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 this work we describe a teaching proposal to calculate the eccentricity of the Moon's trajectory by applying a geometrical technique. The values of the ratios between the Earth-Moon distance and the diameter of the Moon at apogee and at perigee were calculated from a kinematic model associated with a geometrical technique of image analysis. The…

In this paper, we introduce a technology-enhanced pedagogical sequence for supporting lower secondary school students' sense-making of the concept of volume in a non-procedural and non-formula-driven way. Specifically, we illustrate a novel approach of using dynamic geometric environment (DGE) to introduce the meaning of volume and then deriving…

Geometrography, "alias the art of geometric constructions" was proposed by Emile Lemoine between the late 1800s and the early 1900s. It consisted originally of a system to measure the complexity of ruler-and-compass geometric constructions, capable of: designate every geometric construction by a pair of values that manifests its…

This article describes a modeling project the author did with his algebra, geometry, and applications students in grades 9, 10, and 11 as an enrichment and exploration of an extremely relevant real-world phenomenon and introduction to the adult profession of epidemiologist. With discussion and exploration, this project consumed two periods of 88…

This document introduces, describes and exemplifies the technical features of some recently implemented automated reasoning tools in the dynamic mathematics software GeoGebra. The new tools are based on symbolic computation algorithms, allowing the automatic and rigorous proving and discovery of theorems on constructed geometric figures. Some…

A physical pendulum with a magnetic load is proposed for comparison of linear and nonlinear oscillations. The magnetic load is repelled by permanent magnets which are disposed symmetrically relative to the load. It is established that positions of the pendulum and the magnets determine the dependence of restoring force on displacement of the load.…

In fall 2013 the author taught a graduate course entitled Topics in Geometry for Middle and High School Teachers. The majority of the participants had extensive experience teaching mathematics in middle school or high school. The fact that participants were rather comfortable with the basic concepts of school geometry made it both possible and…

The Odd Shape Out task was an open-ended problem that engaged students in comparing shapes based on their properties. Four teachers submitted the work of 116 students from across the country. This article compares various student's responses to the task. The problem allowed for differentiation, as shown by the many different ways that students…

Emperor penguins, the largest of all the penguin species, attain heights of nearly four feet and weigh up to 99 pounds. Many students are not motivated to learn mathematics when textbook examples contain largely nonexistent contexts or when the math is not used to solve significant problems found in real life. This article's project explores how…

Since 1973, the National Assessment of Educational Progress (NAEP) has gathered information about student achievement in mathematics. Results of these periodic assessments, produced in print and web-based formats, provide valuable information to a wide variety of audiences. They inform citizens about the nature of students' comprehension of the…

Motivated by quaint woodcut depictions often found in many late 16th and 17th century ballistic manuals of cannonballs fired in air, a comparison of their shapes with those calculated for the classic case of a projectile moving in a linear resisting medium is made. In considering the asymmetrical nature of such trajectories, the initial launch…

It is a widespread misunderstanding in introductory physics courses that the motion of rolling bodies in general can be calculated using the point of contact as a reference point when equating the rate of change of angular momentum to the torque. In this paper I discuss in general two correct rules to be used instead, in order to derive the…

Learning to reason spatially is increasingly recognized as an essential component of geometry education. Generally taken to be the "ability to represent, generate, transform, communicate, document, and reflect on visual information," "spatial reasoning" uses the spatial relationships between objects to form ideas. Spatial thinking takes a variety…

This article discusses a comparative study, funded by the European Union, of the teaching of mathematics in five European countries, (Flanders, England, Finland, Hungary and Spain) to students in the upper primary (ages 10-12) and lower secondary (12-14) years. These ages were chosen as they represent a time when many students' experiences of…