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## Solutions to the Tipping Point Problem

The problem posed in MT August 2011 (vol. 105, no. 1, pp. 62-66) asked readers to consider the two-dimensional version of tipping a bowl (assumed to be a rectangular prism) to spoon out the last little bit of melted ice cream. Here is the essence of the problem: Given a fluid region of fixed area A contained in a rectangle whose width is W, find a formula for the fluid depth D when the container is tilted through a known angle T that is measured from horizontal.

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## The Back Page: My Favorite Lesson: The Trig Tribe

My favorite lesson is my lighthearted introduction to righttriangle trigonometry. I explain that we are starting a lesson that connects to the social studies curriculum—specifically, the Lewis and Clark expedition.

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## Equivalence Relations across the Secondary School Curriculum

An analysis of problems from state assessments and other sources helps preservice teachers discover analogous mathematical representations.

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## “Tricks” Work!

The use of mnemonic devices, or “tricks,” in the mathematics classroom has been criticized by some authors. However, when used in the proper context, such “tricks” can be extraordinarily helpful in motivating students and helping them remember procedures while understanding concepts and mastering appropriate mathematical vocabulary.

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## Mathematical Lens: A Road for Every Wheel

Can you imagine riding a tricycle with square wheels? Can you imagine that this tricycle would give you as smooth a ride as a traditional tricycle? A New York Times article (Chang 2011) described a tricycle that had square wheels but that could be ridden “smoothly around a circular path ridged like a flower's petals.” It then explained that the ridged surface on which the tricycle rode undulated such that “the tricycle's axles—and the rider—remain in the same height as they move.”

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## Delving Deeper: Chordic vs. CORDIC: How Calculators and Students Compute Sines and Cosines

Students who have grown up with computers and calculators may take these tools' capabilities for granted, but I find something magical about entering arbitrary values and computing transcendental functions such as the sine and cosine with the press of a button. Although the calculator operates mysteriously, students generally trust technology implicitly. However, beginning trigonometry students can compute the sine and cosine of any angle to any desired degree of precision using only simple geometry and a calculator with a square root key.

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## Improving Mathematics Discourse through Action Research

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).

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## Lines as “Foci” for Conic Sections

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.

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## Battleship Trig

This lesson provides an opportunity for students to use mathematical modeling and explore right-triangle trigonometry in the context of protecting battleships.

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## Making STEM Connections

Acting as quality control engineers and service providers, students collaborate to engage, explore, and explain their results.