In a 1999 article in Mathematics Teacher, we demonstrated how graphing systems of linear inequalities could be motivated using real-world linear programming problems (Edwards and Chelst 1999). At that time, the graphs were drawn by hand, and the corner-point principle was applied to find the optimal solution. However, that approach limits the number of decision variables to two, and problems with only two decision variables are often transparent and inauthentic.

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

### Chris Harrow and Lillian Chin

Exploration, innovation, proof: For students, teachers, and others who are curious, keeping your mind open and ready to investigate unusual or unexpected properties will always lead to learning something new. Technology can further this process, allowing various behaviors to be analyzed that were previously memorized or poorly understood. This article shares the adventure of one such discovery of exploration, innovation, and proof that was uncovered when a teacher tried to find a smoother way to model conic sections using dynamic technology. When an unexpected pattern regarding the locus of an ellipse's or hyperbola's foci emerged, he pitched the problem to a ninth grader as a challenge, resulting in a marvelous adventure for both teacher and student. Beginning with the evolution of the ideas that led to the discovery of the focal locus and ending with the significant student-written proof and conclusion, we hope to inspire further classroom use of technology to enhance student learning and discovery.

### Martin Griffiths

I always seek activities that might stretch my students yet would be accessible to them; that might require logical thought yet would contain counterintuitive elements; that might provide the opportunity to venture into new mathematical realms yet would have a simple starting point. This article and the activity that inspired it did indeed arise by way of a relatively straightforward problem that I proposed to one of my classes.

Readers comment on published articles or offer their own ideas.

### Craig J. Cullen, Joshua T. Hertel, and Sheryl John

Technology can be used to manipulate mathematical objects dynamically while also facilitating and testing mathematical conjectures. We view these types of authentic mathematical explorations as closely aligned to the work of mathematicians and a valuable component of our students' educational experience. This viewpoint is supported by NCTM and the Common Core State Standards for Mathematics (CCSSM).

### Elliott Ostler

Processes using linear measurement can be adapted to teach complex topics such as polynomial multiplication, rational exponents, and logarithms.

Regarding the reflection “On the Area of a Circle” by Cheng, Tay, and Lee (MT April 2012, vol. 105, no. 8, pp. 564-65), it is possible to prove that one can arrange infinitely many sectors of a circle into a rectangle to show that a circle's area is π^{2}. However, the authors' derivation is invalid because they assume their conclusion by using the area of the circle within their proof.

### Michael Dempsey

When understood and applied appropriately, mathematics is both beautiful and powerful. As a result, students are sometimes tempted to extend that power beyond appropriate limits. In teaching statistics at both the high school and college level, I have found that one of students' biggest struggles is applying their understanding of probability to make appropriate inferences.

### Maria L. Hernandez and Nils Ahbel

luidMath™ (www.fluiditysoftware.com), a new mathematics software tool for Tablet devices, computers, and interactive whiteboards, can create a dynamic graph or table with a simple gesture and recognize written expressions as the mathematical relationship they intend. The software uses a stylus as its input device. By changing constant values in an equation to parameters, the user can create sliders instantly and see graphs and tables change dynamically. The CAS (Computer Algebra System) functionality allows simplification of algebraic expressions and solution of equations and can perform all the calculations from algebra through calculus. FluidMath uses standard mathematical notation to explore explicitly and implicitly defined functions, parametric functions, polar functions, and recursively defined functions.

### Dick J. Smith and Eric F. Errthum

Many mathematics instructors attempt to insert guided exploration into their courses. However, exploration tasks frequently come across to students as contrived, pertinent only to the most recently covered section of the textbook. In addition, students usually assume that the teacher already knows the answers to these explorations.