# Lesson 7

Simulating Multi-step Experiments

## 7.1: Notice and Wonder: Ski Business (5 minutes)

### Warm-up

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.

### Launch

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.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

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.

## 7.2: Alpine Zoom (15 minutes)

### Activity

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.

### Launch

Arrange students in groups of 3. After students have had a chance to think about an experiment themselves, select groups to share their responses. Give students 5 minutes for partner discussion, 5 minutes to run the simulation, then 5 minutes for a whole-class discussion.

Students using the digital version have an applet on which they can run up to 10 simulation trials.

Representation: Develop Language and Symbols. Eliminate barriers and provide concrete manipulatives to connect symbols to concrete objects or values. Provide access to simulation tools, such as number cubes, papers in bages, and spinners.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Design Principle(s): Optimize output; Maximize meta-awareness

### Student Facing

Alpine Zoom is a ski business. To make money over spring break, they need it to snow at least 4 out of the 10 days. The weather forecast says there is a $$\frac{1}{3}$$ chance it will snow each day during the break.

Use the applet to simulate the weather for 10 days of break to see if Alpine Zoom will make money.

1. Describe a chance experiment that you could use to simulate whether it will snow on the first day of break.
2. How could this chance experiment be used to determine whether Alpine Zoom will make money?
• In each trial, spin the spinner 10 times, and then record the number of 1’s that appeared in the row.

• The applet reports if the Alpine Zoom will make money or not in the last column.

• Click Next to get the spin button back to start the next simulation.

3. Based on your simulations, estimate the probability that Alpine Zoom will make money.

### Launch

Arrange students in groups of 3. After students have had a chance to think about an experiment themselves, select groups to share their responses.
If possible, allow them to use the simulation they have suggested (rolling a number cube, papers in a bag, etc.). If the simulation is not readily available, provide each group with a spinner from the blackline master. Give students 5 minutes for partner discussion, 5 minutes to run the simulation, then 5 minutes for a whole-class discussion.

Students using the digital version have an applet on which they can run up to 10 simulation trials.

Representation: Develop Language and Symbols. Eliminate barriers and provide concrete manipulatives to connect symbols to concrete objects or values. Provide access to simulation tools, such as number cubes, papers in bages, and spinners.
Supports accessibility for: Visual-spatial processing; Fine-motor skills
Design Principle(s): Optimize output; Maximize meta-awareness

### Student Facing

Alpine Zoom is a ski business. To make money over spring break, they need it to snow at least 4 out of the 10 days. The weather forecast says there is a $$\frac{1}{3}$$ chance it will snow each day during the break.

1. Describe a chance experiment that you could use to simulate whether it will snow on the first day of spring break.
2. How could this chance experiment be used to determine whether Alpine Zoom will make money?

Pause here so your teacher can give you the supplies for a simulation.

3. Simulate the weather for 10 days to see if Alpine Zoom will make money over spring break. Record your results in the first row of the table.

day 1 day 2 day 3 day 4 day 5 day 6 day 7 day 8 day 9 day 10 Did they make money?
simulation 1
simulation 2
simulation 3
simulation 4
simulation 5
4. Repeat the previous step 4 more times. Record your results in the other rows of the table.

5. Based on your group’s simulations, estimate the probability that Alpine Zoom will make money.

### Anticipated Misconceptions

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

### Activity Synthesis

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.

• “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.)

## 7.3: Kiran’s Game (15 minutes)

### Optional activity

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.

### Launch

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.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer for recording their results and calculations of probability of winning Kiran’s game.
Supports accessibility for: Language; Organization

### Student Facing

Kiran invents a game that uses a board with alternating black and white squares. A playing piece starts on a white square and must advance 4 squares to the other side of the board within 5 turns to win the game.

For each turn, the player draws a block from a bag containing 2 black blocks and 2 white blocks. If the block color matches the color of the next square on the board, the playing piece moves onto it. If it does not match, the playing piece stays on its current square.

1. Take turns playing the game until each person in your group has played the game twice.
2. Use the results from all the games your group played to estimate the probability of winning Kiran’s game.
3. Do you think your estimate of the probability of winning is a good estimate? How could it be improved?

### Student Facing

#### Are you ready for more?

How would each of these changes, on its own, affect the probability of winning the game?

1. Change the rules so that the playing piece must move 7 spaces within 8 moves.

2. Change the board so that all the spaces are black.

3. Change the blocks in the bag to 3 black blocks and 1 white block.

### Activity Synthesis

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.

