




912 > Mathematics 










Grade level: 912 

Subject: Mathematics 

Duration: One to two class periods 
















Students will
1.

understand how scaling factors can be used to make representations of astronomical distances; 
2.

learn how to write and solve equations that relate real distance measurements to scaled representations of the distances; and 
3.

understand how the use of scientific notation makes calculations involving large numbers easier to manage. 










The class will need the following:
• 
Calculators 
• 
Pencils and paper 
• 
Ruler 
• 
Reference materials about the universe, such as books, magazines, and the Internet 
• 
Copies of Classroom Activity Sheet: Understanding Sizes and Distances in the Universe 
• 
Copies of TakeHome Activity Sheet: Comparing the Sizes of Planets 










1.

Begin by discussing the vastness of the universe. For example, tell students that light travels at the unimaginably fast speed of 300 million meters per second, and yet light takes years to travel to us from the stars and takes thousands or evenmillionsof years to travel the depths of space between galaxies. When we’re dealing with those kinds of distances, 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 smallerscale distances. By putting these quantities in the context of a wellunderstood frame of reference, they begin to have more meaning. 
2.

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. Discuss the different units of measurement that are used to describe 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. Review any unfamiliar units of measurement, such as lightyears 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 get a sense of how distance is measured at each scale.
Place

Units of measurement

Example

Street address 
Feet, meters (within a house) 
A room might be 10 × 14 feet. 
City 
Miles, fractions of miles 
You might drive ½ mile to the grocery store; a town might be about 10 miles wide. 
State 
Tens to hundreds of miles 
The distance from Austin to San Antonio is a little more than 50 miles; Texas is about 600 miles across. 
United States 
Hundreds to thousands of miles 
The distance from New York to Los Angeles is 3,000 miles. 
Earth 
Tens of thousands of miles. 
Earth’s circumference is 25,000 miles. 
Solar System 
Millions to billions of miles, or astronomical units (AU). (An AU is the average distance from Earth to the sun, or 93 million miles.) 
Neptune is 30 AU, or 2.79 billion miles, from the sun. 
Milky Way Galaxy 
Hundreds of thousands of lightyears. (A lightyear is the distance that light travels in one year, or about 6 trillion miles.) 
The Milky Way is about 100,000 lightyears across. 
Local Group (a cluster of about 20 galaxies, including the Milky Way) 
Millions of lightyears 
The Andromeda galaxy is about 2.2 million lightyears away from our Milky Way galaxy. 
Supercluster (a group of clusters) 
Hundreds of millions of lightyears 
The Virgo supercluster of galaxies is about 150 million lightyears across. 
Universe 
Billions of lightyears 
The farthest known galaxy (the edge of the observable universe) is 13 billion lightyears away. 

3.

Explain that one way to put the enormous sizes and distances of space into perspective is to compare them to smaller scales that are easier to grasp. In this activity, students will convert distances and sizes in space to smaller units. To begin, distribute the Classroom Activity Sheet: Understanding Sizes and Distances in the Universe, and have students work in pairs to answer the questions. 
4.

To help students understand how to solve these problems, you may wish to do the following problem together 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 distanced) by setting up the following ratio:
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 Earthmoon distance.”Substituting what we know shows us that:
Remember that it’s important to keep track of the units. If Earth’s diameter had been given in kilometers, it would be incorrect to use 240,000 miles for the Earthmoon 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 isolated, we find thatd = (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.

5.

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. Use the following examples to illustrate the powers of 10:
 1 can be written as 10^{0}(because anything to the power zero is 1).
 10 can be written as 10^{1}(because anything to the first power is itself).
 100 can be written as 10^{2}(because 10 multiplied by itself, or 10 × 10, equals 100).
 1,000 can be written as 10^{3}(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 × 10^{0}.
 99.1 can be written as 9.9 × 10^{1}.
 4,526 can be written as 4.526 × 10^{3}.
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, 10^{5}× 10^{3}= 10^{8}.
 When dividing two numbers with exponents, if the base numbers are the same, subtract the exponents. For example, 10^{4}/10^{2}= 10^{2}.

