The intended unit of study will explore quadratic functions and is meant to familiarize students with their graphs and the components that define a parabola: the vertex, y-intercept, and x-intercept(s). Students will utilize the regression features of the graphing calculator to determine the equation of a parabola given multiple points along its path. This graphical approach to quadratics is meant to solidify students' conceptual understanding of parabolas. In addition, students will hypothesize how changing features of a catapult affects the trajectory of the projectile and ultimately manipulate the coefficients of a quadratic function in standard form.
Many mathematics texts and resources prompt students to explore the effects of changing coefficients on the shape of a quadratic, but few resources place such a task within a tactile framework. In this unit, students are charged with the task of manipulating a catapult in order to launch a given object a certain distance and height. By analyzing the projectile's trajectory, students are encouraged to hypothesize about and experiment with how to alter the catapult's trajectory. To facilitate student understanding of the larger project, students will participate in mini-lessons designed to introduce keys concepts one at a time. Each of these mini-lessons follows the catapult theme so there is a direct relationship between the conceptual ideas and the manipulatives.
Students are introduced to the unit project by showing a clip from the BBC show Top Gear. The episode focuses on building a catapult-like device that can accurately launch a car through the air and hit a designated target. Students will be asked to describe the shape of the projectile's path. What are the key features on which we should focus? The class will then discuss some of the considerations engineers must make when attempting such a project.
Throughout the unit students are provided multiple examples of catapults to deepen their understanding of the types of questions engineers ask themselves when designing a catapult or to emphasize specific considerations that students should make in their own projects. Each example is meant to inspire students to look at their projects from a different perspective.
This unit is designed for implementation at Wilbur Cross High School (WCHS) and is to be used as an extension to the district's Algebra 1 and Algebra 2 curricula, which extensively study linear and quadratic functions, respectively. The initial application of this unit should occur in Algebra 1 as a way of introducing data collection methods and best-fit models. Initially, students will simply compare how altering arm length, the angle of release, and spring torsion affect the horizontal distance a projectile travels. Students will assess whether the data yields a linear relationship or some other type of relationship. Since the New Haven Algebra 1 curriculum focuses very narrowly on linear functions, students will be asked to assess linear relationships and the factors that indicate that the data are not linear. This will reinforce the concepts of linear functions while also foreshadowing non-linear topics of study in later courses like Algebra 2 and Pre-Calculus.
The second opportunity to implement this unit is in Algebra 2, when students begin to analyze quadratic functions and their graphs. Throughout this unit, students will again experiment with altering the catapult's arm length, the angle of release, and the spring torsion. In this application, however, students will focus on specific points along the trajectory path. First, students will experiment with drawing parabolas given only two points. Through analysis, students will discover that a third point is necessary to deduce the shape of a parabola. Students must hypothesize how to collect these three data points in an accurate manner. Ultimately, students will discover that they can measure the initial launch height, the horizontal distance traveled, and another point that will become known as the canyon point. Through analysis of the data, students will strive to draw conclusions about how altering the three components
WCHS is the largest comprehensive high school in the New Haven Public School district and serves one of the most diverse populations in the city. The school's composition is approximately 89% minority and 72% economically disadvantaged. In 2011, approximately 50% of the 2007 freshman class actually graduated. In an article written by Marian Edelman for change.org about student dropouts, Edelman claims that many of the students that choose to drop out can be spotted as early as fourth to sixth grade and asserts that one of the indicators (in addition to attendance and behavior) is student performance in math class.
This unit is designed with the specific aim of engaging students at a new level that will hopefully encourage them to take more interest in their mathematical studies.