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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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## 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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## 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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## 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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## The Popcorn Box Activity and Reasoning about Optimization

Investigating rates of change in volume without calculation leads to an enriched sense of the optimization process and encourages reflection and connection among different approaches.

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## Geometry and the Design of Product Packaging

Simulating a real-world marketing situation, students examine the mathematical calculations that play an integral part in product design.