Lesson 6

This warm-up prompts students to compare four images. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the images in comparison to one another. To allow all students to access the activity, each image has one obvious reason it does not belong. Encourage students to use appropriate terminology (eg The bottom left spinner is the only one with an outcome that has a probability greater than 0.5).

During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the unit.

Arrange students in groups of 2–4. Display the image for all to see. Ask students to indicate when they have noticed which image does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each image doesn’t belong. Follow with a whole-class discussion.

Ask each group to share one reason why a particular image does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as probability. Also, press students on unsubstantiated claims.

In this activity, students estimate the probability of a real-world event by simulating the experience with a chance experiment (MP4). Students see that that multiple simulation methods can result in similar estimates for the probability of the actual event.

Arrange students in groups of 3. Prepare each group with supplies for 1 type of simulation: choosing a slip from a bag, spinning a spinner, or rolling 2 number cubes.  The supplies for each of these simulations include:

• a bag containing a set of slips from the blackline master
• a spinner cut from the blackline master, a pencil and a paper clip
• 2 standard number cubes

Set up the following simulation by telling the students: Diego must cross a busy intersection at a crosswalk on his way to school. Some days he is able to cross immediately or wait only a short while. Other days, he must wait for more than 1 minute for the signal to indicate he may cross the street. We will simulate his luck at this intersection using different methods and estimate his probability of waiting more than 1 minute.

Teacher note: The bag of papers and spinner are designed to have a probability of 0.7 to wait more than 1 minute. The number cubes have a probability of approximately 0.72 to wait more than 1 minute. To the extent that the students are estimating the probabilities, these are close enough to give similar results.

Give students 15 minutes for group work followed by a whole-class discussion.

The purpose of this discussion is for students to understand why simulations are useful in place of actual experiments.

Select at least one group for each of the simulation methods to display the materials they used to run their simulation and explain the steps involved in using their materials.

Ask students, "Why do you think these simulations are more useful than actually doing the experiment many times?" (It would take a lot of time and work for Diego to walk to school more than usual, but it is easy to do the simulation many times quickly.)

Select students to share what they noticed about the graph of the fraction of days Diego had to wait as the simulated days went on. 

In this activity, students have the opportunity to design their own simulations that could be used to estimate probabilities of real-life events (MP4).  Students attend to precision (MP6) by assigning each possible outcome for the real-life experiment to a corresponding outcome in their simulation in such a way that the pair of outcomes have the same probability. In the discussion following the activity, students are asked to articulate how these simulations could be used to estimate probabilities of certain events.

Keep students in groups of 3. Give students 5 minutes quiet work time to design their own experiments, followed by small-group discussion to compare answers for the situations and whole-class discussion.

As students work, monitor for students who are using the same chance events for multiple scenarios (for example, always using a spinner) and encourage them to think about other ways to simulate the event.

Students may think that the number of outcomes in the sample space must be the same in the simulation as in the real-life situation. Ask students how we could use the results from the roll of a standard number cube to represent a situation with only two equally likely outcomes. (By making use of some extra options to count as "roll again.")

The purpose of this discussion is for students to think more deeply about the connections between the real-life experiment and the simulation.

Select partners to share the simulations they designed for each of the situations.

Some questions for discussion:

• "How could a standard number cube be used to simulate the situation with the buses?" (Each bus is assigned a number 1 through 4. If the cube ends on 5 or 6, roll again.)
• "If one of the buses was numbered with your math teacher's favorite number and you wanted to increase the probability of that bus being selected, how could you change the simulation to do this?" (Add more of the related outcome. For example, using the standard number cube as in the previous discussion question, the bus with the favorite number could be assigned numbers 4 and 5 while the other buses are still 1 through 3.)
• Two of the tunnels in the video game lead to a swamp that ends the game. How could you use your simulation to estimate the probability of choosing one of those two tunnels? (Since all of the tunnels are equally likely to lead to the swamp, it can be assumed that "left" and "right" lead to the swamp. Spin the spinner many times and use the fraction of times it ends on "left" or "right" to estimate the probability of ending the game. It should happen \(\frac{2}{3}\) or about 67% of the time.)
• "You and a friend are among the people going to lunch. How could you use the simulation you designed to estimate the probability that you or your friend will be the one to choose the restaurant?" (My friend and I will be represented by 1 and 2 on a number cube. Roll the number cube a lot of times and find the fraction of times 1 or 2 appear, then estimate the probability that we will be the ones selected.)