# What are the Chances?

## Unit Summary

After exploring basic probability concepts, students take on the role of game designers to design a fair game for a toy company. They describe the rules for play, explain how probability affects the fairness of the game, and present their game to the toy company’s board of directors trying to persuade them to sell their game.

## Curriculum-Framing Questions

**Essential Question**

What's fair?**Unit Questions**

Why is understanding probability important when playing a game?

How does probability affect fairness?**Sample Content Questions**

What is probability?

How can you measure the likelihood of an event?

How do you predict probable outcomes?

## Assessment Processes

View how a variety of student-centered assessments are used in the What are the Chances? Unit Plan. These assessments help students and teachers set goals; monitor student progress; provide feedback; assess thinking, processes, performances, products; and reflect on learning throughout the learning cycle.

## Instructional Procedures

**Prior to Instruction**

This unit of study makes use of the *Visual Ranking Tool*. Examine the *Visual Ranking Tool* as you plan instruction to learn about the tool and how to use it with your students.

**Setting the Stage**

Ask students if they have ever been in a situation where they had bad luck or good luck. Pose the Essential Question: *What's fair?* Break students into small groups and have them discuss the Essential Question and record their initial responses. Encourage them to talk about why they think life is fair or unfair, as well as what they mean by fair and what luck has to do with fairness. Ask several students to share their responses to the Essential Question and then tell them that they will begin a unit on probability. In this unit they design a fair game and learn how to use probability to determine how to increase their chance of winning. Introduce students to a learning log. The learning log is used to assess student thinking and give them an opportunity to reflect on activities and important questions.

**Tapping Prior Knowledge**

To address the Content Question, *What is probability?* introduce the idea of probability by discussing the likelihood of events occurring. Encourage students to focus on the language of probability as they use their life experiences to recall events that are certain, impossible, likely, and unlikely to happen. Record these events and introduce students to a probability scale, ranging from zero to one.

**Determining Fairness**

Bring in a variety of mathematical-based games. Discuss the rules of each game and brainstorm a list of characteristics that make the game fair. Post these in the classroom to refer back to later in the unit.

**What are the Chances?** **Activity**

1. Overview of activity:

Students use the probability scale to determine how likely an event is to occur. They use prior knowledge to make inferences about the likelihood of an event.

2. Materials needed:

- Ribbon or string (about ten feet in length)
- Three large index cards with the following titles: 1-CERTAIN; 0-IMPOSSIBLE;1/2
- Trash can and ball of paper

3. Activity procedures:

Stand ten feet away from the trash can and hold the ball of paper in your hand. Ask the students, “What is the likelihood that I will be able to throw the paper into the trash can on my first try?” Focus the discussion on vocabulary such as likely, unlikely, probably, maybe, certain, impossible, and highly unlikely.

Revisit the term “probability” with the class and review its meaning. Probability can be defined as the chance of an event occurring. Ask students to name the instances that they have heard the term used in their everyday lives.

On the board, list the words likely and unlikely. Ask each student if they think it is likely or unlikely you will make the basket and tally their responses. Throw the ball of paper into the trash can. Have a discussion related to the outcome of your throw and if you threw the ball of paper again, *Would it result in the same outcome? Does the probability of it going in the trash can increase or decrease each time?*

To discuss the Unit Question, *How can you measure the likelihood of an event?* tell the class that probability can be expressed on a probability scale. Explain to the students that we will examine how you measure likelihood by using this probability scale. Place a long piece of ribbon or string on the floor representing the scale. Ask the students to name a number that would best represent an event that is impossible (0). Choose a student to stand at this position on the scale and hold a card marked, “0 IMPOSSIBLE”. Ask students to name events that are impossible, for example: there will be 12 hours in the day tomorrow, there will be 13 months in the year next year. List student responses on a piece of chart paper.

Now ask students to name a number that would best represent an event that is certain (1), for example: there will be 24 hours in the day tomorrow, there will be 60 minutes in the next hour. Encourage students to name events that are certain and record their responses. Mark 1/2 on the scale and have a student stand halfway between 0 and 1 and hold the card “1/2”. Ask the students to make a prediction about the weather for tomorrow and come up and stand on the position on the walk-on probability scale that best represents the likelihood of their weather prediction coming true. Students need to explain their reasons for standing at a particular spot.

As a check for understanding, have students create their own graphic organizers, making a probability scale and putting events at designated places along the scale. Also, have students reflect on the following Unit Question in their learning logs: *How do you measure the likelihood of an event?*

**Use Visual Ranking to Rank Events from Most Likely to Least Likely**

Before proceeding with the next activity, click here to set up the What Are the Chances? project in your workspace.

Introduce the *Visual Ranking Tool* using the demonstration space at Try the Tool. Show students how to rank and compare lists, and how to describe items and explain their relative merit using the comments feature.

The following questions are addressed in the Visual Ranking activity:

- What are the chances it will happen?
- How do you predict probable outcomes?

Students prioritize and rank the likelihood of certain personal events. The tool activity should spark lively discussions among group members and apply criteria to evaluate the lists.

