were written so that they can be completed in any order. While
is some interconnectedness between some of the topics, one exploration
does not rely on another. The short summary of the explorations
The four types of rigid motions are first explained and then how these
relate to symmetry is explored. Students are shown how symmetry
defined and how objects can be grouped by their type of
- Star Figures:
The concepts of star figures and star polygons are given.
then explore a number of different properties of these
Polyhedra: Students explore the five regular polyhedra.
why there are only five. They are asked to construct some of
solids and discover Euler’s formula.
Pythagorean Theorem: Various visual proofs of the Pythagorean
are explored. Pythagorean triples are also investigated.
exploration ends with a Pythagorean puzzle.
Golden Ratio: A definition of the golden ratio is given.
It is shown
how this number relates to the golden rectangle, the Fibonacci
sequence, and the logarithmic spiral. Golden triangles and the
pentagram are also explored.
in Perspective: Students are shown how to construct a drawing in
one-point, two-point, and three-point perspective. They are shown
objects can look quite different depending on the viewing
Theory: Five problems or puzzles are given at the beginning of
section. After a number of definitions and examples are given,
puzzles are then solved by the students. Along the way Euler
circuits are explored as well as Euler’s formula and a counting
- Tessellations: Regular and
semi-regular tessellations are quickly explored. It is
shown how Escher-type tessellations can be made using a variety of
techniques. The nonperiodic tessellations (Penrose tiles and
pinwheels) are also investigated.
Students construct and discover properties of the Koch snowflake, the
Sierpinski triangle, and the dragon curve. Fractal dimension is
Knots: Students are first shown how to draw Celtic knots and
asked to explore various mathematical properties of these knots.
We take a mathematical look at how shoes can be laced. We explore
many different types of lacings of various sizes there are as well as
the lengths of some of these lacings.
strips: We explore properties of Möbius strips and
objects including Möbius shorts, trefoil knots, and Möbius
also look at the properties of these objects when they are cut in a
couple of ways.