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## Activities for Students: Rational Functions: A New Perspective

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.

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A hands-on approach to studying quadratic functions emphasizes the engineering design process.

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## Activities for Students: Soap and Slope: Mathematical Adventures in Fluid Dynamics

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.

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## Activities for Students: Connecting Probability and Algebra

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.

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## Exploring Quadratic Functions with Logger Pro

Using technology, students in an extended second year algebra class engage in an activity that introduces them to quadratic functions.

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## How Big is Too Big? Attendance Matters

Short items from the media focus mathematics appropriate for classroom study.

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## Investigating the Effects of Parameters on Quadratic Functions

Technology is in a constant state of flux. As a result, if we seek to use the latest forms of technology in our teaching, our teaching with technology must be ever changing. Edwards and Özgün-Koca (2009) describe an investigation involving the TI-Nspire CAS to understand the effect of b on the graphical representation of a quadratic function of the form f (x) = ax2 + bx + c. Adapting this idea, I show how updates to technology can enhance this investigation to create an even more motivating and compelling experience for students. The investigation will make use of sliders and the spreadsheet capability of the TI-Nspire (the directions provided here apply to the TI-Nspire CX CAS).

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## The Back Page: My Favorite Lesson: All Knotted Up

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.

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## Students as Mathematics Consultants

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.

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## Pirates of the Parametric

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.