Lesson 6

The Addition Rule

6.1: Hats Off, Sneakers On (5 minutes)

Warm-up

The mathematical purpose of this lesson is for students to motivate a conceptual understanding of the addition rule. Monitor for students discussing counting people twice.

Launch

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Student Facing

The 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
  1. 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.
  2. What is the probability that a person selected at random from those in the park is wearing sneakers or wearing a hat?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The goal of this discussion is to use the context as a model to introduce a conceptual understanding of the addition rule. Ask previously identified students to explain why Andre counted some people twice when he stated that the number of people wearing sneakers or wearing a hat is 21.

Here are some questions for discussion.

  • “Why did Andre need to subtract 8 from 21 to get the correct answer?” (It needs to be subtracted because the 8 people that were wearing sneakers and wearing a hat were counted twice. We counted them twice when adding the column total for wearing sneakers and the row total for wearing a hat.)
  • “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?” (\(\frac{12}{25}\))

6.2: State Names (15 minutes)

Activity

The mathematical purpose of this activity is to formally introduce the addition rule, to apply it in context, and to interpret the answer in terms of the model. Listen for students mentioning the concept of substitution.

Launch

Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem. Display the task statement and the first question with the table. After a shared reading, ask students, “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). Clarify the meaning of any unfamiliar phrases or vocabulary. After the second read, ask students to list any quantities that can be counted or measured, (The state has a population greater or less than 4 million people. The state begins with A to M or N to Z). During the third read, reveal the remaining questions, and invite students to discuss possible strategies. 
Design Principle: Support sense-making

Student Facing

Jada has a way to find the probability of a random outcome being in event A or event B. She says, “We use the probability of the outcome being in event A, then add the probability of the outcome being in category B. Now some outcomes have been counted twice, so we have to subtract the probability of the outcome being in both events so that those outcomes are only counted once.”

Jada's method can be rewritten as:

\(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\)

  1. The table of data summarizes information about the 50 states in the United States from a census in the year 2000. A state is chosen at random from the list of 50. Let event A be “the state name begins with A through M” and event B be “the population of the state is less than 4 million.”
      population less than 4 million population at least 4 million
    name begins with A through M 11 15
    name begins with N through Z 13 11

    Alaska is one of the 11 states in the top left cell of the table. California is one of the 15 states in the top right cell of the table. Nebraska is one of the 13 states in the bottom left cell of the table. New York is one of the 11 states in the bottom right cell of the table. For each event, write which of the four states listed here is an outcome in that event.

    1. A or B
    2. A
    3. B
    4. A and B
  2. Find each of the probabilities when a state is chosen at random:
    1. \(P(\text{A or B})\)
    2. \(P(\text{A})\)
    3. \(P(\text{B})\)
    4. \(P(\text{A and B})\)
  3. Does Jada's formula work for these events? Show your reasoning.
  4. Seniors at a high school are allowed to go off campus for lunch if they have a grade of A in all their classes or perfect attendance. An assistant principal in charge of academics knows that the probability of a randomly selected senior having A's in all their classes is 0.1. An assistant principal in charge of attendance knows that the probability of a randomly selected senior having perfect attendance is 0.16. The cafeteria staff know that the probability of a randomly selected senior being allowed to go off campus for lunch is 0.18. Use Jada's formula to find the probability that a randomly selected senior has all As and perfect attendance.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

Priya lists all of the multiples of 3 for whole numbers between 1 and 48  inclusive.

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

Tyler lists the all of the multiples of 4 between 1 and 48 inclusive.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

  1. Use a Venn diagram to display the multiples of 3 and the multiples of 4 between 1 and 48.

  2. If a whole number between 1 and 48 inclusive is selected at random, find each probability:
    1. \(P(\text{multiple of 3})\)
    2. \(P(\text{multiple of 4})\)
    3. \(P(\text{multiple of 3 or 4})\)
    4. \(P(\text{multiple of 3 and 4})\)
  3. Priya and Tyler extend their lists of multiples to include whole numbers between 1 and 96 inclusive. If a whole number between 1 and 96 inclusive is selected at random, find each probability and compare it to the probability in the previous problem:
    1. \(P(\text{multiple of 3})\)
    2. \(P(\text{multiple of 4})\)
    3. \(P(\text{multiple of 3 or 4})\)
    4. \(P(\text{multiple of 3 and 4})\)
  4. If a whole number between 1 and 100 inclusive is selected at random, what do you think \(P(\text{multiple of 3 and 4})\) is? Explain your reasoning.
  5. If a whole number between 1 and \(n\), inclusive, for any whole number value of \(n\), what is \(P(\text{multiple of 3 and 4})\)?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The mathematical purpose of this discussion is to formally introduce the addition rule and discuss how it is applied and interpreted in context.

