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.

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### 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

### Kent Thele

Encourage investigation of the conic-section attributes of focus, eccentricity, directrix, and semi-latus rectum using polar coordinates and projective geometry.

### James Metz, Lance Hemlow, and Anita Schuloff

Explore the relationship between families of quadratic expressions factorable over the integers and Pythagorean triples.

### Arsalan Wares

A paper-folding problem is easy to understand and model, yet its solution involves rich mathematical thinking in the areas of geometry and algebra.

### Lorraine M. Baron

Assessment tools–a rubric, exit slips–inform instruction, clarify expectations, and support learning.

### Debra K. Borkovitz

A few years ago, I encountered two different problems in which the number 3 played surprising roles. I found myself wondering, “Why 3? What's so special about 3?” Further investigation led to continuous extensions involving exponents, logarithms, a parametric equation, maxmin problems, and some history of mathematics. As you read, pause to try the problems and play with the applets (the article's title is a big hint!)

A set of problems of many types.