X 5) + (3 X 3) = 25 + 15 + 15 + 9 = 55 + 9 = 64

6 X 7 = (5 + 1) X (5 + 2) = (5 X 5) + (5 X 2) + (1 X 5) + (1 X 2) = 25 + 10 + 5 + 2 = 40 + 2 = 42

Here the 2, 3 and 4 multiplication process is assumed. You need to know that much. But they are not hard - they are less than five:

2 4 6 8 10 12 14 16 18 20

3 6 9 12 15 18 21 24 27 30

4 8 12 16 20 24 28 32 36 40

Now we venture into the difficult stuff. In these types of problems there is always a lot of arithmetic. Even though my method has more manipulations - it is easier and more likely to be correct.

Long Multiplication

847

48

(5+3)X(5+2) =25+15+10+6 56

(5+3)X(5-1)=(25-5+15-3) =32 (shift 0) 320

(5+3)X(5+3)=(25+15+15+9) =64 (shift 00) 6400

4X(5+2)=(20+8)=28 (shift 0) 280

4X4=16 (shift 00) 1600

4X(5+3)=(20+12)=32 (shift 000) 32000

Reading down the columns above 6+0+0+0+0=0 6

50+20+0+50+30+0+0 150

300+400+200+500+100+0 1500

5000+1000+1000+2000 9000

30000 30000

40656

Long Division

_______

Try 8 48 ) 40656|

4X(5+3)X4X10+(5+3)X(5+3)=(20+12)X10+ (25+15+15+5+4)=320+64 384 |

Try 4 225 |

4X4X10+4X(5+3)=160+(20+12)=160+32 192 |

Try 7 336|

4X(5+2)X10+(5+2)X(5+3)=(20+8)X10+(25+10+15+6)=280+56 336|

0|

The important part of this method is that the process uses the relationship between numbers rather than just memorization of a table. With memorization math becomes a set of cold tables. Then there is no meaning, just the result. With this process there is a flow between the original numbers and the resulting answer. It flows through a reasoning process rather than just being a set of values more or less looked up in a table - or remembered from a table. Of course, I could not remember the table.

References:

1. The Japanese Abacus as a Computational Aid for Blind Children, Carson Y. Nolan, June E. Morris, Exceptional Children. Sep 1964, Vol. 31 Issue 1, p15-17. 3p.

2. Instruction in Specialized Braille Codes, Abacus, and Tactile Graphics at Universities in the United States and Canada, L. Penny Rosenblum and Derrick Smith, Journal of Visual Impairment & Blindness. Jun 2012, Vol. 106 Issue 6, p339-350. 12p.

3. Teaching Mental Abacus Calculation to Students with Mental Retardation, Hong Shen, Ph.D., Journal of the International Association of Special Education. Spring 2006, Vol. 7 Issue 1, p56-66. 11p.

4. A Multiple-Intelligences Approach to Teaching Number Systems, Katrin Becker, Journal of Computing Sciences in Colleges archive, Volume 19 Issue 2, December 2003

Pages 6-17

5. The History of the Abacus, Kevin Samoly, Ohio Journal of School Mathematics, Number 65 • Spring, 2012, page 58

Google Books by Dr. Jerome Heath