Yudu - Connect Volume24 Issue 5 May-June 2011DRM

Egyptian Fractions

I began this exploration by bringing the students the copies of a page from David Eugene Smith’s book, Number Stories of Long Ago (1921).1 A passage from the book seemed to be a good starting place. In this passage, Smith describes, in very simple terms, that Ancient Egyptians used fractions only with a numerator of 1, with the exception of 2⁄3, for which they had a special character. Smith’s passage offers the following example:

“Instead of thinking of ¾, as we do, the priests . . . thought of ½ + ¼ ; and instead of thinking of 7⁄8, they thought of ½ + ¼ + 1⁄8 . . .” (p. 85)

This sentence was puzzling for students. So far, they had experimented with express-

ing quantities that are less than a whole, they had learned the common notations, and they knew that some fractions were equivalent to other fractions (i.e., 1⁄3 = 3⁄9). They knew how to express a fraction using various classroom objects (tangrams, papers, boards, etc.). On the other hand, they had never seen fractions added together, nor had they experimented with adding fractions with different denominators. They could not see a clear connection between 7⁄8 and ½ + ¼ + 1⁄8. The first breakthrough came from a student who, in a flash of inspiration,

managed to demonstrate, using at first a piece of paper and later blocks, that ¾ and ½ + ¼ are indeed equivalent.2 But for most students, 7⁄8 was still quite prob- lematic. Misha, an accomplished inquirer, had a more specific puzzle: “Why write 7⁄8 in this way?” It turned out that he was not arguing against the Egyptian notations, but he wanted to know why the priests had not used a much easier way of describing 7⁄8 by writing 1⁄8 + 1⁄8 + 1⁄8 + . . . until they had enough one-eighths. Building understanding happened slowly, and it took another day before a

convincing explanation of the 7⁄8 problem was offered. Even then, it did not mean that all students were comfortable with the concept. Fraction equivalency is a very difficult topic, and to pass over it too quickly can cause long-lasting damage to the students’ number sense. In the beginning, most of the students tended to use Misha’s method, which was adding the same unit fraction as many times as necessary. The more they practiced with different numbers, however, the more likely they were to use more complex solutions. They also learned from their friends.

Arabic Fractions

Every morning I would write a handful of Arabic fractions on a large piece of paper and leave it to the students to try and express these in Egyptian notations. My writing, for example, would read:

13⁄27 = Immediately one of the students would come up with the first answer, using Misha’s

method: 13⁄27 = 1⁄27 + 1⁄27 + 1⁄27 + . . .

But another student would add 1⁄3 + 1⁄27 + 1⁄27 + 1⁄27 + 1⁄27 as a second solution. Others

would add even more interesting combinations. Later in the day we would discuss the solutions that the students found most puzzling, trying to understand the writer’s reason- ing. As the denominators of the fractions we were translating became larger (99⁄100, for example), the students felt encouraged to think of elegant solutions from the start.

1. One of the very few books attempting to present the history of mathematics to children, written by the most eminent scholar of math history to date. 2. At this point we were using Egyptian hieroglyphics to discuss the numbers, as the students were familiar with Ancient Egyptian numerals.

Using blocks, students could see that ¾ is equivalent to ½ + ¼.

Misha, an accomplished

inquirer, had a more specific puzzle.