By Barbara A. Swartz, posted February 16, 2016 —

For many students of algebra, factoring quadratic equations can seem like a completely arbitrary thing to do. Why would anyone want to know the zeros of a quadratic function? Teachers can set up introductory factoring lessons to provide the rationale through projectile motion. Everybody wants to know, Will we hit the target?

Projectile Motion, the PhET Interactive Simulation by the University of Colorado-Boulder is a great way to pique students’ interest in quadratic functions and their solutions. In this simulation, students are able to manipulate the angle of the cannon to hit the target a specified distance away with a number of fun and different objects (e.g., a Buick, a pumpkin, and even an adult human!). The simulation also draws out the path of the object fired, so that students can see the shape of this “new” function.

Students can make multiple attempts and see them all on the same screen, comparing the shapes of the different graphs, and then can simply click the Erase button when they want to clear the screen. (See the figure below.)

Teachers can probe students’ thinking about how they could model this pathway with a mathematical function and why they might want to do so. Where is the projectile’s height zero? How can that information help find the function’s equation? Can we ensure that the projectile is on an accurate path before firing a single shot?

I encourage teachers to make the connections to multiplying the linear factors of –x and x – 4 or x and –x + 4 (see previous post). Doing so helps set the stage for when students are given the quadratic function and asked to solve it.

Last, teachers and students can use graphing applications, such as GeoGebra, to explore and analyze the function created from the pathway of the projectile (See the figure below).

This simulation itself does not help students with factoring or with solving quadratic functions, but it can help provide some rationale for why anyone would want to do these things—to be sure to hit the target! Admittedly, this is an oversimplification of projectile motion, but using it as a short opening activity to provide some context, motivation, and rationale might help some students make connections between the algebraic manipulations of the equations and what they stand for. Students can start to see why the zeros are important, thus providing some motivation for factoring quadratic equations.

BARBARA A. SWARTZ, bswartz@mcdaniel.edu, is an assistant professor of mathematics education at McDaniel College in Westminster, Maryland. She is interested in mathematics teacher education and teaches mathematics courses for prospective elementary and secondary school mathematics methods.