National Aeronautics and Space Administration andSpace Aeronautics National ilk y W a y Exploring the M Exploring the Milky Way H Exploring the Milky Way Galaxy 1 Math textbooks routinely provide ‘real world’ examples for students using familiar examples drawn from every-day situations, but there are many other areas where proportional relationships and working with large numbers aid in understanding our physical world. This collection of activities is intended for students looking for additional challenges in the math and physical science curriculum in grades 6 through 8, but where the topics are drawn from astronomy and space science. As a science extension, ‘Exploring the Milky Way’ introduces students to mapping the shape of the Milky Way galaxy, and how to identify the various kinds of galaxies in our universe. Students also learn about the shapes and sizes of other galaxies in our universe as they learn how to classify them. The math problems cover basic scientific notation skills and how they apply to working with ‘astronomically large’ numbers. It also provides exercises in plotting points on a Cartesian plane to map the various features of our Milky Way. For more weekly classroom activities about astronomy and space visit the NASA website, http://spacemath.gsfc.nasa.gov Add your email address to our mailing list by contacting Dr. Sten Odenwald [email protected] Original cartoon artwork kindly provided by Nathalie Rattner (Copyright 2012). Ms. Rattner is a highly versatile, award winning Canadian artist whose works have been sold to private collections in North America. Throughout her career, she has been inspired by her love of life, history and the diversity of the world around her. The artwork used in this book was created by her when she was in middle school ! 2 Exploring the Milky Way Galaxy NCTM: Principles and Standards for School Mathematics Grade 6-8 : In grades 6–8 all students should Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation; model and solve contextualized problems using various representations, such as graphs, tables, and equations. make and use coordinate systems to specify locations and to describe paths Common Core Mathematics Standards 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.6.b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms (including scientific notation) as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 8.EE.3 - Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. 8.EE.4 - Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Exploring the Milky Way Galaxy 3 4 Exploring the Milky Way Galaxy We all know a little bit about our nearest star – the Sun! It’s a yellow-colored star, which means it is hotter than ‘red hot’ and sizzles with a temperature of 6,000 Celsius. It also contains a LOT of matter. In fact, there is so much matter in the sun that scientists use ‘scientific notation’ to write this amount numerically. Here’s what it looks like in ordinary decimal form: 1,989,000,000,000,000,000,000,000,000,000 kg ! Review of Scientific Notation: A decimal number is written as 1,230,000.0 or ‘one million, two hundred thirty thousand’. To write it in scientific notation, ask yourself, how many times do I have to multiply 1.23 by 10 to make it into 1,230,000? Your answer would be 6 times. So 10x10x10x10x10x10 = 1,000,000 and 1.23 x 1,000,000 = 1,230,000. We can write this using base-ten exponents so we have 6 1,000,000 = 10 , and then 6 1,230,000 = 1.23x10 . So in scientific notation, how would you write 456.1 ? 2 Answer: 4.561x10 How would you write the mass of the sun in kilograms shown in the decimal form above? Exploring the Milky Way Galaxy 5 Addition and Subtraction: OK, now that you recall the basics of re- writing numbers in scientific notation, let’s review how to manipulate them using the basic operations of arithmetic: Addition, Subtraction, Multiplication and Division. Suppose I gave you the numbers 1.564x107 and 3.41x105 and asked you to add them together. The easiest thing to do is to take the largest number and re- write it to the same exponent as the smaller number: 5 5 156.4x10 and 3.41x10 . Now add the numbers together to get 156.4 + 3.41 = 159.81 5 Put the exponent term back into the number 159.81x10 7 Then re-write it in scientific notation as 1.5981x10 . You can do the same steps for subtraction! Let’s try these two problems: 1) Our sun has a mass of 1.989x1030 kg and the nearby dwarf star Proxima Centauri has a mass of 2.45x1029 kg. What is the difference in masses of these two stars? 2) The proton has a mass of 1.67262158 × 10-27 kilograms and an electron has a mass of 9.1094 × 10-31 kilograms. What is the mass of a hydrogen atom that consists of one electron and one proton? 6 Exploring the Milky Way Galaxy Multiplication: OK, now how about this. Suppose I gave you two HUGE numbers and asked you to multiply them. You have a galaxy like the Milky Way that has 250 billion stars, and 30 each star has a mass of 2.0x10 kilograms. What is the total mass of the galaxy in kilograms? Remember that when you multiply two numbers in scientific notation, first you multiply the two decimal numbers (2.5 x 2.0 = 5.0, then you add the exponents (30+11 = 41) …like this: 11 30 2.5x10 x 2.0 x 10 = 30+11 41 (2.5 x 2.0) x 10 = 5.0 x 10 kilograms. What is the age of the universe in seconds expressed as (3.1 x 7 10 10 seconds/year) x (1.37x10 years)? Division: Now suppose that you wanted to divide two numbers 33 -24 3.2x10 and 1.6x10 . First you divide the two decimals 3.2/1.6 to get 2.0, then you subtract the exponents to get 33 - (-24) = 54 +57 and so the answer is 2.0x10 . How many hydrogen atoms are there in a star expressed as 33 -24 (2.0x10 grams) / (1.6x10 grams/atom)? Exploring the Milky Way Galaxy 7 More Review Problems Convert these numbers into Scientific Notation: 1) Length of a year. 31,560,000.0 seconds = 2) Power output of sun : 382,700,000,000,000,000,000,000,000 watts = 3) Mass of an electron: 0.00000000000000000000000000000091096 kilograms = 4) Radius of hydrogen atom : 0.00000000529177 centimeters = Write these numbers in Standard Form: 11 5) Stars in the Milky Way: 2.5x10 = 21 6) Number of stars in the universe: 9.0x10 = -25 7) Mass of an atom of gold in kilograms: 3.27x10 = Add or subtract these numbers: 14 13 8) 1.34 x 10 stars + 1.3 x 10 stars = 25 23 24 9) 7.523 x 10 kg - 6.30 x 10 kg + 1.34 x 10 kg = Perform these calculations: 10) Tons of TNT needed to make crater 100 km across: 15 3 6 3 9 (4.0 x 10 Joules/m ) x (1.0 x 10 m ) /(4.2 x 10 Joules/ton) = 11) Number of sun-like stars within 200 light years of our solar system: -3 7 (2.0 x 10 stars per cubic light year) x 3.35 x 10 cubic light years = 8 Exploring the Milky Way Galaxy Exploring the Milky Way Galaxy 9 The sun is very far away too! Astronomers have many different ways to measure its distance from Earth. From the sun to Earth it is a distance of 93 million miles. This is also the same as 150 million kilometers if you prefer metric units …which scientists do! Once again, the distance to the sun, 150,000,000 kilometers, is 8 more easily written in scientific notation as 1.5 x 10 kilometers.