• “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.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their estimated probability of winning Kiran’s game, present an incorrect solution based on a misconception that arises when conducting a few trial simulations. For example: “The probability of winning Kiran’s game is zero because nobody in our group won in all of the six games played.” Ask students to identify the error, critique the reasoning, and revise the original statement. As students discuss in partners, listen for students who state that estimating the probability of winning based on only six trials will not result in an accurate estimate. Therefore, more trials must be conducted in order to improve the estimate of the probability of winning. This routine will engage students in meta-awareness as they critique and correct the language used to estimate the probability of winning a game.
Design Principles(s): Cultivate conversation; Maximize meta-awareness

## 7.4: Simulation Nation (10 minutes)

### Activity

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).

### Launch

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.

Conversing: MLR8 Discussion Supports. Invite students to take turns matching a situation to a simulation. Display sentence frames to help students explain their reasoning. For example, "Situation ___ matches with simulation ___ because . . ." Listen for the connection between the numerical quantities in the simulation and the situation. Encourage students to challenge each other when they disagree. This will help students justify how all the parts of the simulation can be used to match accordingly with the situation.
Design Principle(s): Support sense-making; Cultivate conversation

### Student Facing

Match each situation to a simulation.

Situations:

1. In a small lake, 25% of the fish are female. You capture a fish, record whether it is male or female, and toss the fish back into the lake. If you repeat this process 5 times, what is the probability that at least 3 of the 5 fish are female?

2. Elena makes about 80% of her free throws. Based on her past successes with free throws, what is the probability that she will make exactly 4 out of 5 free throws in her next basketball game?

3. On a game show, a contestant must pick one of three doors. In the first round, the winning door has a vacation. In the second round, the winning door has a car. What is the probability of winning a vacation and a car?

4. Your choir is singing in 4 concerts. You and one of your classmates both learned the solo. Before each concert, there is an equal chance the choir director will select you or the other student to sing the solo. What is the probability that you will be selected to sing the solo in exactly 3 of the 4 concerts?

Simulations:

1. Toss a standard number cube 2 times and record the outcomes. Repeat this process many times and find the proportion of the simulations in which a 1 or 2 appeared both times to estimate the probability.

2. Make a spinner with four equal sections labeled 1, 2, 3, and 4. Spin the spinner 5 times and record the outcomes. Repeat this process many times and find the proportion of the simulations in which a 4 appears 3 or more times to estimate the probability.

3. Toss a fair coin 4 times and record the outcomes. Repeat this process many times, and find the proportion of the simulations in which exactly 3 heads appear to estimate the probability.

4. Place 8 blue chips and 2 red chips in a bag. Shake the bag, select a chip, record its color, and then return the chip to the bag. Repeat the process 4 more times to obtain a simulated outcome. Then repeat this process many times and find the proportion of the simulations in which exactly 4 blues are selected to estimate the probability.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Representation: Internalize Comprehension. Use color and annotations to illustrate connections between representations. As students share their reasoning for matching situations with simulations, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing

## Lesson Synthesis

### Lesson Synthesis

• “How are the simulations in this lesson different from the simulations in the previous lesson?” (These have multiple parts for each experiment. Also, it would be difficult to compute the exact probability, so simulations seem more necessary.)
• “The chance that it will be cloudy on a single day is simulated by rolling a standard number cube twice. How many times will the number cube need to be rolled to simulate a week?” (14 times. It is rolled twice for each day and there are 7 days in a week, so 14 rolls are needed.)
• “Each day, a student randomly reaches into a bowl of fruit and picks one for their lunch that day. To simulate the situation, he creates a spinner with 4 equal sections labeled: apple, orange, watermelon, and peach. Why might this simulation not represent the situation very well?” (Usually watermelons are much larger than the other 3 fruits listed, so there is probably not an equal chance of that being selected, so the spinner should probably have a larger wedge for watermelons.)

## Student Lesson Summary

### Student Facing

The more complex a situation is, the harder it can be to estimate the probability of a particular event happening. Well-designed simulations are a way to estimate a probability in a complex situation, especially when it would be difficult or impossible to determine the probability from reasoning alone.

To design a good simulation, we need to know something about the situation. For example, if we want to estimate the probability that it will rain every day for the next three days, we could look up the weather forecast for the next three days. Here is a table showing a weather forecast:

today
(Tuesday)
Wednesday Thursday Friday
probability of rain 0.2 0.4 0.5 0.9

We can set up a simulation to estimate the probability of rain each day with three bags.

• In the first bag, we put 4 slips of paper that say “rain” and 6 that say “no rain.”
• In the second bag, we put 5 slips of paper that say “rain” and 5 that say “no rain.”
• In the third bag, we put 9 slips of paper that say “rain” and 1 that says “no rain.”

Then we can select one slip of paper from each bag and record whether or not there was rain on all three days. If we repeat this experiment many times, we can estimate the probability that there will be rain on all three days by dividing the number of times all three slips said “rain” by the total number of times we performed the simulation.