Distance Measurements and Scientific Notation - Essay Example

Roach, D., Gibson, D., & Weiber, K. (2004). Why is 25 not  5? The Mathematics Teacher, 97 12–13. TOPIC: DISTANCE MEASUREMENTS AND SCIENTIFICNOTATION GRADE LEVEL: 9-12 AIM: To enable students to write and solve equations OBJECTIVES: 1. To explain how to write and solve equations that relate distance measurements to scales representations of distances; and 2. To expound on how to use the scientific notation to make calculations more manageable. PRIOR KNOWLEDGE: Students are expected to: Understand how scaling factors can be used to make representations of astronomical distances; Explain and apply basic and advanced properties of the concepts of numbers, e.g. Understands the properties and basic theorems of roots, exponents (e.g., [bm][bn] = bm+n), and logarithms; Understands and applies basic and advanced properties of the concepts of measurement. MATERIALS / EQUIPMENT: To introduce innovation into teaching methodology, the teacher recommends the use of the following materials / equipment: Calculators Pencils and paper Ruler Published materials about the universe, such as books, magazines, and the Internet Class Activity Sheet on Understanding Sizes and Distances in the Universe Copies of the assignment on Comparing the Sizes of Planets VOCABULARY: Astronomical unit. A unit of length used in astronomy equal to the mean distance of Earth from the sun, or about 93 million miles (150 million kilometers). light-year Light year. The distance that light travels in one year in a vacuum, or about 5.88 trillion miles (9.46 trillion kilometers). Parallax. The angular difference in the direction of a celestial body as measured from two points in Earth’s orbit. Scaling factor. The proportion between two sets of dimensions. DO NOW / START UP TASK: Start by discussing how vast the universe is. For instance, explain to students that light travels at an unbelievably fast speed of 300 million meters per second, and yet light takes years to travel to us from the stars and takes thousands or even millions of years to travel the depths of space between galaxies. When these kinds of distances are dealt with, it’s no wonder that we often think of them as being beyond our grasp. One way to put these distances into perspective is to think of them as multiples of smaller-scale distances. In such a context, they will take on more meaning. MOTIVATION: Help students grasp our place in the enormous universe by reviewing your school’s “galactic address”—beginning with its street address and ending with its place in the universe. Review the units of measurement that are used to represent distances in each part of the galactic address. Give students examples for each step, or have them use reference materials to provide their own examples. Help them appreciate unfamiliar units of measurement, such as light-years and astronomical units. By thinking about their location on a small scale first and then moving out to a much larger scale, students begin to learn a sense of how distance is measured at each scale. This understanding will motivate them to learn through the lesson further. DEVELOPMENT AND INSTRUCTIONAL ACTIVITIES: 1. Tell students that one means of putting these unimaginable sizes and distances into perspective is to compare them to smaller scales that are easier to understand. In this activity, students will convert and sizes in space to smaller units. Distribute the Class Activity Sheet: Understanding Sizes and Distances in the Universe. Instruct students to work in pairs to answer the questions. 2. To assist students in understanding the solution to these problems, you may wish to solve the following problem as a class: Problem: Using a scale in which a quarter represents Earth, what would the distance from Earth to the moon be? Solution: Three pieces of information are needed in order to determine this scale distance to the moon: the diameter of the quarter, Earth’s diameter, and the actual distance from Earth to the moon. Measuring the quarter reveals that it has a diameter of 1 inch. Earth’s diameter is about 8,000 miles. The actual distance from Earth to the moon is an average of 240,000 miles, although this distance can vary with the moon’s orbit around Earth. For these calculations, though, the average can be used. Now that we have these three pieces of information, we can find the fourth piece (the scale distance d) by setting up the following ratio: A diameter of quarter = scale distance (d) Diameter of the Earth average Earth-Moon distance This is equivalent to the statement “The diameter of a quarter is to Earth’s diameter as our scale distance is to the actual average Earth-moon distance.” 