6.

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. Also included for students are constants that provide helpful information to be used in scaling. Students must figure out which information is needed to solve each problem. Students can work with partners to solve the problems, but each student should fill out his or her own sheet. All the questions from the Classroom Activity Sheet and the answers are listed below.
Questions on the Classroom Activity Sheet: Understanding Sizes and Distances in the Universe If Earth were the size of a penny
 how large would the sun be?(81 inches, or 6.7 feet, in diameter)
 how far away would the sun be?(8718.75 inches, 726.5 feet, 242 yards)
 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.)
Helpful Measurements
 A penny is about ¾ inch in diameter.
 Earth is 8,000 miles across.
 The sun has a diameter of 861,000 miles.
 One mile equals 5,280 feet.
 The average distance from Earth to the sun is 93 million miles.
 A basketball is roughly 12 inches in diameter.
 Neptune is 30 AU from the sun, or 2.79 billion miles.
 One lightyear is 6 trillion miles.
 The nearest star, Proxima Centauri, is 4.2 lightyears away.
 The sun’s distance from the center of the Milky Way is about 30,000 lightyears.
 A football field is 100 yards (300 feet) long.
 The Milky Way is about 100,000 lightyears across.
 The distance to the Andromeda galaxy is 2.2 × 10^{6}lightyears.
 The farthest known galaxy is 13 billion lightyears away.

7.

Assign the TakeHome Activity Sheet: Comparing the Sizes of Planets for homework. If time permits, discuss the answers in class. You could have students draw the planets to scale to compare the sizes of different planets visually. 






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Complete the Classroom Activity Sheet as a wholeclass project. Instead of completing all seven problems focus on four or five. Students may also enjoy completing the TakeHome Activity Sheet together as a class.






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1.

Parallax is the apparent change in position of an object when it’s viewed from two different places. Astronomers use this phenomenon to measure the distances to some stars. They assume that the stars are fixed, and as the Earth moves in orbit they take measurements of the apparent shift in position. Then they calculate the distance based on a trigonometric relationship between the parallax angle and the “baseline” (the radius of Earth’s orbit). 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.(Exponents are used to express and calculate large numbers. For example, if you needed to multiply 1,000,000,000,000,000 by 1,000,000,000, instead of dealing with all those zeros, you could write the equation as 10^{15}X 10^{9}and add the exponents to get the answer, 10^{24}. Exponents are also used in business to express large sums of money and in science to express pH levels, the magnitude of earthquakes, and the brightness of stars.)

3.

Vast distances in space are often measured in lightyears. A lightyear is the distance that light travels in one year, or about 6 trillion miles. Altair, a star in the constellation Aquila, is 16.6 lightyears away, which means that the light we see now from that star left its surface 16 years and 219 days ago. Describe what was happening in the world when the light we are seeing from Altair first left that star. How far away is Altair in miles? 
4.

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? 
5.

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? 
6.

Describe how you could measure the height of a mountain without having to climb it. (Hint: Imagine that you’re standing 10 miles from the base of the mountain.) 






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You can evaluate students using the following threepoint rubric:

Three points:active participation in classroom discussions; cooperative work within groups to complete the Classroom Activity Sheet; ability to answer more than three questions correctly

Two points:some degree of participation in classroom discussions; somewhat cooperative work within groups to complete the Classroom Activity Sheet; ability to answer three questions correctly

One point:small amount of participation in classroom discussions; attempt to work cooperatively to complete the Classroom Activity Sheet; ability to solve one problem correctly






