In this article, the author presents a Desmos activity where students adjust the measures of angles in radians to reposition a laser and a mirror so the beam passes through three stationary targets. This Radian Lasers activity can be extended to simulate project-based learning (PBL), a pedagogical approach for applying concepts and skills from…

An original approach using three appropriate Pythagorean triangles is presented for the detailed mathematical analysis of an ideal conical pendulum. The triangles that are used in this analysis relate specifically to the physical dimensions of the conical pendulum, the magnitudes of the forces acting during the conical pendulum motion and a…

In 1936, Albert Einstein wrote a brief article where he suggested the possibility that a massive object acted as a lens, amplifying the brightness of a star. As time went by, this phenomenon--known as gravitational lensing--has become a powerful research tool in astrophysics. The simplest and symmetrical expression of a gravitational lens is known…

Historical accounts of trigonometry refer to the works of many Indian and Arab astronomers on the origin of the trigonometric functions as we know them now, in particular Abu al-Wafa (ca. 980 CE), who determined and named all known trigonometric functions from segments constructed on a regular circle and later on a unit circle (Moussa 2011;…

How do students think about an angle measure of ninety degrees? How do they think about ratios and values on the unit circle? How might angle measure be used to connect right-triangle trigonometry and circular functions? And why might asking these questions be important when introducing trigonometric functions to students? When teaching…

In this article, the author describes the development and implementation of a measurement-based group activity designed to support students in understanding the connection between angle magnitude and the shape of the sine function. She explains that the benefit of this activity is that it allows students to build their trigonometric knowledge…

The study of trigonometry suffers from a basic dichotomy that presents a serious obstacle to many students. On the one hand, there is triangle trigonometry, in which angles are commonly measured in degrees and trigonometric functions are defined as ratios of sides of a right-angled triangle. On the other hand, there is circle trigonometry, in…

Constructing model hot air balloons is an activity that captures the imaginations of students, enabling teachers to present required content to minds that are open to receive it. Additionally, there are few activities that lend themselves to integrating so much content across subject areas. In this article, the authors describe how they have…