Examining the covariation of triangle dimensions and area offers a geometric context that makes analyzing a piecewise function easier for students.

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### Nina G. Bailey, Demet Yalman Ozen, Jennifer N. Lovett, Allison W. McCulloch, and Charity Cayton

Three different technological activities to explore parameters of quadratic functions each has its own pros and cons.

### Kristin Frank

Lessons that focus on a conceptual understanding offer an opportunity for students to learn about mathematical structure, not just computation.

### Thomas Edwards, S. Asli Özgün-Koca, and Kenneth Chelst

A quadratic equation was the basis for activities involving both concrete and technological representations.

### Erin E. Baldinger, Matthew P. Campbell, and Foster Graif

Students need opportunities to construct definitions in mathematics. We describe a sorting activity that can help students construct and refine definitions through discussion and argumentation. We include examples from our own work of planning and implementing this sorting activity to support constructing a definition of linear function.

### Rebecca Vinsonhaler and Alison G. Lynch

This article focuses on students use and understanding of counterexamples and is part of a research project on the role of examples in proving. We share student interviews and offer suggestions for how teachers can support student reasoning and thinking and promote productive struggle by incorporating counterexamples into the classroom.

### Wendy B. Sanchez and David M. Glassmeyer

In this 3-part activity, students use paper-folding and an interactive computer sketch to develop the equation of a parabola given the focus and directrix.

When visitors enter the High Museum in Atlanta, one of the first pieces of art they encounter is Physic Garden, by Molly Hatch (details in **photographs 1 and 2**). Physic Garden consists of 456 handpainted dinner plates arranged to form a rectangle with 24 horizontal rows and 19 vertical columns and extends from the floor to the ceiling of the first floor. The design of the “plate painting” was inspired by two mid-18th-century English ceramic plates from the museum's collection (**photograph 3**).

### Michael Weiss

One of the central components of high school algebra is the study of quadratic functions and equations. The Common Core State Standards (CCSSI 2010) for Mathematics states that students should learn to solve quadratic equations through a variety of methods (CCSSM A-REI.4b) and use the information learned from those methods to sketch the graphs of quadratic (and other polynomial) functions (CCSSM A-APR.3). More specifically, students learn to graph a quadratic function by doing some combination of the following:

Locating its zeros (x-intercepts)

Locating its y-intercept

Locating its vertex and axis of symmetry

Plotting additional points, as needed