**By: Lathiful A., Puri P., Kurnia R.**

**A. ****Introduction**

Fraction is taught in school as an important subject to understand the next topic in mathematics. Many students are familiar with fraction as a notation of number consisting of numerator and denominator. For the students as the learner, definition of basic concepts of fraction has to be frame with care because we find many cases related to fraction in our life. But sometimes they have difficulties in understanding what the meaning of fraction is. They only use the procedural algorithm of fraction that they learn from school without understand what the meaning of fraction itself. When asked to add two fractions and get an integer answer, they added the numerators or the denominators of the two fractions.

To introduce the meaning of fraction we must also investigate the history of the fraction. Given that emphasize, this is our way to teach mathematics inspired by history (based on ICMI study). Therefore, in this case we focus on constructing the teaching and learning process in the class by introducing the students how human develop and learn about fraction. Introducing fraction with the history leads the students understand what the basic concept of fraction is.

Since the students do not recognize the meaning of fraction, it will be introduced through the context that is familiar with them and connect with the history of fraction. In this paper we will use the Egyptian system as starting point to understand the concept of fraction.

In this paper, we will implement the introducing of fraction inspired by The Egyptian System using the following trajectories:

**B. History of Fraction**

Fractional word actually comes from the Latin “fractio” which means break. To understand how the fraction has evolved into a form that we know today, we must step back further to historical time to find out the process of notation of fraction.The history of fraction passed through long time and long journey. There were several notation fractions evolved simultaneously with the finding of number systems in historical times.

**1. ****Egyptian numeral hieroglyphics**

In 1800 BC, the Egyptians were writing fractions. Their number system is base 10, so they have separate symbols for 1, 10, 100, 1000, 10000, 100000, and 1000000. Writing system of ancient Egypt like in the following picture is called *hieroglyphics*:

The Egyptians wrote all the fraction using fractional units (fraction which has 1 as a numerator). They put a picture of the mouth (which means the part of) above the number. For example figure 3 represented the fifth.

They stated other fragments as the number of fractional units, but they are not allowed to repeat a fractional unit on this side. In other words, if we interpret to our number system today is like in the following:

In Egyptian systems, 2/3 or any unit fraction (fraction with one as the numerator, like 1/7) were expressed in a simple, straightforward way. 1/2 had a sign of its own ( ), as did 2/3 ( ). And the other unit fractions were just the symbol (meaning “part”), with the denominator expressed as an integer, under this symbol. I drew the “r” after the symbol, to show that it is pronounced “r”. 1/7 would be that symbol with seven, small vertical lines under it ( ).

The big disadvantage from Egypt system is that it was very difficult to calculate. To try to overcome this, the Egyptians made a lot of tables, so they can find the answers to these problems.

**2. Ancient**** Rome**

In Ancient Rome, fraction is only written using words to explain part of the whole. They are based on the unit of weight which was called “as” in which one “as” was made up of 12 uncia. Therefore, the fraction in that time was centered on twelfths. For example:

112 was called uncia (means 1/12)

124 was called semuncia (means 1/12)

1288 was called scripulum (means 1/288)

However, those words make it very difficult to perform calculations.

**3. The ****Babylonians**

Beside of that, there were the other people, *The Babylonians*, who came up with a more reasonable way to represent fractions. They did it before the Roman method. Their number system based 60. Babylonians extended only to enter their number fractions in sixtieths, as we did for the tenth, hundred etc. However, they do not have the kind of zero or a decimal point. This makes it very confusing because they can be interpreted in different ways. They also made a group of ten.

Here is an example:

Babylonian number 12 and 15

*The Babylonians* has also expressed the notation of fraction in their system. For example they expressed the fraction which is interpreted like in the following table:

Sixtieths X60 Unit Number

Although the Babylonians had a very sophisticated way of writing fractions, it contains the deficiency also. Around 311BC they designed Zero. It made number system become easier. However, they still didn’t recognize about a decimal point, so it was still difficult to distinguish the fraction and integers.

**4. India**

The format which we know today is directly derived from the work of Indian civilization. The success is the way to write fractions as numbers which have three main ideas:

i) Each number has the symbol,

ii) The number depends on the position of digit in whole number,

iii) Zero is no need to interpret and also to fill the place of the missing units

In a fraction of India was written very similar to what we do now, with one number (the numerator) and the other above the other (denominator), but without the line. For example:

**5. Arab**

It was the Arabs who add the line (and sometimes pulled horizontally, sometimes slash) which we now use to separate the numerator and denominator: 3/4.

So here we have the pieces as we now recognize. It’s amazing to think how much thought has been entered into the way we write.

**C. ****The Implementation of Teaching and Learning**

Because the characteristic in our paper (based on ICMI study) is about learning mathematical topics inspired by history, therefore we use the implementation of experiental mathematical activities. An experimental mathematical activity used in learning activity is a method activity. In this case, students are asked to solve the problems in contextual world, the problem about dividing chocolates equally. This problem is similar to the Egyptian system in which they divided food equally. We choose the Egyptian system as a method to introduce fraction concept. We make the lesson plan which will be implemented by teacher in grade 3 primary school. We use the basic concept of fraction implemented through the context. The teaching and learning process was started with a little explanation about The Egyptian System whereas in 3000 BC Egyptians have an interesting way to represent fractions. The teacher also shows to the students the number system of Egyptian System in historical times (like we mentioned on the previous page).

The Egyptian System has a notation for 1/2, 1/3, 1/4 and so on. This is called a unit fraction. In Egyptian System, it does not allow them to write 2/5 or 3/4 or 4/7 as we see today.

Instead, they can write each fraction as the sum of the fractional unit where all the fractional units in a different form.

For example,

3/4 = 1/2 + 1/4

6/7 = 1/2 + 1/3 + 1/42

After the students are introduced with The Egyptian System, the teacher asks students to involve in the mathematical activities such in the following:

1. The teacher brings 5 chocolates. She said that she will share those chocolates to 8 people. She asks the students to find the strategies in sharing 5 chocolates for 8 people equally without using calculator.

2. Our prediction is the students will add together and then divide it become 8 parts. Of course it will be difficult if they don’t have the precise measurement tools. Therefore, they have to find another way which is easier until they find the unit fraction such in the following:

- First of all they divide each chocolate become two equal parts (making half of each)
- When the students come up into the fifth chocolate, we hope the students do the same thing i.e dividing each part become two equal parts until they get 8 equal parts

- After the students do those activities, the teacher asks the students the meaning of sharing 5 things to 8 things. Does it make sense for students or not?

3. When the students have successful in finding the strategy in sharing the chocolates for the certain number equally, the teacher brings other chocolates. For this time, she brings 4 chocolates which will be shared to 7 people. In similar way, the students have to solve this problem and then they have to compare the two problems above. We hope the students use the unit fraction like in the following:

4. From those activities, we hope the students can determine which one is the larger between 5 of 8 and 4 of 7.

**D. Conclusion**

History of fraction is an inspiration for us to understand the fraction. The Egyptian concept about fraction which was using “unit fraction” was an inspiration in introducing fraction concepts to students. By using contextual problems as experienced of Egyptian contextual problems become the starting point for facilitating students to construct their understanding of fraction concepts.

**E. References**

Roger Herz-Fischler. The Shape of the Great Pyramid.

Newman, James. The World of Mathematics. New York: Dover Publications, Inc. 1956.

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