HISTORICAL CONNECTIONS
IN MATHEMATICS
Volume I
Developed and Published
by
TM
AIMS Education Foundation
This book contains materials developed by the AIMS Education Foundation. AIMS (Activities Integrating
Mathematics and Science) began in 1981 with a grant from the National Science Foundation. The non-profit
AIMS Education Foundation publishes hands-on instructional materials that build conceptual understanding.
The foundation also sponsors a national program of professional development through which educators may
gain expertise in teaching math and science.
Copyright © 1992, 2005, 2012, 2013 by the AIMS Education Foundation
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ISBN 978-1-60519-066-2
Printed in the United States of America
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
INTRODUCTION
TABLE OF CONTENTS
CHAPTER 1
Portrait of Pythagoras...........................................................................................................9
Pythagoras: The Master Teacher.........................................................................................10
Activities:
Number Shapes........................................................................................................12
Square, Oblong, and Triangular Numbers.................................................................13
Pythagorean Discoveries..........................................................................................14
Figurate Families......................................................................................................15
The Spider and the Fly.............................................................................................16
A Pythagorean Puzzle...............................................................................................17
Pyramid Puzzles........................................................................................................18
CHAPTER 2
Portrait of Archimedes........................................................................................................19
Archimedes: The Greek Streaker........................................................................................20
Activities:
Archimedes' Mobiles................................................................................................22
A Teeter-Totter Discovery.........................................................................................23
A Balancing Act.......................................................................................................24
Counting Kernels......................................................................................................25
Predicting Float Lines...............................................................................................26
The King's Crown: A Skit..........................................................................................27
CHAPTER 3
Portrait of Napier................................................................................................................29
Napier: The 16th Century Mathemagician..........................................................................30
Activities:
Lattice Multiplication................................................................................................32
Napier's Rods...........................................................................................................33
Russian Peasant Method of Multiplication.................................................................34
Earthquake Mathematics..........................................................................................35
The Magic Rooster: A Skit........................................................................................36
Napier's Magic: A Crossword Puzzle.........................................................................38
CHAPTER 4
Portrait of Galileo................................................................................................................39
Galileo: The Father of the Scientific Method........................................................................40
Activities:
Galileo Drops The Ball..............................................................................................42
Have Gravity: Must Travel.........................................................................................43
How High Can You Throw?.......................................................................................44
How Fast Can You Throw?........................................................................................45
Heartbeats and Pendulums.......................................................................................46
CHAPTER 5
Portrait of Fermat...............................................................................................................47
Fermat: The Marginal Mathematician.................................................................................48
Activities:
Primes and Squares..................................................................................................50
Prime Number "Machines"........................................................................................51
Number Tricks..........................................................................................................52
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© 2012 AIMS Education Foundation
The Proof is in the Pudding.......................................................................................53
Suspicious Sailors....................................................................................................54
CHAPTER 6
Portrait of Pascal................................................................................................................55
Pascal: Launcher of the Computer Age...............................................................................56
Pascal's Triangle.................................................................................................................58
Activities:
Summing up Pascal..................................................................................................59
Pascal Magic............................................................................................................60
MATHEMATICS........................................................................................................61
Pascal's Perimeter....................................................................................................62
A Birthday Surprise..................................................................................................63
CHAPTER 7
Portrait of Newton...............................................................................................................65
Newton: Small But Mighty..................................................................................................66
Activities:
A Tower of Powers of 2.............................................................................................68
Chain Letter Madness...............................................................................................69
A Series Surprise......................................................................................................70
The Binomial Theorem.............................................................................................71
The Short Giant: A Skit............................................................................................72
CHAPTER 8
Portrait of Euler..................................................................................................................75
Euler: The Bridge to Topology............................................................................................76
Activities:
Vertices, Regions, and Arcs......................................................................................78
Traveling Networks...................................................................................................79
Faces, Vertices, and Edges.......................................................................................80
Knight's Move on the Chessboard.............................................................................81
Inside or Outside?.....................................................................................................82
CHAPTER 9
Portrait of Germain.............................................................................................................83
Germain: Mathematics in a Man's World.............................................................................84
Activities:
Four Fours...............................................................................................................86
$1.00 Words.............................................................................................................87
Palindromes.............................................................................................................88
Counting Divisors.....................................................................................................89
Happy Numbers.......................................................................................................90
Midnight Math: A Crossword Puzzle..........................................................................91
CHAPTER 10
Portrait of Gauss.................................................................................................................93
Gauss: The Prince of Mathematicians.................................................................................94
Activities:
Gauss's Challenge....................................................................................................96
Summing Odds........................................................................................................97
Last Digits................................................................................................................98
Average Ability.........................................................................................................99
What's for Lunch?...................................................................................................100
RESOURCES...................................................................................................................101
SUGGESTIONS AND SOLUTIONS.................................................................................103
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
INTRODUCTION AND SUGGESTIONS FOR TEACHERS
“I am sure that no subject loses more than mathematics
by any attempt to dissociate it from its history.”
