Think of some phenomenon that you would like to understand but “seeing” how it works is hidden from you. Scientists have asked many questions like these over the years… How does the body heal a broken bone? Why does water expand when it freezes? How does a tree convert sunlight to energy for its growth? These processes were once a black box -- a metaphor that compares the phenomenon to something happening inside a container that no one can see because the walls are dark. But as scientists dreamed up possible explanations for how these things might work, then tested, revised, and retested their ideas, slowly we all came to “see” inside. How do you figure out what’s going on inside your black box? In this hands-on activity, we will use this metaphor to understand how scientific inquiry works, and also the mathematical basis for why this approach is powerful. This activity is borrowed with permission from Biology in a Box.

Grade Level: 9th-12th grade

Subject Matter: Scientific Method, Probability, Mathematics

### Part 1: What’s in the Box?

**Activity Materials**

• Black and White Boxes: We order tins from www.specialtybottle.com (# TNF8; at the time of publication 85 cents each).

• 11 unique items per White Box: we use rubber band, rubber ball, rubber stopper, poker chip, paperclip, tooth pick, marble, steel ball, crayon, penny, cotton ball

• 2 items (ie a paperclip and a crayon, but each may be different) for each Black Box

• Plastic containers for storing White Box items not in use, to prevent items from rolling off desks

**Vocabulary**

**Theory:** A set of statements or principles that explain a group of facts or natural phenomena that has been repeatedly tested and is widely accepted

**Hypothesis:** A tentative explanation for an observation, phenomenon, or scientific problem that can be tested by further investigation

**Prediction:** A statement of expectation for how something will happen in the future. If you base a prediction on your hypothesis, if you test your prediction and the outcome you expected does not occur, your hypothesis may not be true.

**Scientific inquiry:** The process by which scientists figure out how things work: by asking questions, using logic and reasoning to guess how things work, then test those guesses to see if they are accurate

Science has figured out theories for how many things work at scales too big, too small or too hidden for us to see with our own eyes, but how did they do it? They came up with a way to deduce their theories based on formulating questions, making educated guesses, then acquiring and evaluating evidence: **scientific inquiry.** We’ve not yet run out of unanswered questions, so we still use these methods today. Hypothesis-driven science asks questions and seeks answers through the development of tentative answers (hypotheses). These educated guesses are tested through experimentation and evaluation of results. Hypothesis-driven science can be modeled by a stepwise process that involves observation, hypothesis formation and testing, as shown in the figure “The Scientific Method”.

In In this exercise, you will use the hypothesis-driven scientific method to help you decide on the possible identity of two mystery objects (from a set of 11 potential objects) in your “Black Box”.

**NO PEEKING INTO THE BLACK BOX UNTIL YOU COMMIT YOUR HYPOTHESIS TO PAPER, ANNOUNCE IT TO YOUR CLASS, AND YOU ARE INSTRUCTED TO OPEN IT!**

**What to Do**

- Split into groups of 3-4 students. Each group should take one White Box, one Black Box, and one plastic container (use this to keep White Box objects not in use from rolling off desks/tables).
- Examine the objects in the White Box, listing them on a sheet of paper or on the board at the front of the room so that you may consult the list as your investigation proceeds.
- As a group, consider how you will approach the problem scientifically. For example you might assess the characteristics (sounds, mass, response to the movement of a magnet) of the Black Box itself and record these. Your hypothesis would be that the objects would have to have a particular sound, weight, behavior relative to one another. You might instead examine the behavior of the 11 objects, in pairs, or individually, in the White Box tin, and make predictions about how the objects should sound, respond to magnetism, etc.) if they were in the Black Box.
- Write down the testing procedures you plan to use.
- Start the process, remembering to follow the steps of the scientific method!
- Record the steps you have taken throughout your investigation.
- Record your revised hypotheses as you gain information.
- Finally, record what your team concludes is in the Black Box.

When finished, each group should report the following information to the class:

- Approach taken to the problem
- Hypotheses they have made and modified through the testing process
- The two objects they concluded are in the Black Box and why.

Teams should open the Black Box and check to see if they were correct.

Record the number of items that were determined correctly by each team on the board (0, 1 or 2 items correct). What percentage of the class got both the items correct? What percentage got at least 1 item correct?

**Discussion Questions:**

How does the definition of a theory presented above, which is how a scientist defines a theory, different from the way the word “theory” is used in everyday language?

Do you expect your class would’ve been as successful in determining what is inside the black box if they had merely guessed?

