Lesson 1

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. 

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.

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.

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.

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.

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.

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 \(\frac14\).

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}.)

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.

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.

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.

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.)