**
Exploring Fundamental Concepts In Elementary Mathematics**

This series of
essays represents the bulk of the course I've offered at mathematics camps in
the United States (Math Zoom,
Awesome Math,
Epsilon Camp,
MathWorks), at
AMSAT in Bangalore India and Teacher Circle workshops
sponsored by the American Institute of Mathematics.

Place Value
Problems is an essay on the representation of rational and irrational numbers
in our standard powers of 10 notation. The sister
essay Fusing Dots should be read soon after this one. In PVP we explore
problems with digits. For example why do the two-digit numbers ab and
ba
always differ by a multiple of 9.

Fusing Dots, also
known as Exploding Dots, is the beautiful idea of
James Tanton. We use it here
to create exotic number systems: Fibonacci, base
phi, Cantor (aka factorial), and negative base.

Counting Problems

Bug Problems

Boxes and Cubes
is an essay on visualizing geometric configurations. For example suppose all
six faces on an n x n x n cube composed of n cubed unit cubes are painted red. How
many of the cubes have some red paint on them?

**divisor**
structure of a composite number geometrically using a device known as the Hasse
diagram of the number. I use ZomeTools to
build the Hasse diagrams for numbers like 30, 300, and 63000.