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 general method is presented for evaluating the sums of "m"th powers of the integers that can, and that cannot, be represented in the two-element Frobenius problem. Generating functions are introduced and used for that purpose. Explicit formulas for the desired sums are obtained and specific examples are discussed.

A heuristic approach to explaining of the Black-Scholes option pricing model in undergraduate classes is described. The approach draws upon the method of protocol analysis to encourage students to "think aloud" so that their mental models can be surfaced. It also relies upon extensive visualizations to communicate relationships that are…

In this study, the authors wanted to examine whether success on items testing basic equivalence knowledge, such as the meaning of the equal sign and ability to solve problems such as 3 + 5 = 4 + _, predicted success on items testing more advanced algebraic thinking (i.e. principles of equality and solving equations that use letter variables). This…

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…

The note introduces sequences having M-bonacci property. Two summation formulas for sequences with M-bonacci property are derived. The formulas are generalizations of corresponding summation formulas for both M-bonacci numbers and Fibonacci numbers that have appeared previously in the literature. Applications to the Arithmetic series, "m"th "g -…

University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…

The objective of this research was to better understand the transfer of learning between different variations of pre-algebra problems. While the authors could have addressed a specific variation that might address transfer, they were interested in developing a general model of transfer, so we gathered data from multiple problem types and their…

The extreme interval values and statistics (expected value, median, mode, standard deviation, and coefficient of variation) for the smallest (min) and largest (max) values of exponentially distributed variables with parameter ? = 1 are examined for different observation (sample) sizes. An extreme interval value g[subscript a] is defined as a…

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of "r" natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.

A problem given in the Australian Mathematics Competition for the Westpac Awards was stated as follows: With how many zeros does 2008! end? In this article, the author solves this problem, and provides further discussion on the related problems. These problems form a good model that helps students develop a logical thinking process toward problem…

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

This article explores phenomena related to fitting polynomials with data sets with equally spaced x-values.

The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of…

Reality--real life--is complex, more complex than can be imagined. The majority of our most pressing issues confronted in the management of organizations are overwhelmingly complex. I make the case for a systems design approach as a remedy. Examples introduced in this paper illuminate conceptual tools that advance how people transform their…