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.

## Informing Practice: A Hybrid Perspective on Functions

### research matters for teachers

### Eric Weber

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.

### Scott Steketee and Daniel Scher

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

### Timothy Deis

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.

### Matt B. Roscoe

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

### David A. Yopp

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

### George J. Roy, Vivian Fueyo, Philip Vahey, Jennifer Knudsen, Ken Rafanan and Teresa Lara-Meloy

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.

### Jennifer Suh and Padmanabhan Seshaiyer

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

### Terri L. Kurz, Mi Yeon Lee, Sarah Leming and Wendy Landis

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.

## Quick Reads: Posters: More Than Just for Looks

### a good idea in a small package

### Greisy Winicki-Landman and Christine Latulippe

Posters, commonly employed for decoration, can be used to introduce and practice new concepts and help assess student learning.