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## A Conceptual Approach to Absolute Value Equations and Inequalities

Sitting in the back of Ms. Corey's sixthgrade mathematics class, I enjoyed seeing students enthusiastically demonstrate their understanding of absolute value. On the giant number line on the classroom floor, they counted the steps that they needed to take to get back to zero. The old definition of absolute value of a number as its distance from zero—learned by students and teachers of the previous generation—has long ago been replaced with this algebraic statement: |x| = x if x ≤ 0 or − x if x < 0. The absolute value learning objective in high school mathematics requires students to solve far more complex absolute value equations and inequalities. However, I cannot remember students attacking the task with enthusiasm or having any understanding beyond “make the inside positive.”

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## Technology Tips – Piecewise-Defined Functions on the TI–Nspire

Students generally first encounter piecewise–defined functions in the form of a step function (perhaps the postage stamp function) in an algebra class. Piecewise–defined functions do not play a central role in mathematics before calculus although they can serve as challenging examples in the precalculus curriculum. Before the advent of the TI–Nspire, entering piecewise–defined functions on the calculator was time consuming and not particularly user friendly. That has changed.

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## Love Unlimited

Hearts are the theme of a collection of problems and solutions.

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## Mathematical Lens: Up on the Roof

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month, photographs of ceiling trusses provide a setting for a geometry discussion.

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## Teaching Absolute Value Meaningfully

What is the meaning of absolute value? And why do we teach students how to solve absolute value equations? Absolute value is a concept introduced in first-year algebra and then reinforced in later courses. Various authors have suggested instructional methods for teaching absolute value to high school students (Wei 2005; Stallings-Roberts 1991; Friedlander and Hadas 1988), but here we focus on an investigation that will help students make meaning of the absolute value equation in the context of a practical situation. We connect absolute value to the concepts of rate, time, distance, and slope.

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Interesting solutions and ideas emerge when preservice and in-service teachers are asked a traditional algebra question in new ways.

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## The ABCs of Covariance

While filling three-dimensional letters, students analyzed the relationship between the height of water level and elapsed time.

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## Algebraic Activities Aid Discovery Lessons

Three graphing activities lead students to discover the shapes and properties of the graphs for linear, quadratic, and absolute value functions and inequalities.

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