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Course: Technology Education Submitted By: Tracy Hoskins, CATE TECHED, Tulsa Public Schools Project Name: Discovery of Neptune Situation or Problem: Developments in mathematics often stimulate innovations in science and technology. Students will explain why the development of scientific knowledge depends on mathematical concepts such as the use of measurement, symbols, estimation, accuracy, and the idea of scaling and proportion. They will describe the role of technology in the confirmation of mathematical predictions. Students will address open-ended questions in verbal and written form. This activity will allow them to contribute to a group endeavor by offering useful ideas, supporting the efforts of others, and focusing on the task. They will actively participate in group discussions, ideation exercises, and debates; demonstrate curiosity, exhibit motivation for learning, and use class time effectively; and, exhibit and refine inherent personal qualities such as creativity and resourcefulness. Project Description and Purpose: To familiarize students with the relationship between technology, mathematics, and science by examining mathematical concepts, scientific laws, and the process of innovation. Performance Specifications: Technology Standards 1. Students will develop an understanding of the relationships among technologies and connections with other fields of study. (ITEA-STL 3) 2. Students will develop an understanding of the nature of mathematics. (AAAS-BSL) 3. Students will develop an understanding of the nature of technology. (AAAS-BSL) Mathematics Standards – Unit 2 and Unit 4 4. Students will develop an understanding of mathematics processes. (Process Standard Connections 4.1) (NCTM) 5. Students will develop an understanding of mathematics processes. (Process Standard Communications 2.1 and 2.3) (NCTM) 6. The student will use measurement to solve problems in a variety of contexts. (4.3a) 7. The student will use data analysis and statistics to interpret data in a variety of contexts. (5.1 and 5.3) Language Arts Standards – Unit 1 and Unit 2 8. The student will interact with the words and concepts in a text to construct an appropriate meaning. (3.1a, 3.1b, 3.3a, 3.5a, 3.5c, 3.2a and 3.4a) 9. The student will conduct research and organize information. (5.2d) 10. The student will read, construct meaning, and respond to a wide variety of literary forms. (4.2b) Project Based Activity – Technology Education – Discovery of Neptune - 1 Instructions and Parameters: 1. Explain why the development of scientific knowledge depends on mathematical concepts such as the use of measurement, symbols, estimation, accuracy, and the idea of scaling and proportion. 2. Describe the role of technology in the confirmation of mathematical predictions. 3. Explain that developments in mathematics often stimulate innovations in science and technology. 4. Address open-ended questions in verbal and written form. 5. Contribute to a group endeavor by offering useful ideas, supporting the efforts of others, and focusing on the task. 6. Actively participate in group discussions, ideation exercises, and debates. 7. Demonstrate curiosity, exhibit motivation for learning, and use class time effectively. 8. Exhibit and refine inherent personal qualities such as creativity and resourcefulness. Engagement Engaging question – How do you think the planet Neptune could be discovered using a mathematical process instead of the use of a telescope? 1. The teacher will assess prior knowledge and possible misconceptions related to the relationship between technology, mathematics, and science. 2. The teacher will show students a picture of our solar system and then a picture of the planet Neptune. 3. The teacher will state that the planet Neptune was discovered through the application of mathematical processes and not the use of a telescope. Exploration 1. The teacher will explain that a vignette is a presentation of material providing a description of people, circumstances, processes and/or events that represent an illustrative example within a particular area or topic of study. 2. Students will read “The Discovery of Neptune” vignette found at the end of this lesson. 3. Students will complete the exercises at the end of the reading. Explanation 1. The students will present their answers for the exercise and state their questions. 2. The teacher will define perturbation as a disturbance in the regular elliptic or motion of a heavenly body, produced by some force additional to that which causes its regular motion, as the perturbations of the planets are caused by their attraction on each other. 3. The teacher will explain that: a. The study of perturbations has led to important discoveries in astronomy. b. Within the solar system, the existence and position of Neptune was predicted because of the deviations of Uranus from its computed path. c. Neptune was discovered in 1846, after being predicted by two mathematicians who found that Uranus was not moving in a perfectly elliptical path. d. By calculating the gravitational tug necessary to pull Uranus in its orbit, they were able to tell astronomers where to point their telescopes to discover Neptune. e. Likewise, Pluto was discovered by its effect on Neptune. Project Based Activity – Technology Education – Discovery of Neptune - 2 f. Beyond the confines of the solar system, perturbations in the orbits of stars caused by the gravitational forces of orbiting bodies have led to the discovery of a number of extra solar planetary systems. 