The Infinite Bender Series Problem

thingiverse

In a thought-provoking episode of Futurama, Bender creates an infinite series of himself with decreasing mass, prompting Professor Farnsworth to worry about the Benders taking over Earth. In this engaging activity, students delve into the formula presented in the show and develop a more realistic equation to model the situation. They design robots using geometric shapes and apply the geometric series sum formula to demonstrate that the Bender copies converge. The clip can be found at http://www.cc.com/video-clips/enlpp8/futurama-infinite-benders. Print Settings: Printer Brand: MakerBot Printer: MakerBot Replicator Rafts: Yes Supports: Doesn't Matter Resolution: Standard Infill: 10% How I Designed This: I utilized Tinkercad to design a robot using various geometric shapes. Students can be creative with this aspect and make their robot as simple or complex as they desire. They can attempt to replicate Bender or create an object that resembles a robot. If students do not wish to spend time on the design portion of this lesson, there are figures of Bender available on Thingiverse that they can use and then modify the mass. Once I designed the robot, I placed it in Makerbot Desktop and experimented with scaling until I achieved the desired mass. I conducted this step without any rafts or supports; however, for the actual printing process, I added both. Depending on the shape of your robot, you may require supports. Overview and Background: Students will explore infinite series and the concepts of convergence and divergence. The formula from the television show will diverge, but it does not accurately represent the situation. Students will develop their own formula and demonstrate that it is a geometric series that converges. Objectives: * Students will gain practice in determining whether an infinite series converges or diverges in a real-world scenario. * Students will devise a model that accurately describes a situation. Skills Learned (Standards): * Students will become proficient in identifying whether an infinite series converges or diverges. * Students will develop a model that accurately represents a situation. Lesson Plan and Activity: First, students are presented with a clip from Futurama where they witness an infinite series. Then, students demonstrate that this series diverges but does not accurately depict the situation from the show. Next, students create their own model that is more accurate. Students form groups and each group selects one term in the series to design and print two robots that together have that mass. Finally, students analyze their model to verify that the mass of the Benders actually converges. Duration of Lesson: 2-3 days Preparation: Students should possess knowledge about infinite series and how to determine whether they diverge or converge. They should be familiar with geometric and harmonic series. Rubric and Assessment: Here is a sample rubric for use with the worksheet. Question (points) 1 (5) 2 (10) 3 (5) 4 (5) Total: 25 References: If students wish to learn or refresh their knowledge on infinite series, they can watch the Khan Academy videos on this topic. https://www.khanacademy.org/math/precalculus/seq-induction/infinite-geometric-series/v/deriving-geometric-series-sum-formula