Constructing a model of the solar system



Constructing a model of the solar system

Part 2: The relative distances of the planets from the Sun

Time: approximately 45-50 minutes

Materials:

A few different sized marbles

Several different sized ball bearings, available at most hardware stores

Other round objects of different sizes, including bead, pebbles, etc.

A few grains of sand, poppy seeds, sugar and/or ground black pepper

Index cards (one for each group)

Metric rulers (one for each group)

Calculators (one for each group)

Marker pen (one for each group)

Cellophane tape

Measuring wheel or 100 m measuring tape

Solar system chart from Appendix B

Overview

This lesson continues from Part 1: The relative size of the planets. In this lesson, students will “shrink” the distance from the Sun to Pluto down to about the size of a football or soccer field. They will place their planet in the field at its relative distance from the Sun. Students will need to use a new scale to determine a new diameter for their planet and the distance of their planet from the Sun. In most cases this lesson will need to be performed outdoors.

Purpose

Students will gain an appreciation for how far away the planets of the solar system are from each other. This lesson also involves knowledge of ratios, scale models and measuring.

Standards

A complete list of the standards covered by this lesson is included in Appendix A at the end of the lesson.

Procedure

A good way to begin this lesson is to remind the class of the question that you posed at the end of Part 1 – how can you construct a model of the solar system that can fit into an area about the size of a football field? Ask for ideas from the your students. The amount of coaching they will need to arrive at a workable scale will depend on the particular class. You may need to remind them of the following:

• The area required must accommodate the greatest distance that you plan to represent – the distance between Pluto and the Sun.

• Since a football field is around 100 meters, you want the distance from the Sun to Pluto to be around 100 m. The average distance from the Sun to Pluto is 5,913,520,000 km. If we round this up to 6 billion km, then we can set our scale at 100 m = 6 billion kilometers.

• For flexibility, we can convert our scale to meters, centimeters or millimeters. Dividing each side of our scale by 100 gives us 1 m = 60 million km. Dividing this by 100 on each side gives 1 cm = 600,000 cm, and then dividing each side of this equation by 10 gives 1 mm = 60,000 mm. Students will probably find 100 m = 6 billion km most helpful when calculating relative distance, and either 1 cm = 600,000 cm or 1 mm = 60,000 km most useful when calculating diameter.

You should inform the class that the solar system model that you will be constructing is going to show all the planets lined up on the same side of the Sun. In reality, it is extremely rare for all the planets to be on the same side of the Sun at the same time. A more accurate model would have the planets randomly distributed in their orbits around the Sun, but to do this would require twice as much area. Also, it would be more difficult to see how far away different planets are from each other. So, although you are showing the Sun and all 9 planets, the model you are constructing is actually showing only half the area that the planets move through as they orbit the Sun.

The first step is to have the class determine the size of the object you will use to represent the Sun. They can calculate this with the following equation:

1,400,000 km (diameter of the Sun) = ? cm (diameter of object representing the Sun)

600,000 km 1 cm

The object used for the Sun will be a little under 2.5 cm, or roughly the size of a small rubber “super” ball. Select an object of the appropriate size and hold it up in front of the class. Remind the class of what they found out in Part 1 of this lesson. Ask the students who modeled Mercury, Mars or Pluto what they learned about their planet’s size compared to the Sun, and ask them how big they think their planet will be at this new scale. Remind the class that they will want to lay this model out on the lawn or field outside, and ask them how easy it will be to see their planet at this new size if they lay it down directly on the dirt or grass. Since most planets might be very hard to see in this model, tell the class that you want them to tape their planet onto an index card using clear cellophane tape. They will need to write the name of their planet, it diameter and its average distance from the Sun on the index card that they tape their planet to. For convenience, its is also a good idea to have them write the distance their object will be from the object representing the Sun in the model on the card. This will speed things up when they lay out the model outside.

Tell the students that they will need to calculate the diameter of their planet and its distance from the Sun using the new scale. Distribute index cards, solar system charts, rulers and calculators. If needed, you can put the following 2 formulae on the board:

Diameter of planet:

? km (diameter of their planet) = ? mm (diameter of object representing their planet)

60,000 km 1 mm

Distance from Sun:

? km (distance from the Sun) = ? m (distance of object from model Sun)

60,000,000 km 1 m

At these scales, Jupiter will be a little over 2 mm, Saturn about 2 mm, Uranus and Neptune a little under 1 mm, and the rest of the planets the size of grains of ground pepper or dust from soil or sand. Mercury, Mars and Pluto will actually be fractions of a dust grains, so students modeling these planets may have to say the grain of pepper they are using to represent their planet is actually 10x larger (or whatever number is appropriate) than what their planet should be.

