If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.

The fire piston is just a sealed syringe containing a small amount of tinder. When the plunger is forced downwards, the air inside is compressed and heats up, setting fire to the tinder. It has been used as a convenient and portable way of starting fires "over a wide area from northern Burma and Siam through the Malay Peninsula and the Malayan…

In this article, the author offers a "Pierce-ing," rational dissection of the notion of cultural equivalence that exposes the hypocrisy of its purveyors. He tackles three major argument lines--the Probabilities Argument, the Internal-Division Argument, and the Change Argument--and proves that it is simply very improbable that culture, in any…

The formulae for the convergents of continued fractions are always given recursively rather than in explicit form. This article derives explicit formulae for the convergents of the continued fraction expansions for square roots.

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. An approximate formula is obtained for one of the characteristics of the motion--the vertical…

The optical quality and properties of over 200 telescopes residing in museums and private collections have been measured and tested with the goal of obtaining new information about the early development of the achromatic lens (1757-1770). Quantitative measurements of the chromatic and spherical aberration of telescope objective lenses were made…

We examined the degree of dependence between standardized-mean-difference effect sizes in multiple-treatment studies in meta-analysis in terms of the correlation formula provided by Gleser and Olkin (1994). To explore the impact of group size and the values of the true multiple-treatment effect sizes, we simplified the formula for the correlation…

In this article, by using Rolle's theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment ["An integral mean value theorem", Math. Gazette 54 (1970), pp. 300-301] and Sahoo ["Some results related to the integral mean value…

PEMDAS is a mnemonic device to memorize the order in which to calculate an expression that contains more than one operation. However, students frequently make calculation errors with expressions, which have either multiplication and division or addition and subtraction next to each other. This article explores the mathematical reasoning of the…

In this article, we report two studies investigating what mathematicians value in a pedagogical proof. Study 1 is a qualitative study of how eight mathematicians revised two proofs that would be presented in a course for mathematics majors. These mathematicians thought that introductory and concluding sentences should be included in the proofs,…

This paper justifies the need for, and offers some suggestions on, the selection and implementation of mathematical problems known as dynamic solution exercises (DSEs). The intent of this article is to help provide insight into how mathematics teachers can go about making "vertical articulation" a cooperative and tangible part of the…

The great gorilla jump is an activity designed to allow calculus students to construct an understanding of the structure of the Riemann sum and definite integral. The activity uses the ideas of position, velocity, and time to allow students to explore familiar ideas in a new way. Our research has shown that introducing the definite integral as…

This paper reports on the nature of the conceptual understandings developed by Year 12 Victorian Certificate of Education (VCE) physics students as they made the transition from the essentially deterministic notions of classical physics, to interpretations characteristic of quantum theory. The research findings revealed the fact that the…

This classroom note is presented as a suggested exercise--not to have the class prove or disprove Goldbach's Conjecture, but to stimulate student discussions in the classroom regarding proof, as well as necessary, sufficient, satisfied, and unsatisfied conditions. Goldbach's Conjecture is one of the oldest unsolved problems in the field of number…

Using modern technology to examine classical mathematics problems at the high school level can reduce difficult computations and encourage generalizations. When teachers combine historical context with access to technology, they challenge advanced students to think deeply, spark interest in students whose primary interest is not mathematics, and…