The standard approach to finding antiderivatives of trigonometric expressions such as sin(ax) cos(bx) is to make use of certain trigonometric identities. The disadvantage of this technique is that it gives no insight into the problem, but relies on students using a memorized formula. This note considers a technique for finding antiderivatives of…

The Boltzmann factor is the basis of a huge amount of thermodynamic and statistical physics, both classical and quantum. It governs the behaviour of all systems in nature that are exchanging energy with their environment. To understand why the expression has this specific form involves a deep mathematical analysis, whose flow of logic is hard to…

We explain how to simulate both univariate and bivariate raw data sets having specified values for common summary statistics. The first example illustrates how to "construct" a data set having prescribed values for the mean and the standard deviation--for a one-sample t test with a specified outcome. The second shows how to create a bivariate data…

North Korea is one of the most closed nations in the world. It is difficult to access information about their education system. North Korea's economic system was reorganized on July 1, 2002, as well as the education system. This change has impacted North Korea's mathematics education. With limited information on North Korean mathematics textbooks,…

This article describes one classroom activity in which the author simulates the Newtonian gravity, and employs the Euclidean Geometry with the use of new technologies (NT). The prerequisites for this activity were some knowledge of the formulae for a particle free fall in Physics and most certainly, a good understanding of the notion of similarity…

This contribution reports about a seven-month long video-based study in two regular Flemish sixth-grade mathematics classrooms. The focus is on teachers' approaches towards problem solving. In our analysis we distinguished between a paradigmatic-oriented (focus on the mathematical structure) and a narrative-oriented (focus on the contextual…

This article presents a mathematical function for measuring the congruence among the strand representation in the enacted curriculum, state assessments, and district pacing guides. Using the Manhattan distance as its basis, this method allows researchers to incorporate the strand coverage of the enacted curriculum (i.e., the curriculum actually…

An introductory approach to black holes is presented along with astronomical observational data pertaining to the presence of a supermassive black hole at the center of our galaxy. Concepts of conservation of energy and Kepler's third law are employed so students can apply formulas from their physics class to determine the mass of the black hole…

This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem. (Contains…

Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length c and height h (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving c and h that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian…

A method is derived for the numerical evaluation of the error term arising in some Gauss-type formulae modified so as to approximate Cauchy Principal Value integrals. The method uses Chebyshev polynomials of the first kind. (Contains 1 table.)

The importance of visualisation and multiple representations in mathematics has been stressed, especially in a context of problem solving. Hanna and Sidoli comment that "Diagrams and other visual representations have long been welcomed as heuristic accompaniments to proof, where they not only facilitate the understanding of theorems and their…

Infinity and infinitely small numbers pique the curiosity of middle school students. From a very young age, many students are intrigued, interested, and even fascinated by extremely large or extremely small numbers or quantities. This article describes an activity that takes this curiosity about infinity into the domain of adding the numbers of an…

In this note, based on the Binet formulas and the power-reducing techniques, closed forms of generating functions for the powers of Fibonacci sequences are presented. The corresponding results are extended to some other famous sequences as well.

As extended by Ginsberg, Midi's theorem says that if the repeated section of a decimal expansion of a prime is split into appropriate blocks and these are added, the result is a string of nines. We show that if the expansion of 1/p[superscript n+1] is treated the same way, instead of being a string of nines, the sum is related to the period of…