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## Seeing Algebraic Structure: The Rubik's Cube

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

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).

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## Algebra Homework: A Sandwich!

Given two slices of bread—a problem and the answer—students fill in the fixings: their own mathematics reasoning.

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## Developing the Area of a Trapezoid

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

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## Mysterious Subtraction

Conjecturing is central to the work of reasoning and proving. This task gives fourth and fifth graders a chance to make conjectures and prove (or disprove) them.

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## Changing the Rules to Increase Discourse

A second-grade teacher challenges the raise-your-hand-to-speak tradition and enables a classroom community of student-driven conversations that share both mathematical understandings and misunderstandings.

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## Bird Boxes Build Content Area Knowledge

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

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## Develop Reasoning through Pictorial Representations

Solutions coupled with drawings can illustrate students' understandings or misunderstandings, particularly in the area of proportional reasoning.

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## Improving Student Reasoning in Geometry

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

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## Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.