Few high school students associate mathematics with playfulness. In this paper, we offer a series of lessons focused on the underlying algebraic structures of the Rubik's Cube. The Rubik's Cube offers students an interesting space to enjoy the playful side of mathematics, while appreciating mathematics otherwise lost in routine experiences.

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## Mathematical Explorations: A New Twist on Collaborative Learning

### classroom-ready activities

### Stephanie M. Butman

Research on students' learning has made it clear that learning happens through an interaction with others and through communication. In the classroom, the more students talk and discuss their ideas, the more they learn. However, within a one-hour period, it is hard to give everyone an equal opportunity to talk and share their ideas. Organizing students in groups distributes classroom talk more widely and equitably (Cohen and Lotan 1997).

### Agida G. Manizade and Marguerite M. Mason

When calculating the area of a trapezoid, students use a range of problem-solving strategies and measurement concepts.

### Sherri Ann Cianca

Communicating reasoning and constructing models fold nicely into a geometry activity involving the building of nesting boxes.

### Bobson Wong and Larisa Bukalov

Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof.

### Tutita M. Casa

This instructional tool helps students engage in discussions that foster student reasoning, then settle on correct mathematics.

## Informing Practice: Examples as Tools for Constructing Justifications

### research matters for teachers

### Kristen N. Bieda and Jerilynn Lepak

Research explores how to help students build from, instead of building with, examples when justifying mathematical ideas.

### Colin Foster

Exploring even something as simple as a straight-line graph leads to various mathematical possibilities that students can uncover through their own questions.

### Carol J. Bell

Reasoning and Proof is one of the process standards set forth in NCTM's *principles and standards for school mathematics* (2000).