L e s s o n 5 (MATHEMATICS) 71 J o u r n e y s i n F i l m : H i d d e n F i g u r e s The Math of Space Travel: Orbits and Conic Sections Enduring Understandings • In many cases, the orbits of planets and of spacecraft can be described as ellipses or hyperbolas. • Circles and ellipses are related by scaling. Circles, ellipses, parabolas, and hyperbolas can be generated by slicing certain 3D figures. • Distances in space are enormous, far beyond everyday human experience. Scientific notation is an effective way to make calculations with the large numbers inherent to orbital calculations. • Essential Questions • How can the orbits of planets and manmade satellites be represented mathematically? What physical representations can we make of orbital paths? • How do dilations and scale factors relate geometric shapes to each other and also enable the visualization of large distances? • How can the large numbers inherent in orbital calculations be handled in an efficient manner? Notes to the Teacher Mathematics is at the heart of the film Hidden Figures . It is mathematics that supports the ambitions of the three principal characters, mathematics that fills their days in West Computing, and mathematics that brings John Glenn and later astronauts back from their missions. Calculating orbits was particularly important, and so was the ability to handle immensely large numbers in a manageable way. This lesson gives students the opportunity to strengthen their skills in both calculating orbits and managing huge numbers. Lesson 5 begins with an investigation of calculations that involve exponents, and from there leads students to apply those rules to calculations involving large numbers. The intention is that students will see the need for scientific notation when dealing with astronomical distances and develop an intuitive understanding of the notation through several mental arithmetic exercises. Exponential arithmetic and scientific notation are together in the same lesson in order to highlight the connections between the two. When students struggle with scientific notation, it is often because they have learned exponential arithmetic by rote, and because they have not been led to see the connections between scientific notation and exponents. The first part of the lesson is a sequence of problems that lead the students through some (but not all) of the calculation rules for exponential expressions. Scientific notation is nominally a middle school standard in most schools. However, discomfort with exponents, and in particular with scientific notation, is widespread at all ages. Scientific notation is used
72 J o u r n e y s i n F i l m : H i d d e n F i g u r e s The second part of the lesson is designed as a “from first- principles” investigation of ellipses and how they relate to German mathematician and astronomer Johannes Kepler’s first and second laws. The goal in many cases is not to construct a mathematical proof of the claims made, but only to examine why they might be reasonable. For a more
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- Winter '17
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