This article describes how we used the context of mailing a package with seventh-grade students to explore and connect the different representations of step and linear functions. (Contains 2 tables and 7 figures.)

Four activities consisting of puzzles are introduced, which help students to recognize the strength of algebraic generalizations. They also assist them to comprehend algebraic concepts, and enable them to develop their individual puzzles and games.

We report some findings of the Longitudinal Proof Project, which investigated patterns in high-attaining students' mathematical reasoning in algebra and in geometry and development in their reasoning, by analyses of students' responses to three annual proof tests. The paper focuses on students' responses to one non-standard geometry item. It…

Packing an infinite number of cubes into a box of finite volume is the focus of this article. The results and diagrams suggest two ways of packing these cubes. Specifically suppose an infinite number of cubes; the side length of the first one is 1; the side length of the second one is 1/2 , and the side length of the nth one is 1/n. Let n approach…

Typically, secondary and college algebra students attempt to utilize either completing the square or the quadratic formula as techniques to solve a quadratic equation only after frustration with factoring has arisen. While both completing the square and the quadratic formula are techniques which can determine solutions for all quadratic equations,…

In this paper we have converted the Laplace transform into an integral equation of the first kind of convolution type, which is an ill-posed problem, and used a statistical regularization method to solve it. The method is applied to three examples. It gives a good approximation to the true solution and compares well with the method given by…

Equivalence relations with a non-transparent transitive property are scarce in textbook examples. The equivalence relation given here is not only one with an interesting transitive property to prove, but also one that is instructive for students who are encountering proofs and relations for the first time.

Technology is a powerful tool in assisting students in problem solving by allowing for multiple representations. The vignette offered in this article provides insight into ways to solve open-ended problems using multiple technologies.

This article describes Comer's Method, a student's discovery of an alternative approach to solving the determinant of a 3x3 matrix.

For two given vectors of the three-dimensional Euclidean space we investigate the problem of identifying all rotations that transform them into each other. For this purpose we consider three types of rotation matrices to obtain a complete characterization. Finally some attention is paid to the problem of obtaining all rotations taking two vectors…

Two visual proofs illustrating a cubic Fibonacci number identity are given. These provide a basis for further activities and consideration of the use of visual approaches to other identities.

Real numbers are often a missing link in mathematical education. The standard working assumption in calculus courses is that there exists a system of "numbers", extending the rational number system, adequate for measuring continuous quantities. Moreover, that such "numbers" are in one-to-one correspondence with points on a "number line". But…

An easy way to present the uniqueness of the square root of a positive semidefinite matrix is given.

A model for car following on a closed loop is defined. The stability of the solutions of the model is investigated by considering the evolution of the roots of the corresponding characteristic equation in the complex plane. The solution provides a motivation for investigating the behaviour of the roots of a simple class of algebraic equation.…

The authors hope to show how a geometric insight can add to the richness of our students' experiences when they first encounter the solutions to two equations in two unknowns.