*Teacher’s Note: Options for Expanding the Lesson*

- A set of Black Boxes are permanently sealed (we drill screws into the tins). This represents how the scientific method can never absolutely verify something as true, the Black Box cannot be opened. This drives students crazy, too.
- The same two items go in each group’s Black Box. After each group has reached a conclusion, the class comes together to discuss and try to reach a consensus. This highlights the importance of collaboration and sharing of information within the scientific community, not only increasing the overall knowledge base, but also helping reduce the influence of bias in interpreting results.
- Groups are given access to magnets and balances to determine their black box items. This illustrates how, while the scientific method cannot completely verify something as true, new techniques, tools may come along to shed new light on the black box.

### Part 2: What are the odds if you’d just guessed?

Suppose that instead of using the scientific method, you just guessed which items were in your Black Box without performing any experiments or observations. Would you have done as well? Probably not! Although the scientific method may not have completely determined which objects were in your tin, it did help narrow the list of possibilities -- increasing the probability that your final choices would be correct. Probability can mathematically explain why applying the scientific method to the Black Box problem beats merely guessing the contents.

**Vocabulary**

**Sample space:** a set of *possible outcomes*. For example, the sample space of rolling a single die can be represented as follows: {1,2,3,4,5,6}, with each of the numbers in the brackets representing all of the possible results of the roll.

**Event:** a subset of the sample space, containing some (or all) of an experiment’s outcomes. For example, the set *E*={2,4,6} is an event, representing rolling an even number on the die.

**Elementary events:** an event such as {2}, which contains a single outcome.

**Mutually exclusive events:** two events that contain none of the same elementary events. For example, {2} and {4,6} are mutually exclusive events

**Probability:** the likelihood that a chosen event will occur. This can be expressed in many ways, such as a fraction, a decimal, or a percentage, and represents the chances for that particular occurrence divided by the total chances of any occurrence.

**Got that? Try these …**

1. Suppose that you roll a six sided die. Let *E* denote the event that you roll less than a five. Write down all of the elements that belong to the event *E*.

2. Let *B* be the event that you roll 1,4,or 6, that is, let *B*={1,4,6}. Are *B* and *E* mutually exclusive? If not, which elementary events belong to both *B* and *E*?

Axioms (Rules) of Probability

IfEis an event, then we denote the probability thatEoccurs byP(E)

- The sample space
Sof an experiment is the set of all possible outcomes. One of the outcomes in the sample space will definitely occur, so if you add up all of their individual probabilities you will get 1 or 100%.- The likelihood of a particular event ranges from impossible to absolutely definite. (
P(E) = 0, or 0% likely, to 1, or 100% likely.)- If an outcome can be one of two alternatives (but not both), the probability of either event occurring is equal to the sum of the likelihood of each event’s occurrence.

A couple of very useful rules follow directly from the axioms of probability…

Rule 1:

When every outcome in a set of possible outcomes is equally likely to occur, the probability that a specific outcome occurs is equal to one divided by the number of possible outcomes.

For example, one would calculate the probability of rolling a 6 on a six-sided die (6, possible outcomes {1,2,3,4,5,6} and 1 desired outcomes {6}, like this:

Rule 2:

When every outcome in a set of possible outcomes is equally likely to occur, the probability that a specific event occurs is equal to the number of outcomes in the event divided by the number of possible outcomes.

For example, one would calculate the probability of rolling an even number on a six-sided die (6, possible outcomes {1,2,3,4,5,6} and 3 desired outcomes {2,4,6}, like this:

**Got that? Try these...**

- What is the probability of rolling a 2 on a 6-sided die?
- What is the probability of rolling a 4 or a 6 on a 6-sided die?
- If your group had only one mystery object in the black box, what is the probability that you could have randomly guessed the mystery object?

But you had 2 items in your box, and therefore two possible mutually exclusive elementary events! So what is the probability you would have randomly guessed both correctly? To answer this we need to learn a little more. First, how many possible outcomes were there?

The Counting Principle

If there arexways to perform one task, andyways to perform a second task, then there arexyways to perform both tasks.Suppose for example that we flip two coins. There are 2 outcomes to flip the first coin (heads or tails) and 2 outcomes to flip the second coin, so there are 2×2=4 possible outcomes to flip both coins.

For the Black Box, the possible outcomes are the pairs of objects that could be present. Any pair of objects is equally likely. Therefore, the probability that the Black Box contains a marble and a cork (or any other pair of objects) is equal to one divided by the number of possible pairs.