Glashier
Our goal in this book is to provide a collection of resources to make it easy for
teachers to integrate the history of mathematics into their teaching. While mathematics
history textbooks abound, there are not many sources which combine concise biographical information with activities to use in the classroom. We hope that the problem
solving experiences, the portraits, and the anecdotal stories will facilitate a broad,
natural linkage of human elements and mathematical concepts.
The value of using history in teaching mathematics is currently gaining emphasis.
Providing a personal and cultural context for mathematics helps students sense the
larger meaning and scope of their studies. When they learn how persons have discovered and developed mathematics, they begin to understand that posing and solving
problems is a distinctly human activity.
Using history in the mathematics classroom is often a successful motivational tool. Especially when combined with manipulatives, illustrations, and relevant
applications, historical elements have the power to make mathematics “come alive”
as never before. By viewing mathematics from a historical perspective, students learn
that the process of problem solving is as important as the solution.
This book can be used in many ways. The teacher may choose to read or share
biographical information and anecdotes as an introduction to one or more of the
activities in a particular section. Portraits may be posted or distributed, and puzzles or
skits may be used independently. It may be most effective, however, to focus on one
mathematician at a time. A wide range of activities may be incorporated into a unit on
a specific mathematician, allowing the teacher to make cross-disciplinary connections
with social studies, language arts, and science.
Mathematicians may be selected for emphasis according to the concepts being
introduced in the mathematics curriculum or may be used at random for enrichment.
While some of the activities do not replicate the exact problems the mathematicians
worked on, they represent the areas of interest of those mathematicians.
Activities have been chosen to appeal to a wide range of interests and ability levels.
Complete solutions and suggestions for use are included in the back of this book.
Wilbert Reimer
Luetta Reimer
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
Pythagoras
c. 560 - c. 480 B. C.
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
PYTHAGORAS
THE MASTER TEACHER
Anecdotes:
Biographical Information:
Pythagoras (pi-THAG-uh-rus) of Samos
(c.560-c.480 B.C.) was a Greek philosopher
and religious leader responsible for important
developments in mathematics, astronomy, and
music theory. Little is known about Pythagoras’s
early life, except that he was born on the island
of Samos and, as a young man, traveled extensively. His followers became a “secret brotherhood” which focused on religious rites as well
as intellectual pursuits.
There are several legends about Pythagoras’s
death: one says he was slain by enemies in the
presence of his young wife; another says he was
burned in a fire during a political riot.
Hiring a Student
Pythagoras was excited about his mathematical discoveries. He wanted to share them
with someone, but no one would listen. Finally,
in desperation, he cornered a young boy in the
marketplace and offered to teach him the arithmetic he had discovered. The boy refused. He
had no time for such frivolity. He had to work
to help provide for his family. “Tell you what,”
Pythagoras implored. “I’ll pay you daily wages if
you’ll just listen to me and try to learn.” It was a
deal; Pythagoras had started his first school.
Eventually, Pythagoras ran out of money.
By then, his student was so intrigued that he
offered to pay Pythagoras to continue teaching
him. Eventually, the teacher’s initial investment
was returned!
Contributions:
The Pythagoreans:
- were the first to use letters on geometric
figures.
- provided the first logical proof of the theorem a2+b2=c2.
- represented whole numbers as geometric
shapes.
- divided all numbers into even and odd.
- demonstrated the construction of the five
regular solids.
- asserted that the earth was round.
Quotations by Pythagoras:
“Number rules the universe.”
“Everything is arranged according to number
and mathematical shape.”
“Number is the origin of all things, and the
law of number is the key that unlocks the secrets
of the universe.”
“Be silent, or say something better than
silence.”
