The study of asymptotes constitutes one of the earliest and most significant encounters a high school mathematics student can have with infinity. The study of horizontal asymptotes, in particular, contributes to a student's later understanding of limits and the notion of “arbitrarily close to.” Typically, students are exposed to horizontal asymptotes in the contexts of exponential and rational functions. Real–life applications of exponential functions abound, but real–life applications of rational functions do not. In this article, we present a hands–on activity that allows students to explore horizontal asymptotes of graphs of rational functions visually and mathematically and that has been used successfully with intermediate algebra, precalculus, college algebra, and mathematics methods students.
Anna A. Davis, Ronald E. Zielke and Jessica L. Lickeri
A hands-on approach to studying quadratic functions emphasizes the engineering design process.
Flor Jacqueline Alarcón Mejía
Algebra, probability, and sequences—all important curricular material—can be connected by a question that will challenge students: What is the probability P that the function f(x) = x 2 + rx + s has real zeros when r and s are real numbers between 0 and 9, inclusive? This problem involves an infinite sample space, making it more interesting for students who have worked on probability problems with only finite sample spaces.
The mathematical concept of slope can be made real through a set of simple, inexpensive, and safe experiments that can be conducted in the classroom or at home. The experiments help connect the idea of slope with physical phenomena related to surface tension. In the experiments, changes in surface tension across the surface of the water, which correspond to greater slopes on the graph, lead to increased motion of the fluid. The mathematical content, targeted to middle school and high school students, can be used in a classroom or workshop setting and can be tailored to a single session of thirty to ninety minutes.
Jennifer L. Jensen
Five problems—relating to gas mileage, the national debt, store sales, shipping costs, and fish population—require students to use functions to connect mathematics to the real world.
Jamie-Marie L. Wilder and Molly H. Fisher
Our favorite lesson is a hands-on activity that helps students visually “tie” (pun intended) the concepts of rate of change and y-intercept together in a meaningful context using strings and ropes. Students tie knots in ropes of various thicknesses and then measure the length of the rope as the number of knots increases. We provide clothesline, twine, bungee cord, and other ropes found at local crafts, sporting goods, and home stores. We avoid very thin string, such as thread or knitting yarn, because the knots are small and the string length does not change enough to explore a rate of change. A variety of thicknesses is important because this allows for variability in the rates of change.
Seán P. Madden and Dean Allison
Some children enjoy playing with the spring–loaded, plastic toy cannon that accompanies many model pirate ships. This cannon pivots in such a way that it can be fired at any angle from about −15 degrees to 195 degrees when viewed from a position directly in front of the cannon. The spring provides each fired projectile with approximately the same initial velocity. This toy, together with a digital camera, can be used with high school students to demonstrate the influence of angle on parabolic trajectories and, simultaneously, to explore the underlying parametric equations. In particular, students can discover and verify that the vertices of projectile motion paths for a given initial speed and a range of launch angles trace an ellipse.
Students analyze a photograph to solve mathematical questions related to the images captured in the photograph. This month the editors consider photographs of African bowls made of recycled telephone wire. The mathematics involves trigonometry, parametric equations and their graphs, and linear regression.
Virtual worlds, such as the one inhabited by the players of World of Warcraft, can serve as sampling grounds for students who are video gamers.
Margaret Cibes and James Greenwood
Short items from the media focus mathematics appropriate for classroom study.