Using the fact that the sum of the first n odd numbers is n[superscript 2], we show visually that n[superscript 2] is the same as 0 (mod 3) when n is the same as 0 (mod 3), and n[superscript 2] is the same as 1 (mod 3) when n is the same as plus or minus 1 (mod 3).

Using the fundamental theorem of calculus and numerical integration, we investigate carbon absorption of ecosystems with measurements from a global database. The results illustrate the dynamic nature of ecosystems and their ability to absorb atmospheric carbon.

What is the area of the (inner) square obtained by slicing the corners off a larger square? This visual proof avoids algebra by considering the area of a parallelogram.

We produce a continuum of curves all of the same length, beginning with an ellipse and ending with a cosine graph. The curves in the continuum are made by cutting and unrolling circular cones whose section is the ellipse; the initial cone is degenerate (it is the plane of the ellipse); the final cone is a circular cylinder. The curves of the…

Using calculus only, we find the angles you can rotate the graph of a differentiable function about the origin and still obtain a function graph. We then apply the solution to odd and even degree polynomials.

We show that inside every triangle the locus of points satisfying a natural proportionality relationship is a parabola and go on to describe how this triangle-parabola relationship was used by Archimedes to find the area between a line and a parabola.

Given a family of "p" greater than or equal to 3 points in the plane, some three of them have the property that the smallest circle encompassing them encompasses all "p" points. Similarly, we show that for "p" greater than or equal to 3 circles, there are three of them such that the smallest circle encompassing them…

Measures of centre (the mean, median and mode) are fundamental to the discipline of statistics. Yet previous research shows that students may not have a thorough conceptual understanding of these measures, even though these statistics are easy to calculate. This study describes the findings of a study of pre-service teachers' ideas of measure of…

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the…

FOIL is a well-known mnemonic that is used to find the product of two binomials. We conduct a large sample (n = 252) observational study of first-year college students and show that while the FOIL procedure leads to the accurate expansion of the product of two binomials for most students who apply it, only half of these students exhibit conceptual…

In this article, we share a model of flipped instruction that allowed us to gain a window into our students' mathematical thinking. We depict how that increased awareness of student thinking shaped our mathematics instruction in productive ways. Drawing on our experiences with students in our own classrooms, we show how flipped instruction can be…

This paper presents an extended project that offers, through American football, an application of concepts from enumerative combinatorics and an introduction to proofs course. The questions in this paper and subsequent details concerning equivalence relations and counting techniques can be used to reinforce these new topics to students in such a…

Our goal with this paper is three-fold. We want to increase awareness of inquiry-based learning by presenting the strategy we use to develop and implement lessons and activities. We describe our approach to structuring lessons in mathematics in a way that engages the students by using language and constructs with which they are familiar from other…

This paper presents results from three teaching experiments intended to guide students to reinvent truth-functional interpretations for mathematical disjunctions. The initial teaching experiments revealed that students' emergent strategies for assessing disjunctions did not entail or facilitate the development of a relevant partitioning of example…

This article outlines a class activity where students prepare croissants to get an intuitive feeling for the nature of a strange attractor. In particular during the preparation of the pastry dough, students investigate the effects of stretching and folding. These physical processes force trajectories (in this case the pastry dough) to remain…