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## Delving Deeper: The Diver Problem: The Surfer Problem in 3D

Mathematics is not a “handbag of tricks” but rather a discipline of inquiry and creativity, as Nabb (2010-11) notes, and he has shared his methods and excitement for the inquiry approach. By engaging calculus students in a search for examples of infinite series that meet certain conditions, or arguments that such series do not exist, Nabb appropriately aligns his teaching with the Standards for Mathematical Practice found in the Common Core State Standards (CCSSI 2010). In particular, three Standards for Mathematical Practice come to mind: (a) “make sense of problems and persevere in solving them”; (b) “reason abstractly and quantitatively”; and (c) “model with mathematics” (CCSSI 2010, pp. 6-8).

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## Open-ended Questions and the Process Standards

Educating students—for life, not for tests—implies incorporating open-ended questions in your teaching to develop higher-order thinking.

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## An Application of Mathematics to Computer Programming: Connecting Translation Vectors, the Minkowski Difference, and Collision Detection

Translation by a vector in the coordinate plane is first introduced in precalculus and connects to the basic theory of vector spaces in linear algebra. In this article, we explore the topic of collision detection in which the idea of a translation vector plays a significant role. Because collision detection has various applications in video games, virtual simulations, and robotics (Garcia-Alonso, Serrano, and Flaquer 1994; Rodrigue 2012), using it as a motivator in the study of translation vectors can be helpful. For example, students might be interested in the question, “How does the computer recognize when a player's character gets hit by a fireball?” Computer science provides a rich context for real-life applications of mathematics-programmers use mathematics for coding an algorithm in which the computer recognizes two objects nearing each other or colliding. The Minkowski difference, named after the nineteenth century German mathematician Hermann Minkowski, is used to solve collision detection problems (Ericson 2004). Applying the Minkowski difference to collision detection is based on translation vectors, and programmers use the algorithm as a method for detecting collision in video games.

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## Back to Treasure Island

Some twenty years ago, when I was a university student, one of my lecturers presented a problem that he called Treasure Island. At first glance, the problem appeared to be unsolvable. After students made some futile attempts, the lecturer presented the surprising solution, without providing any explanation or even a hint. I spent the rest of the lecture thinking about the problem and trying to discover a solution.

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## Connecting Quadratics, Line Segments, Continued Fractions, and Matrices

One of the many benefits of teaching mathematics is having the opportunity to encounter unexpected mathematical connections while planning lessons or exploring ideas with students and colleagues. Consider the two problems in figure 1.

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## Technology-Based Geometry Activities for Teaching Vector Operations

Students connect operations in linear algebra with geometric representations.

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## Triangles from Three Points

Using technology to solve triangle construction problems, students apply their knowledge of points of concurrency, coordinate geometry, and transformational geometry.

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## Delving Deeper: Geometry of the Fibonacci Matrix

Diagonalization of the Fibonacci matrix ties the sequence to geometry.

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## Angry Birds Mathematics: Parabolas and Vectors

Vector properties and the birds' frictionless environment help students understand the mathematics behind the game.

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## Using PhET Simulations in the Mathematics Classroom

Imagine that you and your language arts colleagues are teaching Edgar Allan Poe's short story, “The Pit and the Pendulum.” This thrilling story takes us to the Inquisition during which a prisoner is surrounded by hungry rats and bound to a table while a large pendulum slowly descends. The prisoner believes that the pendulum is 30-40 feet long and estimates that it should take about 10-12 swings before he is hit, leaving him with about a minute or a minute and a half to escape. Are his estimations correct? If so, will he make it out in time?