Tell students that “Jada's formula is called the addition rule.” Display the addition rule “\(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\)” for all to see and refer students to the table used previously.

  wearing sneakers not wearing sneakers total
wearing a hat 8 2 10
not wearing a hat 3 12 15
total 11 14 25

Here are some questions for discussion.

  • “The addition rule can be applied to the question, What is the probability that a person selected at random from those in the park is wearing sneakers or a hat? What is A ? What is B?” (A refers to the event of wearing sneakers. B refers to the event of wearing a hat)
  • “What is the \(P(\text{wearing sneakers})\)?” (\(\frac{11}{25}\))
  • “What is the \(P(\text{wearing a hat})\)?” (\(\frac{10}{25}\))
  • “What is the \(P(\text{wearing sneakers and wearing a hat})\)?” (\(\frac{8)}{25}\))

Substitute the appropriate values into the addition rule and display for all to see: \(\frac{13}{25} = \frac{11}{25}+\frac{10}{25} - \frac{8}{25}\). Ask, “Why is \(\frac{8}{25}\) being subtracted?” (It is subtracted because the 8 people that were wearing sneakers and wearing a hat were counted twice. We counted them twice when adding the column total for wearing sneakers and the row total for wearing a hat.)

6.3: Coffee or Juice? (10 minutes)

Activity

The mathematical purpose of this activity is for students to apply the addition rule.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of two. Give students 2 minutes of quiet time to work the first question and then pause for a brief whole-class discussion about the application of the addition rule.

Representation: Access for Perception. Read the first question to students aloud as they follow along. Reread the question as needed and give students a few additional minutes to work the problem. Read questions two and three aloud when students are ready to proceed. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing

Student Facing

  1. At a cafe, customers order coffee at the bar, and then either go to another table where the cream and sugar are kept, or find a seat. Based on observations, a worker estimates that 70% of customers go to the second table for cream or sugar. The worker also observes that about 60% of all customers use cream for their coffee and 50% of all customers use sugar. Use the worker's estimates to find the percentage of all customers who use both cream and sugar for their coffee. Explain or show your reasoning.

    A photo of a cup of coffee
  2. At the grocery store, 70% of the different types of juice come in a bottle holding at least 400 milliliters (mL) and 40% of the different types of juice come in a low-sugar version. Only 25% of the juice varieties are in bottles holding at least 400 mL and have a low-sugar version. What percentage of the different types of juice come in a bottle holding at least 400 mL or are low-sugar? Explain your reasoning.
  3. Complete the table showing the number of cans and bottles of low-sugar and regular juice in one of the shelves at a grocery store. You may not use zero in any of the empty spaces. 
      less than 400 mL at least 400 mL total
    low-sugar available     80
    no low-sugar available      
    total 60 140  

    Use the table to find the probabilities for a juice chosen from the shelf at random.

    1. \(P(\text{less than 400 mL})\)
    2. \(P(\text{no low-sugar available})\)
    3. \(P(\text{no low-sugar available or less than 400 mL})\)
    4. \(P(\text{no low-sugar available and less than 400 mL})\)

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

For the third question, students may not know that they need to use the information from the second question to complete the table. Prompt them to use the information provided in the second question to complete the table.

Activity Synthesis

The goal of this discussion is to assess student understanding of the addition rule.

Here are some questions for discussion.

  • “How did you know you needed to use the addition rule to do the first question?” (I knew to use the addition rule because I was asked to find the percentage who used ‘cream or sugar’ and was given percentages involving the cream, the sugar, and the cream sugar. It made sense to substitute these values into the addition rule. I thought about using a Venn diagram, but it seemed easier just to use the addition rule.)
  • “How did you figure out \(P(\text{no low-sugar available and less than 400 mL})\) for question 3?” (I substituted the probabilities from part a, b, and c into the addition rule.)
  • “If 100% of the class is wearing sneakers or jeans, 80% of the class is wearing sneakers, and 70% of the class is wearing jeans, what percentage is wearing sneakers and jeans?” (50%)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

Lesson Synthesis

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 \(0.8 = 0.7 + 0.6 - P(\text{club and sport}\)). 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.)

6.4: Cool-down - Math in Science Class (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

The addition rule is used to compute probabilities of compound events. The addition rule states that, given events A and B, \(P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})\).

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 \(0.70 + 0.35 = 1.05\).

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 we do this. To fix this double counting, we should subtract the probability that the chosen student is in both categories so that these students are only counted once.

The addition rule then shows that the probability the student bought a medium shirt or a blue shirt is 0.90 since \(P(\text{medium or blue}) = P(\text{medium}) + P(\text{blue}) - P(\text{medium and blue})\) or \(P(\text{medium or blue}) = 0.35 + 0.70 - 0.15\), which is 0.90.