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How Many Miles Are There in a Light Year?
Tell students that we know that light travels at a speed of 186,000 miles per second. Have them use that information to figure out how many miles are in a lightyear. Then have them try to figure out a scaling factor to help make the large distance easier to understand.
Answer: Multiply the speed of light by the length of time in one year. The tricky part, however, is that the speed of light is expressed in miles per second. Because you have to keep the units the same, you must figure out the length of time in one year in seconds. To do that, multiply the number of days in one year (365) times the number of hours in one day (24) times the number of minutes in one hour (60) times the number of seconds in one minute (60). The answer is 31,536,000
seconds. To find out how many miles are in a lightyear, multiply 186,000 miles per second by 31,536,000 seconds. The answer is about 6 trillion miles. To understand what this means, think of Earth, which has a diameter of 8,000 miles, as being the size of a pea, which has a diameter of ¼ inch. Using that scale, a lightyear would be about the distance across the United States.
Selecting the Right Units
Explain to students that establishing the most appropriate unit of measurement is critical when it comes to measuring distances. The right choice of unit can make calculations simpler and help ensure accuracy. Have them imagine trying to measure a mountain in centimeters or an ant in miles. Ask students to research the items in the list below and arrange them in order of scale from smallest to largest. Then have them write down, next to each item, the unit best suited for measuring it. When the list is complete, have the class discuss the tools and procedures that could be used to make each measurement.
 an electron
 the Olympus Mons (a volcano on Mars)
 the Amazon River
 the Great Red Spot (a hurricanelike feature in Jupiter’s atmosphere)
 a #2 pencil
 Mt. Everest
 a school flagpole
 the Ring Nebula (estimated to be ½ lightyear across)
 an EP3E reconnaissance plane
 Ganymede (the largest moon of Jupiter)
 the Empire State Building
 a grain of wild rice






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The Astronomy Café: 365 Questions and Answers from "Ask the Astronomer"
Sten Odenwald. W. H. Freeman, 1998. Using a questionandanswer format, the author covers a wide range of information about space. Each chapter is preceded by a short description of the questions most people have about astronomy. Many questions relate to the distances between the stars and planets and their relative sizes. A few color plates, along with charts and graphs, add to the text.
Milestones of Science
Curt Suple. National Geographic, 2000. Filled with the luscious photographs that are National Geographic's trademark, this title traces the history of science from prehistory to the present. Each chapter covers a particular time span and the development of man’s understanding of the universe. Especially exciting are the chapters that cover the development of modern astronomy, from Galileo to today's space flights.






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astronomical unit
Definition: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).
Context:In expressing planetary distances, multiples of the castronomical unit
—the average distance from Earth to the sun—are often used.
lightyear
Definition:A unit of length in astronomy equal to the distance that light travels in one year in a vacuum, or about 5.88 trillion miles (9.46 trillion kilometers).
Context:Many astronomers prefer to uselightyearsto measure stellar distances because they are easier to work with than other units.
parallax
Definition:The angular difference in the direction of a celestial body as measured from two points in Earth’s orbit.
Context:After measuring the star’sparallax, astronomers were able to determine that the star was much closer than previously thought.
scaling factor
Definition:The proportion between two sets of dimensions.
Context:The map indicated ascaling factorof 1 inch to every 10 miles.






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This lesson plan may be used to address the academic standards listed below. These standards are drawn from Content Knowledge: A Compendium of Standards and Benchmarks for K12 Education: 2nd Edition and have been provided courtesy of theMidcontinent Research for Education and Learningin Aurora, Colorado.
Grade level:912
Subject area:Mathematics
Standard:
Understands and applies basic and advanced properties of the concepts of numbers.
Benchmarks:
Understands the properties and basic theorems of roots, exponents (e.g., [b^{m}][b^{n}] = b^{m+n}), and logarithms.
Grade level:912
Subject area:Mathematics
Standard:
Understands and applies basic and advanced properties of the concepts of measurement.
Benchmarks:
Selects and uses an appropriate direct or indirect method of measurement in a given situation (e.g., uses properties of similar triangles to measure indirectly the height of an inaccessible object).
Grade level:912
Subject area:Mathematics
Standard:
Understands and applies basic and advanced properties of the concepts of measurement.
Benchmarks:
Uses unit analysis to solve problems involving measurement and unit conversion (e.g., between the metric and U.S. customary systems and in foreign currency conversions).






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Chuck Crabtree, freelance curriculum writer. 





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