We present an algorithm for a matrix-based Enigma-type encoder based on a variation of the Hill Cipher as an application of 2 × 2 matrices. In particular, students will use vector addition and 2 × 2 matrix multiplication by column vectors to simulate a matrix version of the German Enigma Encoding Machine as a basic example of cryptography. The…

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

In Brazil we have identified a predilection of the authors of Mathematical History books for the discussion of the fundamentals of Differential and Integral Calculus. On the other hand, when we consider the teaching of Mathematics in the school context, it is essential to know the teaching of the historical and dynamic evolution of the concepts,…

This article aims to illustrate a process of making connections, not between mathematics and other activities, but within mathematics itself--between diverse parts of the subject. Novel connections are still possible in previously explored mathematics when the material happens to be unfamiliar, as may be the case for a learner at any career stage.…

Usually a student learns to solve a system of linear equations in two ways: "substitution" and "elimination." While the two methods will of course lead to the same answer they are considered different because the thinking process is different. In this paper the author solves a system in these two ways to demonstrate the…

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show…

The aim of this article is to characterize the 2 x 2 matrices "X" satisfying X[superscript 2] = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers. The recommendations regarding the teaching of the identities given in this article can be presented in two cases. The first is related to the pedagogical aspect. The…

This article demonstrates how within an educational context, supported by the notion of hidden mathematics curriculum and enhanced by the use of technology, new mathematical knowledge can be discovered. More specifically, proceeding from the well-known representation of Fibonacci numbers through a second-order difference equation, this article…

We use the sum property for determinants of matrices to give a three-stage proof of an identity involving Fibonacci numbers. Cassini's and d'Ocagne's Fibonacci identities are obtained at the ends of stages one and two, respectively. Catalan's Fibonacci identity is also a special case.

This article presents several activities (some involving graphing calculators) designed to guide students to discover several interesting properties of Fibonacci numbers. Then, we explore interesting connections between Fibonacci numbers and matrices; using this connection and induction we prove divisibility properties of Fibonacci numbers.

The sample assessment items in this volume are sorted according to the strands of number and operations, algebra, geometry, measurement, and data analysis and probability. Because one goal of assessment is to determine students' abilities to communicate mathematically, the writing team suggests ways to extend or modify multiple-choice and…