Using Clairaut's historic-dynamic approach and dynamic geometry tools in middle school can develop students' conceptual understanding before they encounter formal proof in geometry.

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### Dick J. Smith and Eric F. Errthum

Many mathematics instructors attempt to insert guided exploration into their courses. However, exploration tasks frequently come across to students as contrived, pertinent only to the most recently covered section of the textbook. In addition, students usually assume that the teacher already knows the answers to these explorations.

### Katie L. Anderson

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes a set of lessons where sixth graders use virtual pattern blocks to develop proportional reasoning. Students' work with the virtual manipulatives reveals a variety of creative solutions and promotes active engagement. The author suggests that technology is most effective when coupled with worthwhile mathematical tasks and rich classroom discussions.

### Irina Lyublinskaya and Dan Funsch

Symbolic geometry software, such as Geometry Expressions, can guide students as they develop strategies for proofs.

### Jenny Livingstone and Julian F. Fleron

Google SketchUp is free, powerful and widely used Computer Aided Design (CAD) software that can have a transformative impact on the teaching of geometry. This article introduces Google SketchUp to readers through lessons that can be integrated into geometry classrooms and also provides additional resources for readers interested in learning more.

### Samuel Obara and Zhonghong Jiang

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

### Günhan Caglayan

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.

### Nicholas H. Wasserman and Itar N. Arkan

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).

### Patricia O'Donnell and Amanda Frick

Do you remember clever, energetic Speedy Gonzales, “the fastest mouse in all Mexico,” one of the animated characters in the Warner Brothers Looney Tunes cartoon series? This month our “math by the month” activities, reminiscent of the spirited Speedy, will have your students calling ¡Ándale! ¡Ándale! ¡Arriba! ¡Arriba! (colloquial Spanish for “Come on! Hurry up!”) as they ask for more problem-solving scenarios based on this month's racing theme.