When young children copy a vertical line that protrudes from the midpoint of a pre-drawn oblique baseline, they tend to perpendicularise this line. We measured this bias for standard figures and for symmetrical angular figures. The 5- and 6-year-old participants produced the expected bias for the standard figures. However, these same children…

Students' difficulty with reasoning and justification in the context of proofs in geometry has been well documented. Studies reveal that many of the ideas that students are asked to prove are known to be true, and therefore, the process of such proof is likely to remain meaningless and purposeless. The implication of this claim is that students…

The location of the number "c" arising from Cauchy's Average Value Theorem is described when the size of the interval is small. This article discusses various generalizations of theorem 1, to the context of Cauchy?s Average Value Theorem--but without appealing to theorem 1. Obviously, hypotheses involving the functions "f" and "g" will be…

This note provides a self-contained introduction to conics as loci of points equidistant from circles, lines and points, including a study of the loci of points equidistant from two circles, separated, intersecting or touching. (Contains 1 table and 8 figures.)

Nature is full of different crystals and many of them have shapes of regular geometric objects. Those in which the fractal structure of a geometric object can be recognized are especially unusual. In this paper a generalization of one of these shapes is described: a formation, based on an n-dimensional simplex. The construction of an n-dimensional…

How to iterate geometric shapes to construct Baravelle spirals and Pythagorean trees is demonstrated in this article. The "Surfing Note" sends readers to a site with applets that will generate fractals such as the Sierpinski gasket or the Koch snowflake.

The study of the rotational symmetries of the regular polyhedra is important in the classroom for many reasons. Besides giving the students an opportunity to visualize in three dimensions, it is also an opportunity to relate two-dimensional and three-dimensional concepts. For example, rotations in R[superscript 2] require a point and an angle of…

In this short note is presented an easy and instructive proof of l'Hopital's Rule in both the 0/0 and [infinity]/[infinity] cases, obtained by translating the theorem into a simple geometric statement about paths in the plane. (Contains 4 figures.)

A previous note described how complex numbers can be used for elementary analytic geometry in the plane, describing lines, circles and their intersections using complex Cartesian equations. In the present note, a description of elementary plane transformations, namely homotheties and translations, their group structure and their operations on…

In this article, an attempt is made to find all linear transformations that map a standard Pythagorean triple (a Pythagorean triple [x y z][superscript T] with y being even) into a standard Pythagorean triple, which have [3 4 5][superscript T] as their fixed point. All such transformations form a monoid S* under matrix product. It is found that S*…

A study is conducted regarding the change in teaching process in the geometry of transformations, teaching by small steps rather than by a complicated method. In the process of making mathematics teaching better, many challenges are faced, and several risks are taken to solve the questions asked by the students.

This article contains investigative activities to assist students in constructing formulas out of an understanding of the area of geometric shapes. Included with this article are a "Finding Areas on Square Dot Paper Activity Sheet" and a "Finding Areas on Triangle Dot Paper Activity Sheet." (Contains 1 table and 11 figures.)

For a given orbital period and eccentricity, we determine the maximum time lapse between the winter solstice and the spring equinox on a planet. In addition, given an axial precession path, we determine the effects on the seasons. This material can be used at various levels to illustrate ideas such as periodicity, eccentricity, polar coordinates,…

Polish students learn about Platonic solids in the final year of secondary grammar school, but in such a way that few perceive the beauty of this branch of mathematics. An average student associates polyhedra with calculating lengths and angles in a cube, cuboid or pyramid. But for the author, 3-D geometry is the branch of mathematics that shows…

This paper, the first of a two-part article, follows the trail in history of the development of a graphical representation of the complex roots of a function. Root calculation and root representation are traced through millennia, including the development of the notion of complex numbers and subsequent graphical representation thereof. The…