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What's Inside the Cube? Students' Investigation with Models and Technology

Applying Zometool, vZome software, and The Geometer's Sketchpad to tetrahedrons nested in cubes enhances students' spatial visualization skills.

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Activities for Students: The Hands Project

The author presents an activity in which the lines in students' hands are analyzed, with curves and lines fit to each one.

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Mathematical Lens: A Mathematical Pyramid Scheme

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month, the photographs are of a pyramid in Egypt, and students are asked to compute volume, slant height, and the ratio of the base of the pyramid to its height.

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Technology Tips: Constructing the Square Root of a Complex Number

The author uses The Geometer's Sketchpad first to construct the square root of an arbitrary real number and then to construct the square root of a complex number.

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Using Technology to Understand Rate of Change

The following series of learning activities are from an afterschool math club called Go Go Gizmos that focuses on modeling mathematics with the use of technologies. This account describes how a classroom teacher and a math educator taught and assessed students' understanding of the rate of change using a variety of technologies. In particular, we chose data collection probeware called Go!Motion, which is a stand-alone motion-data-collection device from Vernier that sends data to the computer for analysis and simulation applets from http://explorelearning.com. The Go!Motion device can be connected to a computer and displays an interactive real-time spreadsheet with graphing capabilities. The objectives in the unit were for students to investigate physical representations of slope as a rate of change in mathematics and as velocity in science and the y-intercept as the initial condition, or starting position. In these investigations, students and teachers become partners in developing mathematical ideas and solving math problems (NCTM 2000).

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Welcome to Volume 18

Share news about happenings in the field of elementary school mathematics education, views on matters pertaining to teaching and learning mathematics in the early childhood or elementary school years, and reactions to previously published opinion pieces or articles.

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Delving Deeper: Deepening Understanding of Transformation through Proof

In the introductory geometry courses that we teach, students spend significant time proving geometric results. Students who conclude that angles are congruent because “they look that way” are reminded that visual information fails to provide conclusive mathematical evidence. Likewise, numerous examples suggesting a particular result should be viewed with skepticism. After all, unfore–seen counterexamples render seemingly valid conclusions false. Inductive reasoning, although useful for generating conjectures, does not replace proof as a means of verification.

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Technology Tips: An Archimedean Walk

The circle, so simple and yet complex, has fascinated mathematicians since the earliest civilizations. Archimedes, a well–known Greek mathematician born in 287 BCE, began to unravel part of the mystery involving π by applying iteration to the circle. Building on Euclid's postulates and theorems, Archimedes used iterations of inscribed and circumscribed regular polygons to find upper and lower bounds for the value of π. These bounds are close approximations of the value of π, and one is still used today: 22/7 differs from π only in the third place to the right of the decimal (see fig. 1).

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Flower Power – Sunflowers as a Model for Logistic Growth

Logistic growth displays an interesting pattern: It starts fast, exhibiting the rapid growth characteristic of exponential models. As time passes, it slows in response to constraints such as limited resources or reallocation of energy (see fig. 1). The growth continues to slow until it reaches a limit, called capacity. When the growth describes a population, capacity is defined as “the maximum population that the environment is capable of sustaining in the long run” (Stewart 2008, p. 628).

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An Excel–lent Card Trick

Card tricks based on mathematical principles can be a great way to get students interested in exploring some important mathematical ideas. Bonomo (2008) describes several variations of a card trick that rely on nested floor functions, but these generally go beyond the reach of beginning algebra students. However, a simple spreadsheet implementation shows students why the card trick works and allows them to explore several variations. As an added bonus, students are introduced to composite functions, the floor function, and iteration, and they learn how to use formulas and the INT function in Microsoft Excel. The depth of the mathematical explanation can be varied according to students' background.