# Lesson 1

Up to Chance

## 1.1: Which One Doesn’t Belong: Spinners (5 minutes)

### Warm-up

This warm-up prompts students to compare four images. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

### Student Facing

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may disregard the size of the regions in the spinners and state that the probability is 1 divided by the total number of regions. Prompt them to look at the size of each region. Display spinner A and spinner D and emphasize that the spinner is more likely to land in the region where the arrow points for spinner A than where the arrow points for spinner D because of the size of the region.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as probability or outcomes. Also, press students on unsubstantiated claims.

## 1.2: You’re Saying There’s a Chance? (10 minutes)

### Optional activity

The mathematical purpose of this activity is for students to develop probability models and use them to determine probabilities of events. This lesson is optional because it revisits below grade-level content learned in grade 7 about chance events and probability. Listen for students mentioning probability, outcome, event, or sample space.

### Launch

Arrange students in groups of 2 to 4. Give students 5 minutes of quiet time to work the questions and then pause for a whole-class discussion.

Conversing: MLR2 Collect and Display. Listen for and collect the language students use to identify and describe the probability each outcome. Capture student language that reflects a variety of ways to describe probability, outcomes, and event. Write students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Access for Perception. Read the information about Elena’s Spanish class and each of the subsequent questions aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing

### Student Facing

In Elena’s Spanish class, they have a quiz every two weeks.

• For the first quiz of the year, Elena takes time to study and understands the material very well. The quiz involves 20 multiple choice questions with possible answers A, B, C, or D. Elena tries her best to answer the questions correctly.

• For the second quiz of the year, Elena has been absent a lot and does not understand the material at all. The quiz involves 20 multiple choice questions with possible answers A, B, C, or D. Elena fills in the answer sheet without even looking at the questions.

• For the third quiz of the year, Elena is still lost in the class and has not come in for any help. The quiz involves 20 true or false questions. Again, Elena fills in the answer sheet without even looking at the questions.
1. Based on the description, rank the quizzes in order from worst expected grade to best.
2. For each of the 3 quizzes, explain why you think chance played a large or small role in determining Elena’s score.
3. For the second and third quizzes, Elena did not look at the questions. Explain why you think she might do better on one than the other.
4. What percentage of the questions do you think Elena will get right on each quiz? Explain your reasoning.
5. For the second quiz, the teacher made a mistake and each of the questions had two correct answers. The teacher accepted either of the correct answers for full credit. What percentage of the questions do you think Elena will get right on the second quiz with the new scoring? Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may think that the score when guessing is zero or close to zero. Prompt the class to guess a letter from A, B , C and D and ask each student to record their guess. Tell students that the answer is B. Show students that some students guessed correctly. Emphasize that the probability of guessing the correct answer is $$\frac14$$.

### Activity Synthesis

The purpose of the discussion is to help students understand the total number of outcomes in the sample space and the number of outcomes that are in an event.

Ask, “Elena actually got a 90% on the first quiz and a 95% on the third quiz. Is this impossible, unlikely, or expected?” (Unlikely, but still possible. She may have made many lucky guesses on the third quiz.)

If it doesn’t come up, define the word outcomes as the possible results for each thing that happens. The collection of all possible outcomes is called the sample space. Ask, “For each question on the second quiz, what is the sample space?” (It consists of 4 outcomes: A, B, C, and D).

An event is a group of outcomes from the sample space. Here are some questions for discussion.

• “What event is Elena concerned with in this task?” (Elena is concerned with the event that she will get the question correct.)
• “What was the outcome of the event before the teacher’s mistake was realized?” (For each question the event contained a single outcome.)
• “What were the outcomes of the event after the mistake was realized?” (For each question each event had two outcomes in the event.)

A numerical value that represents the chance of an event occurring is called the probability of that event. Probabilities are given as either numerical values between 0 and 1 or a percentage. Here are some questions for discussion.

• “On the second quiz, what was the original probability of Elena getting a correct answer for each question?” (Elena originally had a probability of 0.25 of getting the answer correct for each question.)
• “What did the probability change to when the teacher’s mistake was realized?” (It was raised to 0.5 when the teacher’s mistake was realized. Similarly, Elena had a 50% probability of getting the answer correct for each question on the third quiz.)

If time permits, discuss questions such as . . .

• “What is the sample space for each question on quiz three?” (It consists of 2 outcomes: True and False.)
• “If there were 5 choices for each multiple choice question and only one of the choices is correct, what is the probability that Elena would get one right by guessing?” (0.2 or 20%)
• “What is an example of an event and a sample space that you have encountered previously when studying probability?” (One example is to pick a whole number between 1 and 10 and find the probability that it is odd. The sample space is {1,2,3,4,5,6,7,8,9,10} and the event is {1,3,5,7,9}.)

## 1.3: A Fair Game (15 minutes)

### Optional activity

The mathematical purpose of this activity is for students to compare probabilities from a model to observed frequencies, and to further understand the role of chance when investigating probability. This lesson is optional because it revisits below grade-level content learned in grade 7 about chance events and probability.

