Modeling with Random Devices (7.SP.8c)
Domain:
Statistics and Probability
Big Idea (Cluster):
Investigate chance processes and develop, use, and evaluate probability models
Common Core Standards:
7.SP.8c 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: For example, 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?
Mathematical Practice(s):
MP 1 Make sense of problems and persevere in solving them
MP 2 Reason abstractly and quantitatively
MP 3 Construct viable arguments and critique the reasoning of others
MP 4 Model with mathematics
MP 5 Use appropriate tools strategically
MP 7 Look for and make use of structure
Content Objectives:
Language Objectives:
Students will be able to describe how a random
device could be used to simulate a real-life
situation.
Talk with your group about key words in a real-life
situation
Students will be able to determine if a specific
random device could be used to simulate a real-life
situation.
Vocabulary:
Equally Likely
Model
Likelihood
Random Device
Experimental Probability
Write a reason a model will simulate a real-life
situation
Write numbers and words to explain your thinking
Prior Knowledge: Concepts students need to
know
Outcome
Probability
Trial
Simulation
Random
In Grade 7, students write the same value represented
as a fraction, decimal or percent.
Beginning the 2014-2015 school year, Grade 7 will
be the only K-8 grade level where probability is
taught. Deep understanding will need to be the focus
of instruction.
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Modeling with Random Devices (7.SP.8c)
Questions to Develop Mathematical Thinking:
Common Misconceptions/Challenges:

Challenge: Students often struggle to identify a
model to simulate a given real-life situation.
How would you identify the number of
possible outcomes given the probability in
percent, decimal, or fraction form?
 What type of tool could be used to help
simulate the desired event if you know the
number of outcomes?
 Would a spinner, area model, number cube, or
coin have similar outcomes as your real-life
situation?
 Has your group identified the “key
component” and “assumptions” in the real-life
situation?
 What are the “assumptions” in the real-life
situation that might help you determine a
model to run a simulation?
 How might the number of outcomes help you
determine the number of trials to run in the
simulation?
Strategy: First, have the students write the
probability in fraction form. Next, identify the
number of outcomes for the given real-life probability
situation based on the denominator. Then, have the
students discuss equivalent forms of the fraction in
order to choose an appropriate model to simulate a
real-life situation.
Spinners are a useful choice since the number of
sections may be adjusted to match probabilities.
Coins and two-colored chips are useful for
probabilities of ½. A number cube is useful for
probabilities that are multiples of 1/6. A standard
deck of playing cards are useful for multiples of ¼.
ASSESSMENT:
Observe student work and listen to student discussions for:




Accurate identification of the “key component” the “assumptions” for each real-life situation
Descriptions of the model having same number of outcomes as the probability of the desired event
Mathematical evidence supporting how the random device could or could not be used to simulate
the real-life situation
Connections between the number of trials that should be run in the simulation and the “assumption”
in the real-life situation
Use B1 as a formative assessment. Students should realize a coin only has two possible outcomes and you
cannot model fourths with a coin.
MATERIALS:
Copies of “Understanding a Simulation” notes
Copies of “Modeling with Random Devices”
Optional: Random Devices (number cube, coin, spinners)
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Modeling with Random Devices (7.SP.8c)
Launch: (10-15 minutes)
Ask students if they have ever run an experiment at home or in a class. Have students describe the types of
experiments they have run before. Ask students what are some of the important details to pay attention to
when you are running an experiment. Students might respond with variables, accurate measurements,
data, predictions and conclusions. There are also “experiments” in a math class. We call them
simulations. A simulation is a model of an experiment used to find the likelihood of an event. A
simulation is used to help answer questions about real-life situations by using probability experiments that
are similar to the real-life situation.
Students will need some general background knowledge on simulations and the process of designing a
simulation. Hand out the “Understanding a Simulation” notes and review the process of designing a
simulation. The “Key Components” and “Assumptions” will help students make a decision about the type
of model that may be used to simulate a real-life probability situation. The “key component” is the desired
event. The “assumptions” identify the desired number of outcomes based on a predicted probability. The
“assumptions” also identify the number of trials that should be run in the simulation. The number of
outcomes determines the type of model or random device students will use to simulate the real-life
situation.
Have students practice identifying the “key component” and “assumptions” in their groups with the
following example:
In a group of 5 people, what is the chance that two were born in the same month?
The “key component” is month of birth. The “assumptions” are all 12 months are equally likely and you
have a group of 5 people. The simulation model must have an option of 12 outcomes. A student would
run trials in multiples of 5 for the simulation. A spinner with 12 equal sections, each identified with a
different month, could be used to simulate the real-life situation.
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Modeling with Random Devices (7.SP.8c)
Explore: (20 minutes)
Hand out the “Modeling with Random Devices” student page and have students read the directions in their
groups. Have students share out to the class or group student actions the teacher and students should see
and hear during the task. For example, we should see all students leaning in to discuss with the group and
writing their response with mathematical evidence. We should hear all students discussing “key
components” and “assumptions,” describing the random device, and providing mathematical evidence for
their reasoning.
Have random device models available to students during exploration. If students need an extension, they
may use a random device and run a simulation to see if the goalie will save half of the shots or if the
basketball player will hit 75% of the free-throws.
When I observe students:






