# Selected Topics in Astronomy and Space Studies

## CONTENTS OF CURRICULUM UNIT 96.06.11

## An Interdisciplinary Approach to Kepler’s Laws

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## LESSON PLAN

## Lesson # 1 | Grade 7 |

*Aim*: How to define and apply the units of Light Year and Astronomical Units.

*
Instructional Objectives
*
: Students should be able to provide a working definition of the units of Light Year and Astronomical Units. They should be able to use these units in defining distances in space.

*
Activity
*
: Use a globe and a small light bulb on a stand to demonstrate rotation and revolution. Students should be asked to tell the difference between a rotation and a revolution. Students should be given calculators to assist in mathematical calculations.

*
Development
*
: If there were no obstruction and we turn on a very powerful light here in New Haven, how long would it take for if to get to California (students should be allowed to determine the necessary information they need to get the answer)?

If light travels at a speed of 186,000 miles per (150,000 kilometers per second), how far would it travel in (a) 10 seconds (b) 20 seconds (c) 1 minute (d) 1 hour (e) 1 day?

What is a light second?

What is a light hour?

What is a light day? What is a light year When do we use light year? (students should be provided with vast distances in the Universe where light year is used).

How far is the Sun from the Earth?

What name could we give for a distance that is equal to the distance of the sun from the Earth?

What is the distance of 3 AU in miles?

What would be the same distance in Kilometers?

*
Application
*
: Students should be given various values in light years and they should be able to estimate the distances in light years They should be given various distances which they should be able to convert to AU or from AU to miles and kilometers.

## Lesson # 2 | Grade 7 |

*Aim*: How to evaluate a formula

*Instructional Objectives*: Students should be able to substitute numerical values for the variables or symbols in a formula. They should be able to define and calculate velocity, distance and time.

*Material*: Calculator

*
Development
*
: The recipe of a cake can be used to demonstrate the nature of a formula. Here the cake represents the subject of the recipe or formula. Average velocity V,is equal to the distance d, divided by the time t: V =
*
d t
*

What is the subject of the formula?

If d is 60 miles and t is 2 hours how would we know the average velocity?

The earth orbits at a velocity of about 66,000 miles an hour. What does that statement really mean? How far a distance would the earth travel in two hours?

What did we do to get that answer?

How could we put that in a formula where V is velocity, d is distance and t is time?.

If the earth travels at a Velocity of 66,000 miles per hour how long will it take to travel (a) 33,000 miles (b) 213,000 miles? Using the same letters: What formula can we develop for finding the time?

*
Application
*
: Using the developed formulas student should be given problems requiring them to find velocity, distance and time.

## Lesson # 3 | Grade 7 |

*Aim*: How to find the square root of a number

*
Instructional Objectives
*
: Student should be able to find the square root of a given value.

*
Activities
*
: Students can use grid paper to draw several squares of varying dimensions. The dimensions can then be expressed in an exponential form 2 x 2 equals two to the second power; 3 x 3 is three to the second power.

Students are then given examples of perfect squares and are required to figure out the dimensions which produce the squares. This is later defined as the root of the number. The perfect squares can then be expressed in the form of a mathematical statement: d to the second power is 81 what is the value of d?

Students are then introduced to the square root sign: M = 144. What is M?
*
Application
*
:

- 1. Students should provide examples of perfect squares and their roots.
- 2. Students should calculate the square roots of given values using a calculator.

## Lesson # 4 | Grade 7 |

*Aim*: To draw ellipses of various shapes.

*Instructional Objective*: students should be able to draw ellipses of various shapes by manipulating the foci. They should be able to identify the basic elements of an ellipse.

Background Information:

Students should be provided with the information of Kepler’s discovery of the elliptical orbits of the planets.
*
Activities
*
: Students could work in a cooperative setting. They should make a loop with a piece of string which is about six inches long. Draw a horizontal line of about 10 inches in length on a sheet of construction paper. Stick two thumb tacks about one inch apart on the line. Place the string around both tacks and use a pencil to gently pull the string while using the loop as a guide to draw a closed curve.

Students should label the points of the thumb tacks as the foci. Half way between the foci is the center. The major and minor axes should be draw and identified.

*
Application
*
: Students should use the same loop to draw several ellipses by moving the foci further apart. Students should then draw several ellipses by bringing the foci closer and closer together.

*
Evaluation
*
: When the foci move further and further apart, what effect did it have on the shape of the curve? When the foci were moved closer to each other, how did it affect the shape of the curve?

What comment would you make about a circle?

## Lesson # 5 | Grade 7 |

*Aim*: How to estimate area

*Instructional Objective*: Applying Kepler’s second law students should be able to estimate the area of sweep in a planetary orbit for a constant time period.

*
Activities
*
: Students could work in a cooperative setting.

- 1. Student will draw on a grid paper a planetary orbit (an ellipse) with the Sun as one focus. The Sun will be closer to one section of the orbit.

*Development*: Why do the planets move around the Sun? Which planets would you expect to move faster, the ones closer in or the ones farther out?

Why ?

As the planet orbits in its elliptical orbit what kind of effect will the sun have on its velocity?

*
Application
*
: Shade in a section from the area nearest to the sun. Shade in an equal area in the section furthest from the sun. Repeat this activity on an ellipse of a different size.

## Lesson # 6 | Grade 7 |

*Aim*: To calculate the period of rotation and the distance from the Sun

*
Instructional Objectives
*
: Students should be able to apply Kepler’s third law in finding the orbital period and the distance of the planet from the Sun.

*
Definitions
*
: Sidereal Period, Astronomical Unit
*
Material
*
: Calculators, copy of the planetary orbits of the Solar System.

*
Development
*
: Which planet would you expect to have the shortest orbital period ?

Why would that be true?

Which planet would you expect to have the longest orbital period?

If the distance of a planet from the Sun is twice the distance of the Earth from the Sun, how many AU would it be?

If a planet takes three Earth years to orbit the Sun, how many Sidereal periods does it represent?

*
Application
*
: Use the formula to calculate the Sidereal Period: P
^{
2
}
= D
^{
3
}
; where P represents the Sidereal Period and D is the distance from the sun in Astronomical Units.

Use the formula D =
^{
3
}
P
^{
2
}
to find the distance of the planet from the Earth in AU