What would be the probability of correctly guessing that the Black Box contains a marble and a cork? To answer this, we first need to count the number of possible pairs. We’ll start by counting the number of ways there are to form a pair.

In our case of 11 objects, there are eleven ways to choose the first object, but only ten unique ways to add the second member of the pair, since each Black Box tin will contain two different objects. Therefore, according to the counting principle, there 11 × 10, or 110 ways to form a pair.

**Got that? Try these ...**

- Suppose that you have a marble, a metal ball, and a penny. Imagine that you form a pair by choosing two objects from this set. Use the counting principle -- how many ways can you form a pair?
- Write down all the possible unique pairs of objects that could be formed from a marble, a metal ball, and a penny. Why doesn’t this equal the number you got from using the counting principle?

Although there are 110 different ways to form a pair from the set of eleven objects present in the White Box in our experiment, there are actually only 55 different pairs. This is because there are two ways to form every pair. For example, we could form the pair with a metal ball and a rubber ball by choosing the metal ball and then the rubber ball, or by choosing the rubber ball and then the metal ball. Thus, it follows that there are half as many pairs as there are ways to form a pair (110/2 = 55).

Now we can figure out the probability of getting the identity of both objects in the Black Box correct by guessing alone, without the use of the scientific method!

Since there are 55 different pairs of the 11 objects in the White Box tin, and you are equally likely to choose each pair, the probability that you choose a correct pair is:

That is, you make approximately 2 correct guesses for every 100 guesses you make. Thus, you can see that the probability of getting the identity of your mystery objects correct by guessing alone is very low!

**Discussion Question:**

Compare this probability to the proportion of Black Box pairs your teams correctly identified. How did the scientific method help improve your success?

**Extra Challenge**

- How many pairs in our Black Box experiment are in an event that has a penny?. Hint: Remember our definition of an event. Here the outcomes are pairs of objects, and the event of interest is a set of pairs that have pennies.
- What is the probability that you choose a pair that has a penny? Any pair that has a penny is in this event.
- What is the probability that your Black Box has a pair consisting of a penny or a marble, but not both a marble and a penny?
- What is the probability that, by guessing alone, you correctly guess the identity of one (but not both) of the objects in your Black Box?

**Answers:**

- {1,2,3,4}
- B and E are not mutually exclusive because both of them contain elementary events {1} and {4}
- 1/6 or .17 or 17%
- 1/3 or .33 or 33%
- 1/11 or .09 or 9%
- 3 (first item) and 2 (second item), so 3 * 2 = 6
- There are six ways to form a pair if we consider the order that the objects are added: marble and metal ball, marble and penny, metal ball and penny, penny and metal ball, penny and marble, metal ball and marble. But, there are only three distinct pairs of objects, as each unique pair is duplicated.

**Challenge Answers**

- Every object that isn’t a penny can be used to make a pair with a penny. Since there are ten objects that are not pennies, there are ten pairs that have a penny: {penny, poker chip}, {penny, rubber ball}, {penny, rubber band}, {penny, metal ball}, {penny, marble}, {penny, tooth pick}, {penny, paper clip}, {penny, cotton ball}, {penny, cork}, {penny, crayon};
- Since there are ten pairs that have a penny, and 55 possible pairs, the probability that you choose a pair with a penny is:

- Since there are nine pairs that contain a penny but do not contain a marble, and nine pairs that contain a marble but do not contain a penny, there are eighteen pairs in this event. Since there are 55 possible pairs, the probability that this event occurs is

- This is simply a generalization of the previous question. Since the presence of a particular object in the tin is random with respect to other types of objects, the probability that you find a pair that contains one of the objects in your tin but not the other object of interest is the same as the probability that you find that you have a pair of objects in your tin with a marble or a penny, but not both, or

**Extended Links:**

Carl Sagan describes how a scientist figured out the circumference of the earth 2200 years ago through experimentation.

http://www.youtube.com/watch?v=1dBipPfFQJM&feature=fvwrel

They Might Be Giants – Put it to the Test (music video)

http://www.youtube.com/watch?v=9kf51FpBuXQ

___________________

*In Biology by Numbers, learn about the ways math can solve biological problems. Produced by the National Institute for Mathematical and Biological Synthesis (NIMBioS). NIMBioS brings together researchers from around the world to collaborate across disciplinary boundaries to investigate solutions to basic and applied problems in the life sciences. NIMBioS is sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture with additional support from The University of Tennessee, Knoxville.*

## Discussion