Mathematics 115: Mathematics through its History

Main Topics

## Interim 2002

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 Euclid's 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

## Interim 2000

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 Euclid's 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

# Interim 1999

Number systems from various cultures

Using the abacus

Pythagorean Theorem

Alternative algorithms for arithmetic

Egyptian mathematics

Geometry in Euclid's 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 California)

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 New World)

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:     Euclid's 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 Samos - Brian

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

Hypatia's astrolabe - Wendy

Video:        The Tunnel of Samos

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 Africa and the South Pacific, Konigsberg Bridge Problem

- 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 U.S. census:  counting vs. statistics

Voting methods:  Condorcet, de Borda