Project Master
Name
Date
PROJECT
1 Doing the Project
3
Time
An Ancient Multiplication Method
Thousands of years ago, the Egyptians developed one of the earliest multiplication
methods. Their method uses an idea from number theory.
Every positive whole number can be expressed as a sum of powers of 2.
▶ Exploring an Ancient Method
SMALL-GROUP
ACTIVITY
of Multiplication
22
23
24
25
26
2
4
8
16
32
64
1 + 2 + 16
19 =
2.
67 =
1 + 2 + 64
Follow the steps below to use the Egyptian method to multiply 19 ∗ 62.
19 º 62 =
Step 1 List the powers of 2 that are less than the first factor, 19.
Read and discuss the introduction and Problems 1 and 2 on Math
Masters, page 383 as a class. On the Class Data Pad, list the
powers of 2 from the table on the Math Masters page. Model the
steps for the Egyptian method of multiplication on the board or a
transparency. Allow partners time to discuss Problem 3. Then ask
volunteers to share their responses. Some students may have
analyzed the Egyptian algorithm in Part 3 of Lesson 2-9; consider
having these students lead the discussion.
Step 2 List the products of the powers of 2 and the second
factor, 62. Notice that each product is double the
product before it.
Step 3 Put a check mark next to the powers of 2 whose sum is
the first factor, 19.
Step 4 Cross out the remaining rows.
3.
Finally, have students look at the first two problems at the bottom
of Math Masters, page 384, which have been solved using a
variation of the Egyptian algorithm. Ask students to solve the
third problem by the same method. Recreational mathematics
books call this method the Russian Peasant Algorithm. It is
performed by repeatedly halving the number in the left column,
ignoring any nonzero remainders, and doubling the number in the
right column. All rows that have an even number in the left
column are crossed out, and the remaining numbers in the right
column are added. The sum of these partial products is the
answer to the multiplication problem.
Explain why you don’t have to multiply by any number other
than 2 to write the list of partial products when you use the
Egyptian method.
(Math Masters, p. 385)
When students have completed their comparison charts, bring
them together to share preferences. Have students support their
choices with examples.
4
8
248
496
16
992
✓ 1
✓ 2
124
4
8
248
496
✓ 16
62
992
Math Masters, p. 383
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Project Master
Name
PROJECT
3
4.
Have students consider the multiplication algorithms they know
and record the advantages and the disadvantages of each on
Math Masters, page 385. This should help students decide which
algorithm works best for them. However, this need not be the
only algorithm they use. It is also important to emphasize that a
paper-and-pencil algorithm might not be the most efficient way to
solve a problem. Mental computation, a calculator, or an estimate
might be a better choice.
62
124
Once you begin the list, think of the powers of 2 as
repeated-factor expressions. So each time you multiply, it’s
the previous product multiplied by 2. This is because each
time you are multiplying by one more factor of 2. For
example, 23 ∗ 62 = 2 ∗ 2 ∗ 2 ∗ 62, or 8 ∗ 62, and 24 ∗ 62
can be thought of as 8 ∗ 62 ∗ 2. So if you know that
23 ∗ 62 = 496, then 24 ∗ 62 = 496 ∗ 2.
INDEPENDENT
ACTIVITY
Algorithms
1
2
19 º 62 = 1,178
Step 5 Add the partial products that are not crossed out.
62 + 124 + 992 = 1,178
So 19 ∗ 62 = 1,178
Have students solve the three multiplication problems at the top of
Math Masters, page 384 using the Egyptian algorithm.
▶ Comparing Multiplication
21
1
Write a number sentence to show each of the numbers below as the sum of powers
of 2. For example, 13 = 1 + 4 + 8.
1.
(Math Masters, pp. 383 and 384)
20
Date
Time
An Ancient Multiplication Method
cont.
Try to solve these problems using the Egyptian method.
85 ∗ 14 =
1
2
✓ 4
8
✓ 16
32
✓ 64
✓
1,190
14
28
56
112
224
448
896
1,190
1,634
38 ∗ 43 =
43
1
✓ 2
86
✓ 4
172
8
344
16
688
✓ 32 1,376
1,634
45 ∗ 29 =
1
2
✓ 4
✓ 8
16
✓ 32
✓
1,305
29
58
116
232
464
928
1,305
Try This
5.
Here is another ancient multiplication method, based on the Egyptian method.
People living in rural areas of Russia, Ethiopia, and the Near East still use this
method. See whether you can figure out how it works. Then try to complete
the problem in the third box, using this method.
