In this article, the authors share their favorite "Construct It!" activity, which focuses on rate of change and functions. The initial approach to instruction was procedural in nature and focused on making use of formulas. Specifically, after modeling how to find the slope of the line given two points and use it to solve for the…

This paper discusses the mathematical and didactical problems of teaching indefinite integral in the context of the ubiquitous availability of online integral calculators. The symbol of indefinite integral introduced by Leibniz, unfortunately, does not contain an indication of the interval on which the antiderivatives should be calculated. This…

The purpose of this article is to provide an explicit formula for the bounds of integration of the regular simplex centred at the origin. Furthermore, this article rigorously proves that these integration bounds recover the volume of the regular simplex. To the authors' knowledge, this is the first time that such integration bounds have been…

This article presents a simple, fast, and equally accurate technique for measuring the area of a circle and of an ellipse without using geometric formulas. This therefore, together with the known radius of the circle and the semi-major and semi-minor axes of the ellipse, allows for the calculation of [pi]. The experiment is easy, thrilling, and…

Recently, Lima lamented on the 'time-consuming and tiresome' pedagogical nature of repeated integration by parts, throwing down the challenge of providing a pencil-and-paper solution to a related problem in a few seconds. Lima put forth a simple formula as an alternative. In this work, I offer a response to Lima's challenge and his formula.…

Since the content of the Theory of Interest course in an actuarial science program is tied to an external exam, instructors may be hesitant to attempt to use inquiry-based learning. In this paper, I outline how and why I did so, including descriptions of the materials that I wrote. I found that student performance on the final exam improved…

A formula is derived for a 'two-dice horse race', in which two ordinary dice are thrown repeatedly and each time the sum of the scores determines which horse (numbered 2 to 12) moves forward one space. This paper answers a question posed in a former "Teaching Statistics" article, and demonstrates the value of simulation.

We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses.

This article explains how explorations into the quadratic formula can offer students opportunities to learn about the structure of algebraic expressions. In this article, the author leverages the graphical interpretation of the quadratic formula and describes an activity in which students derive the quadratic formula by quantifying the symmetry of…

A visual aid for teaching the formula for the Mann-Whitney U statistic is proposed, inspired by 'proof without words', a didactic exercise from the mathematics literature that uses a visual approach to justify mathematical statements, including formulas and theorems.

Error analysis consumes much of the focus in introductory physics labs. Catastrophic cancellation is a spike in error that occurs when subtracting two measurements of roughly equal magnitude. Often termed "loss of significance" or "subtractive cancellation," this effect can easily relegate experimental results to utter…

Nitrate is a crucial parameter for assessing water quality, owing to its dual function in ecological systems. These functions can be beneficial or detrimental depending on whether nitrate concentrations are low or high, respectively. The ultraviolet spectrophotometric method (standard as 4500-NO[subscript 3]--B) is a classic method for determining…

The formula for the variance of a binomial distribution is both concise and elegant. However, it is often taught without reference to the underlying reasoning. That being the case, is it important, or useful, to understand why this formula can be used to calculate the requisite result? In this article, the author demonstrates a teaching sequence…

All students need to discuss mathematics to develop and deepen understanding. However, for English Learners (ELs), peer dialogue is imperative and indispensable to conceptual understanding as they participate in mathematical practices and engage in increasingly sophisticated uses of language (Heritage, Walqui, and Linquanti 2015). As ELs share…

The fundamental theorem of calculus (FTC) plays a crucial role in mathematics, showing that the seemingly unconnected topics of differentiation and integration are intimately related. Indeed, it is the fundamental theorem that enables definite integrals to be evaluated exactly in many cases that would otherwise be intractable. Students commonly…