Ohio Standards Connections: 5. Data Analysis and Probability: Students pose questions & collect, organize, represent, interpret & analyze data to answer those questions. Students develop & evaluate inferences, predictions & arguments that are based on data. Benchmark K. Make predictions based on theoretical probabilities & experimental results. English/Language Arts: 9. Research Standard: Students define & investigate self-selected or assigned issues, topics & problems. Benchmark: B. Evaluate the usefulness & credibility of data & sources. C. Organize information from various resources & select appropriate sources to support central ideas, concepts & themes. 6. Writing Applications. Students need to understand that various types of writing require different language, formatting and special vocabulary. Data Analysis: Benchmark I. Design an experiment to test a theoretical probability, and record & explain results. Technology: 5. Technology & Information Literacy: Students engage in information literacy strategies, use the Internet, technology tools & resources, & apply information-management skills to answer questions & expand knowledge. Benchmark B: Apply a research process model to conduct research & meet information needs. Library 5: Effective school library media programs provide information literacy skills instruction. Benchmark D: Utilizes a research model to locate, use, & evaluate information.

Lesson Summary: This lesson is an introduction to our in-depth study of probability. The purpose is to explore the difference between theoretical and experimental probability through the Monty Hall Problem (see attached). Estimated Duration: 5 -54 minute periods Commentary: This lesson helps show students that probability is not always what it appears to be. Most make guesses regarding the probability of winning and are very surprised when they begin experimenting and then eventually learn the way to calculate the probabilities. Pre-Assessment: Explain a number of probability experiments and ask students to draw conclusions. Each student should write out their answers and their explanations to be read by the teacher for an initial assessment on the students’ ability and understanding. Fill a bag with 2 colors of marbles and ask the probability of pulling out a marble of a particular color. Ask the students to determine how many different ways 4 kids could sit in a row at the movie theater. Given “BANANA,” how many distinguishable ways of ordering the letters can they find? If you flip a coin three times, what’s the probability that at least 2 of them are “heads?” Scoring Guidelines: These written responses must be read and the teacher needs to use them to gauge the background information the class has a theory of probability. Often, an extra couple of days are needed to be sure there is an understanding that is sufficient to understand the Monty Hall Problem and how to calculate the probability in the end. Post-Assessment: Each student is required to write an essay that compares and contrasts the experimental results of the student with the theoretical results found in articles. Students will use databases and/or the internet to locate these articles. Students should reach the correct conclusion at the end of the lesson that it is better to switch curtains; should be able to explain why this is the better choice. Scoring Guidelines: The essay is worth 10 points: Up to 3 points are given for the student’s use of experimental data to justify an initial hypothesis. Up to 2 points will be given for a discussion of how probabilities must add up to 1. Up to 4 points will be given for the theoretical explanation that the student read and discussed with others. 1 point is given for general neatness and following directions. Instructional Procedures: 1. Discuss the “Lets Make a Deal” Game and play the 3 curtain game 2 or 3 times as a class. 2. At this point, have the students make an initial guess, in paragraph form, to explain whether they think it is better to stay with their first curtain, switch or if there is no difference. They must explain why they took this position. 3. Each student should play the curtain game at least 20 times with a partner, recording the results of each game. 4. The students should now predict which choice they think is the better choice. Again, this should be in paragraph form and their 20 games should be used to support their conclusions. 5. The data of the entire class will now be combined. This will allow them to see how many times “switching” leads to winning and “not switching” leads to winning. There will be over 400 data points to work with and this sample size should help students to predict the “right” answer. 6. Students need to write a third essay based on the larger sample. Arguments should be clear and answers should be backed up by the data. 7. Visit the library to use the databases or internet to begin researching the “Monty Hall problem.” 8. Students will read articles and get help when needed with interpreting the article. 9. Students are to write two final paragraphs. The first paragraph discusses the theoretical probability learned from the articles and the second compares their data to the theoretical data. They must answer the question “did the data represent the probability?” Differentiated Instructional Support Students on IEPs (for writing disabilities) can verbally explain each of the above steps to a teacher or to a partner. A search of the internet will find various articles on the Monty Hall problem with a broad range of reading levels. An appropriate level must be found for each student. Extension This unit is the beginning of the study of probability that meets all the probability strands at the ninth grade level. Homework Options and Home Connections Homework will depend on the level of independence and the ability of the class. In this particular class, the lesson was completed in class. Students were using homework time to review past lessons. Interdisciplinary Connections Every student studies probability in genetics in their Life Sciences courses at Rocky River High School. Essay writing and comparing and contrasting data are essential skills in all content areas. Materials and Resources: Students have the option of searching the EBSCOhost or Gale databases offered through the library. They are also able to use search engines such as Google. The use of Ask Jeeves is discouraged as results are less than satisfactory. Key Vocabulary Experimental data, experimental probability, theoretical probability, hypothesis Technology Connections Students will be using the library computers and/or the mobile cart of laptops to complete their research. Students will be doing word processing for the creation of their essays. Enrichment programming for the TI-82 graphing calculators can be used as well. Research Connections Students will use the internet for research as it is the easiest, and the quickest, way for them to locate material on the Monty Hall problem. Databases proved too limited in their coverage of this game and the articles located on mathematical probabilities were too esoteric for the purpose of the assignment. Due to these difficulties the assistance of the teacher and the librarian are essential to student success. General Tips It is important that a teacher often checks the progress of each student for understanding. Each paragraph that is written should be read before moving on so that certain misconceptions can be dealt with during the process. Attachments See attached for an article explaining the Monty Hall Problem as well as a list of websites students can use to understand the problem.