Four variations of the Koch curve are presented. In each case, the similarity dimension, area bounded by the fractal and its initiator, and volume of revolution about the initiator are calculated. A range of classroom exercises are proved to allow students to investigate the fractals further.

The area concept may appear in a wide range of problem situations. The representations used (geometrical, numerical and algebraic) in such situations entail particular practices that condition the partial area meanings. Thus, this study aims to characterize the partial area meanings mobilized in the resolution of area tasks. Each of the area…

Creativity and problem solving are considered to be twenty-first-century competencies, therefore promoting mathematical creativity should be an important part of school mathematics. The study presented in this paper is inspired by the notion of mathematical creativity and the utilization of multiple solution tasks (MSTs) to investigate students'…

The aim of this paper is to present a teaching proposal for the theoretical part relating to the first- and second-order linear difference equations with constant coefficients suitable for the first-year students at various types of universities. In contradistinction to the methods often applied (memorization of algorithms without a proper…

The purpose of this note is to describe several challenging problems in Euclidean geometry. The note also contains author's solution sketches to the two problems.

There are many problems whose solution requires proof that a quadrilateral is cyclic. The main reason for writing this paper is to offer a number of new tools for proving that a particular quadrilateral is cyclic, thus expanding the present knowledge base and ensuring that investigators in mathematics and teachers of mathematics have at their…

Windmill images and shapes have a long history in geometry and can be found in problems in different mathematical contexts. In this paper, we share and discuss various problems involving windmill shapes and solutions from geometry, algebra, to elementary number theory. These problems can be used, separately or together, for students to explore…

In this article, we present an educational proposal designed to work on spatial geometry and three-dimensional thinking with students in the last years of Primary Education. The proposal is comprised of several activities that combine the use of a classroom videogame and working with manipulative materials to create a representation of the…

This study was performed to determine preservice elementary mathematics teachers' solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers'…

We present action research of a problem posed as part of a multi-participant national (Israeli) test checking the mathematical knowledge of high school students at the ages of 16-17, where some of those who solved this problem made an error by using the converse to a well-known theorem, where the converse is not true. In order to examine the…

The purpose of this note is to introduce two challenging problems in geometry that were inspired by some well-known problems in high school geometry. These problems can be explored in the dynamic geometry environments of computer construction programmes.

This study investigates middle school students' reasoning about parallelism and perpendicularity of two line segments. Data were collected from 83 middle school students through an identification task consisting of various examples and non-examples of the parallelism and perpendicularity of two line segments. One-to-one interviews were also…

Often, straightforward notions from one mathematical domain, when altered even slightly, can become rich and rewarding investigations involving numerous additional domains -- particularly when the investigation includes rigorous proof. This study begins with a familiar high school geometry problem (namely finding the circumcentre of a triangle),…

This paper presents research into closed orbits parallel to quadrilaterals inscribed in various geometric shapes that can be represented by mathematical functions: straight lines, circles, ellipses, parabolas, and hyperbolas. Mathematical proofs have been given for the existence of an infinite number of parallel orbits for each of these shapes.…

These notes discuss several related problems in geometry that can be explored in a dynamic geometry environment. The problems involve an interesting property of hexagons.