The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

Young adolescents build notions of figures through experiences. When young adolescents begin middle school mathematics, they already have prior assumptions and conceptualizations about shapes (Herbst, Fujita, Halverscheid, & Weiss, 2017). Over years of experiences, interactions, and exposures with a particular concept, these students develop…

The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

A Steiner chain is defined as the sequence of n circles that are all tangent to two given non-intersecting circles. A closed chain, in particular, is one in which every circle in the sequence is tangent to the previous and next circles of the chain. In a closed Steiner chain the first and the "n"th circles of the chain are also tangent…

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the "Elements" of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in…

Art uses mathematics in many ways. The author's teaching and involvement with mathematics and art in the Bridges Conference connecting the two have convinced him more and more that concepts for understanding mathematics can be achieved by the use of art. The author states that one reason he believes in teaching mathematics through art is that it…

Making constructions with paper is called "origami" and is considered an art. The objective for many fans of origami is to design new figures never constructed before. From the point of view of mathematics education, origami is an interesting didactic activity. In this article, the authors propose to help High School students understand new…

The Department of Mathematics and Statistics at Utah State University offers two courses for preservice secondary school mathematics teachers that complement each other in working toward a vision for school mathematics in which "all students have access to high-quality, engaging mathematics instruction." Through these courses, students…

Many times teachers hear students say: "Why are we learning this? Why do we have to know this? When are we going to use this outside of class time?" These common questions are probably familiar to most high school teachers. An 11th-grade English teacher attended a university-school district professional development (PD) program on…

This note presents geometric and physical interpretations of the sufficient condition for a critical point to be a strict relative extremum: f[subscript xx]f[subscript yy] - f[superscript 2][subscript xy] greater than 0. The role of the double derivative f[subscript xy] in this inequality will be highlighted in these interpretations. (Contains 14…

In this note, a simple proof of the Generalized Ceva Theorem in plane geometry is presented. The approach is based on the principle of equilibrium in mechanics. (Contains 2 figures.)

A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.

The study investigated the effect of using mathematics laboratory in teaching on students' achievement in Junior Secondary School Mathematics. A total of 100 JS 3 Mathematics students were involved in the study. The study is a quasi-experimental research. Results were analyzed using mean, standard deviation and analysis of covariance (ANCOVA).…

This paper describes aspects of 13 year-olds activity in mathematics as emerged during the implementation of proportional geometric tasks in the classroom. Pupils were working in pairs using a piece of software specially designed for multiple representation (symbolic and graphical) of the variation in parametric procedures with dynamic…

Investigates the idea of the center of mass of a polygon and illustrates centroids of polygons. Connects physics, mathematics, and technology to produces results that serve to generalize the notion of centroid to polygons other than triangles. (KHR)