A first-year algebra student's curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics. In this article, the authors, math teachers in Massachusetts, examine how to solve for factorials of negative numbers and discuss how they taught…

KenKen® is the new Sudoku. Like Sudoku, KenKen requires extensive use of logical reasoning. Unlike Sudoku, KenKen requires significant reasoning with numbers and operations and helps develop number sense. The creator of KenKen puzzles, Tetsuya Miyamoto, believed that "if you give children good learning materials, they will think and learn and…

Many students find proofs frustrating, and teachers struggle with how to help students write proofs. In fact, it is well documented that most students who have studied proofs in high school geometry courses do not master them and do not understand their function. And yet, according to NCTM's "Principles and Standards for School Mathematics"…

Proof is a central component of mathematicians' work, used for verification, explanation, discovery, and communication. Unfortunately, high school students' experiences with proof are often limited to verifying mathematical statements or relationships that are already known to be true. As a result, students often fail to grasp the true nature of…

In a course on proofs, a number of problems deal with identities for Fibonacci numbers. Some general strategies with examples are used to help discover, prove, and generalize these identities.

The purpose of this article is to present a collection of problems that allow students to investigate magic squares and Latin squares, formulate their own conjectures about these mathematical objects, look for arguments supporting or disproving their conjectures, and finally establish and prove mathematical assertions. Each problem is completed…

A square-dance number is defined as an even number which has the property that the set which consisted of the numbers one through the even number can be partitioned into pairs so that the sum of each pair is a square. Theorems for identifying square-dance numbers are discussed. (YP)

Describes generalized Fibonacci sequences that satisfy many elegant identities and possess curious properties. Provides physical applications and connections to various branches of mathematics. (KHR)

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.