1 inch = (d) 8,000 miles 24,000 miles It’s imperative to keep track of the units. If the Earth’s diameter had been given in kilometers, it would be incorrect to use 240,000 miles for the Earth-moon distance. We would need to convert that distance to kilometers, too. Because both diameters are given in miles, they cancel each other and can be crossed out of the equation. In this problem, we should expect our result to be in inches, the same unit as the quarter’s diameter. By multiplying both sides of the equation by 240,000 miles to isolate d, we find that d = (240,000 miles) × (1 inch/8,000 miles) = 30 inches So, at this scale, the distance between Earth and the moon would be 30 inches. 3. Before students start working on the problems, it may be useful to go over scientific notation, which is a helpful way to deal with large numbers. Give the following examples to illustrate the powers of 10: 1 can be written as 100 (because anything to the power zero is 1). 10 can be written as 101 (because anything to the first power is itself). 100 can be written as 102 (because 10 multiplied by itself, or 10 × 10, equals 100). 1,000 can be written as 103 (because 10 multiplied three times, or 10 × 10 × 10, equals 1,000). Explain that we can use these powers of 10 to represent decimal places, too: 3.4 can be written as 3.4 × 100. 99.1 can be written as 9.9 × 101. 4,526 can be written as 4.526 × 103. Review the properties of exponents to make scientific notation even more useful: When multiplying two numbers with exponents, if the base numbers are the same, just add the exponents. For example, 105 × 103 = 108. When dividing two numbers with exponents, if the base numbers are the same, subtract the exponents. For example, 104/102 = 102. 4. Have each pair of students solve the problems listed below, which also appear on the Classroom Activity Sheet: Understanding Sizes and Distances in the Universe. You may wish to provide helpful measurement tips to help them solve these problems. a. If Earth were the size of a penny how large would the sun be? (81 inches, or 6.7 feet, in diameter) b. how far away would the sun be? (8718.75 inches, 726.5 feet, 242 yards) c. What is located about that distance from your classroom? (Answers will vary.) If the sun were the size of a basketball how far away would Neptune be from the sun? (3237 feet, or 0.6 miles) how far away would the nearest star, Proxima Centauri, be from the sun? (5,538 miles) Find two places on a world map that are about this distance apart. (Answers will vary.) how far would it be to the center of the Milky Way? (36,538,218 miles) About how many trips to the moon does this distance equal? (152) If the Milky Way were the size of a football field how far away would the Andromeda galaxy be? (6,600 feet, or 1.25 miles) how far would it be to the farthest known galaxy? (39 million feet, or 7,386 miles) Find two places on a world map that are about this distance apart. (Answers will vary.) HOMEWORK: Have them accomplish the assignment: Comparing the Sizes of Planets for homework. If time permits, discuss the answers in class. You may have them illustrate the planets to scale to compare the sizes of different planets visually ASSESSMENT: The effectiveness of teaching may be evaluated using a 3-point scale, as follows: 3 points – The class actively participates in the discussion and activities. There is apparent cooperation within and among groups for the completion of the Class Activity Sheet. More than three questions were answered correctly. 2 points – The class participates in discussions and activities to a significant extent. There is observable cooperation within and among groups to complete the Class Activity Sheet. Exactly three questions were answered correctly. 1 point – There is slight or weak participation in classroom discussions and activities. There is some attempt to cooperate for the completion of the Class Activity Sheet. One problem has been solved correctly. SUMMARY: 1. Considering that the more distant an object is, the smaller the angle it will make, why would parallax measurements be better suited for stars than for galaxies? 2. What is the value of using exponents? Give some examples of when they are commonly used. 3. Explain why it would be impossible for scientists to measure stellar distances that are accurate to within a few feet. Why is it not critical to attain such accuracy when dealing with astronomical distances? 4. Does knowing how to use a scale on a map help you understand how to use scale to measure distances in the universe? How are they similar? How are they different? 5. Explain how you could measure the height of a mountain without having to climb it. Through the approach presented in the lesson plan, students will appreciate the mathematics learning process more. The examples are more innovative, and are expected to spur more excitement from them. Moreover, doing the exercises in dyads is a good first step at encouraging them to solve problems. Solving problems as a class also helps ease their fear of failure, and encourages them to take on complex problem solving tasks. The clear explanation of vocabulary will promote consistent and thorough mathematical expression even at their stage. Finally, the exercises and the homework are both designed to emphasize both critical and mathematical and logical thinking. Read More