Have students log in to their *Visual Ranking* team space. Review with students the prompt for this project: *Which of the events are more likely to occur and why?* Rank the following events with the one that is most likely to occur on top. As students rank their events remind them to explain their reasoning for each item by using the comments feature of the tool. As students sort their lists, listen to their discussions and ask questions to help groups negotiate, make choices, and express their thinking. Questions such as the following can prompt students to elaborate on their thinking:

*Why have you ranked the bottom 3 and the top 3 the way you have?**What are the chances the event will happen?**How did you decide that this event is more likely than that one?*

After students finish the exercise, have them compare their lists with the lists that were ranked by the other student groups. Direct students to read each other’s comments about the relative merit of each factor. Have students discuss why their lists are alike and different. Suggest that they identify the groups that ranked items most and least like they did. Have similar and dissimilar groups meet to discuss their rankings and rationale behind the order. Encourage groups to revise their thinking based on the things they learn from other groups.

## Examine the Visual Ranking Activity

*The Visual Ranking Tool* space below represents one team’s ranking on this project. The view you see is functional. You can roll over the white icon to see the group’s comments and click the compare button to see how different groups ranked the items.

**Project Name:** What are the Chances? (Click here to set up this project in your workspace)

**Question:** Which of the events are more likely to occur and why?

Explore an** interactive demo.**

Whole Group Discussion

Whole Group Discussion

Using a projector system and networked computer display the lists and discuss general themes that appear. Ask students to consider: *Is any factor consistently in the top of the ranking? At the bottom of the ranking? How is where the factors are located (at the top or bottom) related to their degree of likelihood?*

Students have gained some understanding of how likely or unlikely an event is to occur. To gauge prior knowledge, have students respond in their learning log to the questions, *How can you be sure you’ve placed the factors in the correct order?* and, *How do you think you measure the likelihood of an event?* Review the entries and differentiate instruction based on student responses.

**Secret Spinners: What are the Chances?**

Tell students that today’s lesson will focus on the exploration of the Content Question: *How do you predict probable outcomes?* In this activity students will be making inferences to predict outcomes and drawing conclusions about possible results. Create a spinner like the example below (you can use card stock, a brad (brass tack) and a paper clip) and ask them to name all of the possible outcomes (green, red, blue). Have students predict what color the spinner is most likely to land on and justify their responses.

Put students into groups of three or four and give each group a “secret spinner” (that you have created) in an envelope. Tell the students these are NOT to be shared with other groups, as they are top secret. Give each student a copy of the Secret Spinner handout.

Examples of secret spinners:

After predicting the results of their upcoming experiment, instruct the students to spin the spinner 30 times and keep track of their results on a frequency table.

Each group then shares the results of their probability experiment with the rest of the class. Based on the data presented, the class will predict what they think the group’s secret spinner looked like. The group then reveals their secret spinner for the class to see. Discuss the results.

Ask students to conduct a think-pair-share to address the following question: *How do you predict probable outcomes on such things as spinners?*

Collect the Secret Spinner handouts. Have students respond in their learning logs to the following questions; *How do you predict probable outcomes? *and *How can understanding probable outcomes help you change your luck?* Review the entries to assess students' acquisition of key concepts and modify instruction as necessary.

**Rolling Dice: What are the Chances?**

To reinforce inferring skills and allow students to experience another probability lesson, conduct the following activity. Introduce the activity by displaying a large square number cube that has the numbers one to six. Ask students to name all of the possible outcomes that they could roll using the number cube (1,2,3,4,5,6). Ask students to determine if one number has a better chance than another when rolled and to explain their reasoning. Tell students that they will continue with the exploration of the Content Question, *“How do you predict probable outcomes?”* by examining dice in today’s lesson.

Use the computer software program at What are your Chances?*

This program simulates the rolling of a number cube. Display the software program (see Internet resources) so that students can look at 1,000 rolls of a number cube and analyze the results.

Then ask what the possible sums are if the two dice are rolled. To tap prior knowledge, present the following scenario:

Mario and Amanda are playing a dice game. Each time the dice are rolled, they find the sum of the dots. Mario gets a point every time a 10 is rolled. Amanda gets a point every time an 8 is rolled. Mario thinks he will win because he predicts a 10 will occur most often. Amanda disagrees and thinks she will win because she thinks an 8 will occur most often.

Ask students to write in their learning log whether they agree with Mario or Amanda or if they think some other sum will occur most often. Ask them to give their reasoning for their prediction.

Then have students work in groups to investigate the chances for rolling a particular sum. Have each person in the group create a number line for the possible sums (2,3,4,5,6,7,8,9,10,11,12) and place “x’s” each time the sum is rolled. Have students roll the dice 15 times. Create a classroom frequency distribution graph (a number line with the “x’s” to represent how many times each sum occurred). Ask students to compare their own group data to the whole class data. Ask students if the chances are the same for all of the sums, and if not, which ones are more likely to occur and which ones are least likely to occur.

