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.

This article describes activities that introduce basic trigonometric functions in a smooth and natural way. The only requirements are basic algebra, the Pythagorean theorem, and the concept of area.

When students take General Chemistry there are substantially fewer molecular images than they will encounter in Organic Chemistry. The molecular images Organic Chemistry students see in their textbooks are ones that use dashes and wedges to represent 2D and semi 3D views, ball and spoke, ball and wire, and structural formulas, to name just a few.…

The paper details a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers…

This article describes how a series of lessons might be used to allow students to discover the size of the Earth, the distance to the Moon, the size of the Moon, and the altitude of Mount Piton on the Moon. Measurement with a sextant, principles of geometry and trigonometry, and historically important scientists and mathematicians are discussed.

This paper describes an attempt to facilitate students' learning of scientific concepts by using detectors that take as input physical information and output an instantiation of the concept. The principle hypothesis was that students would have a better understanding of the concept of image if they were taught to use a simplified, runnable model…

The majority of research on mathematics practice in everyday situations within cultures has investigated the use of arithmetic and geometry concepts and processes. To extend this research to a situation using measurement ideas, this paper investigates the mathematics practice of a group of carpet layers in an effort to detail how ordinary people…

The historical development of the concepts of ratio and proportion is discussed. (SD)

What happens in the limiting cases of a parabola when the distance between the focus and directrix approaches zero and when the distance approaches infinity is discussed. (LS)

Six films produced by the College Geometry Project at the University of Minnesota are reviewed; titles of six others are listed. (DT)

THIS PROJECT PROVIDES A PROGRAM FOR THE EDUCATION OF NINTH GRADE STUDENTS IN GENERAL MATHEMATICS. THE PURPOSE OF THE PROJECT IS (1) TO WRITE UNITS OF WORK THAT INCORPORATE PRACTICAL APPLICATIONS FROM THE EVENTS AND AFFAIRS OF URBAN LIFE, (2) TO SEEK OUT PROBLEMS FROM LOCAL INDUSTRY, AND (3) TO PROVIDE DIFFERENTIATED ACTIVITIES AND EXERCISES THAT…

THIS BOOKLET, ONE OF A SERIES, HAS BEEN DEVELOPED FOR THE PROJECT, A PROGRAM FOR MATHEMATICALLY UNDERDEVELOPED PUPILS. A PROJECT TEAM, INCLUDING INSERVICE TEACHERS, IS BEING USED TO WRITE AND DEVELOP THE MATERIALS FOR THIS PROGRAM. THE MATERIALS DEVELOPED IN THIS BOOKLET INCLUDE (1) THE HISTORY AND MEANING OF LINEAR MEASURE, (2) FINDING THE…