Lesson 5

The purpose of this warm-up is to elicit the idea that Venn diagrams can be used to represent events of a sample space, which will be useful when students describe events as a subset of a set of a sample space using characteristics of the outcomes, and as unions, intersections, or complements of other events in a later activity. While students may notice and wonder many things about these images, understanding how "and," "or," and "not" are used when discussing sample space and probability are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that traits that belong to both bats and birds are listed where the two circles overlap, all of the traits in the circles of the Venn Diagram belong to bats or birds, and the trait listed outside of both of the circles does not belong to birds or bats.

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the Venn Diagram for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the Venn diagram. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If understanding how "and," "or," and "not" are used when discussing sample space and probability does not come up during the conversation, ask students to discuss this idea.

Tell students that when we use "or" in mathematics it is inclusive. Here are some questions for discussion.

• “How many traits belong to birds or bats?” (6)
• “How many traits belong to birds and bats?” (2)
• “How many traits do not belong to birds or bats?” (1)
• “What is an “or” question that you could ask involving something in this room? How would you answer it? Tell your partner.” (I could ask, “How many people have a pencil or pen on their desk?” I would answer it by counting the number of people with a pen or a pencil on their desk.)
• “Ask a question similar to your previous one but using “and” instead of “or.” How is the question different than the previous one?” (My new question is, “How many people have a pencil and a pen on their desk?’ It is different because this would include only the students with both a pen and a pencil.)
• “Is it possible for the answer to the second question to be greater than the answer to the first question? Explain your reasoning.” (No. It is not possible because the “or” question includes all of the possibilities and the second question would have to be equal to or less than the quantity of all possibilities.)

If time permits, ask students to think about this situation. You have three people with bowls of ice cream and one has chocolate, one has vanilla, and one has both chocolate and vanilla.” Here are some questions for discussion.

• “How many people have vanilla ice cream?” (2)
• “How many people have chocolate ice cream?” (2)
• “How many people have chocolate and vanilla ice cream?” (1)
• “Is the correct answer to, ‘How many people have chocolate or vanilla ice cream’ 2 or 3? Explain your reasoning.” (It is 3 because everyone has either chocolate or vanilla?)
• “How is this related to the birds and bats questions?” (It is related because when you ask, ‘Which of these traits belong to birds or bats?’ it is everything but the 6 legs trait.)

The mathematical purpose of this activity is to describe events as a subset of a set of sample space using characteristics of the outcomes, and to determine the probabilities of some events using the subsets of the sample space.

Arrange students in groups of two. Tell students, “The small dots indicate that the name listed in the diagram is a country.” Give students quiet work time and then time to share their work with a partner.

Some students may struggle with the difference between "and" and "or" when used in a situation. Ask students in the class who play a spring sport to stand up and remain standing. Then ask students who play an instrument to stand. Tell students that "the people standing represent the people who play a spring sport or who play an instrument." Ask students who play a spring sport and who play an instrument to raise their hand. Tell students that "the people with their hands raised represent the people who play a spring sport and who play and instrument." Emphasize that "and" statements require all parts of the statement to be true and "or" statements require at least one one part of the statement to be true.

The goal of this discussion is to help students make the connections between subsets of sample spaces and probability, and to give them additional practice using “and,” “or,” and “not” in mathematics.

Here are some questions for discussion.

• “True or false: all the countries on the diagram are crown dependencies or in the British Isles” (True)
• “How many countries are part of the United Kingdom or are crown dependencies?” (7)
• “How many countries are part of the United Kingdom and are crown dependencies?” (0)
• “If a country is chosen at random, what is the probability that it is part of the Isle of Ireland?” (\(\frac{2}{8}\))

This info gap activity gives students an opportunity to determine information about subsets of a sample space. The subsets are then used to find a probability of an event. To complete the tables, students must ask for information from a partner using mathematical language.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Tell students they will continue to work with sample spaces. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What was challenging about solving the problem?” (The challenging part was having to figure out how to use the information from my partner to fill in the table because the information only gave me clues to find the numbers rather than the numbers themselves.)

• "What percentage of students in your table are in 11th grade?" (37.2% for probroblem card 1 and 47.5% for problem card 2.)

Highlight for students how a table can be used to represent sample space and to calculate probabilities and percentages.

The mathematical purpose of this activity is to compare probabilities calculated through experimentation to probabilities calculated by using the sample space. The activity is optional to offer students additional practice with creating sample spaces, calculating probability, and comparing hands-on experiments to theoretical probabilities. 

Some students may not know how to represent probability as a percentage. Prompt students to multiply the value they obtained to represent probability (between 0 and 1 inclusive) by 100 to get the percentage. Emphasize that a percentage is a quantity described by a rate per 100.

The goal of this discussion is to make sure students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent. A secondary goal of the discussion is to informally assess student understanding of how “and,” “or,” and “not” are used when finding probabilities.

Here are some questions for discussion.

• “Why is the sample space 36 outcomes for rolling two number cubes?” (It is 36 outcomes because there are 6 choices for the first cube and 6 choice for the second cube, 6 times 6 is 36. You can see this in a tree diagram from a previous lesson.)
• “According to the sample space there are 36 different outcomes for rolling two number cubes. You only rolled the cubes 20 times. Raise your hand if you rolled the same outcome more than once when you rolled 20 times. Why is this possible?” (It is possible to repeat the same roll more than once because rolling the dice is random. Just because you rolled double sixes on one turn does not reduce your chance of rolling double sixes on the next turn.)
• “How are the events “rolling doubles” and “rolling doubles and rolling a 6 on the first cube” different? Explain your reasoning.“ (The difference between the two questions is that the first question included all of the double rolls, but the second question only counts the rolls when both cubes are six.)
• “What is the difference between the probability of “rolling doubles” and the probability of “rolling doubles and rolling a 6 on the first cube”? Explain your reason” (It is 0.15. The probability of “rolling doubles” was 0.25 and the probability of “rolling doubles and rolling a 6 on the first cube is 0.10, 0.25 minus 0.10 equals 0.15.)
• “For your 20 rolls, what percentage of them have the first die equal to 1 or 2?” (Sample response: 0.4 or 40%)
• “Using the sample space, what is the probability of rolling a 1 or a 2 on the first die?” (\(\frac{12}{36}\) or 33.3%)