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…

The cognitive relationship between intuition and proof is complex and often students struggle when they need to find mathematical justifications to explain what appears as self-evident. In this paper, we address this complexity in the specific case of open geometrical problems that ask for a conjecture and its proof. We analyze four meaningful…

In a two-step synthesis starting with Ni(CH[subscript 3]COO)[subscript 2] ·4H[subscript 2]O and salicylaldehyde, students prepare Nethyl and N-isopropyl Ni(II) Schiff base (imine) complexes. After isolation and recrystallization, the products are characterized by room-temperature magnetic susceptibility measurements, FTIR spectroscopy, and…

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°.

Folk wisdom of the state of Montana asserts that Flathead Lake, located in the state's northwest corner, is the largest natural freshwater lake west of the Mississippi. But what does it mean to be the "largest?" And, how does one respond to Californians' claims that Lake Tahoe, located on the California-Nevada border, is the rightful…

By using high-leverage models to connect student learning experiences to overarching concepts in mathematics, teachers can anchor learning in ways that allow students to make sense of content on the basis of their own prior experiences. A rectangular area model can be used as a tool for understanding problems that involve multiplicative reasoning.…

The Gauss-Lucas Theorem is a classical complex analysis result that states the critical points of a single-variable complex polynomial lie inside the closed convex hull of the zeros of the polynomial. Although the result is well-known, it is not typically presented in a first course in complex analysis. The ease with which modern technology allows…

This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of…

The authors designed a course--Origami and Geometry--and describe one of the activities in which origami was used to fold a regular dodecahedron. They find that origami can provide students with concrete material for observing geometrical structures and give them opportunities to examine their thoughts on geometry.

The solution of problems and the provision of proofs have always played a crucial part in mathematics. In fact, they are the heart and soul of this discipline. Moreover, the use of different techniques and methods of proof in the same mathematical field, or by combining fields, for the same specific problem, can show the interrelations between the…

One way to ease the solution of physics problems is to visualize the situation. However, by visualization we do not mean the pictorial representation of the problem. Instead, we mean a sketch for the solution of the problem. In this paper a new approach to solving physics problems, based on decomposing the problem into with and without gravity, is…

For 500 years, dream catchers have been cultural symbols of intrigue worldwide. The most common folkloric design is a 12-point dream catcher. According to Native American legend, the first dream catcher was woven by a "spider woman" to catch the bad dreams of a chief's sick child. Once the bad dreams were caught, the chief's child was…

Valid proofs need not be in the traditional two-column format. This classroom activity allows students to explore, discuss, and use specialized facts to create a general statement of truth.

In this article, we describe one possible way to modify volume tasks in textbooks to promote conceptual understanding of the volume formula: length × width × height. From the results of previous studies regarding students' understanding of volume measurement, we identified three conceptual ideas key to understanding volume: filling…

This article discusses GeoGebra's geometry capabilities to assist the teaching issues and learning of geometry and measurement content in grades 5-9. Within this geometry context, the authors focus on using learning goals to select applets from a large public database. Also mentioned are some common pitfalls to avoid, such as error-prone applets.