### Launch

Arrange students in groups of 2. Give students 2 minutes of quiet time to work the first question and then pause for them to share work with a partner. Give students quiet work time and then time to share their work for the last question with a partner.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 2–3 partners to share and get feedback on their response to the first question. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “Why did you put _____ first?” and “What did you look at on the spinner when deciding the order?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students understand the probability of different situations by communicating their reasoning with a partner.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to work in pairs and each select 2 methods to focus on. Students can work separately on question 2 and collaborate on questions 1 and 3.
Supports accessibility for: Organization; Attention; Social-emotional skills

### Student Facing

Han, Clare, Mai, and Kiran are inventing a game for the county fair. Players will spin a spinner and if it points to the section labeled B, then the player will win a prize.

Han says, “I think this spinner is a good one. What do you think?”

Clare says, “I like the spinner, but I think people should be able to spin again if they don’t win the first time.”

Mai says, “What if we just make 3 sections like this?”

Kiran says, “I think it might make more sense if we just do two sections like this.”

1. Put the proposals in order of the probability that a player will win using that method from least to greatest. Explain your reasoning. Share your explanation with a partner.
2. Each student writes a computer program to play the game using the method they suggested. The computer runs the program for a short time and reports the number of wins and losses. Use the results to estimate the probability of winning using each method.
• Han's method: 2,513 wins; 7,516 losses.
• Clare’s method: 876 wins; 1,127 losses.
• Mai’s method: 2,026 wins; 3,984 losses.
• Kiran’s method: 322 wins; 3,621 losses.
3. By talking to their friends, they figure out that a good probability for winning is about $$\frac{1}{5}$$ since it will let enough people win to draw in customers, but not cost them too much for prizes. Which method fits this best?
4. Before they settled on a spinner game, they considered other things at their booth. Which of these suggestions would be considered chance experiments?
• A watermelon-eating contest. The fastest to eat a wedge of watermelon wins the prize.
• Two cubes have one face labeled “Win!” If both cubes land with the “Win!” side facing up, the player wins a prize.
• A ball is placed under one of five cups. The cups are shuffled around under a cover so the player cannot see how they are moved. The player chooses one of the cups and wins a prize if it has the ball under it.
• Players push a button that starts lighting up different regions on a board. The game is rigged so that every fifth person wins.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

1. Draw a spinner with 3 or more sections that are not all the same size. 2. Your spinner is spun 1,000 times. Estimate the number of times you would expect the spinner to land on each section. Explain your reasoning.

3. A different spinner with only two sections is spun 1,000 times. It landed on one section 136 times and the other section 864 times. Draw a spinner that is likely to produce similar results if spun 1,000 times. Explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may divide the number of wins by the number of losses to estimate the probability of winning instead of dividing the number of wins by the total number of wins and losses. Prompt students to look at the computer program results for Andre's method. Show students that the probability of winning is 2,513 divided by the sum of 2,513 and 7,516 or approximately 0.25.

### Activity Synthesis

The purpose of this discussion is to help students make the connection between probabilities from a model and observed frequencies, and to discuss the role of chance in experiments.

Here are some questions for discussion.

• “Why did the computer generate 2,513 wins and 7,516 losses for Andre’s method instead of 2,513 wins and exactly three times as many losses (7,539)?” (The model for Andre’s method predicts that there will be three times as many losses as wins, but when we do experiments, we expect there to be roughly 3 times as many losses as wins, not exactly three times as many.)
• “What would a spinner look like that has a $$\frac{1}{5}$$ chance of winning?” • “Why isn’t the watermelon eating contest a chance experiment?” (It is not a chance experiment because there is skill involved so the outcome should always be the same if the same contestants compete.)

## Lesson Synthesis

### Lesson Synthesis

Flip a fair coin for the class and do not show them the result. Poll the class to guess what they think the coin shows. Record the results of the poll on the board.

Here are some questions for discussion.

• “What is the sample space for the coin flip?” (There are two outcomes: heads and tails)
• “What is the probability that the coin shows heads?" (0.5 or 50%)
• “What is the probability that the coin shows tails?" (0.5 or 50%) Display the result of the coin toss and ask, “What is the probability that a person selected at random in this class correctly guessed the result of the coin toss?” (The answer is the ratio of the number of students who correctly guessed the side shown to the total number of students who made a guess.)

## 1.4: Cool-down - What Affects Probability? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Few things are certain in life, but we can look more closely at uncertain processes, called chance experiments, to get a better idea of the likelihood of a particular thing happening. Lots of times, it is possible to recognize several different results that may happen. These results are called outcomes. The collection of all possible outcomes is called the sample spaceAn event is a group of outcomes from the sample space.

For example, flipping a coin is a chance experiment that has possible outcomes of landing with heads facing up or tails facing up. Sometimes, one outcome is more likely than another. For example, when walking outside in a thunderstorm, it is much more likely you will get hit by a raindrop than a lightning bolt.

To put a number to this likelihood is to find or estimate its probability. The probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. It is estimated that the probability of getting hit by lightning in the United States in one year is about $$\frac{1}{700,000}$$, which means that, each year in the United States, a person has a 0.00014% ($$1 \div 700,\!000 \approx 0.0000014$$) chance of getting hit by lightning.