Let students find the connecting thread to create, illustrate, and share word problems to bridge school math and real-life math.

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### Thomas G. Edwards and Kenneth R. Chelst

Connecting the formula to the graphic representation of quadratic functions makes the mathematics meaningful to students.

### Christopher W. Parrish, Ruby L. Ellis, and W. Gary Martin

NCTM identified eight Mathematics Teaching Practices within its reform-oriented text, Principles to Actions: Ensuring Mathematical Success for All (2014). These practices include research-informed, high-leverage processes that support the in-depth learning of mathematics by all students. Discourse within the mathematics classroom is a central element in these practices. The goal of implementing the practice facilitate meaningful discourse is to give students the opportunity to “share ideas and clarify understandings, construct convincing arguments regarding why and how things work, develop a language for expressing mathematical ideas, and learn to see things from other perspectives” (NCTM 2014, p. 29). To further support implementing meaningful discourse, mathematics educators must become adept at posing questions that require student explanation and reflection, hence, pose purposeful questions, which is another of the eight practices. Posing purposeful questions allows “teachers to discern what students know and adapt lessons to meet varied levels of understanding, help students make important mathematical connections, and support students in posing their own questions” (NCTM 2014, pp. 35-36).

MT's letters to the editor department. Readers comment on published articles and share their mathematical interests.

### Wayne Nirode

One of my goals, as a geometry teacher, is for my students to develop a deep and flexible understanding of the written definition of a geometric object and the corresponding prototypical diagram. Providing students with opportunities to explore analogous problems is an ideal way to help foster this understanding. Two ways to do this is either to change the surface from a plane to a sphere or change the metric from Pythagorean distance to taxicab distance (where distance is defined as the sum of the horizontal and vertical components between two points). Using a different surface or metric can have dramatic effects on the appearance of geometric objects. For example, in spherical geometry, triangles that are impossible in plane geometry (such as triangles with three right or three obtuse angles) are now possible. In taxicab geometry, a circle now looks like a Euclidean square that has been rotated 45 degrees.

### Casey Hawthorne and Bridget K. Druken

Examples of solving equations and inequalities, analyzing quadratic expressions, and reasoning with functions show three ways to engage students in this mathematical practice.

### Claudia M. Bertolone-Smith and Linda Gillette-Koyen

Avoid off-task behavior, such as horseplay, rolling on the floor, and meowing, with a reliable routine that promotes students' thinking, communication, and social safety in sharing their ideas.

### Edited by Anna F. DeJarnette and Stephen Phelps

A monthly set of problems is aimed at a variety of ability levels.

### Kelly W. Remijan

These four activities connect mathematics to science, technology, engineering, and art.