Exploring and deriving proofs of closed-form expressions for series can be fun for students. However, for some students, a physical representation of such problems is more meaningful. Various approaches have been designed to help students visualize squares of sums and sums of squares; these approaches may be arithmetic-algebraic or combinatorial…

Develops the formulas for the sum of the numbers from 1 to n and for the squares of the numbers from 1 to n geometrically by utilizing the formulas for the area of a triangle and the volume of a pyramid. (MDH)

Discusses the hydrodynamic reasons why a riverbed meanders through a plain. Describes how water movement at a bend in a river causes erosion and changes in the riverbed. Provides a mathematical model to explain the periodic shape of meanders of a river in a plain. (MDH)

Discusses the use of scaling test scores for an algebra class. Provides example data, several equations used in scaling, and graphs. (YP)

Presented is a method for factoring quadratic equations that helps the teacher demonstrate how to eliminate guessing through establishment of the connection between multiplication and factoring. Included are examples that allow the student to understand the link between the algebraic and the pictorial representations of quadratic equations. (JJK)

Discusses the use of mathematical modeling. Describes types, examples, and importance of mathematical models. (YP)

This section provides mathematical activities in reproducible formats appropriate for students in grades 8-10. The activity is designed to provide an experience in model building while developing the concept of slope. (PK)

Discusses the mathematical model presented by Vito Volterra to describe the dynamics of population density. Discusses the predator prey relationship, presents an computer simulated model from marine life involving sharks and mackerels, and discusses ecological chaos. (MDH)

Discusses three types of bridges to determine how best to model each one: (1) drawbridge; (2) balance bridge; and (3) bascule bridge. Describes four experiments with assumptions, analyses, interpretations, and validations. Provides several diagrams and pictures of the bridges, and typical data. (YP)

Presents two methods of calculating the expected value for a participant on the television game show "The Wheel of Fortune." The first approach involves the use of basic expected-value principles. The second approach uses those principles in addition to infinite geometric series. (MDH)

Emphasizes a real-world-problem situation using sine law and cosine law. Angles of elevation from two tracking stations located in the plane of the equator determine height of a satellite. Calculators or computers can be used. (LDR)

Explores the problem of combining algebraic terms from the students' point of view and suggests changes in certain traditional teaching practices. (PK)

Describes software, hardware, and devices that were designed to provide students with an environment to experiment with basic ideas of mechanics, including nonlinear dynamics. Examines the behavior of a Lorenzian water wheel by comparing experimental data with theoretical results obtained from computer-based sensors. (MDH)

Following a brief historical review of the mechanism of mathematical modeling, examples are included that associate a mathematical model with given data (changes in sea level) and that model a real-life situation (process of parallel parking). Also provided is the rationale for the curricular implementation of mathematical modeling. (JJK)