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## Connecting Slope, Steepness, and Angles

Students must be able to relate many representations of slope to form an integrated understanding of the concept.

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## Activities for Students: Connecting Spatial Reasoning Ideas in Mathematics and Chemistry

Stereochemistry and three-dimensional analysis constitute significant parts of this student activity.

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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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## Is That Square Really a Circle?

Considering circles in taxicab geometry helps students with Euclidean concepts.

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## Visible Thinking in High School Mathematics

Chalk Talk and Claim-Support-Question are two routines for developing students' ability to use multiple representations and encouraging classroom discussion.

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## Exploring Quadrilaterals through Flexible Approaches

Rich mathematical tasks—here, finding and categorizing the quadrilaterals that can be made with vertices on a 4 × 4 grid—can promote adaptability in any classroom.

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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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## A River Runs through Math Class

Let's go wading! Students connect fundamental mathematics concepts in this real-world, problem-solving field experience.

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## The Back Page: My Favorite Lesson: A Sweet Way to Investigate Volume

This article describes a series of “food labs” designed to help calculus students make sense of abstract volume concepts. The methods include slicing (e.g., disk, washer, cross-section) and shell methods, and the author discusses how to use them in the classroom.

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## Delving Deeper: Boundaries in Higher Dimensions and Their Kinship with Combinatorics

This is a description of a collaborative investigation by mathematics teachers into the numbers of dimensional boundaries for n > 2. Functions are fit to the patterns observed, and a relationship to Pascal's triangle is noted.