Lesson 7

The purpose of this warm-up is to elicit ideas that will be useful in the discussions in this lesson. While students may notice and wonder many things about these images, the business side of skiing and its dependance on weather are the important discussion points.

Arrange students in groups of 2. Tell students that they will look at two images, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the images for all to see. Ask students to give a signal when they have noticed or wondered about something. 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 image. After each response, ask the class whether they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If the dependence of ski businesses on weather does not come up during the conversation, ask students to discuss this idea.

In this activity, students continue to model real-life situations with simulations MP4), but now the situations have more than one part. Finding the exact probability for these situations is advanced, but simulations are not difficult to run and an estimate of the probability can be found using the long-run results from simulations (MP8). If other simulation tools are not available, you will need the blackline master.

Students may be confused by the phrase “at least 4 days.” Explain that in this context, it means 4 or more.

The purpose of this discussion is for students to understand the connection between the results of their simulation and the real-life situation.

Ask each group for the number of times Alpine Zoom made money in their simulations.

Consider asking these discussion questions:

• “Using the class’s data, estimate the probability that Alpine Zoom will make money.” (Theoretically, this should be close to 45%.)
• “Do you anticipate Alpine Zoom will make money this spring break?” (It’s not likely, but it’s possible.)
• “Over the next 10 years, if the weather patterns continue to be the same, do you anticipate Alpine Zoom will make money over that time or not?” (This is even less likely. There is less than a 50% chance it will make money each season, so over 10 years, it will probably lose money more than make money.)
• “Is this a business you would invest in? Explain your reasoning.”
  (I would not invest in it because it is unlikely to make money over the years.)

Since this activity is mainly included for practice and may take some additional time to complete, it is included as an optional task and including it is up to the teacher’s discretion.

In this activity, students practice doing many trials of multi-step situations to estimate the probability of an event. In the discussion following the activity, students construct arguments (MP3) about how changes to the game might affect the probability of winning.

Arrange students in groups of 3. Provide each group with a paper bag containing 2 black blocks and 2 white blocks inside. If black and white blocks are not available, instruct students on their color equivalents. Give students 5 minutes to run the simulation, 5 minutes for partner discussion, then have a whole-class discussion.

The purpose of the discussion is for students to think about how changing the rules of the game might change the probability of winning. 

Collect the data from the class for the number of wins and display the results for all to see.

Consider asking these discussion questions:

• “Based on the class’s data, estimate the probability of winning the game.” (The theoretical probability of winning is \(\frac{3}{16} \approx 0.19\).)
• “Does the game seem too easy or hard to win? If so, how could Kiran change the game slightly to make it harder or easier?” (If it is too hard, move the pawn closer to the end or allow more than 5 moves to win.)
• “The bag contained 2 black and 2 white blocks. If the bag had 4 blocks of each color, would that make it easier or harder to win?” (Neither. It would be the same difficulty since there is still an equal chance to get each color.)
• “Do you think the estimate from the class’s data is a better estimate than the one you got on your own?” (Since there is more data from the entire class, it should be a better estimate than one individual’s.)

In this activity, students practice what they have learned about simulations by matching real-life scenarios to simulations. In the discussion, students are asked to explain their reasoning for their choices and think about other valid choices that could be made (MP3).

Keep students in groups of 3. Give students 5 minutes of small-group time to take turns matching the items and discussing their reasoning, followed by whole-class discussion.

Students may not see the connection between the standard number cube and the situation with 3 doors. Remind students it is important that the probabilities match, but not necessarily the outcomes. Since the simulation matches 2 of the outcomes to one door, the probabilities will match.

The purpose of this discussion is for students to articulate the reasons they chose to match the items they did.

For each situation, select students to explain why the simulation should go with it. Although some students may have just looked at a portion of the situation and simulation, encourage students to explain all of the parts of the simulation. Consider the problem with fish; 25% is mentioned and the spinner is the only option that also has a 25% chance associated with it. Prompt students for more details by asking,

• “Why do we need to spin the spinner 5 times?” (A fish is selected from the lake 5 times.)
• “Why does the number need to show up 3 or more times?” (We want a probability that three or more fish are female.)
• “What do the numbers 1 through 4 represent when doing a trial with the spinner?” (Each section represents a \(\frac 14\) probability. The section labeled ‘4’ is the 25% chance that a fish will be female, while sections labeled 1–3 are the 75% chance that a fish will not be female.)
• “Could the spinner have 8 sections? If so, how would you label the sections? What would each label represent?” (Yes, labels vary. Sample response: Label sections 1–8, where sections 7–8 represent the 25% chance that a fish will be female, while sections labeled 1–6 are the 75% chance that a fish will not be female.)

For each of the scenarios, ask students if any part of it could be changed and still result in the simulation working. For example, there could be 4 blue chips and 1 red chip in the bag for simulation D. For simulation C, we could count the fraction of times when 3 tails appear rather than heads.