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Taking Polynomials For a Ride

When roller coaster designers plan rides, they use polynomial functions to design steep rises, stomach-churning drops, and the points where a coaster dips below ground to fly through a dark tunnel. For their final winter term project, Algebra 2 students designed their own roller coasters by incorporating all they had learned about third-degree or higher polynomials this term. 

Students began by working in groups to solve basic roller coaster-related polynomial problems like graphing the polynomial function for the height of a roller coaster after a given amount of time and then finding its height in feet at different points in the ride. Once they had some practice graphing the polynomials, finding the maxima and minima of a function, and finding the zeros for each function, they were ready to start their design challenge.  Each student was tasked with designing their own coaster and graphing it in Desmos, an online graphing calculator. They were also asked to produce a hand-drawn side view of the ride. Each design had to include at least four functions, two of which are degree three or higher, could include circles, ellipses, or trigonometric functions. The coaster also was required to go below ground level at least twice. With their final design, students had to identify all of the local maxima and minima, state the domain and range of each function’s part of the roller coaster, find the zeros using two specific strategies, and ensure that their ride starts and ends at the same point. 

While both classes are composed of talented math students, this project challenged these advanced mathematicians in various ways. “The open-ended nature of the assignment combined with the fact that they are using a relatively new function was challenging,” notes Algebra 2 teacher and Math Department Chair Cassandra Papalilo. In math, students are often given the problem and the strategy they should use to solve it. This project required students to incorporate some of the skills that they have learned in design thinking to problem solve and change strategies when needed throughout. “They were discovering the limitations of the methods they had learned in the book, and it was a cool lesson that sometimes you need to find another strategy to fit specific criteria,” says Math Teacher Sam Poland, who teaches the second section of Algebra 2. Unlike the problems they had encountered in their book, students also quickly discovered that not all of the calculations in their roller coaster design would come out to nice whole numbers. They had to get comfortable with incorporating decimals and fractions in their calculations. Finally, students had the creative challenge of designing a structure that incorporated all the required elements and was also realistic enough that it could plausibly exist at an amusement park somewhere in the world.
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