Drake+Equation

//Meagan Morscher, GK-12 Fellow, Northwestern University// //(taken from// http://gk12.ciera.northwestern.edu/classroom/lessonplans.html)

//Key concepts: probability, complex estimates, Drake Equation//

The purpose of the activity is to teach students the basic rules of probability theory and how to apply the theory in order to make complex estimates in the real-world and in science.

This is an activity in which students develop the skill of making complex estimates using probability theory. There are four parts to the activity.
 * Overview**

Part I is an introduction to probability using examples of games of chance, like dice rolling and the lottery. Here, simple probability theory is reviewed, and students learn how to evaluate the probability of different outcomes.

In part II, students apply the knowledge and procedure of part I to do a real-world estimate: the total number of carpeted rooms in the households of all students in their grade level, combined.

In part III, students develop their own procedure for calculating the number of advanced civilizations in the Milky Way that might have the ability to communicate with us. After brainstorming about the information that they need to know about the Milky Way galaxy and about life in order to make the estimate, they put the information together in the form of an equation.They are essentially developing a version of what has become known as the //Drake Equation,// developed by Dr. Frank Drake in 1961//.// In this part of the activity, the focus is not on the actual //numbers// that go into the equation, but rather the //procedure// of combining information in a probabilistic way to get an estimate of the likelihood of an event or outcome.

Finally, in part IV, students watch part of a video about the Drake Equation, and the scientist Drake himself explains his estimate. Students then use an interactive web applet that allows them to learn about the different quantities in the equation, and scientists’ best estimates of the values. They can assess the components of the equation themselves and vary any numbers that they think are unreasonable, and they see how this changes the estimate, and how it compares to Drake’s original estimate.

The activity should involve a lot of class discussion, especially for the second half, since students are not expected to have any sort of background astronomy knowledge. Students come up with their own procedures for making the estimates, and they answer assessment questions throughout, relating to probability, estimates, and assessing their results, and the significance of this estimation procedure.


 * Learner Objectives**
 * Is there Life in Other Worlds?**

Students will understand how to calculate simple probabilities Students will understand the principles behind making complex estimates

If asked to make a particular estimate, students are able to determine the important quantities necessary for making the estimate, and put them together in equation form

Students will be able to explain how complex estimates can be used in both science and in everyday life


 * AAAS Benchmarks**
 * 2C/M2b.** 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.
 * 12B/H1*.** Use appropriate ratios and proportions, including constant rates, when needed to make calculations for solving real-world problems.
 * 12B/H2*.** Find answers to real-world problems by substituting numerical values in simple algebraic formulas and check the answer by reviewing the steps of the calculation and by judging whether the answer is reasonable.
 * 12B/H3*.** Make up and write out simple algorithms for solving real-world problems that take several steps.

2 class periods (can vary depending on level of students, and in particular students’ experience with probability and equations).
 * Time**

High school, grades 9-10. Originally used in a biology class, because of the obvious connection: //life// on other planets. However, activity can be used in any science course, not just biology. It would even be well suited for a basic mathematics course.
 * Level**

No prior knowledge of astronomy or biology is required.


 * Materials and Tools**


 * Activity Packet** (includes introduction, instructions, and assessment questions)

Rubric

http://www.youtube.com/watch?v=UzRirEcx-GQ&NR=1 "Introduction to the possibility of life outside of Earth "Interview with Dr. Frank Drake, where he explains the history of the Drake Equation and the actual numbers he plugged into the equation, and the resulting estimate.
 * Video** - The Search for Life: The Drake Equation part 1/4 (first 10 minutes only)

"View descriptions of each of the quantities in Drake Equation "See Drake’s numbers and his estimate "Learn what scientists currently believe are reasonable estimates for the quantities in the equation "Vary each of the quantities and see how the estimate changes "
 * Interactive Drake Equation**: http://www.pbs.org/wgbh/nova/space/drake-equation.html

//*////Requires student access to computers//

Print activity packets, get set up to play YouTube video to class. Teacher may want to consider using some means to assess students’ prior knowledge about doing probability calculations to know how much guidance to give in part 1 (Probability warm-up) and to predict how long the activity will take. Teacher should know a bit about the history and significance of the Drake Equation in order to be able to lead a meaningful discussion with students and to address questions and concerns. Teacher may also want to consider the background of students (socio-economic status, religious, etc.) in order to predict what challenges may come up, especially with the topic of life on other planets and the vastness of the Milky Way galaxy, etc.
 * Preparation**

None
 * Prerequisites**

In 1961, Dr. Frank Drake organized a conference with all the leading scientists interested in extra-terrestrial life. At the conference, the scientists addressed the following question: what information do we need to know in order to estimate the number of civilizations in the Milky Way galaxy that might be trying to communicate with us? They determined that there were seven important numbers that when multiplied together, give an estimate of the number of communicating civilizations in the Milky Way; this later became known as the Drake Equation, which is
 * Background**
 * where**
 * N=Rxf****p** **xn****e** **xf****l** **xf****i** **xf****c** **xL**
 * N** = number of communicating civilizations in the Milky Way
 * R** = star formation rate in the Milky Way
 * f****p** = fraction of stars that form planets
 * n****e** = average number of planets per star suitable for life
 * f****l** = fraction of suitable planets where life actually begins
 * f****i** = fraction of those planets life-bearing planets where //intelligent// life develops
 * f****c** = fraction of civilizations that develop interstellar communication
 * L** = average lifetime of communicating civilizations

