This book, photographically reproduced from its original 1942 edition, is an extended essay on one of the three problems of the ancients. The first chapter reduces the problem of trisecting an angle to the solution of a cubic equation, shows that straightedge and compasses constructions can only give lengths of a certain form, and then proves that…

Problems involving the generation and counting of isosceles triangles interior to a given isosceles triangle are described. (SD)

Describes how many of the geometrical and physical properties of the binary elastic collision of two particles can be represented by an isoceles trapezoid in the velocity space. (BB)

An activity is presented which involves making a stadia tube from a cardboard mailing tube and using it to measure horizontal and vertical distances using the properties of similar triangles. (MN)

The use of basis matrix methods to rotate axes is detailed. It is felt that persons who have need to rotate axes often will find that the matrix method saves considerable work. One drawback is that most students first learning to rotate axes will not yet have studied linear algebra. (MP)

Some ways of thinking and acting geometrically are described which are related to the approach used by ancient humans. The focus is on intuitive geometric imagery, an attempt to resurrect a way of describing possible viewpoints of geometry outside of those commonly accepted. (MP)

Activities designed to lead pupils through the process of using the basic measuring and drawing devices of geometry are presented and move to the discovery of several surprising generalizations about arbitrary triangles. (MP)

Describes an activity in which students observe a simple situation, explore conjectures, and generate conclusions from their exploration of the conjectures to develop the essential triangle-congruence theorems. Includes teacher's guide. (KHR)

Presents Heron's original geometric proof to his formula to calculate the area of a triangle. Attempts to improve on this proof by supplying a chain of reasoning that leads quickly from premises to the conclusion. (MDH)

Research has shown that certain ways of teaching can make a difference in whether students learn standards-based content. Many strategies have proven to be effective in teaching literacy, mathematics, science and social studies. These strategies have facilitated blending academic and career/technical subjects to make learning more meaningful for…

Before the 19th century the idea of more than three dimensions was exceptional. During the 19th century, however, geometry was revolutionized and new branches were developed. This revolution also created the idea of the possibility of a n-dimensional geometry or space; flatland, i.e. n = 2, was a consequence of this new thinking. In 1884 the…

This reprinted "Mathematics Teacher" article gives a brief history of the mathematics of Escher's art. (Contains 9 plates.)

Mathematics begins in human experience thousands of years ago as empirical and intuitive experiences. It takes the deliberate naming of concepts to help crystallize and secure those observations and intuitions as abstract concepts, and to begin separating the concept of number from specific instances of objects. It takes the creation of compact…

As a result of attending a course on moving images, the authors consider the balance between visualising and verbalising geometry. In thinking about the balance between visualising and verbalising (and verbal reasoning), the authors began to appreciate more than before that geometric visualisation is greatly aided by what they know--if everyone…