## Prime Time Probability: Using Computer Games to Facilitate Finding Probability of Independent Events

- Lesson

This lesson
integrates finding probability and strategic play in the Calculation Nation^{®}
game, Prime Time. Students will work in groups to determine the best
movement option, rolling a die, spinning a spinner or flipping a coin, for
their first move of the game. Students will calculate the probability of events
and use that information as well as logic and reasoning to defend their choice
for the best movement option for their first turn in Prime Time.

This
lesson can be used effectively as an activity prior to a unit on probability as
well as a culminating assessment for a unit on probability. Begin your unit on probability by allowing
students to play the game Prime Time on Calculation Nation^{®}. Once they’ve had this opportunity to
experience probability informally, you will begin teaching them how to find the
probability of events through tree diagrams, area models and/or formulas. Once your unit on probability is finished, you
can introduce this activity again and ask students to use probability to find
the “best” random number generator.

Calculation Nation: Prime Time

Prior to your unit on probability, have students access the game Prime Time on Calculation Nation. Allow students to familiarize themselves with the rules of the game and show them each of the random number generators (dice, spinners, coins) available for use in the game. As students play the game, ask them to identify which random number generator is best and explain their reasoning. During your probability unit, you can refer back to this game to motivate students.

After you’ve taught the unit on probability, allow students to play Prime Time again. As students play, ask which random number generator is their favorite. Students will be participating in a think-pair-share. This gives students an opportunity to develop their own ideas and then share them with a partner. This share time will allow students to validate their own ideas but also learn about a different point of view that may give them an opportunity to modify their thinking. Give students about 45 seconds to think about their favorite random number generator (RNG), and then have them share with a partner for 45 seconds. After partners are finished sharing, ask for volunteers to share with the whole class.

At the beginning of the game, there are many primes on which students may land and for which they can collect points (2, 3, 5, 7, and 11). Introduce the Prime Time Probability Activity Sheet to students and ask them to complete it individually.

__Prime Time Probability Activity Sheet__

Prime Time Probability Answer Key

The Prime Time Probability activity sheet begins by asking students to identify the prime numbers they can land on with their first move. Once students have identified primes, they will need to choose a random number generator (RNG) and determine the probability of landing on a prime with that specific RNG on their first turn. Once students have found the probability with one RNG they are asked to determine the probability of landing on prime with your first turn using a different RNG. Then students are asked to determine which RNG is a better option and they should defend their choice.

Once all students have completed the task, divide students into groups of 3 or 4. When grouping students, ensure that several different RNG options are represented in each group. The goal is for students to have a rich discussion about which method of moving is “best,” and students should have approached the problem from a variety of ways. Once students are grouped, ask them, “Is there a best option for your first move. What is it? Why is it the best? Use probability and reasoning to defend your answer.” As a group, students will be asked to determine which RNG is best. While students are working in their groups, circulate around the room to ask and answer questions, and you may need to mediate arguments among group members. Students may not agree on which random number generator is best, but they will need to come to a census. You can ask questions of students such as, why do you think this RNG is better than that one? Ask the group whether they agree or disagree with that student’s reasoning. If students truly cannot come to a consensus, have them use a RNG (such as a coin or a die) to determine which RNG they will represent on the poster.

As a closing activity, have them present their work to the class. Make sure you sequence their presentations in a logical way. You may want to start with students who used pictures or graphs to show their work and move to students who calculated probabilities. You can also sequence from the least complex RNG to the most complex RNG. While students are presenting, make sure to allow students to make connections between each body of work. To do this, ask questions such as “How are ___’s choice and ____choice similar?” or “How are ____’s representation and _____’s representation are alike or different?”

### Ideas for Differentiation

- For students who finish early, you may ask them to choose an additional way to move and determine the probability of landing on a prime. You may also ask students to show you an alternative way to find the probability of an event (for example, if students made a list ask them to draw a diagram).
- For students that need more of a challenge encourage them to find probability of the more complex events. You should steer them towards finding the probability of landing on a prime when rolling the die and adding the numbers, rolling the die and subtracting the numbers or spinning the uneven spinner.

