Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a…

Large gender gaps in performance on questions involving projectile motion have been observed at high school and university level, even amongst high-achieving students. This gap is particularly problematic because projectile motion is typically one of the first topics formally taught in physics, and this may give girls an inappropriately negative…

Early in their chemistry education, students learn to do empirical formula calculations by rote without an understanding of the historical context behind them or the reason why their calculations work. In these activities, students use paperclip "atoms", construct a series of simple compounds representing real molecules, and discover,…

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

This article describes a hands-on mathematics activity wherein students peel oranges to explore the surface area and volume of a sphere. This activity encourages students to make conjectures and hold mathematical discussions with both their peers and their teacher. Moreover, students develop formulas for the surface area and volume of a sphere…

Many workshops and meetings with the US high school mathematics teachers revealed a lack of familiarity with the use of transformations in solving equations and problems related to the roots of polynomials. This note describes two transformational approaches to the derivation of the quadratic formula as well as transformational approaches to…

While teaching in New York City public schools, the author began planning collaboratively with a team of mathematics teachers. Collaborative planning enabled them to consider a range of issues that may not have emerged had they been planning independently. In this article, the author first describes their work through excerpts of conversations…

Introduces the method of ray tracing to analyze the refraction or reflection of real or virtual images from multiple optical devices. Discusses ray-tracing techniques for locating images using convex and concave lenses or mirrors. (MDH)

A common misconception among students setting up force-acceleration problems is to think of the expression "mass times acceleration" as a force itself. Presents a new formula to express the relationship between force, mass, and acceleration, and discusses its benefits. (MDH)

Presents simple experiments using the flow of water from bell jars that can provide an easily visualized introduction to exponential decay. (Author)

Discusses solutions to the problem of maximizing the range of a projectile. Presents three references that solve the problem with and without the use of calculus. Offers a fourth solution suitable for introductory physics courses that relies more on trigonometry and the geometry of the problem. (MDH)

Calculates an approximation to Avagadro's number for one mole of water by assuming the mole to be in cubical form and then halving the cube three times, thereby doubling the surface area of the original cube. The calculations are derived from the work necessary to perform these divisions. Presents calculated values for several liquids. (MDH)

Discusses formal-economical/epistemic-powerful features of physics formulas: the first puts constructs in logical relations to each other, reducing constructs to symbols; the second describes the correspondence which links the construct to natural events. Implications of the dual nature of formulas (considered as efficient learning tools) for…

Describes the modern hot-air balloon and the physics of ballooning. Proposes that students construct their own hot-air balloon and presents an experiment calculating the time needed for a balloon to rise to the ceiling of a gymnasium. (MDH)

Describes two simple methods of error analysis: (1) combining errors in the measured quantities; and (2) calculating the error or uncertainty in the slope of a straight-line graph. Discusses significance of the error in the comparison of experimental results with some known value. (MDH)