Unit 8, Lesson 2
Playing with Probability
Integrated Math 2
2.1
Warm-up
Standards Alignment
Building On
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.
Addressing
Building Toward
HSS-CP.A.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
Instructional Routines
NoneActivity Narrative
In this activity, students collect data and think about the relationship between relative frequency and probability. Students draw names from a bag and analyze their results in a later activity.
Each group should draw at least 15 names from the bag. If the groups are smaller, more rounds may be needed.
Launch
Arrange students into groups of 3–4.
Tell students that it is important for the activity that they do not at any time examine the entire contents of the bag they are given. Read the instructions for the activity together, and clarify any questions that students have about what they should be doing.
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.
Explain to students that the bags with the students’ names represent a reward system from their teacher. Every time a student in the class is notably helpful, the teacher puts that student’s name on a slip of paper and puts it into a bag. If the same student is helpful more than once, the student’s name will be entered multiple times. At the end of the month, the teacher draws several names for prizes.
Activity
NoneYour teacher will give your group a bag containing slips of paper with names on them. It is important not to open the bag to reveal all of the slips at any time. Record your group's data in writing.
Follow these steps to collect data about the names in the bag:
- Shake the bag, then draw out only 1 slip of paper.
- Read the name you drew out loud so that everyone in the group can keep track of the results in the table.
- Return the slip of paper to the bag and pass the bag to the next person in the group.
- Repeat these steps until each person in the group has had a chance to draw at least 3 names or the group has drawn at least 15 times.
| Clare | Mai | Priya | Elena | Jada | Han | Andre | Diego | Noah |
|---|---|---|---|---|---|---|---|---|
- How many times did your group draw a name from the bag?
- What name did you draw most frequently?
- For the name you drew most frequently, estimate the probability of drawing that name on the next draw.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to make sure that students understand the relationship between relative frequencies and probability. Invite 3–4 groups to share their answers for the last question and to explain their reasoning.
Here are some questions for discussion.
- “Why did the groups not get the same answer?” (Our bags might have different names in them.)
- “Even if all of the groups had identical bags, do you expect the answers to the last question to be the same?” (Not exactly the same. There is some randomness to what names are drawn from the bag.)
2.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
Activity Narrative
In this activity, students analyze their results from the Warm-up to estimate the probability of drawing a name from their bag based on the relative frequency of drawing the name. They also reason that additional trials can more accurately estimate probability.
Launch
Students remain in the same group as for the Warm-up activity and use the data they collected in the Warm-up activity to answer the questions.
MLR8 Discussion Supports. At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their estimates of the relative frequency given the estimated probabilities in the Activity Synthesis. Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Advances: Speaking, Conversing, Representing
2.3
Activity
Materials
NoneActivity Narrative
In this activity, students use probabilities of events to suggest possible events that result in the given probabilities. The events in this activity refer to the probability of encountering letters or categories of letters such as vowels. Students think of words for which selecting a random letter from the word results in the given probabilities. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
Launch
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, and then ask them to explain their reasoning. ( because 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. ( because M, T, and H are all consonants, so 3 of the 4 outcomes are consonants.)
Ask students to think of a word that has these probabilities when selecting a letter from the word at random.
- Probability of selecting an A is 0.2.
- Probability of selecting a vowel is .
- Probability of selecting a Z is 0.4.
If it is not suggested, PIZZA is a word that fits the clues.
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.
MLR8 Discussion Supports. Display sentence frames to support discussion in groups. Examples:
- “_____ works because _____”
- “I noticed _____, so a word is _____”
Advances: Speaking, Conversing
Representation: Internalize Comprehension. Use multiple examples and non-examples to reinforce understanding of probability. For example, provide a list of words such as, "eat, mass, says, shy, tag, too, top, toss, tote, tree, and zap" and ask students whether any of the words on the list fit the probabilities given.
Supports accessibility for: Conceptual Processing, Attention
Supports accessibility for: Conceptual Processing, Attention
Lesson Synthesis
Draw and replace cut-up slips from the paper bag containing the letters PIZZAPIZZA. Record them as you choose them. Do this 15 times.
Here are some questions for discussion.
- “What do you think is in the bag?” (The letters in the word PIZZA)
- “Based on the 15 draws, what is the relative frequency of the letter Z?” (The number of Zs drawn divided by 15.)
Display the contents of the bag.
- “If a letter is selected at random, what is the probability that it is the letter Z?” (, 0.4, or equivalent because there are 4 Zs in the bag and 10 letters)
- “If letters were drawn and replaced 100 times instead of 15 times, how many times would you expect the letter Z to be drawn? (I would expect it to be drawn about 40 times but probably not exactly 40 times.)
- “The bag contains the letters in the word PIZZA twice. If it contained the letters for the word PIZZA only once, would you expect the frequency of the letter Z to change if letters were drawn and replaced 100 times? Explain your reasoning.” (It would still be about forty times because the ratio of the number of the letter Z to the total number of letters is the same when you use the letters in the word PIZZA only once.)
Explain that probability and results from repeated experiments can be used to estimate a ratio of outcomes, but cannot be used to figure out exactly what is in the bag. No matter how many times the experiment is repeated, we can not determine how many copies of the word PIZZA are in the bag.
Student Lesson Summary
Some probabilities are estimated by doing an experiment, or sometimes by simulating the experiment many times and collecting data about how often outcomes come up. For example, a radio show holds a contest in which callers are entered for a chance to win a ticket to a concert in town. The probability of each caller winning is estimated by considering previous similar contests and comparing the number of callers to the number of ticket winners. If a previous contest had 327 callers and 5 ticket winners, then the probability of winning a ticket can be estimated by:
or
This means that each caller has about a 1.5% chance of winning a ticket to the concert.
Other probabilities can be determined by recognizing the expected relative likelihood of outcomes among all possible outcomes. For example, we know that the probability of rolling a 2 on a standard number cube is because there are 6 equally likely outcomes in the sample space for each roll, and the event of rolling a 2 is one of those outcomes. This can be written as . Similarly, or because there are 2 outcomes in the event.