## Who Would Have Figured?

### Melanie Malone

#### Description

Students discover what happens when a coin is tossed a few times versus when a coin is tossed many times. They discover the answer to -What is the probability of heads, and does it change as you toss the coin more times?-

#### Objectives

The student compares experimental results with mathematical expectations of probabilities.

#### Materials

-Copies of tables for each pair of students (See attached file.)
-Pennies: a set of 10 for each pair of students
-Pencil
-Paper for writing results
-Optional teacher resource: HISTORICAL CONNECTIONS IN MATHEMATICS (Volume 1), by Luetta and Wilbert Reimer, 1992, AIMS Education Foundation
-Rubric (see attached file)

#### Preparations

1. Duplicate copies of tables (worksheet).
2. Set up teams for each class.
3. Put together sets of 10 pennies for each pair of students (Students can help with this).

#### Procedures

1. Lesson introduction: Begin discussion of probability with a talk about Pierre de Fermat and Blaise Pascal. The AIMS Historical Connections in Mathematics is a great source for information about mathematicians. Volume 1 has information about both in it. Another option is to do a search of the Internet for information about these gentlemen. Make transparencies of the information and then go over it with the class. Include a few pertinent questions about those people on the next test given to the class. Students are responsible for writing down the contributions of each mathematician discussed.

2. Group students or have students group themselves in pairs. Give out the pennies and worksheets. Before students begin, discuss the Law of Large Numbers and basic coin toss probability. The probability of tossing a heads on any given coin toss is:

P(heads) = 1/2 =50%

*Note: Students need to be completely familiar with computing percentages.

3. Have students toss their coins 10 times and fill in the first line of the worksheet. Have them continue this until they have done 5 sets of 10 coin tosses (50 tosses).

4. Next have one of the pair share their information with another team. Both teams should exchange their information for 50 coin tosses and this information should be added together and listed under the 100 coin toss section of the worksheet.

5. If time permits, have each pair exchange their information for 100 coin tosses with other groups and try to get 1000 toss information (table 2). What students should discover is the Law of Large Numbers. That is, if a coin is tossed a large number of times, heads should occur about one-half of the time. The more coin tosses the class can collect, the more obvious this becomes.

6. At this point, the teacher can have each team prepare a short essay or a paragraph or two about their findings. Remind students to start with the first sets of coin tosses (10 times each) and compare this data with the larger number of coin tosses. Share assessment criteria with them before they begin, so they are aware of teacher expectations.

#### Assessments

Student group papers should reflect the process they used (which groups they visited for additional data, what that data was). The teacher can set up the papers in a scientific manner so that the group needs to give an initial hypothesis and then give a concluding hypothesis. The teacher can also have students make observations after 10 tosses, 50 tosses, 100 tosses, etc. This way students are forced to see how the data can change. Another fun aspect of this is to have them create a group graph of the 10 tosses and see if students can determine a trend. Base the grade on the following questions:
1. What was the initial hypothesis?
2. What was the concluding hypothesis or conclusion?
3. What conjectures were made during various intervals of the data collection process?
4. What data supports your conclusion?
A rubric for assessing this activity is included in the attached file.

#### Attached Files

Both of the tables listed under Procedures and a grading rubric to use as is or edit for use.     File Extension: pdf