Some of the quantities in the equation are fairly well constrained by astronomical observations, such as the star formation rate in the galaxy. Other terms, however, are quite uncertain. For example, for all the planets with right conditions to support life, on what fraction of them does life //actually// begin? Since we have only one example of a planet where life exists, our own Earth, it is difficult to speculate about the likelihood of life beginning elsewhere. Despite the uncertainty in the estimate, it is still a useful calculation to do because it can act as a guide for scientists interested in extraterrestrial life. What if the estimate we make is extremely tiny? This would imply that, based on our current understanding, it is not very likely that there are other communicating civilizations in our galaxy, and perhaps we should direct our efforts toward studying other things. Or perhaps our current understanding is wrong? On the other hand, what if we estimate that there may be MANY such communicating civilizations? Is this enough to convince ourselves that the topic is in interesting one to pursue?

This leads to interesting questions: If there are many civilizations out there, why have we not found them yet? Before getting into the estimates about civilizations in the Milky Way, it is important to first review the basic rules of probability. How do we calculate the probability of an event or outcome? What if the outcome depends on a number of different quantities, and we have to put together a set of intermediate probabilities to get the total probability of the quantity of interest - how do we combine the intermediate probabilities? We can draw from prior experience in rolling dice and the lottery to discover how to calculate compound probabilities, or probabilities that involve multiple events occurring simultaneously.

We can also use probability to estimate things in the real-world. Suppose you wanted to estimate the number of red cars in Chicago? You might start with the population of Chicago, but then you must consider that not all Chicago residents have cars. What fraction of people in Chicago have cars? Finally, out of all those cars, what fraction of them are red? To figure this out, you might go out onto a busy street, counting all the cars that go by, while also noting how many of them are red. Thus you get an estimate of the number fraction of all cars that are red. Putting these things together, you can estimate the total number of red cars in Chicago. As a real-world example, in part II of the activity, we will follow a similar thought process to estimate the total number of carpeted rooms in the households of all the 9th grade students in the school. After reviewing how to do simple probability calculations and applying the technique to a familiar, real-world estimate, we go on to do the much more complex estimate of the number of communicating civilizations in the galaxy. The technique of doing complex estimates can be applied to a broad range of examples in everyday life and in science. It is a powerful technique that can be used when we cannot simply measure or count the quantity of interest. This is the significance of the Drake Equation - it is the best that we can do, since we do not have the means to simply count the extra-terrestrial civilizations.

Many useful resources can be found on the web. Here are a few examples: http://www.astrosociety.org/education/publications/tnl/77/77.html http://www.seti.org/page.aspx?pid=336

It is likely that some of the students may have thought about life on other planets. But many may have never thought about it, and quite possibly think it is extremely unlikely. Students may not have an idea of just how many other stars similar to our Sun there are in the Milky Way! You will want to make a point of describing how big the galaxy is, while also being sensitive toward students who do not think that there could be other life- bearing planets. After all, we do not know of any extra-terrestrials - why should we assume that they do exist? Would we not know of them if they did exist?
 * Teaching Notes**

The activity works well when there is a lot of discussion among students and also the teacher. Here are discussion questions for the activity. These can be addressed as an introduction to the activity (before doing the probability calculations) or when you get to part III of the activity.

Who has ever wondered whether there is life on other planets in our galaxy? !Who thinks that it us unlikely that there is another civilization trying to communicate with us right now?

How can we learn about whether there are other civilizations in our galaxy?

What difficulties might we encounter in trying to find extra-terrestrials?

How can we apply what we know about our Solar System and life on Earth to speculate about life in other worlds?

Do you think life in other worlds will be similar to life on Earth?

For most of the packet, students should work in pairs or small groups. As students complete the different sections, the class can compare their results. The teacher should make sure that all students fully understand each section before going on to the next.

In the version of the activity packet provided, students do not have to actually calculate their estimate for N using the procedure that they designed (they only give actual numbers for the Drake Equation). To do this, students would need additional time to gather the information/numbers that they need based on the equation that they wrote. There may be overlap in the variables that they chose to use in their equation and the ones that they used in the Drake Equation (in which case, they could use the suggestions provided in the Interactive Drake Equation applet). Otherwise, they will have to do their own web searching to find estimates for the numbers that they need.

A nice way to give closure to the activity is to give a homework assignment where students summarize the procedure of making complex estimates, using the Drake Equation as an example. Students can also come up with other examples of how the estimation procedure can be used in everyday life. Here are some sample homework questions.

Think of another real-world example in which you could apply the procedure of doing a complex estimate. Explain how you would do the estimate, write out an equation with variables and key, and do a calculation to give an actual number for your estimated quantity.
 * Complex estimates in the real-world**

Write a few paragraphs reflecting on the procedure of doing a complex estimate to speculate about the number of communicating civilizations in our galaxy. Briefly outline the procedure we used to estimate //N//, and then address the following questions: why did we use this estimation procedure to calculate //N//? Can you think of any other way to estimate //N//? Do you think this type of calculation is useful? Why or why not? Do you have any other thoughts about this part of the activity?
 * Reflection on Drake Equation**

There are many questions throughout the packet to get the students to think critically and understand the main points, and for the teacher to use in assessing student understanding of the material. A performance rubric is provided for use as a guide when assessing student responses. Students earn between 0 and 5 points for different parts of the activity, where questions are sometimes categorized together when they are closely related.
 * Assessement**