Egyptian Unit Fractions and the Rhind Papyrus

This week we studied the first record of advanced mathematics called the Rhind Papyrus. It is not a theoretical treatise, but a list of practical problems encountered in administrative and building works. The text contains 84 problems concerned with numerical operations, practical problem-solving, and geometrical shapes. The majority of literate Egyptians were scribes and they were expected to undertake various tasks. These must have demanded some mathematical as well as writing skills. The Rhind Mathematical Papyrus is also important as a historical document, since the copyist noted that he was writing in year 33 of the reign of Apophis, the penultimate king of the Hyksos Fifteenth Dynasty (about 1650-1550 BC) and was copied after an original of the Twelfth Dynasty (about 1985-1795 BC). It was copied by the scribe Ahmes.

The papyrus was acquired by the Scottish lawyer A.H. Rhind during his stay in the ancient city of Thebes (modern day Luxor,Egypt) in 1858. The Rosetta Stone, also in the British Museum, is probably the reason that we are able to decipher the Rhind Papyrus.

All the children explored working with sectors (these are portions of circles like the 1/8th slice of pizza). They had to identify the numerator (top number) of the fraction which is the number of pieces shaded in and the denominator (bottom number) of the fraction which is the number of pieces it takes to make a whole.

I also encouraged them to simplify their fractions. For example, if they identify 4/4 this is one whole so write “=1”; if they identify 2/4=1/2 and so on. I also gave them a document with which they can cut out sectors of 1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10, 1/12. These should be colored differently for each unit fraction. Please put them in a zip lock bag.

The 3rd-6th graders explored the Egyptian formula for splitting unit fractions into two or more distinct unit fractions different denominators (Egyptian mathematics would not allow numerators to have numbers other than one and would not allow an equation to repeat a denominator such as 1/2 = ¼ + ¼).

These students worked with sectors of 1/2, 1/3, 1/4, 1/6 and 1/12 and were able to see that 1/2 = 1/3 + 1/6 and that 1/3 = 1/4 + 1/12. Some of them made a conjecture that 1/4 = 1/5 + 1/20. They were able to see that 1/n = 1/(n+1) + 1/n(n+1). This was discovered by these ancient Egyptians. They worked on creating 1/2 with 2 fractions, then 4, then 8, then 16. (see attached).

Children should use the Chinese Stick Multiplication method or standard algorithm for multiplication if they know it. Otherwise, they can use distributive property. If they are not able to do the multiplication, they should just show which two numbers need to be multiplied.