Ryan Gerwitz and Doug Schwartz
MAED 417
Methods for Teaching Middle and Secondary Mathematics
Professors Cox and Howard
“COUNT LIKE AN EGYPTIAN”
The purpose of this lesson plan is to reinforce the use of “duplation” in our schools today. Duplation
is an algorithm that was developed by the Ancient Egyptians that allowed them to multiply and
divide numerals by doubling.
Professional Standards Addressed:
This lesson addresses the following NYS-MST standards:
K.N.13 Determine sums and differences by various means.
5.PS.2 Understand that some ways of representing a problem are more efficient than
others.
5.PS.21 Explain the methods and reasoning behind the problem solving strategies used.
This lesson addresses the following NCTM standards:
- Understand numbers, ways of representing numbers, relationships among numbers, and
number systems.
Instructional Objectives:
Following the completion of this lesson, students should be able to:
-Multiply and “divide” numbers using the “duplation” algorithm.
-Perform calculations using Egyptian hieroglyphics.
Instructional Protocol/Itinerary:
Start with the “hook” by introducing Egyptian numerals. Then introduce students to the
concept of duplation using hieroglyphics. Have another teacher, or someone who understands
duplation, help you demonstrate to the class. Have a “race” with your fellow instructor to see who
finds the correct answer faster. Explain the difference between the traditional method of
multiplication and duplation. Next, use Hindu-Arabic numerals (modern numbers) with duplation.
Make sure to multiply and divide using duplation.
COUNT LIKE AN EGYPTIAN
Every student learns common arithmetic (addition, subtraction, multiplication, and division)
in our schools today. Unfortunately, in most cases, there are very few methods in which it is taught
(differentiated instruction). Also, with the use of calculators, arithmetic by “hand” has become
obsolete. The purpose of this lesson is to introduce the students to the different methods of
Egyptian mathematics… without the use of calculators.
Luxor (formerly Thebes) is the place where most of our knowledge about the Ancient
Egyptians originated from. Specifically, this knowledge was found in the Rhind papyrus around 1650
BCE. The Ancient Egyptians had several forms of writing material available to them. These
included stone, papyrus (paper made from strips of pulp of a water reed), wood, and pieces of
pottery. The Egyptian writing system was split into the hieratic (religious) and the demotic
(popular). The hieratic was a cursive script derived from the hieroglyphic, while the demotic was a
(less formal) later form of the hieratic.
The Egyptian numeral system was a base 10 simple grouping system that involved symbols
known as “hieroglyphics.” The hieroglyphics were written (most commonly) from left to right, (but
also right to left), with the lower values starting on the left. Some hieroglyphics are listed below:
Suppose the Egyptians wanted to multiply or “divide” two numerals. The Egyptians would use a
process known as “duplation”: an algorithm in which calculations are commonly carried out by using
multiplication and division. To avoid confusion, let’s look at how we would multiply two numbers using
duplation in Hindu Arabic.
Multiplication using Duplation
Suppose we wanted to multiply 47 and 895. We need two columns in which the left column
represents powers of two, starting at 2 = 1. We stop listing powers of two if we were to exceed
the smaller number (in this case, 47). In the right column, we start with the larger number (in this
case, 895), and double it. Each number in the right column is doubled until we reach our highest
power of 2 in the left column:
0
20 = 1
21 = 2
22 = 4
23 = 8
2 4 = 16
2 5 = 32
895
1790 = (895)(2)
3580 = (1790)(2)
7160 = (3580)(2)
14320 = (7160)(2)
28640 = (14320)(2)
Doubling each term in the right column is equivalent to multiplying each power of 2, in the left
column, by 895.
Continuing, let’s underline powers of two that sum to 47 in the left column:
1
2
4
8
16
32
895
1790
3580
7160
14320
28640
Let’s now sum those numbers which correspond to the underlined numbers:
895 + 1790 + 3580 + 7160 + 28640 = 42065. Thus, (47)(895) = 42065.
This will always work! Now we ask the question… why? (This may be a good homework problem!)
Fact: 47 can be represented, in powers of two, as: 32 + 8 + 4 + 2 + 1. Thus,
47895  32  8  4  2  1895  32895  8895  4895  2895  1895  42065
Fact: All whole numbers can be represented as a sum of powers of 2.
Example:
31  2 4  23  2 2  21  20
579  29  2 6  21  2 0
Division with Duplation
Division works very similarly with Duplation. Suppose we wanted to perform: 255  5.
The Egyptians would interpret this, in words, as the following: “Multiply 5 so as to get 255”,
because they didn’t think in terms of division. We start, in the right column, with the smaller of
the two numbers we want to divide; e.g. If we want to divide 255 and 5, we place 5 in the right
column. We then double until, if we were to double again, we would exceed the higher number (in
this case 255). Then we list the powers of two in the left column again:
1
2
4
8
16
32
5
10
20
40
80
160
Continuing, let’s underline the numbers in the right column that sum to the larger number (255):
1
2
4
8
16
32
5
10
20
40
80
160
Now sum the powers of two in the left column that correspond to those in the right column:
1 + 2 + 16 + 32 = 51. Thus, 255  5 = 51.
Do we always get “nice” quotients when we divide? What if we have: 514  36?
1
2
4
8
36
72
144
288
If we sum 72, 144, and 288, we obtain 504. Since we can not exceed 514, we can not sum 36 and
504. Thus, we have 2 + 4 + 8 = 14 as our quotient and 10 as our remainder. The Egyptians, of
course, would add an extra row with 10 in the right column and
10
in the left column (remember
36
that the Egyptians needed to multiply some number by 36 to obtain 10). Thus, we have the
following:
1
2
4
8
36
72
144
288
10
36
10
Now, we see that 514 = 72 + 144 + 288 + 10. So, if we sum the numbers in the left column that
correspond with those in the right, we obtain: 2 + 4 + 8 +
14 +
10
. Thus, we obtain our answer of
36
10
.
36
If we were to multiply the following 2 numbers using Egyptian numerals, this is how we would do it
(recall that the Egyptians numeral system was a base 10 simple grouping system):
The final answer (in Egyptian) should be equivalent to 3072 (in Hindu-Arabic):
Conclusion: It does not matter what numeral system we are using to multiply if we use the duplation
algorithm.
Possible “Homework” problems for students
1)
Multiply
and
using
duplation. Use only Egyptian hieroglyphics throughout.
2)
Add
and
3)
Multiply 1000 and 597 using duplation.
4)
Divide 765 by 89 using duplation.
5)
Give an explanation of why duplation always works when multiplying and dividing two
numbers.
Bibliography
Hieroglyph Image found on: http://homepage.mac.com/shelleywalsh/MathArt/EgyptDivide.html.
Image revised 11/8/2010.
Howard, C. A. (2009). Mathematics Problems From Ancient Egyptian Papyri. Mathematics Teacher,
103(5), 332 – 339.
Seppala-Holtzman, D. N. (2007). Ancient Egyptians: Russian Peasants Foretell the Digital Age.
Mathematics Teacher, 100(9), 632 – 635.
Smith, D. E. (1953). History of Mathematics. New York, NY: Dover Publications, INC.