Computing common logarithms on the basis of a few successive square roots of 10 reinforces rules of logarithms and powers for secondary school students.
Rose Sinicrope and Daniel V. Bellittiere
The orbits of planets about the sun and satellites about the earth are elliptical. The shape of the orbit can be described by its eccentricity and can be modeled algebraically and graphically. The exploration of orbits enriches our understanding of the mathematical representations, definitions, and connections for ellipses.
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.
Mara G. Landers
A measurement-based activity can help students struggling to understand trigonometric functions.
Sheldon P. Gordon
In the climactic scene in The Wizard of Oz, Toto draws back the curtain to expose the Wizard of Oz, and Frank Morgan admits, “I am really a very good man but just a poor wizard.” This statement is reminiscent of Arthur C. Clarke's famous third law: “Any sufficiently advanced technology is indistinguishable from magic” (Clarke 1962, p. 36). For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry.
Sheldon P. Gordon
We tell students that mathematical errors should be avoided, but understanding errors is an important tool in developing numerical methods.
Thomas J. Bannon
Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.
When each student has become part of an ordered pair, we go outside to the school parking lot, where I chalk the outline of a Cartesian plane.
Nicholas H. Wasserman
The practice of problem posing is as important to develop as problem solving. The resulting explorations can be mathematically rich.