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 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 in a mathematically precise way when discussing events and probability are the important discussion points (MP6).

Ask students to share the things they noticed and wondered. Record and display their responses, without editing or commentary. If possible, record the relevant reasoning on or near the Venn diagram. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If understanding how "and," "or," and "not" are used when discussing events 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. Tell students to answer the questions according to the diagram.

• “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)

If time permits, ask students to think about this situation. "You have three people with towers made of colored building blocks. One has a red tower, one has a yellow tower, and one has a tower that is both red and yellow.” Here are some questions for discussion.

• “How many people have a tower with yellow blocks?” (2)
• “How many people have a tower with red blocks?” (2)
• “How many people have a tower with red and yellow blocks?” (1)
• “Is the correct answer to, ‘How many people have a tower with red or yellow blocks?’ 2 or 3? Explain your reasoning.” (It is 3 because all three people used either red or yellow blocks in their tower.)

In this activity, students describe events as a subset of the sample space, using characteristics of the outcomes. Then they determine probabilities for combined events based on the number of outcomes in the event. Students must reason abstractly and quantitatively to make connections between the countries in the real world and their representation in the diagram (MP2).

This activity gives students an opportunity to determine and request the information needed to understand 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 Information 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 to ask increasingly more precise questions until they get the information they need (MP6).

This optional activity provides additional practice understanding combined events. Students roll number cubes to collect data and then analyze the outcomes using language describing characteristics of some outcomes. Then students estimate probabilities based on their collected data.

In the digital version of the activity, students use an applet to simulate rolling number cubes before answering questions about combined events. The applet allows students to collect data quickly, using a number generator. Use the digital version if physical number cubes are not available or if time is limited and students can readily access the applet on a device.