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## Using Covariation Reasoning to Support Mathematical Modeling

Table representations of functions allow students to compare rows as well as values in the same row.

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## Adding It All Up: Reconceiving the Introduction of the Integral

An applied approach to understanding the integral—using a burst pipe—involves physical quantities and helps deepen the concept for students.

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## The Back Page: My Favorite Lesson: Modeling Exponential Decay

Our favorite lesson, an interactive experiment that models exponential decay, launches with a loud dice roll. This exploration engages students in lively data collection that motivates interest in key components of the Common Core State Standards for Mathematics: functions, modeling, and statistics and probability (CCSSI 2010).

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## Spaghetti Sine Curves: Virtual Environments for Reasoning and Sense Making

The Spaghetti Sine Curves activity, which uses GeoGebra applets to enhance student learning, illustrates how technology supports effective use of physical materials.

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## Galileo, Gauss, and the Green Monster

Using ideas of Galileo and Gauss but avoiding calculus, students create a model that predicts whether a fly ball will clear the famous left-field wall at Fenway Park.

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## Power Indices and U.S. Presidential Elections

The 2000 presidential election provides an ideal backdrop for introducing the electoral voting system, weighted voting, and the Banzhaf and Shapley-Shubik Power Indices.

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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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## Activities for Students: Ellipses and Orbits: An Exploration of Eccentricity

The orbits of planets about the sun and satellites about the earth are elliptical. The shape of the orbit can be described by its eccentricity and can be modeled algebraically and graphically. The exploration of orbits enriches our understanding of the mathematical representations, definitions, and connections for ellipses.

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## Errors in Mathematics Aren't Always Bad

We tell students that mathematical errors should be avoided, but understanding errors is an important tool in developing numerical methods.