The students modeling Earth should also calculate the diameter of the Moon and its distance from Earth (not its distance from the Sun). At this scale, the Moon would be about 6.5 mm away from the Earth, so they should show both the Earth and the Moon, appropriately shaped from each other, on their index card with both appropriately labeled.

The distances that the objects will be from the Sun in the model are:

Mercury 97 cm (about 1 M)

Venus 1.8 m

Earth 2.5 m

Mars 3.8 m

Jupiter 13 m

Saturn 23.8 m

Uranus 47.8 m

Neptune 75 m

Pluto 98.5 m

Once all the groups have (1) determined the diameter of their planet and its distance from the Sun, (2) selected something to represent their planet, (3) taped it to their index cards, and (4) written the required information on the card, (it is a good idea the check their work to make sure they have the correct distances), take the class outside and begin arranging the model. Start by placing the “Sun” on the ground. Tell the students that as you move out from the Sun you want them to keep track of the distance and let you know when to stop so they can mark the position of their planets. Begin measuring the distance as you move outward. At 97 cm have someone from the Mercury group lay down the Mercury card at the correct spot. At 1.8 m have the Venus card placed on the ground, and continue on outward in a similar fashion. When Earth is placed in position have someone from the group point out how close the Moon is to Earth at this scale. Point out that the Moon is the only extraterrestrial object that a human has ever landed on. As you continue out from Earth, remind the students that NASA has recently been charged by the president to develop plans for sending a human to Mars. Ask them to keep the distance from Earth to the Moon in mind as you mark off the distance to Mars. This will help them gain an appreciation for how challenging a manned mission to Mars would be. After you have placed Mars in position, remind the class that Mercury, Venus, Earth and Mars are often referred to as the inner planets. Have them keep that in mind as you mark off the five outer planets. You may also want to remind them that the asteroid belt lies between the orbits of Mars and Jupiter. If you are using a tape measure that is less than 100 meters long you can have a student mark the position of the end of the tape measure as you shift it outward.

If the weather is windy you may need to have one member from each group stand on a corner of their card to keep it from blowing away. If this is the case, you should have another member from their group replace them at this time so that they can walk around and see the positions of the other planets. Once all the planets have been marked off and all the students have had an opportunity to walk around and see them, you can bring the class back inside or, in the students and the weather are both cooperating, have a discussion outside. A good way to start off is to ask the students what they noticed about the arrangement of the planets after constructing the model. Hopefully they noticed how far apart the outer planets are compared to the inner planets and how far apart the inner planets are compared to the distance between the Earth and the Moon. You should point out that right now all the planets are at their closest position to each other. Most of the time they are at different positions in their orbits around the Sun and therefore much farther away from each other, even though their distance from the Sun is about the same. You might also point out that Pluto has the most eccentric orbit of the 9 planets. As it is currently represented in the model it is at its average position. Sometimes its orbit takes it even farther away than it is depicted, and sometimes it is closer, occasionally even crossing over the orbit of Neptune so that Pluto briefly becomes the 8th planet out from the Sun and Neptune the 9th.

If time permits, you can let the students see how large the Sun appears on different planets. Start by having them stand near Earth and hold up the object representing the Sun. At that distance it should look similar in size to the Sun in the sky. As they move in closer it will look larger, but what is even more dramatic is to have them move out to see how small it looks on the different outer planets.

Appendix A

Standards Addressed

Benchmarks (Grades 3 through 5)

2C – Mathematical Inquiry

Numbers and shapes-and operations on them-help to describe and predict things about the world around us.

In using mathematics, choices have to be made about what operations will give the best results. Results should always be judged by whether they make sense and are useful.

3A – Technology and Science

Measuring instruments can be used to gather accurate information for making scientific comparisons of objects and events and for designing and constructing things that will work properly.

4A – The Universe

Planets change their positions against the background of stars.

The earth is one of several planets that orbit the sun, and the moon orbits around the earth.

4B – The Earth

Like all planets and stars, the earth is approximately spherical in shape. The rotation of the earth on its axis every 24 hours produces the night-and-day cycle. To people on earth, this turning of the planet makes it seem as though the sun, moon, planets, and stars are orbiting the earth once a day.

