MULTIPLICATION USING HISTORICAL METHODS Joanne Peeples El Paso Community College [email protected].

MULTIPLICATION USING HISTORICAL METHODS

Joanne PeeplesEl Paso Community [email protected]

Increasingly students are entering college unable to

multiply or divide integers without using a calculator

(hence having little number sense). Rather than

repeat multiplication methods they have learned,

why not try Egyptian arithmetic, multiplication per

gelosia (“by jealousy”) or Napier’s bones? History

plus math is often a winning strategy.

EGYPTIAN NUMBERS

EGYPTIAN NUMBERS

23 ∩∩│││ or │∩│∩│

Add 23 ∩∩│││ 23 ∩∩│││ ______________ ∩∩∩∩││││││

MULTIPLICATION

26 X 4

___26____________

1 4 2 8 4 16 8 32 16 64 32 128 Too Big! 32 is bigger than 26

26 = 16 + 10 = 16 + 8 + 2

Add: 64 + 32 + 8 = 104

WHY DOES THIS WORK?

424242 134

4222 34

104426

DIVISION (NO REMAINDER)

161 = 112 + 28 + 14 + 7 Therefore 161 7 = 16 +4+2+1 = 23

161 7

_____161___

1 7 2 14 4 28 8 5616 11232 224 Too Big! 224 is bigger than 161

DIVISION (WITH REMAINDER)

84 11

_________84___

1 11

2 22

4 44

8 88 Too Big! 88 is bigger than 84

84 = 44 + 22 + 11 + 7 Therefore 84 11 = 4 + 2 + 1 = 7

with a remainder of 7 We would write 84 11 = 7 + 7/11

EGYPTIAN FRACTIONS

││││ Four

││││ one-fourth

Write = =

3.14 = 3 + │││ ∩ ∩∩│││││

│││ ∩│││││

EXAMPLE

Is this approximately 4, 3, or 2?

1310

1

2

13

1310

4586

FRACTIONS DECREASE IN SIZE (SO THEY CAN BE USED FOR ESTIMATION.)

In a sense, Egyptian fractions are closer to “our” decimal system than “our” fractions.

141.325

1

10

13

FRACTIONS NEVER REPEAT.

Writing as is wrong, since would be a bad approximation.

Writing as is acceptable since is a good approximation.

FRACTIONS MUST NOT BE “CLOSE” TO EACH OTHER. Writing as is not acceptable, since and are

“too close” to each other.

Writing as is acceptable since an

approximation of is reasonable.

A FRACTIONAL REPRESENTATION MAY NOT BE UNIQUE. Note : 1.75 = 175% = = so our representation of numbers is not unique.

AND AS YOU MAY HAVE SEEN, ALL THE FRACTIONS ARE UNIT FRACTIONS.

HOW DO YOU FIND A FRACTIONAL REPRESENTATION? 4 ÷ 7. Think of something you can divide, like bread.

each part is of a whole

So

loaf left

loaves left

loaf left

of loaf

So

OR

45

45

Do not cut!

Whole loaf of loaf

Half loaf of loaf

So

WHAT IF THE AREA OF A TRIANGLE WHOSE BASE IS 37 UNITS AND HEIGHT IS 12 UNITS?

A = x base x height

x base

1 37 = 20 + 16 + 1

10 + 8 + = 18

2

22

(Write ││ as ) 2

2

2

( base) height

Area of triangle is 24 + 192 + 6 = 222 sq. units.

18

1 12

2 24

4 48

8 96

16 192

6

MULTIPLICATION PER GELOSIA 2,314 X 157 =

363,298 2 3 1 4

1

5

7

0 0 0 0

2 3 1

41 1 0 2

0 5 5

01 2 0 2

4 1 7

8

1 1

0

3

6

3 2 9 8

NAPIER’S BONES

BIBLIOGRAPHY

Reimer, David, Count Like an Egyptian, A Hands on Introduction to Ancient Mathematics. Published by the Princeton University Press, 2014.

Ifrah, Georges, The Universal History of Numbers. Published by John Wiley and Sons, Inc., 2000  Napier, John, Rabdology, translated by William Frank Richardson.Volume 15 in the Charles Babbage Institute Reprint Series for the History of Computing. Published by the Massachusetts Institute of Technology and Tomash Publishers, 1990

THANK YOU!

[email protected]

PowerPoint Artist – Amie Stoker