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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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## Six Principles for Quantitative Reasoning and Modeling

Modeling the motion of a speeding car or the growth of a Jactus plant, teachers can use six practical tips to help students develop quantitative reasoning.

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## The Circle Approach to Trigonometry

A connected introduction of angle measure and the sine function entails quantitative reasoning.

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## Graphing Inequalities, Connecting Meaning

tudents often have difficulty with graphing inequalities (see Filloy, Rojano, and Rubio 2002; Drijvers 2002), and my students were no exception. Although students can produce graphs for simple inequalities, they often struggle when the format of the inequality is unfamiliar. Even when producing a correct graph of an inequality, students may lack a deep understanding of the relationship between the inequality and its graph. Hiebert and Carpenter (1992) stated that mathematics is understood “if its mental representation is part of a network of representations” and that the “degree of understanding is determined by the number and strength of the connections” (p. 67). I therefore developed an activity that allows students to explore the graphs of inequalities not presented as lines in slope-intercept form, thereby making connections between pairs of expressions, ordered pairs, and the points on a graph representing equations and inequalities.

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## If Only Clairaut Had Dynamic Geometry Tools

Using Clairaut's historic-dynamic approach and dynamic geometry tools in middle school can develop students' conceptual understanding before they encounter formal proof in geometry.

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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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## The Footprint Problem: A Pathway to Modeling

By exploring an open-ended investigation involving proportional reasoning, students were able to walk through both problem solving and modeling.

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## Redeem Reasoning

### readers speak out

This opinion piece discusses how simplicity, ease, and efficiency—in the guise of shortcuts, tips, and packed procedures—kill mathematical reasoning.

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## Model Eliciting Activities: A Home Run

An aluminum bat activity supports goals of STEM learning by engaging students in resourceful problem solving.

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## Developing Formal Procedures through Sense Making

Allowing students to reason and communicate about integer operations, or any idea, before these ideas are formalized can be an important tool for fostering deep understanding.