Lesson 2

The mathematical purpose of this activity is for students to collect data and think about the relationship between relative frequency and probability. Each group should draw at least 15 names from the bag. If the groups are smaller, more rounds may be needed.

Arrange students into groups of 3 or 4.

Tell students that it is important for the activity that they not examine the contents of the bag they are given at any time.

Distribute paper bags with slips of paper containing the names from the blackline master.

• Bag 1: Clare x3, Mai x5, Jada x7
• Bag 2: Andre x1, Diego x6, Elena x8
• Bag 3: Noah x10, Elena x3, Priya x2
• Bag 4: Clare x10, Mai x3, Jada x2
• Bag 5: Andre x9, Diego x4, Elena x2

You may have more than one of each bag depending on the number of groups. Collect the bags after the students have collected data.

The goal of this discussion is to make sure that students understand the relationship between relative frequencies and probability. Here are some questions for discussion.

• “Which name did you draw most frequently?” (Sample response: Jada)
• “What was the relative frequency of the name you drew most frequently?” (Sample response: In our group, we picked 15 times and got Jada 9 times. The relative frequency was 0.6)
• “Estimate the probability of drawing the name you drew most frequently?” (Sample response: 0.6)

The mathematical goal of this activity is for students to estimate the probability of a chance event by collecting data on the chance process that produces it and predict the estimate relative frequency given the probability.

Students remain in the same group as the warm-up activity and use the data they collected in the warm-up activity to answer the questions.

Some students may have difficulty understanding why the data they collect from 15 draws from the bag does not match exactly to the contents of the bag. Prompt students to look at the contents of Bag 1 (Clare x3, Mai x5, Jada x7) and to think about a situation where Clare's name is drawn on the first 3 draws. Emphasize to students that the slip of paper with Clare's name on it is returned to the bag after each draw so it is possible that Clare's name can be drawn a fourth time. If her name is drawn a fourth time, then it is not possible for Mai to be drawn 5 times and Jada to be drawn 7 times because only 11 draws remain.

The purpose of this discussion is for students to use their investigation of a chance process to estimate the relative frequency given the estimated probability. Tell the students that the bags were:

• Bag 1: Clare x3, Mai x5, Jada x7
• Bag 2: Andre x1, Diego x6, Elena x8
• Bag 3: Noah x10, Elena x3, Priya x2
• Bag 4: Clare x10, Mai x3, Jada x2
• Bag 5: Andre x9, Diego x4, Elena x2

Ask the groups to guess which bag they drew from and explain their reasoning.

Tell students that a convention for writing probability of an event is to write P(event). For example, P(drawing Clare’s name from the bag) would represent the probability of drawing Clare’s name from the bag. When the meaning is clear, it can be shortened to P(Clare). So, for bag 1, P(Clare) = \(\frac{3}{15}\) or \(\frac{1}{5}\) or 0.2.

Here are some questions for discussion.

• “Are you sure which bag you have?” (I am sure it was bag 1 because Jada was our highest probability and she only appears in bag 1 and bag 4. If it was bag 4, I would have expected Clare to be the most frequent.)
• “Would drawing slips out of the bag 100 times make you more sure of which bag you have?” (It would make me more confident that I have bag 1, but I was fairly convinced already.)
• “Do you think your list of 10 names from question 4 matches the list of 10 names created by the other people in your group? Explain your reasoning.” (I do not think it will match because we are not sure which names the teacher will draw, but I would expect that Mai or Clare would be chosen the most frequently.)
• “For question 4, is it possible for the teacher to draw Jada’s name twice if she only draws ten times?” (Yes, it is possible, but not probable.)

The mathematical purpose of this activity is to describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes. In this activity students take turns with a partner creating words that satisfy given characteristics based on probabilities. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

Arrange students in groups of 2.

Tell students that vowels include the letters A, E, I, O, and U. Consonants are all the other letters (the first four consonants are B, C, D, and F). Demonstrate how to use probabilities with words. Write the word MATH on the board and ask students about the probability of picking the letter M when selecting a letter from the word at random, then explain their reasoning. (\(\frac{1}{4}\) since there are 4 letters that each have an equal chance of being chosen and one of them is the letter M.) Ask students for the probability of selecting a consonant from the word when selecting one at random. (\(\frac{3}{4}\) since M, T, and H are all consonants so 3 of the 4 options are consonants.)

Ask students to think of a word (pizza) that has the following probabilities when selecting a letter from the word at random.

• probability of selecting an A is 0.2.
• probability of selecting a vowel is \(\frac{2}{5}\)
• probability of selecting a Z is 0.4.

Tell students that for each question, one partner finds a word that meets the given criterion and explains why they think it meets the criterion. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next question, the students swap roles. If necessary, demonstrate this protocol before students start working.

Some students may have difficulty creating their own word to meet the given criteria. For students who need help, consider providing a page from a newspaper or book to help think of ideas. Prompt students to think about what probability tells them about how many total letters are in the word, how many of the letters are consonants, and how many of the letters are vowels. Emphasize that there are many words that can meet the given criteria.

The goal for this discussion is for students to become familiar with using probability to describe chance events. It also provides an opportunity to informally assess how students think about probabilities where “or” or “not” is used.

Display the word KANGAROOS for all to see. Here are some questions for discussion.

• “If a letter is selected at random, what is the probability that it is a vowel?” (\(\frac{4}{9}\))
• “If a letter is selected at random, what is the probability that it is a consonant?” (\(\frac{5}{9}\))
• “If a letter is selected at random, what is the probability that it is an S?” (\(\frac{1}{9}\))
• “If a letter is selected at random, what is the probability that it is an A?” (\(\frac{2}{9}\))
• “If a letter is selected at random, what is the probability that it is an S or an A?” (\(\frac{3}{9}\))
• “If a letter is selected at random, what is the probability that it is not a K?” (\(\frac{8}{9}\))

If time permits, ask…

• “Whose partner came up with some great clues for problem 5?” (Sample response: My partner came up with two great clues. \(P(\text{vowel}) = \frac{1}{4}\). \(P(\text{J}) = \frac{1}{4}\)))
• “What are some words that satisfy the given probabilities?” (Sample responses: just, jump, jack, jars)
• “For the first question, why is FLOWER a valid word?” (It has 6 letters with 2 vowels and 4 consonants. Since \(\frac{2}{6} = \frac{1}{3}\) and \(\frac{4}{6} = \frac{2}{3}\), the probabilities are the same.)