In the same chapter of his book "Opera Geometrica," Torricelli published two discoveries: 1) initial velocity of a jet from a container increases with the square root of the depth of the hole; 2) he drew the pattern of jets from three openings at the wall of a container filled with water to constant level "H" and determined the…

The integration of mathematics and science in teaching facilitates student learning, engagement, motivation, problem-solving, critical thinking, and real-life application. Although curriculum integration is theoretically desirable for many educators, what to integrate and how to integrate are often the big questions facing teachers working within…

An important step in learning to use math in science is learning to see symbolic equations not just as calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper, I propose "anchor equations" as a…

In this article we present and discuss some theoretical and methodological elements of a wider research that analyzes the teaching practice of two teachers novice and expert from a semiotic theoretical approach. The conceptual framework of this work allows for addressing the epistemological and didactic dimensions of the teaching practice. Three…

This manuscript shares the implementation of an after-school literacy in STEM (science, technology, engineering, and mathematics) program designed for middle grades students to increase their interest in science and mathematics learning. This program was conducted at our local Boys and Girls Club facilities where students learned about four…

Augmented reality (AR) is one of the emerging technologies that has demonstrated to be an efficient technological tool to enhance learning techniques. In this paper, we describe the development and evaluation of an AR system for teaching Euclidean vectors in physics and mathematics. The goal of this pedagogical tool is to facilitate user's…

Quadratic friction involves a discontinuous damping term in equations of motion in order that the frictional force always opposes the direction of the motion. Perhaps for this reason this topic is usually omitted from beginning texts in differential equations and physics. However, quadratic damping is more realistic than viscous damping in many…

We propose an analytic solution to the problem of the mechanical paradox consisting of a sphere rolling upwards on two diverging inclined guides as devised by Gardner. The presence of an unstable equilibrium point is highlighted and the analytic solution is found by means of elementary calculus concepts. (Contains 4 figures and 3 footnotes.)

Ah, springtime, and young people's thoughts turn to... baseball, of course. But this column is not about "how" to throw a curveball, so you'll have to look that up on your own. Here, the focus is on the "why" of the curveball. There are two different things that cause a spinning ball to curve. One is known as the "Bernoulli effect" and the other…

We study the turning point problem of a spherical pendulum. The special cases of the simple pendulum and the conical pendulum are noted. For simple initial conditions the solution to this problem involves the golden ratio, also called the golden section, or the golden number. This number often appears in mathematics where you least expect it. To…

The algebraic concepts and major ideas that govern Newton's laws of motion can often be a challenge for the majority of ninth-grade students. Therefore, to make learning these concepts less task-oriented and more enjoyable, the author developed lessons that allow students to construct and express their understanding of these ideas through…

In this paper we describe an interdisciplinary course on dynamics that is appropriate for nonscience majors. This course introduces ideas about mathematical modeling using examples based on pendulums, chemical kinetics, and population dynamics. The unique emphasis for a nonmajors course is on chemical reactions as dynamical systems that do more…

Uses projectile motion to explain the two roots found when using the quadratic formula. An example is provided for finding the time of flight for a projectile which has a negative root implying a negative time of flight. This negative time of flight also has a useful physical meaning. (MVL)

Describes an activity in which students try to determine, by using physics calculations, when to drop a raw egg from a tower to hit their physics teacher who is walking beneath it. (WRM)

Explains the strange effect a crosswind has on motion. (WRM)