Beacon Lesson Plan Library

The Mathematical Fingerprint of Our Solar System

Richard Angelini Sr.

Description

An integration of science and math in the study of the locations of all planets in our solar system. Students learn the beautiful mathematical model unique to our solar system. A minimal knowledge of mathmatics is necessary.

Objectives

The student determines the appropriate precision unit for a given situation.

The student determines the greatest possible error of a given measurement and the possible actual measurements of an object.

The student knows the relative sizes of the planets, Sun, Solar System, galaxy, and universe.

The student understands the distances of the planets and the asteroid belt from the Sun are vast.

Materials

Black board and chalk.
Student calculators, if you have them but are not necessary.

Preparations

Make copies of the pre-test-post-test sheet and the 'orbits of the planets' sheet found in the attached file, one copy per student.
Review the astronomical unit of measurement and the planets in our solar system. An astronomical unit is the average distance between the center of our sun and the center of planet Earth. All distances within all solar systems are measured in astronomical units. Miles as a unit of measurement are too small and light years are way too big. In this lesson a mathematical pattern in our solar system will be demonstrated. To date, scientists have discovered several solar systems, but ours is the only one with this mathematical pattern; Hence the name Mathematical Fingerprint of Our Solar System.

Procedures

Introduction:
1. Say to the Students: Today, we are going to experiment with the planets in our solar system. But, first, can anyone tell me why I say “our” solar system and not “the” solar system? The correct answer is: in the last five years, scientists have discovered more than 15 other solar systems. This exercise demonstrates the unique quality of our solar system.

2. Tell the students: Scientists know of many solar systems, ours is only one of many. But, ours has features that are very unusual. Can anyone name one? (The correct answers are: life, our home, nine planets, etc.)

3. Say to the students: I am going to show you the one thing about our solar system that is unique to our solar system. (The correct answer is not life, scientists do not know if there is life in places other than our solar system) Distribute the pre-test. Allow 4 minutes for the test.

Exercise:

Say to the students: Ten years before America’s Declaration of Independence, a scientist, Titus, noticed a pattern to our solar system. Later a mathematician, Bode, wrote a formula to express that pattern. The pattern helped predict the existence of the asteroid belt and the planet Uranus. It also points out the deviation of Pluto and Neptune, leading to the planet X theory. It is a simple progression of numbers with a factor. (Distribute to each student a copy of the orbits of the planets found in the attachments.)

4. Write of the board the following numbers:
0 3 6 12 24 48 96 192 384 768

Ask the students: Can you see any pattern in these numbers? The correct answer is, we double the number to get the next number.

5. Next, explain to the students astronomical units. Say: An astronomical unit is the average distance from our Earth to our Sun. It is the unit of measurement used inside all solar systems. In these numbers are the orbits of the planets. Can you see the orbits? (The students will not be able to see the orbits, not yet.) Tell the students to do the math. Tell them to use their calculators, if they have some, to find the answers quickly.

6. Next, add 4 to each number. The result is:

4 7 10 16 28 52 100 196 388 772
Can you see the pattern? No? Next, divide each number by 10. The results are:

.4 .7 1.0 1.6 2.8 5.2 10.0 19.2 38.8 77.2
Can you see a pattern now? Tell the students: Compare these numbers to the distance in astronomical units of the orbit of the planets in our solar system. The numbers are almost identical to the astronomical unit distance to our sun for each of the planets until you reach the farthest planets from our sun. You will notice that a deviation begins in the orbits of the planets Uranus and becomes greater at Neptune, and Pluto. Current theory speculates that some force knocked Pluto out of orbit. The same force would have greatly affected Neptune and to a lesser degree Uranus. Today, this pattern is called Bode's Law because he was the first scientist to publish it way back in 1788!

7. Tell the students: The orbit of Pluto sometimes falls within the orbit of Neptune. In fact, Pluto is very different. It is not what it should be. The outer planets are all giant gas planets. All are round, you know: spheres. They are all on the same orbital plane, except Pluto. Like Mickey Mouse’s dog, our Pluto seems to have gotten itself into trouble. Current theory holds: Pluto was forced out of orbit into the current orbit by some unknown force. Maybe it came from outside our solar system. Or, maybe it is an escaped asteroid. Whatever it is, it is a strange planet. The same force would have affected Neptune and Uranus to a lesser degree.

8. Explain to the students how to compare the actual orbits of the planets with the predicted orbits of the planets. Tell them: Look at 2.8. It is the asteroid belt. It was predicted to be there 110 years before it was discovered. Instruct the students to compare all other actual and predicted orbits. Instruct the students to calculate the percent error in each of the predicted locations of the planets and the actual locations. This can be accomplished by taking the actual result minus the predicted result,dividing by the predicted result to get the percent error. ( Remember, percent error can be either positive or negative.) % Error = Actual Value - True Value(Predicted) / True Value(Predicted) x 100%

9. Administer the post-test.

Assessments

Formative:
1.Take the preceeding number and double it, after 3. Answer: 5 points
2. In step number six, if a student has the right answer, (astronomical unit distance of all planets in our solar system to our sun, approximately). Answer: 5 points

Take the pre-test, give 3 points for each correct answer. Have the students trade papers and grade the pre-test.

Give points for student participation.
4. 5 points for identifying pattern in step #4 of the procedure.
5. 5 points for identifying pattern in step #6 of the procedure as the orbits of most of the planets.

Summative:
Take the post-test. Compare answers with the pre-test.

Extensions

Advanced or gifted learners: extend to the study of the Planet “X” Theory, other solar systems recently discovered, or the deviation of the orbit of Pluto. Also: the plane of the elliptic is different for Pluto. Ask a student to explain this difference in the elliptic plane of Pluto.

Web Links

Web supplement for The Mathematical Fingerprint of Our Solar System
Bodes Law

Web supplement for The Mathematical Fingerprint of Our Solar System
Umbrella Physics

Web supplement for The Mathematical Fingerprint of Our Solar System
Extrasolar

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