- Computers with internet access
- Chart Paper
- Markers
- Paper
- Pencils
__Prime Time Probability Activity Sheet__- Prime Time Probability Answer Key

**Assessment Options **

- After students hear and present all the arguments, have them write in their math journal reflecting on which option they think is best for their first move and why.
- Have students map out their second move of the game. Each student has chosen a first option. Have them begin the game with that first step, record where they land and then choose best movement option for landing on a prime number with their second turn. They must defend their choice with a probability argument and use reason where necessary.

**Extensions **

- Students can compare theoretical and experimental probabilities. Students have already determined the theoretical probabilities of their favorite options. Choose one of those options, have all the students complete their first move with that option and then calculate the experimental probability. Compare the theoretical and experimental probabilities as a class. Repeat for the same movement option and discuss what happens as the students increase the number of trials.
- Review the probability of getting each number with each RNG and make sure it is posted around the room. Have students work in pairs to complete the game and use this probability information for each turn. For example if students land on a 3 after their first turn, they should determine what their goal is for their next turn [Sample Answer: I want to land on 5 or 7] and determine the method that they should use to meet that goal [Sample Answer: I should spin the even spinner because I can get a 2 which will take me to 5 or a 4 which will take me to 7. That gives me a 50% chance of landing on a prime]. Have students repeat this process for the entire game.
- Have students explore a complete strategy for the entire game. Two possible strategies they might consider are:
- Attempt to complete the game in as few turns as possible so as to collect the 250–point bonus at the end of the game.
- Attempt to land on all of the primes and coins, so use random number generators that give small numbers on each turn.
- Have students investigate which random number generators are best for each of the strategies.

**Questions for
Students **

- Which random number generator do
you like best? Why?

[Answers Vary] - Is that random number generator
likely to get you to a prime number? Why
or Why Not?

[Example One: I chose to flip one coin, I either don’t move because I got a zero or I move one space and land on 1. I’m not going to land on a prime. Example Two: I chose to spin the even spinner and I changed my numbers to show 2, 3, 4, and 5. I will land on a prime if I spin a 2, 3 or 5 so I will most likely land on prime number] - Show me another way to find the probability of landing on a prime
with this movement option.

[Answers Vary] - Which of the two options you chose gives a higher
probability of landing on a prime number?
How do you know?

[I think spinning the even spinner (with numbers 2, 3, 4, and 5) is better than rolling the die because the probability of landing on a prime with that spinner is ¾ and the probability of landing on a prime with a die is ½ and ¾ is greater than ½] - How are Groups A’s work and Group
B’s work similar? How are the different?

[Answers Vary] - After hearing every group’s
argument, did anyone change their mind about which movement option is
best? Why or why not?

[Answers Vary]

**Teacher Reflection **

- Was students’ level of enthusiasm/involvement high or low? Explain why.
- Which methods did students use to determine probability? Area models, tree diagrams, pictures, formulas?
- Were the students actively engaged in mathematical discourse? How could I have facilitated their conversations better?
- What adjustments were made during the lesson to encourage full participation from all students?

### What Are My Chances?

### Probably Graphing

### Learning Objectives

Students will:

- Calculate probability of independent events.
- Use logic and probability to defend the choices they would make in a game.

### NCTM Standards and Expectations

- Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.

- Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

### Common Core State Standards – Mathematics

Grade 7, Stats & Probability

- CCSS.Math.Content.7.SP.C.5

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Grade 7, Stats & Probability

- CCSS.Math.Content.7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP2

Reason abstractly and quantitatively.

- CCSS.Math.Practice.MP3

Construct viable arguments and critique the reasoning of others.

- CCSS.Math.Practice.MP6

Attend to precision.