The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described…

Diophantine equations constitute a rich mathematical field. This article may be useful as a basis for a student math club project. There are several situations in which one needs to find a solution of indeterminate polynomial equations that allow the variables to be integers only. These indeterminate equations are fewer than the involved unknown…

The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

In Pre-Calculus courses, students are taught the composition and combination of functions to model physical applications. However, when combining two or more functions into a single more complicated one, students may lose sight of the physical picture which they are attempting to model. A block diagram, or flow chart, in which each block…

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…

The Pythagorean Theorem, arguably one of the best-known results in mathematics, states that a triangle is a right triangle if and only if the sum of the squares of the lengths of two of its sides equals the square of the length of its third side. Closely associated with the Pythagorean Theorem is the concept of Pythagorean triples. A "Pythagorean…

One of the main problems in undergraduate research in pure mathematics is that of determining a problem that is, at once, interesting to and capable of solution by a student who has completed only the calculus sequence. It is also desirable that the problem should present something new, since novelty and originality greatly increase the enthusiasm…

The problem "Given three planar points, find a point such that the sum of the distances from that point to the three points is a minimum" is discussed from several points of view. A solution that uses only calculus and geometry is examined in detail. (MNS)

We sometimes teach our students a method of finding all integral triples that satisfy the Pythagorean Theorem x[squared]+y[squared]=z[squared]. These are called Pythagorean triples. In this paper, we show how to solve the equation x[squared]+ky[squared]=z[squared], where again, all variables are integers.

Finding a fractional number equal to an infinite decimal is solved by two usual methods. Then a third method is discussed that allows students to avoid having to confront the idea of an attained infinity of symbols. (MNS)

Results obtained from investigating number properties are discussed, along with six points that are felt, in general, to be the ingredients necessary for a successful learning experience. Two programs written in BASIC designed to aid in aspects of Number Theory are included. (MP)

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

Advantages and disadvantages of common ways to justify the answer to a probability problem are discussed. One explanation appears superior to the others because it is easy to understand, mathematically rigorous, generalizes to a broader class of problems, and avoids the deficiencies of the other explanations. (MNS)

This article examines the benefits and teaching procedures resulting from a classroom activity in student problem posing. The benefits included an enhancement of student reasoning and reflection and a heightened level of engagement. "Principles and Standards for School Mathematics" indicates that one of the critical requirements for successful…

The focus is on how line graphs can be used to approximate solutions to rate problems and to suggest equations that offer exact algebraic solutions to the problem. Four problems requiring progressively greater graphing sophistication are presented plus four exercises. (MNS)