9A – Numbers

When people care about what is being counted or measured, it is important for them to say what the units are (three degrees Fahrenheit is different from three centimeters, three miles from three miles per hour).

Measurements are always likely to give slightly different numbers, even if what is being measured stays the same.

9B – Symbolic Relationship

Tables and graphs can show how values of one quantity are related to values of another.

9C – Shapes

Graphical display of numbers may make it possible to spot patterns that are not otherwise obvious, such as comparative size and trends.

11B – Models

Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail.

11C – Constancy and Change

Things change in steady, repetitive, or irregular ways-or sometimes in more than one way at the same time. Often the best way to tell which kinds of change are happening is to make a table or graph of measurements.

12B – Computation and Estimation

Add, subtract, multiply, and divide whole numbers mentally, on paper, and with a calculator.

Use fractions and decimals, translating when necessary between decimals and commonly encountered fractions-halves, thirds, fourths, fifths, tenths, and hundredths (but not sixths, sevenths, etc.).

Judge whether measurements and computations of quantities such as length, area, volume, weight, or time are reasonable in a familiar context by comparing them to typical values.

12D – Communication Skills

Use numerical data in describing and comparing objects and events.

Benchmarks (Grades 6 through 8)

2C – Mathematical Inquiry

When mathematicians use logical rules to work with representations of things, the results may or may not be valid for the things themselves. Using mathematics to solve a problem requires choosing what mathematics to use; probably making some simplifying assumptions, estimates, or approximations; doing computations; and then checking to see whether the answer makes sense. If an answer does not seem to make enough sense for its intended purpose, then any of these steps might have been inappropriate.

4A – The Universe

Nine planets of very different size, composition, and surface features move around the sun in nearly circular orbits. Some planets have a great variety of moons and even flat rings of rock and ice particles orbiting around them. Some of these planets and moons show evidence of geologic activity. The earth is orbited by one moon, many artificial satellites, and debris.

4B – The Earth

We live on a relatively small planet, the third from the sun in the only system of planets definitely known to exist (although other, similar systems may be discovered in the universe).

9A – Numbers

Computations (as on calculators) can give more digits than make sense or are useful.

9B – Symbolic Relationship

Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation.

9C – Shapes

The scale chosen for a graph or drawing makes a big difference in how useful it is.

11B – Models

Different models can be used to represent the same thing. What kind of a model to use and how complex it should be depends on its purpose. The usefulness of a model may be limited if it is too simple or if it is needlessly complicated. Choosing a useful model is one of the instances in which intuition and creativity come into play in science, mathematics, and engineering.

12B – Computation and Estimation

Estimate distances and travel times from maps and the actual size of objects from scale drawings.

Decide what degree of precision is adequate and round off the result of calculator operations to enough significant figures to reasonably reflect those of the inputs.

12C – Manipulation and Observation

Use calculators to compare amounts proportionally.

12D – Communication Skills

Read simple tables and graphs produced by others and describe in words what they show.

Benchmarks (Grades 9 through 12)

2C – Mathematical Inquiry

Much of the work of mathematicians involves a modeling cycle, which consists of three steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas. If the match is not considered good enough, a new round of abstraction and manipulation may begin. The actual thinking need not go through these processes in logical order but may shift from one to another in any order.

4A – The Universe

Mathematical models and computer simulations are used in studying evidence from many sources in order to form a scientific account of the universe.

4B – The Earth

Life is adapted to conditions on the earth, including the force of gravity that enables the planet to retain an adequate atmosphere, and an intensity of radiation from the sun that allows water to cycle between liquid and vapor.

9B – Symbolic Relationship

Any mathematical model, graphic or algebraic, is limited in how well it can represent how the world works. The usefulness of a mathematical model for predicting may be limited by uncertainties in measurements, by neglect of some important influences, or by requiring too much computation.

Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

11B – Models

The basic idea of mathematical modeling is to find a mathematical relationship that behaves in the same ways as the objects or processes under investigation. A mathematical model may give insight about how something really works or may fit observations very well without any intuitive meaning.

11C – Constancy and Change

Graphs and equations are useful (and often equivalent) ways for depicting and analyzing patterns of change.

12B – Computation and Estimation

Use ratios and proportions, including constant rates, in appropriate problems.

National Standards (Grades 5-8)

Understandings about Scientific Inquiry

Different kinds of questions suggest different kinds of scientific investigations. Some investigations involve observing and describing objects, organisms, or events; some involve collecting specimens; some involve experiments; some involve seeking more information; some involve discovery of new objects and phenomena; and some involve making models.

