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## Delving Deeper: Twists on the Tower of Hanoi

A t a party that I attended, the hosts gave their guests the Tower of Hanoi puzzle with alternating dark and light discs and a challenge to move the 7 discs to a new post. (I disqualified myself because I knew how to solve the challenge.) However, the hosts' son and daughter-in-law misunderstood the directions and moved the dark discs to one side post and the light discs to the other side post. I immediately wondered, “How many moves did they take, assuming that they made the most efficient moves? How can their interpretation of the problem be generalized to n discs?”

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## Activities for Students: Soap and Slope: Mathematical Adventures in Fluid Dynamics

The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.

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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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## Activities for Students: Dealing Cards with Confidence

When understood and applied appropriately, mathematics is both beautiful and powerful. As a result, students are sometimes tempted to extend that power beyond appropriate limits. In teaching statistics at both the high school and college level, I have found that one of students' biggest struggles is applying their understanding of probability to make appropriate inferences.

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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.