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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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## 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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## 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.

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## Flying High with the Bird Tetrahedron

An origami activity can lead to rich tasks in several branches of mathematics.

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## Teaching Geometry to Visually Impaired Students

Teaching a visual subject to a visually challenged student inspires strategies that benefit all students in a class.

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## Thinking outside the Cube

Mathematical Lens uses photographs as a springboard for mathematical inquiry and appears in every issue of Mathematics Teacher. all submissions should be sent to the department editors. For more background information on Mathematical Lens and guidelines for submitting a photograph and questions, please visit http://www.nctm.org/publications/content.aspx?id=10440#lens.

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## Using Disks as Models for Proofs of Series

Students use balls and disks to prove the general formulas for sums of squares and cubes.

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## Mathematical Lens: Mathematics and the Art of Sol LeWitt

Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month, the art of sculptor and painter Sol LeWitt is analyzed. Counting, combinatorics, and spatial visualization are among the mathematical themes evinced.

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## Don't Fence Me In!

This method for counting lattice octagons strengthens students' counting skills and geometrical thinking.

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## Activities for Students: Filling Bottles with Water

In this activity, students gradually fill bottles of different shapes and graph the water level (height) versus the volume of water inside the bottles. Then they explore the relationship between the shape of a bottle and the resulting graph.