Mathematics is important in all aspects of scientific inquiry.

Earth in the Solar System

The earth is the third planet from the sun in a system that includes the moon, the sun, eight other planets and their moons, and smaller objects, such as asteroids and comets. The sun, an average star, is the central and largest body in the solar system.

National Standards (Grades 9-12)

Understandings about Scientific Inquiry

Mathematics is essential in scientific inquiry. Mathematical tools and models guide and improve the posing of questions, gathering data, constructing explanations and communicating results.

Indiana Standards

Grade 5

Mathematics – Problem Solving

5.2.6 – Use estimation to decide whether answers are reasonable in addition, subtraction, multiplication, and division problems.

5.7.1 – Analyze problems by identifying relationships, telling relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.

Science – Computation and Estimation

5.2.1 – Multiply and divide whole numbers mentally, on paper, and with a calculator.

Earth and the Processes That Shape It

5.3.7 – Describe that, like all planets and stars, Earth is approximately spherical in shape.

Numbers

5.5.1 – Make precise and varied measurements and specify the appropriate units.

Systems

5.6.1 – Recognize and describe that systems contain objects as well as processes that interact with each other.

Grade 6

Mathematics – Number Sense

6.1.6 – Use models to represent ratios.

Computation

6.2.5 – Solve problems involving addition, subtraction, multiplication, and division of positive fractions and explain why a particular operation was used for a given situation.

6.2.6 – Interpret and use ratios to show the relative sizes of two quantities. Use the notations: a/b, a to b, a:b.

6.2.9 – Use estimation to decide whether answers are reasonable in decimal problems.

Science – Communication Skills

6.2.6 – Read simple tables and graphs produced by others and describe in words what they show.

The Universe

6.3.1 - Compare and contrast the size, composition, and surface features of the planets that comprise the solar system, as well as the objects orbiting them. Explain that the planets, except Pluto, move around the sun in nearly circular orbits.

6.3.2 - Observe and describe that planets change their position relative to the background of stars.

6.3.3 – Explain that Earth is one of several planets that orbit the sun, and that the moon, as well as many artificial satellites and debris, orbit around Earth.

Models and Scale

6.7.2 – Use models to illustrate processes that happen too slowly, too quickly, or on too small a scale to observe directly, or are too vast to be changed deliberately, or are potentially dangerous.

Grade 7

Mathematics – Measurement

7.5.2 – Use experimentation and modeling to visualize similarity problems. Solve problems using similarity.

Problem Solving

7.7.1 – Analyze problems by identifying relationships, telling relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.

7.7.11 – Decide whether a solution is reasonable in the context of the original situation.

Science – Models and Scale

7.7.2 – Use different models to represent the same thing, noting that the kind of model and its complexity should depend on its purpose.

Grade 8

Mathematics – Measurement

8.5.3 – Solve problems involving scale factors, area, and volume using ratio and proportion.

Problem Solving

8.7.11 – Decide whether a solution is reasonable in the context of the original situation.

Science – Manipulation and Observation

8.2.3 – Use proportional reasoning to solve problems.

8.2.4 – Use technological devices, such as calculators and computers, to perform calculations.

The Universe

8.3.1 – Explain that large numbers of chunks of rock orbit the sun and some of this rock interacts with Earth.

Earth and Space Science

The Universe

ES.1.7 – Describe the characteristics and motions of the various kinds of objects in our solar system, including planets, satellites, comets, and asteroids. Explain that Kepler’s laws determine the orbits of the planets.

Appendix B: Solar System Chart

|Object |Diameter |Distance from Sun |

| | | |

|Sun |1,390,000 km | |

| | | |

|Mercury |4,880 km |57,910,000 km |

| | | |

|Venus |12,104 km |108,200,000 km |

| | | |

|Earth |12,756 km |149,600,000 km |

| | | |

|Mars |6,794 km |227,940,000 km |

| | | |

|Jupiter |142,984 km |778,330,000 km |

| | | |

|Saturn |120,526 km |1,429,400,000 km |

| | | |

|Uranus |51,118 km |2,870,990,000 km |

| | | |

|Neptune |49, 532 km |4,504,000,000 km |

| | | |

|Pluto |2,274 km |5,913,520,000 km |

| |Diameter |Distance from Earth |

|Earth’s Moon |3,475 km |384,400 |

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