Listen for students making sense that the “key component” is saving the shot on goal and the
“assumptions” are twelve shots are attempted and the goalie saves ½ of the attempted shots (MP 1
and MP 7)
When students struggle to find the difference between the two concepts, have students identify the
outcomes of the probability situation. Then, have students identify the desired event. The desired
event is the “key component” and the probability value is the “assumption.”
Listen for descriptions of the model having same number of outcomes as the probability of the
desired event (MP 2)
Observe student work for mathematical evidence supporting how the random device could or could
not be used to simulate the real-life situation (MP 3 and MP 5)
Listen and observe student work for connections between the number of trials that should be run in
the simulation and the “assumption” in the real-life situation (MP 2)
When students struggle to make connections between the number of trials that are appropriate,
revisit the How Likely Is It investigations where the experimental and theoretical probabilities
became closer together when you increased the number of trials or combined the data from all
groups.
Questions to Develop Mathematical Thinking as you observe:






How would you identify the number of possible outcomes given the probability in percent, decimal,
or fraction form?
What type of tool could be used to help simulate the desired event if you know the number of
outcomes?
Would a spinner, area model, number cube, or coin have similar outcomes as your real-life
situation?
Has your group identified the “key component” and “assumptions” in the real-life situation?
What are the “assumptions” in the real-life situation that might help you determine a model to run a
simulation?
How might the number of outcomes help you determine the number of trials to run in the
simulation?
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Modeling with Random Devices (7.SP.8c)
Summarize: (10 minutes)
Have the students share out their different strategies for using the random device to simulate the real-life
situation. Encourage students to come up with multiple ways to use the random device. Encourage
students to come up with a model that is not identified in the examples they worked on during class. Could
a cup of marbles be used for either of the situations? Colored cubes? Standard deck of cards? Slips of
paper? People? Calculator (random number generator)?
As groups are sharing out their reasoning, check for accurate identification of the “key component” and
“assumptions.” Finally, use B1 as a way to check for understanding of how a random device could or
could not be used to simulate a real-life situation. Students should realize a coin only has two possible
outcomes and you cannot model fourths with a coin.
Solutions:
A1.
Answers will vary. Let heads represent saving a shot on goal and tails represent missing one.
A2.
Answers will vary. Let the numbers that are factors of 4 represent saving a shot on goal and the
numbers not a factor of 4 represent missing one.
A3.
Answers will vary. Let odd numbers represent saving a shot on goal and even numbers represent
missing one.
A4.
You should run a multiple of 12 trials since the goalie might have twelve attempted shots in a
game. You should run more than twelve trials to better predict the actual probability of the event. As you
run each trial, record the outcome of each attempt.
B1.
You cannot use a coin as the random device to simulate this real-life situation. A coin has two
equally-likely outcomes. The outcomes, in this case hit or miss, are not equally-likely. A hit is ¾ and a
miss is ¼.
B2.
You can use a six-sided number cube as the random device to simulate this real-life situation. Let ¾
of the numbers (for example 1, 2, 3, and 4) represent making a free-throw, and let the other numbers (for
example 5 and 6) represent missing one.
B3.
You can use the blocks as the random device to simulate this real-life situation. Let the 6 blue
blocks represent making a free-throw and the 2 yellow blocks represent a miss. Then, blindly draw blocks
out of a container and replace block at the end of each draw.
B4.
You should run a multiple of 20 trials since the basketball player might have twenty attempted freethrows in a game. You should run more than twenty trials to better predict the actual probability of the
event. As you run each trial, record the outcome of each attempt.
Feedback for lesson improvement:
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