October Sky is a film based on the book Rocket Boys that chronicles the childhood and teen years of Homer Hickam, who grew up in a rural West Virginia town where education beyond secondary school was rare. Because of the launch of the Russian rocket Sputnik in the 1950s, Hickam, who was a child then, became inspired to build rockets. He began building rockets with five friends, and they were so successful they went on to win a prestigious award at the 1960 National Science Fair.
In October Sky, science is clearly important because the boys theorize, build, experiment, optimize, theorize, rebuild—following scientific principles of experimentation. They apply mathematical principles to their endeavors. It was particularly important, for example, that they (or at least one of them) learn enough calculus to be able to make the rocket go fast. Without calculus, they would not have been able to calculate shape and angle necessary for maximum thrust and efficiency. They also needed to calculate nozzle-throat area and thrust coefficient.
This film (and the more detailed memoir Rocket Boys) is a great example of how experimentation and math interacted to produce groundbreaking science. The boys had to go beyond their normal high school math class to learn advanced math, or they would not have built a successful rocket.