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• Author or Editor: Michael Todd Edwards
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## Symbolic Manipulation in a Technological Age

As an algebra teacher and an avid calculator user, I often struggle to engage my students in technologically meaningful activities without undermining their conceptual understanding of problem solving. With only a handful of button presses, students using graphing calculators can solve most of the equations that they encounter in any conventional high school algebra textbook. Despite the apparent efficiency of calculator-based methods, traditional step-by-step solution strategies continue to play an important role in the classes that I teach. Through my varied teaching experiences, I have come to realize that traditional practice with pencil and paper—writing out individual steps to solve a problem —strengthens my students' understanding of such basic algebraic concepts as equivalency, inverse, domain, range, and function in a way that most calculator-based methods do not.

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## Visualizing Transformations: Matrices, Handheld Graphing Calculators, and Computer Algebra Systems

I first encountered matrices as tools for solving systems of linear equations as a mathematics major at a small Midwestern university. Although I became proficient at manipulating matrices—adding and multiplying them or calculating their determinants and inverses—I understood matrices as little more than convenient containers for numerical data. When I became a high school mathematics teacher, I was surprised to learn that matrices could be interpreted graphically, as well as symbolically. Along with my students, I found that matrix multiplication could be used to reflect, rotate, or change the size of geometric shapes. In addition to helping one calculate inverses, determinants are useful when investigating areas of polygons.

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## Technology Tips: Visualizing DeMorgan's Laws Using Logic Simulation Software

This month's tips illustrate an interactive, inquiry-based approach for exploring logic operators and equivalence of various Boolean expressions using technology. Specifically, I illustrate the use of several logic simulation tools to perform the following tasks: (1) build and “run” a virtual logic circuit illustrating xy and (2) verify the validity of DeMorgan's laws.

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## Promoting Understanding of Linear Equations with the Median Slope Algorithm

In recent years, the influence of technology on school mathematics curricula has become increasingly pronounced. Today, for instance, calculators seem ever–present in most mathematics classrooms. In fact, a recent study of nearly 4,000 mathematics teachers and 50,000 students reports that approximately 96 percent of eighth–grade students in the United States have access to calculators for use in mathematics classrooms (IEA 2001). The computational power of handheld calculators—particularly those equipped with graphing capabilities—has significantly accelerated the introduction of numerous mathematical topics within school mathematics. For example, before the early 1990s, leastsquares regression was not found in most algebra textbooks. Because by–hand methods for calculating a least–squares regression line required students to minimize the sum of squared residuals—a technique requiring differentiation—the procedure was rarely studied before calculus. Now, however, any student with a graphing calculator can perform a leastsquares linear regression by pressing only a few buttons. Not surprisingly, linear regression is now commonly found in the school algebra curriculum.

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## Activities for Students: Using Overhead Projectors to Explore Size Change Transformations

Through the study of transformation, a wide range of mathematical concepts may be introduced to secondary school students.

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## Fostering Mathematical Inquiry with Explorations of Facial Symmetry

AS A SECONDARY SCHOOL MATHEMATICS TEACHER, I have the opportunity to explore a wide range of mathematical topics with a vast array of students. Student variability–from student to student and from class to class–routinely challenges my instructional ingenuity. Although the mathematical concepts presented in such courses as first-year algebra or geometry are more basic than those presented in upper-level classes, entry-level courses are among the most challenging that I teach.

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## Who Was the Real William Shakespeare?

Connecting language arts and mathematics, students use data analysis and readability measures to identify the Bard.

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## Virtual Miniature Golf

Current standards place significant emphasis on transformations in school geometry: “Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these,” and “dynamic geometry environments provide students with experimental and modeling tools that allow them to investigate geometric phenomena” (CCSSI, 2010, p. 74). With these aims in mind, we share a favorite classroom activity—virtual miniature golf. Building on the work of Coxford and Usiskin (1991) and Powell et al. (1994), this activity provides geometry students with a real-world context for exploring reflection and reflection composition in technology-rich settings.

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## A Tale of Two Sliders

When you hear talk of “sliders,” your mouth may water as you think of miniature burgers that “slide” down your throat. If you are a student or teacher of mathematics, the word slider may also conjure up visions of interactive mathematics tools that enable exploration of parameters and graphs. In the paragraphs that follow, we discuss ideas for using Desmos, a freely available online calculator, to investigate a scenario about tasty sliders. By automating the construction of sliders, Desmos makes their use more accessible for students and teachers alike.

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## Repeated Square Roots

An inquiry-oriented investigation illustrates how students in algebra classrooms may explore limit meaningfully.