# The Physics, Astronomy and Mathematics of the Solar System

## CONTENTS OF CURRICULUM UNIT 07.03.06

## Astronomy: The Mathematician's Perspective

Your feedback is important to us!

After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

## Classroom Activities

Sample lesson plans.

### Lesson 1

This lesson is the introduction to the astronomical unit for the instruction of right triangles in geometry. It will lead to lesson 2 - the Pythagorean Theorem.

**
Objective:
**
Students will become motivated to learn about the next concept, the Pythagorean Theorem, have a greater understanding that mathematics is important in comprehending our daily lives, and have a deeper appreciation of what mathematics can accomplish to that comprehension.

**
Procedure:
**

- 1. Students will place astronomical discoveries in the order of occurrence.
- 2. Students will determine what is common to all discoveries.
- 3. Students will discuss how mathematics plays a part in each discovery.
- 4. Discussion will include limited history on astronomy and how mathematics played a part in it, and a few examples of misconceptions about the solar system.
- 5. Students will evaluate a problem and note that it is unsolvable unless they learn the next concept-the Pythagorean Theorem. This problem will be revisited after the Pythagorean Theorem is discussed.

**
Materials:
**
Two student worksheets and one teacher discussion notes sheet (see Appendix B).

**
Assessment:
**
Student questions and student participation in the discussion.

### Lesson 2

Students will use the Pythagorean Theorem to find distances. (Note: Lesson 2 can also be used to teach trigonometric ratios by focusing on angle measurements and by substituting solving right triangles by trigonometric ratios wherever Pythagorean Theorem appears.)

**
Objective:
**
Students will use the Pythagorean Theorem, the Earth-Moon parallax, and the Sun parallax to find distances between two points on a right triangle.

**
Procedure:
**

1. State and explain the Pythagorean Theorem. (Power point or overhead may be used.)

2. Demonstrate the Pythagorean Theorem by using the Earth-Moon parallax.

(See Appendix C)

3. Reinforce the concept by having the students apply the Pythagorean Theorem using the Sun parallax. (Use the figure in Appendix C and substitute the word Sun for the word Earth and the word Earth for the word Moon and elongate the lines. The distance from the Earth to the Sun is 149.6*10
^{
6
}
km and the Sun's radius is

6.95*10
^{
5
}
km.

4. Students will also solve for distances using the following astronomical situations.

This can be presented to the students in worksheet form.

- a. Two people live in two different cities. They are both looking through a
- telescope at the Hubble space station 579 km above Earth. How far are they
- looking if they are equal distances apart from each other and how far are they
- looking if the distances from each other vary? (Draw a triangle with the two
- people as two vertices on the ground and the space station as the third angle.
- Drop a perpendicular from the space station to the ground to form two right
- triangles. Solve for distances. Other measurements for distances may be
- selected from a textbook or another resource. Offer different scenarios to solve
- for the legs of the right triangle.)
- b. Two people in two different cities are talking to each other using a cell phone.
- The cell phone satellite
^{ 12 }is 643 km above Earth. What is the distance from- the satellite to the ground where the people are using the cell phones? What is
- the distance when the people's locations change as they drive their cars to other
- destinations? (Use the same type of triangle as in a. above and create scenarios
- to solve for the legs of the right triangle.)
- c. Revisit the problem presented in lesson 1 and solve.

**
Materials:
**
Projector for power point or overhead projector for concept presentation, worksheets for the Earth-Moon parallax, the Sun parallax, and the above problems, and rulers.

**
Assessment:
**
Class participation and submitted worksheets.

### Lesson 3

**
Objective:
**
Students will determine the measures of arc lengths and the areas of sectors.

**
Procedure:
**

- 1. Kepler's Laws will be explained (power point or overhead projector may be used).
- 2. Students will works in groups of two or three.
- 3. Students will create a model of their planet's orbit around the Sun.
- 4. Students will map out their planet's seasonal orbit around the Sun.
- 5. Students will calculate the arc length and area of the sector of their season using formulas.
- 6. Students will compare their calculations to the other planets.
- 7. Students will make conjectures about orbital paths and orbital speed of planets.
- 8. Additional exercises: Students will find the measure of an arc length and the area of a sector between two locations on a globe and compare the measure of the arc length to the same distance on the equatorial circumference of the Earth and to the distance listed on a map. Since the Earth is not completely round and all circumferences taken around the Earth may not all be equal, there will be some discrepancies in answers when only the equatorial circumference of the Earth is utilized as the Great Circle and comparisons made. Examples of locations and angles determined: the distance between the North and South Pole will yield a 180
^{ o }angle, the distance between the North Pole and the equator will yield a 90^{ o }angle, the distance between the South Pole and a location in Patagonia in South America will yield a 45^{ o }angle, etc. (See Table 2 to determine the radius of the Earth.)

**
Materials:
**
Data sheet listing orbital speeds and planet distance from the sun (see Table 3 and Appendix D), comparison of measure of arc lengths and area of sectors of planets according to position from the Sun worksheet, heavy poster board, small knife to carve out the arc of the season, balls for planets and Sun, protractors, rulers, and globes.

**
Assessment:
**
Construction of model and completion of worksheet.

**
**