Unit 8, Lesson 6
The Addition Rule
Integrated Math 2
6.1
Warm-up
Standards Alignment
Building On
Addressing
HSS-CP.B.7
Apply the Addition Rule, \$, and interpret the answer in terms of the model.
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students use a two-way table of data to find the number of individuals in an event. Students must critique the reasoning of a fictional student to discover that adding the totals for each characteristic can count some individuals twice (MP3). This leads to a connection to probability in a later activity.
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Activity
NoneThe table displays information about people at a neighborhood park.
| wearing sneakers | not wearing sneakers | total | |
|---|---|---|---|
| wearing a hat | 8 | 2 | 10 |
| not wearing a hat | 3 | 12 | 15 |
| total | 11 | 14 | 25 |
- Andre says the number of people wearing sneakers or wearing a hat is 21, because there are a total of 10 people wearing a hat and a total of 11 people wearing sneakers. Is Andre correct? Explain your reasoning.
- What is the probability that a person selected at random from those in the park is wearing sneakers or wearing a hat?
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is to use the context as a model to introduce a conceptual understanding of the addition rule. Invite a student to explain why Andre’s count is wrong.
Here are some questions for discussion.
- “What is the difference between the number Andre got and the actual number of people wearing a hat or sneakers? How can we see this difference in the table?” (Andre’s count is 8 too high. The 8 people who are wearing a hat and sneakers were counted twice.)
- “How could the number of people wearing sneakers or wearing a hat be represented using a Venn diagram?” (A Venn diagram could be made using two overlapping circles, one for wearing sneakers and one for wearing a hat. There would be 8 people in the middle where the two circles overlap. There would be an additional 3 people in the wearing sneakers circle representing those wearing sneakers but not a hat. There would also be an additional 2 people in the circle representing those wearing a hat but not sneakers.)
- “What is the probability that a person selected at random is not wearing sneakers and not wearing a hat?” ()
6.2
Activity
Instructional Routines
MLR6: Three ReadsMaterials
NoneActivity Narrative
In this activity, students are formally introduced to the addition rule. They apply it in a context to verify that it holds.
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the questions.
- For the first read, read Jada’s method, her equation, and the description of the table aloud, and then ask, “What is this situation about?” (Jada came up with a rule for determining the probability of a random outcome being in event A or B. A table of data summarizes information about the United States from a census.) Listen for and clarify any questions about the context.
- After the second read, ask students to list any quantities that can be counted or measured. (Each state has a population greater or less than 4 million people. Each state begins with A to M or N to Z.)
- After the third read, reveal the bulleted information and this statement: “For each event, write which of the four states listed here is an outcome in that event.” Then ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referring to quantities from the second read. (We could highlight the row and column corresponding to the events A and B, and then focus on the 4 states and where they fit.)
6.3
Activity
Materials
NoneActivity Narrative
In this activity, students practice using the addition rule in different situations.
Monitor for groups who use these strategies. They are listed in order of efficiency:
- Make a two-way table or relative frequency table
- Draw a Venn diagram
- Use the addition rule
Launch
Arrange students in groups of 2. Read the first situation together. Give students 2 minutes to work on the first question, and then pause for a brief whole-class discussion about the application of the addition rule.
Select students with different strategies, such as those described in the Activity Narrative, to share later. Aim to elicit both key mathematical ideas and a variety of student responses, especially from students who haven’t shared recently.
Activity
Lesson Synthesis
Here are some questions for discussion.
- “Why is the addition rule useful?” (In some situations you know the probability of event A and the probability of event B, but you might not know the probability of A or B, or the probability of A and B. When this is the case, using the addition rule can help you to find the solution.)
- “Describe a situation where you could apply the addition rule.” (You know that 80% of the students in the senior class are in a club or play a sport. You know that 70% of the students are in a club and 60% of the students play a sport. How many students are in a club and play a sport?)
- “How do you solve the situation using the addition rule?” (I would use the equation ). The solution is 0.5.)
- “What does a Venn diagram look like that represents this situation?” (The portion that overlaps is 50%. That leaves 20% in a club but not playing a sport, and 10% playing a sport but not in a club.)
Student Lesson Summary
The addition rule is used to compute probabilities of compound events. The addition rule states that, given events A and B, .
For example, the student council sold 100 shirts that are either gray or blue and in sizes medium and large.
| medium | large | total | |
|---|---|---|---|
| gray | 20 | 10 | 30 |
| blue | 15 | 55 | 70 |
| total | 35 | 65 | 100 |
A student who bought a shirt is chosen at random. One way to find the probability that this student bought a shirt that is blue or medium is to begin with the probabilities for shirts sold in that color and size. The probability that the student bought a blue shirt is 0.70 since 70 out of the 100 shirts sold were blue. The probability that the student bought a medium shirt is 0.35 since 35 out of the 100 shirts sold were medium.
Because we are interested in the probability of a blue shirt or a medium shirt being purchased, we might think we should add these probabilities together to find .
This doesn’t seem to work since it is saying the probability is greater than 1.
The problem is that the 15 students who bought shirts that are both medium in size and blue are counted twice when the probabilities are added. To fix this double counting, we should subtract the probability that the chosen student is in both categories so that these students are counted only once.
The addition rule then shows that the probability the student bought a medium shirt or a blue shirt is 0.90 because , or , which is 0.90.