The Pythagorean School
When Pythagoras was about 50 years old,
he selected approximately 300 wealthy persons
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
from the city of Croton to constitute his first official “school.” The curriculum consisted of four
mathemata, or studies: arithmetica (number
theory), harmonia (music), geometria (geometry), and astrologia (astronomy).
Pythagoras divided those who attended into
two groups. The acoustici were permitted only
to listen to the master from behind a curtain.
After three years of silent obedience, such students could be initiated into the mathematici,
the advanced students who could actually see
Pythagoras in action.
As the curtain opened on the classroom stage,
Pythagoras appeared, dressed in a flowing white
robe with a golden wreath on his head and gold
sandals on his feet. After stating a problem, he
would withdraw, allowing time for the students
to work on the problem. His attendants provided
soft background music on their instruments.
After an interval, Pythagoras would reappear
and demonstrate the solution with visual aids
such as colored sand or pebbles.
their sacred symbol—the pentagram, a fivepointed star. They emphasized virtuous living
and friendship, and believed that “Knowledge
is the greatest purification.”
A Pythagorean Celebration
Legend says that Pythagoras was so excited
when he discovered the Pythagorean theorem
(a2 + b2 = c2) that he prepared an unusually generous sacrifice. He offered to the gods not one
but a hundred oxen. For centuries, mathematicians have admired the beauty of this theorem,
but most everyone agrees that Pythagoras got
a little carried away in his celebration.
The Great Cover-Up
Pythagoras taught adamantly that everything
in the world depended upon whole numbers.
When one of his group discovered that some
lengths can not be represented as rational
numbers, that is, they cannot be expressed as
a whole number or the ratio of two whole numbers, the Brotherhood was scandalized. These
new numbers, like √ 2, were called irrational
numbers.
At first, every effort was made to keep this
shocking discovery of irrational numbers secret.
Members were warned not to breathe a word
about it. Eventually, the truth “leaked” out, but
not without consequence. Hippasus, apparently
guilty of talking, mysteriously fell off a boat and
drowned.
The Secret Brotherhood
The Pythagoreans were a religious and political organization as well as a school. There were
many unusual requirements of members. They
shared all things in common, but were strictly
forbidden to discuss their discoveries outside
the Brotherhood. They were vegetarian—deeply
respectful towards animals, whom they felt were
sometimes their friends reincarnated. They
would not wear wool, drink wine, eat beans,
pick up anything that had fallen, or stir a fire
with an iron poker. On their clothing they wore
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
NUMBER SHAPES
The Pythagoreans represented whole
numbers as geometric shapes, often with
pebbles on the sand. The following definitions
reflect this concept.
Pythagoras demonstrated many number
relationships using number shapes. Use
practice golf balls and a glue gun to build these
shapes and show the relationships.
A square number is the number of pebbles
in a square array.
The sum of two consecutive triangular
numbers is a square number.
=
+
The square number 9
A triangular number is the number of pebbles
in a triangular array.
Two times a triangular number is an oblong
number.
=
+
The triangular number 6
An oblong number is the number of pebbles
in a rectangular array having one more column
than rows.
The oblong number 12
An even number is the number of pebbles in
a rectangle having two rows.
Eight times any triangular number plus one
is a square number.
+
+
+
+
+
+
+
+
=
An odd number plus an odd number is an even
number. =
+
The even number 10
An odd number is the number of pebbles in
a rectangle having two rows with one extra
pebble. The odd number 11
HISTORICAL CONNECTIONS IN MATHEMATICS
An even number plus an odd number is an
odd number.
+
=
An even number plus an even number is an
even number.
+
12
=
© 2012 AIMS Education Foundation
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
• •
• •
4th
• • • •
• • • •
• • • •
• • • •
4th
5th
6th
50th
nth
_______
_______
_______
_______
_______
4th
5th
6th
50th
nth
_______
_______
_______
_______
_______
_______
6
2
Complete the table to find the number of
dots in the nth oblong number.
3rd
_______
3rd
Complete the table to find the number of
dots in the nth square number.
2nd
4
2nd
1st
Oblong
Number
Number of
Dots
• • • • •
• • • • • • • • •
• • • • • • • • • • • •
•• • • • • • • • • • • • •
1st 2nd
3rd4th
1
Number of
Dots
3rd
• • •
• • •
• • •
Oblong numbers are numbers which can be
represented by dots in a rectangle having one
dimension one unit longer than the other. The
first four oblong numbers are pictured below.