Introduce students to the idea that a table can be a useful tool in showing the possible outcomes (mathematically) of the sums of two dice. After modeling how to fill in the table, have students complete it:

Ask students the following questions as you circulate through the room making observations and taking notes:

*How many times does each sum appear in the table?**Which sum is most likely to occur on the next roll of dice? Least likely?**How many total possible outcomes? (36)**What does this tell us?**Who is most likely to win the dice game? What number would you pick?*

Introduce the probability game, Is this Game Fair?

Ivan and Rhonda found some chips with odd markings and decided to make up a game using them. They played the game a few times, but Rhonda said it wasn’t fair. Play their game and then decide if you think it is fair (each player has an equal chance of winning).

**Materials:**

1 chip with an **A** side and a **B** side

1 chip with an **A** side and a **C** side

1 chip with a **B** side and a **C** side

**Rules:**

Flip all 3 chips at once.

**Score:**

Ivan gets a point if there is a match.

Rhonda gets a point if there is no match.

Break students into pairs and have them play the game and tally the points in a T-chart. Ask each pair to share their tallies with the whole class and record them on a large sheet of paper. The students will decide that the game is unfair after seeing the class results. Then combine the pairs of students into groups of four or five and ask them to make up a fair game using these three chips so that Ivan and Rhonda would agree that they each have an equal chance of winning. Have the groups share their revised games with the whole class.

Now that students have had additional experiences with probability and gained new knowledge, have them revisit these questions in their learning log:

*Why is understanding probability important when playing a game?**How do you predict probable outcomes?**How can understanding probable outcomes help you change your luck?*

**Create and Present Your Own Game**

Students apply what they have learned as they take on the role of game designers to create a new game for children ages eight through ten. Create an environment that fosters cooperation and decision making, by inviting local business owners to share in the product development process and by having students give feedback to one another. Providing opportunities for students to receive input from others will allow them to investigate alternatives they may not have considered themselves. Each team of designers creates a game using spinners, number cubes, or chips, and

- describes the fair rules for play
- explains why the game is fair
- uses probability language
- lists the possible outcomes for each player

Hand out and discuss the project rubric and the student checklist. Check for student understanding of project expectations and make sure students are using the checklist to guide the creation of the project.

Once students have designed and tested their games they need to create a multimedia presentation that will explain their game to the audience at Game Night. In the presentation students address the Curriculum-Framing Questions, *How does probability affect fairness?, How can you measure the likelihood of an event?,* and *What's fair?*

To help students with the planning and implementing of their game idea and multimedia presentation remind them to use the student checklist to monitor their progress and the project rubric to assess their work. Check for student understanding of project expectations and make sure students are using the checklist to guide the creation of the project. To help students become self-directed learners, pose the following questions to guide their work:

*What information do I need?**What resources do I have?**What are the smaller tasks within this big project?**What do I have to do in a particular order and what can I do any time?**What problems might come up and how do I handle them?*

**Game Night**

Invite parents, school faculty, and the toy and business representatives to attend a Game Night to recognize student work and learning. Students present their slideshows to the participants and then have time to play the games. Ask guests to give students feedback about their game.

**Wrapping- Up**

Return to the Essential Question: *What's fair?* Ask students to think about how they responded to the question at the beginning of the unit. Have them write their thoughts about fairness, chance, and probability in their learning logs. Encourage them to write about what they have learned about these things over the course of the unit and to provide as much detail and examples as possible. Have students complete the self-reflection about their work on the project.

## Prerequisite Skills

- Basic fractions
- Addition
- Familiarity with multimedia presentation software

## Differentiated Instruction

**Resource Student**

- Make modifications as dictated in the student’s IEP.
- Provide visual aids and examples (documents, visuals and examples from this Unit Plan can be helpful).
- Supply an outline of the tasks and timeline for the final project (including milestones).
- Select group best suited to work with this student.
- Provide extra time as needed to complete individual assignments.

**Gifted Student**

- Students can bring to class games of their own and explain how they demonstrate the concept of fair games. Have the students consider whether it is a game of strategy or chance and explain their reasoning. Have students analyze the game for fairness and probability.
- Students can investigate games from different cultures and analyze them for fairness and probability, strategy or chance.

**English Language Learner**

- Provide visual aids and examples (documents, visuals and examples from this Unit Plan can be helpful).
- Try to use example games from student’s native culture in an effort to link the unit to student’s prior knowledge and experiences.
- Utilize the hands-on work of spinning spinners and rolling dice as well as the visual organizers (frequency graph, tables, visual reasoning) in helping students understand and conceptualize the content.

## Credits

A teacher contributed this idea for a classroom project. A team of educators expanded the plan into the example you see here.

#### At a Glance

**Grade Level:** 3-5

**Subject:** Mathematics **
Topics:** Probability, Statistics

**Higher-Order Thinking Skills:**Implementation, Prediction

**Key Learnings:**Degrees of Likelihood; Predicting Skills; Understanding Probability; Determining Fairness

**Time Needed:**Seven 45-minute lessons