4. Students will restate any questions that were not answered in the teacher’s lecture and clarify any unresolved issues. Lesson 1. Students will read the vignette “Gravitational Perturbations and the Prediction of New Planets” and complete the exercise at the end. 2. The teacher will review the exercise questions with the class. Evaluation Students’ knowledge, skills, and attitudes will be assessed using selected response items and rubrics for class participation, group work, brief constructed responses, and extended constructed responses summarizing the lesson. The rubrics will be presented in advance of the activities to familiarize students with the expectations and performance criteria. They will also be reviewed during the activities to guide students in the completion of assignments. The teacher may wish to develop a collection of annotated exemplars of student work based on the rubrics. The exemplars will serve as benchmarks for future assessments and may be used to familiarize students with the criteria for assessment. Project Based Activity – Technology Education – Discovery of Neptune - 3 Vignette 1 The Discovery of the New Planet Neptune In 1846, the planet Neptune was discovered after its existence was predicted because of discrepancies between calculations and data for the planet Uranus. Astronomers found the new planet almost exactly at the position predicted by the calculations. The situation is illustrated schematically in the adjacent diagram. The dominant interaction between Uranus and the Sun is indicated with the heavy line, but some perturbations associated with other masses are indicated by thin lines. By using Newton’s laws to calculate the perturbations on the orbit of Uranus by a hypothesized new planet, mathematicians were able to predict where the planet had to be in order to cause the observed deviations in the position of Uranus. Once astronomers took this calculation seriously, they found the new planet within hours of turning their telescopes on the region of the sky implicated by the calculations. This precise prediction of the new planet and its location was striking confirmation of the power of Newton’s theory of gravitation. Later, similar calculations on supposed perturbations of the orbits of Uranus and Neptune suggested the presence of yet another planet beyond the orbit of Neptune. Eventually, in 1930, a new planet, Pluto, was discovered, but we now know that the calculations in this case were also in error because of an incorrect assumption about the mass of the new planet. It is now believed that the supposed deviations in the orbits of Neptune and Uranus were errors in measurement because the actual properties of Pluto would not have accounted for the supposed perturbations. Thus, the discovery of Pluto was a kind of accident. Exercise 1. When was Neptune discovered? 2. What do you think the word perturbation means? 3. List three questions you have about this vignette. Project Based Activity – Technology Education – Discovery of Neptune - 4 Vignette 2 Gravitational Perturbations and the Prediction of New Planets Computing the orbit of the earth as an ellipse around the center of mass for the earthsun system assumes that they are the only two masses in the universe. In reality, the Universal Law of Gravitation implies that the earth interacts gravitationally not only with the sun, but with every other mass in the Universe: the moon, the other planets, asteroids and comets, the distant stars. The Two-Body Approximation However, from the form of the gravitational force, we see that the interactions are largest when two situations are fulfilled: 1. The product of the masses of the two objects is large, which maximizes the numerator of the expression for the strength of the gravitational force and, 2. The objects are near each other, which minimizes the denominator of the force equation. The two-body approximation that the orbits of the planets are determined only by the gravitational interaction between the sun and the planet is possible because: a. The sun is so massive compared with every other object in the solar system. b. Objects outside the solar system, such as stars, are so distant that the distance squared factor in the denominator renders their gravitational interactions with the planets negligible. For example, the sun is about 300,000 times more massive than the earth, and about 1000 times more massive than the largest planet. Thus, the product of the mass of a planet and the mass of the sun is always much larger than the product of the masses of any two planets, and it is a good initial approximation to neglect all interactions except that of the planet and the sun. Gravitational Perturbations However, the small deviations from this ideal picture have consequences if careful measurements are made. These small deviations from the simplified picture are called perturbations. They can be calculated systematically using Newton’s laws of motion and gravitation from the positions of the known masses in the solar system. If we account carefully for all known gravitational perturbations on the motion of observed planets, and the motion of the planet still deviates from the prediction, there are two options: 1. Newton’s Law of Gravitation requires modification. 2. There is a previously undetected mass that is perturbing the orbits of the observed planets. Exercise 1. What happens to the gravitational force if the distance between objects is increased? 2. The interactions between two bodies is greatest when what two situations are fulfilled? 3. Compare the products of the masses of the Sun and Mars with Project Based Activity – Technology Education – Discovery of Neptune - 5 the product of the masses of Mars and the earth. Project Based Activity – Technology Education – Discovery of Neptune - 6