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## Revisit Pattern Blocks to Develop Rational Number Sense

A classic manipulative, used since the 1960s, continues to offer opportunities for intriguing problem solving involving proportions.

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## Radical Thoughts on Simplifying Square Roots

A geometric approach using exact square manipulatives can promote an understanding of the algorithm to dismantle radical expressions.

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## Building Bridges to Spatial Reasoning

Three days of using building blocks significantly enriched second graders' thinking about multiple dimensions.

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## Why Is the Midpoint an Average?

The relationship between a midpoint and an average showcases the interplay between procedural knowledge and conceptual knowledge in learning mathematics for teaching.

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## UDL for Geometric Length Measurement

A proactive preschool teacher differentiates instruction by using the Universal Design for Learning framework to decrease barriers that limit students' access to classroom learning.

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## Virtual manipulatives to assess understanding

Three preservice teachers used virtual manipulatives during clinical interviews with students of elementary school age. The technology exposed students' problem-solving strategies and mathematical understanding, promoting just-in-time teaching about the target content. The process of completing and reflecting on the interviews contributed to growth of the preservice teachers' technological pedagogical content knowledge.

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## An Appetite for Fractions

Using candy bars as models gives students a taste for learning to represent fractions whose denominators are factors of twelve.

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## Core Conversations with Educative Dragging

The Measure-Trace-Algebratize (MTA) approach allows students to uncover algebraic relationships within familiar geometric objects.

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## Pattern-block frenzy

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.

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## Three-Dimensional Printing: A Journey in Visualization

Using simple materials, a Mathematica software application, and their knowledge of function transformations, students design and create real mathematical sculptures.