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Kevin Phillippi

This department showcases students' in-depth thinking and work on previously published problems. The August 2012 problem scenario leverages back-to-school shopping advertisements for this real-life scenario about making purchases using discount coupons. To access the full-size activity sheet, go to http://www.nctm.org/tcm, Back Issues.

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Readers comment on published articles or offer their own ideas.

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Craig J. Cullen, Joshua T. Hertel, and Sheryl John

Technology can be used to manipulate mathematical objects dynamically while also facilitating and testing mathematical conjectures. We view these types of authentic mathematical explorations as closely aligned to the work of mathematicians and a valuable component of our students' educational experience. This viewpoint is supported by NCTM and the Common Core State Standards for Mathematics (CCSSM).

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Elliott Ostler

Processes using linear measurement can be adapted to teach complex topics such as polynomial multiplication, rational exponents, and logarithms.

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Ann Bremner

Games and videos lend themselves to student engagement and learning, but how do teachers determine what is most beneficial to student learning? How can we effectively incorporate technology into our teaching of mathematics?

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Terri L. Kurz

Free, Web-based balance tools can help students visualize the concept of the equal sign as the pivot point of an equation.

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Brandt S. Lapko

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K–grade 6 classrooms. This article describes how students can use available technology to communicate and share their thinking in popular media formats.

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Kurt J. Rosenkrantz

Students say some amazing things. Back Talk highlights the learning of one or two students and their approach to solving a math problem or prompt. Each article includes the prompt used to initiate the discussion, a portion of dialogue, student work samples (when applicable) and teacher insights into the mathematical thinking of the students. In this month's episode, a six-year-old rising first grader uses a computer simulation to understand addition and subtraction on the number line.

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Jennifer Orr and Jennifer Suh

Teachers share success stories and ideas that stimulate thinking about the effective use of technology in K—grade 6 classrooms. One way to keep young students engaged and interested in practicing counting is to involve them in using cameras. This article explains how first graders capture 100 images, use Windows MovieMaker or PhotoStory to turn the still images into a video, and then narrate a story using precise math vocabulary to explain their mathematical thinking.

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Regarding the reflection “On the Area of a Circle” by Cheng, Tay, and Lee (MT April 2012, vol. 105, no. 8, pp. 564-65), it is possible to prove that one can arrange infinitely many sectors of a circle into a rectangle to show that a circle's area is π2. However, the authors' derivation is invalid because they assume their conclusion by using the area of the circle within their proof.