OBLONG
NUMBERS
1st
Square
Number
1st2nd
•
Square numbers are numbers which can be
represented by dots in a square array. The
first four square numbers are pictured below.
SQUARE
NUMBERS
•
• •
_______
_______
_______
_______
_______
_______
3
1
Complete the table to find the number of
dots in the nth triangular number.
nth
50th
6th
5th
4th
3rd
2nd
1st
4th
•
• •
• • •
• • • •
Number of
Dots
3rd
• • •
• • •
Triangular
Number
1st2nd
•
Triangular numbers are numbers which can be
represented by dots in a triangular array. The
first four triangular numbers are pictured
below.
TRIANGULAR
NUMBERS
PYTHAGOREAN DISCOVERIES
The Pythagoreans discovered many relationships between triangular,
square, and oblong numbers. Use this table to find some of these
relationships.
Triangular
1
3
6
10
15
21
28
36
Square
1
4
9
16
25
36
49
64
Oblong
2
6
12
20
30
42
56
72
1. The sum of two consecutive triangular numbers is a(n)
__________ number.
2. Two times any triangular number is a(n) __________ number.
3. Eight times a triangular number plus one is a(n)
__________ number.
4. Three times any triangular number plus the next triangular
number is a(n) __________ number.
5. An oblong number plus the corresponding square number is a(n)
__________ number.
6. The sum of two consecutive oblong numbers is twice a(n)
__________ number.
7. A triangular number plus the corresponding square number minus
the corresponding oblong number is a(n) __________ number.
•
A triangular number is the number of dots in a triangular array.
••
•••
•••
A
square number is the number of dots in a square array.
•••
•••
••••
An oblong number is the number of dots in a rectangular array having one more
••••
•••• column than rows.
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
FIGURATE FAMILIES
Numbers that can be represented by dots arranged in specific
geometric shapes are called figurate numbers. These numbers can be
divided into “families” according to their shapes.
Discovering the relationship between these number families can be
as much fun as making a family tree!
Complete the table below. Note the many horizontal and vertical
relationships. Take advantage of these patterns as you work.
Family
Rank of Family Members
1st
2nd
3rd
4th
Triangular
1
3
6
10
Square
1
4
9
Pentagonal
1
5
12
Hexagonal
1
6
Heptagonal
1
5th
6th
7th
8th
Octagonal
PENTAGONAL NUMBERS
1st2nd
3rd
HEXAGONAL NUMBERS
4th
HISTORICAL CONNECTIONS IN MATHEMATICS
1st 2nd
15
3rd
4th
© 2012 AIMS Education Foundation
The Spider
and
the Fly
Ceiling
30ft
Sidewall
Floor 12ft
12ft
A room is 30 feet long, 12 feet wide, and 12
feet high. At one end of the room, 1 foot from
the floor, and midway from the sides, is a fly.
At the other end, 11 feet from the floor, and
midway from the sides, is a spider. Determine
the shortest path by way of the floor, ends,
sides, and ceiling, the spider can take to
capture the fly. How long, in feet, is this path?
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
A PYTHAGOREAN PUZZLE
The Pythagorean theorem says that the sum of the areas of the
squares on the two legs of a right triangle is equal to the area of the
square on the hypotenuse.
In the puzzle shown, notice that the two squares on the legs of the
right triangle are made up of five pieces.
Cut out the puzzle and arrange the five pieces to make one square on
the hypotenuse. This illustrates the Pythagorean theorem!
HISTORICAL CONNECTIONS IN MATHEMATICS
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© 2012 AIMS Education Foundation
PYRAMID PUZZLES
A significant contribution of the
Pythagoreans is the representation of
whole numbers as geometric shapes.
Imagine Pythagoras taking a number
of round pebbles and stacking them to
make a triangular pyramid.
Numbers which take this shape are called tetrahedral numbers.
Two popular puzzles which involve tetrahedral numbers can easily be
made with practice golf balls and an electric glue gun.
The objective of both puzzles is to put the pieces together to form a
triangular pyramid. One puzzle uses 6 pieces and the other 4 pieces.
Use your glue gun to construct the individual pieces and then solve
the puzzle!
PUZZLE NO. 1
These six pieces can be assembled to form a triangular pyramid.
+
+
+
+
+
=
PUZZLE NO. 2
These four pieces can be assembled to form a triangular pyramid.
+
+
HISTORICAL CONNECTIONS IN MATHEMATICS
+
18
=
© 2012 AIMS Education Foundation