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.
Thérèse Cozzo and Joseph Cozzo
This lesson provides an opportunity for students to use mathematical modeling and explore right-triangle trigonometry in the context of protecting battleships.
Determining exact values of trigonometric ratios remains an integral part of the high school mathematics curriculum. Students learn to use 45-45-90° triangles and 30-60-90° triangles to determine exact function values of angles of 30°, 45°, and 60°. Such exact-value ratios can help to determine trigonometric ratios for nonstandard angle measures when trigonometric identities and algebra are used. In this lesson, students apply a geometric approach to determine exact-value trig ratios for angle measures of 22.5°, 67.5°, 15°, and 75°. Some students can extend that approach to other nonstandard angle measures.
Alyson E. Lischka, Natasha E. Gerstenschlager, D. Christopher Stephens, Jeremy F. Strayer, and Angela T. Barlow
Select errors to discuss in class, and try these three alternative lesson ideas to leverage them and move students toward deeper understanding.
Charles F. Marion
Anyone who is looking for insights into the problem-solving process in mathematics is well advised to start with two books that have been in print for more than seven and three decades, respectively: How to Solve It (Pólya), first published in 1945; and The Art of Problem Posing (Brown and Walter) in 1983.
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.
Ann E. West
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.
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.
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.”
Bryan C. Dorner
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.