In this Warm-up, students consider a situation and propose questions related to the situation. While students may ask many different kinds of questions, the important discussion points are about understanding the context and the relationship to simulation and probability.
Launch
Tell students to close their books or devices (or to keep them closed). Arrange students in groups of 2. Introduce the context of businesses built around skiing. Use Co-Craft Questions to orient students to the context and elicit possible mathematical questions.
Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation before comparing questions with a partner.
Student Lesson in Spanish
Activity
None
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.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite several partners to share one question with the class and record responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “probability” and “simulation.”
If it does not come up, ask how students might find the probability of snow in the next few days (look at a weather forecast).
7.2
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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). Provide access to a variety of simulation tools such as number cubes, paper and bags, and protractors for spinners for students to use (MP5). If other simulation tools are not available, you will need the blackline master.
In the digital version of the activity, students use an applet to spin a spinner to simulate the situation. The applet allows students to spin a spinner and record the outcome of the spin. The digital version may be preferable if students would benefit from running and understanding a simulation rather than designing and using their own simulation, or if there is not enough time to create and run the simulations by hand.
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. 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.
Activity
None
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 chance it will snow each day during the break.
Describe a chance experiment that you could use to simulate whether it will snow on the first day of spring break.
How could this chance experiment be used to estimate a probability that Alpine Zoom will make money?
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
Repeat the previous step to simulate the entire break 4 more times. Record your results in the other rows of the table.
Based on your group’s simulations, estimate the probability that Alpine Zoom will make money.
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.
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 it will make.)
“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
Activity
Optional
Standards Alignment
Building On
Addressing
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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.
Representation: Internalize Comprehension. Provide students with a graphic organizer, such as a two-column table, to record the results of the simulation and probability of winning. Supports accessibility for: Visual-Spatial Processing, Organization
Activity
None
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.
A game board with one row of alternating white and black squares. There are five squares in total and the board begins and ends with a white square. A playing piece is located on the first white square.
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.
Take turns playing the game until each person in your group has played the game twice.
Use the results from all the games your group played to estimate the probability of winning Kiran’s game.
Do you think your estimate of the probability of winning is a good estimate? How could it be improved?
Student Response
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Building on Student Thinking
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.
Consider asking these discussion questions:
“Based on the class’s data, estimate the probability of winning the game.” (The theoretical probability of winning is .)
“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 piece closer to the end or allow more than 5 moves to win.)
“The bag contains 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.)
7.4
Activity
Standards Alignment
Building On
Addressing
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
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. Then follow with a whole-class discussion.
MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frame for all to see: “I noticed , so I matched . . . .” Encourage students to challenge each other when they disagree. Advances: Speaking, Listening
Activity
None
Match each situation to a simulation.
Situations:
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?
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?
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?
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:
Toss a standard number cube 2 times and record the outcomes. To estimate the probability, repeat this process many times and find the proportion of the simulations in which a 1 or 2 appears both times.
Make a spinner with four equal sections labeled 1, 2, 3, and 4. To estimate the probability, 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.
Toss a fair coin 4 times and record the outcomes. To estimate the probability, repeat this process many times, and find the proportion of the simulations in which exactly 3 heads appear.
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. To estimate the probability, repeat this process many times and find the proportion of the simulations in which exactly 4 blues are selected.
Student Response
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Building on Student Thinking
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 them to explain all of the parts of the simulation. For example, in the situation involving 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 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. 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 coding and annotations to highlight connections between representations in a problem. For example, color code matching situations and simulations. Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
Consider asking these discussion questions:
“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 reaches into a bowl of fruit and randomly selects 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
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 1 slip of paper from each bag and record whether or not the simulation predicts that there will be 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 say “rain” by the total number of times we perform the simulation.
Standards Alignment
Building On
Addressing
Building Toward
7.SP.C.8.c
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?
