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## Talking about the Greek Cross

Meaningful discourse occurs when tasks are chosen carefully and when the teacher steps back and allows students to move to the forefront of their own learning.

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## Informing Practice: A Hybrid Perspective on Functions

### research matters for teachers

Formal notions of function, which appear in middle school, are discussed in light of how teachers might complement the input-output notion with a covariation perspective.

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## A Geometric Path to the Concept of Function

Transformations using dynamic software can provide a unique perspective on a common topic.

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## Tiling the Cube

The Tiling Tubs task is a middle school activity published in NCTM's Navigating through Algebra in Grades 6-8. Students examine a drawing that shows a square hot tub with side length s feet. A border of square tiles surrounds the tub, with each border tile a 1-foot square. Students determine the number of border tiles required to surround the tub and express that number in as many ways as they can, thereby creating various equivalent algebraic representations. A valuable component of Tiling Tubs is its requirement that students contextually justify their algebraic representations. Students also realize that more than one correct representation and justification exists.

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## A Repeat Look at Repeating Patterns

Working with repeating patterns is important for K–grade 2 students because of the connections they will make in later grades with related mathematical ideas.

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## Visualizing Equations Using Color Tiles

Algebraic reasoning is often promoted through an analysis of and generalizations about patterns that appear in mathematics, in nature, or in everyday situations (Driscoll 1999; Kieran 2006; Lee 1996). In accordance with this tendency, the Common Core (CCSSI 2010) emphasizes finding patterns and expressing such regularity in repeated reasoning as an important mathematical practice. NCTM (2000) also recommends that students participate in patterning activities by asking them to describe numeric and geometric patterns; generalize patterns to predict what comes next while providing a rationale for their predictions; and represent patterns in multiple ways, including drawings, tables, symbols, and graphs.

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## A Vehicle for Bivariate Data Analysis

Top-selling cars in America can be the catalyst that drives an analysis of data.

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## Dilations and the Equation of a Line

Track students' understanding of proportional reasoning by combining transformational geometry, similar-triangle reasoning, and linear relationships.

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## Connecting Representations: Using Predict, Check, Explain

Although educators agree that making connections with the real world, as advocated by Principles to Actions: Ensuring Mathematical Success for All (NCTM 2014), is important, making such connections while addressing important mathematics is elusive. We have found, however, that math content coupled with the instructional strategy of predict, check, explain can bridge such real-world contexts. In so doing, this procedure supports the research-informed teaching practices of using evidence of student thinking and aiding meaningful mathematical discussion.

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## Informing Practice: Mathematical Practices That Promote Twenty-First Century Skills

Skills that students will need in the twenty-first century, such as financial literacy, are explored in this classroom-centered research article.