1,634
45 ∗ 29 =
25
38
43
45
6
50
19
86
22
58
3
100
9
172
11
116
1
200
4
344
5
232
325
2
688
2
464
1
1 ,376
1
13 ∗ 25 =
325
1,305
38 ∗ 43 =
13
1 ,634
29
928
1,305
Math Masters, p. 384
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Project Master
Name
Date
PROJECT
Time
2 Extending the Project
Comparing Multiplication Algorithms
3
Think about the advantages and disadvantages of each multiplication method
that you know. Record your thoughts in the chart below.
Sample answers:
Algorithm
Advantages
Lattice
4
3
1
2
2
1
4
0
0
8
6
2
6
6
▶ Using Ancient Numerals in
I must be
sure not to
confuse 60s
and 40s with
6s and 4s.
(Math Masters, p. 386)
Problem 1 on Math Masters, page 386 shows how the ancient
Egyptians might have used their hieroglyph numerals and
algorithm to multiply two numbers. Ask students to solve Problem
1 using hieroglyph numerals. When most students have finished,
have volunteers share their solutions on the board.
6
It’s easy to
work with
doubles of
numbers.
Egyptian
43
º
62
✓ 1
62
✓ 2
124
4
248
✓ 8
496
16
992
✓ 32
1,984
2,666
It takes
too long to
double
numbers.
Problem 2 asks students to use the Egyptian algorithm to multiply
with Roman numerals. Ask students to solve Problem 2 using
Roman numerals. It is sometimes said that “multiplication with
Roman numerals was impossible,” but, at least for smaller
numbers, it seems possible with this algorithm. Have volunteers
share their solutions on the board.
Math Masters, p. 374
385
374-409_EMCS_B_MM_G5_Proj_576973.indd 385
PARTNER
ACTIVITY
Multiplication Algorithms
I multiply only I have to
line up the
1-digit
numbers very
numbers.
carefully.
6
8
Disadvantages
I can do it in
easy steps.
Partial Products
43
∗ 62
60 [40s] = 2,400
60 [3s] =
180
2 [40s] =
80
2 [3s] =
6
2,666
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12
Hindu-Arabic 0
1
2
3
4
5
6
7
8
9
10
Babylonian
✓
Egyptian
∗
XII
XV
I
XV
II
XXX
IV
LX
✓ VIII
Mayan
A
Greek
B
9
D
E
F
Z
H
15
U
CXX
CLXXX
Roman
Ask the class if they feel it would be possible to multiply Roman
numerals for larger numbers using the Egyptian algorithm. Allow
partners time to explore multiplying Roman numerals for larger
numbers.
Project Master
Name
Date
PROJECT
3
1.
Time
Ancient Math Symbols
▶ Learning More about
The ancient Egyptians used picture symbols, called hieroglyphs,
to write numbers. Here is how they might have multiplied 11 ∗ 13
using the algorithm you learned in this project.
=1
✓
= 10
✓
Number Systems
(1 º 13)
(2 º 13)
Much information about Egyptian mathematics is found on a scroll
called the Rhind papyrus. This scroll, copied about 1650 B.C., is
named after the man who purchased it in Egypt in A.D. 1858. It is
now in the British Museum in London.
(4 º 13)
= 100
= 1,000
✓
= 10,000
(8 º 13)
= 100,000
(11 º 13)
INDEPENDENT
ACTIVITY
= 1,000,000
Invite students to use the Internet to learn more about different
numeral systems, such as the Egyptian, Roman, and Babylonian
numeral systems. (The Babylonian numeral system uses a base
of 60.) Consider having students make Venn diagrams to compare
the similarities and differences between the base-10 number
system and a different numeral system of their choice.
On the back of this sheet, try to multiply 21 ∗ 16 using the Egyptian algorithm
and Egyptian numerals.
2.
Do you know any Roman numerals? They were used in Europe for centuries
until Hindu-Arabic numerals replaced them. Today, Roman numerals appear
mainly in dates on cornerstones and in copyright notices.
It is sometimes said that “multiplication with Roman numerals was impossible.”
Is that true? See whether you can multiply 12 ∗ 15 using Roman numerals
and the Egyptian algorithm. Use the back of this sheet.
Examples of Roman Numerals:
I=1
II = 2
III = 3
IV = 4
V=5
VI = 6
IX = 9
X = 10
XX = 20
XL = 40
L = 50
LX = 60
C = 100
D = 500
M = 1,000
The Web site below offers links to a variety of Internet sites on
number systems: http://mathforum.org/alejandre/numerals.html
Math Masters, p. 386
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An Ancient Multiplication Algorithm
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