This article suggests the introduction of the concepts of areas bounded by plane curves and the volumes of solids of revolution in Pre-calculus. It builds on the basic knowledge that students bring to a pre-calculus class, derives a few more formulas, and gives examples of some problems on plane areas and the volumes of solids of revolution that…

Students were asked to find all possible values for A so that the points (1, 2), (5, A), and (A, 7) lie on a straight line. This problem suggests a generalization: Given (x, y), find all values of A so that the points (x, y), (5, A), and (A, 7) lie on a straight line. We find that this question about linear equations must be resolved using the…

Arithmetic tests for divisibility of an integer by another integer are well known. This article states and proves conditions for divisibility in binary form.

The mathematics for finding and plotting the locations of stars and constellations are available in many books on astronomy, but the steps involve mystifying and fragmented equations, calculations, and terminology. This paper will introduce an entirely new unified and cohesive technique that is easy to understand by mathematicians, and simple…

To achieve the vision of mathematics set forth in "Crossroads" ("AMATYC," 1995), students must experience mathematics as a sensemaking endeavor that informs their world. Embedding the study of mathematics into the real world is a challenge, particularly because it was not the way that many of us learned mathematics in the first place. This article…

The orders of presentation of pre-calculus and calculus topics, and the notation used, deserve careful study as they affect clarity and ultimately students' level of understanding. We introduce an alternate approach to some of the topics included in this sequence. The suggested alternative is based on years of teaching in colleges within and…

This is the third and final article in this series. The first article reviewed the literature for research studies on successful developmental programs and students. The second article reported on the qualitative research methods and results documented from a purposive sample of twenty successful developmental mathematics students 3-5 years after…

How might one define a functional operator D[superscript I]f(x), say for f(x) = 1 + x[superscript 2] + sin x, such that D[superscript +1](1 + x[superscript 2] + sin x) = 2x + cos x and D[superscript -1](1 + x[superscript 2] + sin x) = x + x[superscript 3]/3 - cos x? Our task in this article is to describe such an operator using a single formula…

This article shows how to use six parameters describing the International Space Station's orbit to predict when and in what part of the sky observers can look for the station as it passes over their location. The method requires only a good background in trigonometry and some familiarity with elementary vector and matrix operations. An included…

Beyond Crossroads addresses many of the challenges facing American higher education and offers members of AMATYC an ideal opportunity to respond energetically and constructively to these challenges. Notwithstanding the many efforts to improve secondary schooling, most students enter postsecondary education well behind where they should be in…

In this article, Steenken conveys some thoughts about when parents and our citizenry will believe that today's students know mathematics. It will be when students have significant Quantitative Literacy (QL) skills as set forth in AMATYC's new standards, Beyond Crossroads. He supports the strong call for QL to be embedded across all curricula and…

In this paper, departments are urged to consider implementing the type of changes proposed in Beyond Crossroads in College Algebra. The author of this paper is chair of the Curriculum Renewal Across the First Two Years (CRAFTY) Committee of the Mathematical Association of America. The committee has members from two-year colleges, four-year…

For various reasons there has been a recent trend in college and high school calculus courses to de-emphasize teaching the Partial Fraction Decomposition (PFD) as an integration technique. This is regrettable because the Partial Fraction Decomposition is considerably more than an integration technique. It is, in fact, a general purpose tool which…

PreCalculus students can use the Completing the Square Method to solve quadratic equations without the need to memorize the quadratic formula since this method naturally leads them to that formula. Calculus students, when studying integration, use various standard methods to compute integrals depending on the type of function to be integrated.…

This paper considers rules for multiplying exponential and radical expressions of different bases and exponents and/or roots. This paper demonstrates the development of mathematical concepts through the application of connections to other mathematical ideas. The developed rules and most of the employed connections are within the realm of secondary…