**Mathematics 115: Mathematics
through its History**

**Main Topics **

Number systems from various cultures

Using
the abacus

Alternative
algorithms for arithmetic

Pythagorean
Theorem

Pascal's
Triangle

Egyptian
mathematics (doubling, unit fractions, false position)

Pythagorean
pebble arithmetic (triangular, oblong, and square numbers)

Geometry in
*Elements *(axiomatic system,
constructions, 2-column proofs)

Circles and approximations
of π in various civilizations

Geometric
algebra of Greeks and Arabs (including completing the square)

Women in
mathematics

Topology: Platonic solids & Euler’s
formula, map coloring, Moebius strips

Number systems from various cultures

Using
the abacus

Alternative
algorithms for arithmetic

Pythagorean
Theorem

Egyptian
mathematics (doubling, unit fractions, false position)

Pythagorean
pebble arithmetic (triangular, oblong, and square numbers)

Geometry in
*Elements *(axiomatic system,
constructions, 2-column proofs)

Approximations
of π in various civilizations

Geometric
algebra of Greeks and Arabs (including completing the square)

Perspective
drawing

Topology: Platonic solids & Euler’s
formula, networks & Euler paths, map coloring,

Women in
mathematics Moebius strips

Pascal's
Triangle

Number systems from various cultures

Using
the abacus

Pythagorean
Theorem

Alternative
algorithms for arithmetic

Egyptian
mathematics

Geometry in
*Elements*

Approximations
of π

Geometric
algebra

Pascal's
Triangle

Early North
American arithmetic texts

Perspective
drawing

Division
algorithm and ID numbers

Knot theory

Graph theory
and map coloring

Women
in mathematics

**Topics and
More Topics**

Throughout the semester (at least
weekly):

Mathematical
games (include magic squares here) - Bree (Fridays)

Special
problems such as the river crossing, weighted coin, or cannibal problems

Women
mathematicians - Michaelene (References: *Math
Equals,* Teri Perl, more)

Topics/activities: Earliest evidences of counting: Ishango bone, etc.

Begin number
systems and their development (see tomorrow's topic)

Topics/activities: Number systems and their development (Babylonian/Mesopotamian, Egyptian, Greek, Maya, Chinese, Indo-Arabic, Inca quipus) - Nicole and Bree

Topics/activities: Abacus and counting board arithmetic (with
candy!) - Nicole and

Michaelene

Addition and
subtraction: alternate algorithms for subtraction, checking

by casting out 9's

*Tablas** para los
ninos* (first arithmetic printed in

Robert Recorde and the = sign (1500s)

Topics/activities: Multiplication and division: Egyptian doubling and halving - Bree

Other algorithms for
multiplication and division - Bree

*Sumario** Compendioso*
(first math book published in

Mathematics
education throughout history

Topics/activities: Egyptian unit fractions and their use in
practical problems - Bree

Decimals, Simon Stevin (1500s) - Alissa

Practical
mathematics: Egyptian formulas for areas
and volumes,

including volume of a frustum of a pyramid - Michaelene and Rachel

What constitutes /
qualifies as a mathematical formula?

Topics/activities: Figurate numbers (Pythagorean pebble
arithmetic--with candy, of

course), perfect and amicable numbers, primes, Sieve of

Eratosthenes,
some additional Greek number theory - Michaelene

and Wendy

Do humans invent
or discover mathematics? Do animals do
math?

Topics/activities: *Elements:* geometry, including basic definitions;
constructing

triangles, squares, pentagons, and hexagons; quadrature -
Wendy

What constitutes a
mathematical proof?

Further
applications of Greek geometry, including Eratosthenes'

measurement of the circumference of the Earth - Wendy

Tunnel of

Archimedes' volume
formulas (method of the lever) - Brian

Hypatia's
astrolabe - Wendy

Video: The Tunnel of

Topics/activities: Estimates of π by Eudoxus, Archimedes,
and others - Michaelene and

Rachel

Video: The Story of Pi

Topics/activities: Origins of the Pythagorean Theorem,
including picture proofs and

applications by early Chinese mathematicians; also Mesopotamian,

Egyptian,
Indian, and Greek versions - Michaelene and Rachel

Pythagorean
triples and Fermat's Last Theorem, including Sophie

Germain's
contributions to FLT

Videos: The Theorem of Pythagoras, The Proof

Topics/activities: Moebius strip and cutting one hole in a
sheet of notebook paper to form

a ring through which Brian can fit his body - Brian
(obviously!)

Building 3D models
of the Platonic solids (regular polyhedra) and

discovering Euler's formula - Rachel

Four-Color Theorem
- Rachel

Topics/activities: Graphs from

- Brian

Knot theory -
Nicole and Jen

Video: Life by the Numbers 3

Topics/activities: Solving linear equations - method of false
position, samples of problems

from different cultures and societies, negative numbers,
irrational

(incommensurable) numbers

Do you believe in
negative numbers? irrational numbers? imaginary nos.?

History of
arithmetic (again) and algebraic notation: how more succinct

notation made mathematics easier (and life harder?)

Topics/activities: Methods for finding square roots, including
Babylonian algorithm

- Rachel

Greek geometric
algebra (again), then geometric algebra in the Arab

world: al-Khwarizmi
and completing the square, quadratic formula

(give idea of solution of the cubic equation by 16th Century
Italian

algebraists by providing picture of completing the cube) -
Wendy,

Mike

Are pictures
better than words?

Topics/activities: Infinite series (including Zeno's
paradoxes, geometric series, Oresme's

series, geometric progressions, Gauss sum, harmonic series)
- Brian

Fibonacci numbers
- Bree

Topics/activities: "Pascal's" Triangle; Indian and
other combinatorics, esp. Bhaskara

Probability -
Native American dish game; gambling and Pascal - Brian

Topics/activities: Perspective in art and drawing, visualizing
the fourth dimension

- Nicole and
Rachel

And you thought
arithmetic was a boon to capitalism!

Video: Life by the Numbers 1

Topics to be
included in 4-unit version of course:

Modular
arithmetic (congruence), including bar codes and cryptography; begin with Maya

and other calendars
- Mike

Greek logic
and Lewis Carroll logic games - Alissa

Divisibility: Division Algorithm, Euclidian Algorithm

The parallel postulate and
non-Euclidian geometry

Symmetries and
tiling in various cultures

From tracing
out curves (e.g. witch of Agnesi) to coordinate geometry (Agnesi, Descartes)

Florence
Nightingale and graphical statistics (to aid in the prevention of disease)

The

Voting
methods: Condorcet, de Borda