Lesson 4

Tables of Relative Frequencies

4.1: Notice and Wonder: Dog City (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that two-way tables can be used to determine relative frequencies, which will be useful when students use relative frequencies as estimates of probabilities in a later activity. While students may notice and wonder many things about these images, the relative frequency is the important discussion point. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that by dividing the values in the cells by a total allows them to calculate the relative frequency.

Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

Picture of an orange tabby cat
photo of a dog

This two-way table summarizes data from a survey of 200 people who reported their home environment (urban or rural) and pet preference (dog or cat).

  urban rural total
cat 54 42 96
dog 80 24 104
total 134 66 200

What do you notice? What do you wonder?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If relative frequency does not come up during the conversation, ask students to discuss this idea.

Here are some questions for discussion.

  • “What is the relative frequency of dogs?” (\(\frac{104}{200}\))
  • “How do you change this two-way table into a relative frequency table?” (You divide each of the values by 200)
  • “If a person surveyed is chosen at random, what is the probability that their home environment is urban and their pet preference is cat?” (\(\frac{54}{200}\) or 27%)

4.2: Rolling into Tables (15 minutes)

Activity

The mathematical purpose of this activity is to construct and interpret two-way frequency tables of data, and to use the table as a sample space to estimate probabilities. Monitor the groups to determine when to tell the students to stop rolling. When each group has rolled at least 50 times tell the students to stop rolling.

Launch

Arrange students in groups of 2. 

Student Facing

Decide which person will be partner A and which will be partner B.

The result of partner A’s roll is represented by the values on the left side of the table. The result of partner B’s roll is represented by the values on the top of the table.

Use the number generator to roll and record the result. For example, if partner A rolls a 3 and partner B rolls a 5, then record 3,5. Repeat this process until your teacher tells you to stop.

Use the table to summarize the results. For example, if 6,6 appears on your list a total of five times, write a 5 in the bottom right cell of the table.

  1 2 3 4 5 6
1            
2            
3            
4            
5            
6            
  1. Do the values in the table match your expectation? Explain your reasoning.
  2. Based on the table, how many times did partner A roll a 5?
  3. How many times did you both roll the same number?
  4. What percentage of the rolls resulted in the same number from both partners?
  5. What percentage of the rolls resulted in partner A rolling a 3 and partner B rolling a 6?
  6. Based on the table, estimate the probability that partner A will roll a 2 and partner B will roll a 4. Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Launch

Arrange students in groups of 2. Provide each student with a standard number cube.

Student Facing

Decide which person will be partner A and which will be partner B.

The result of partner A’s roll is represented by the values on the left side of the table. The result of partner B’s roll is represented by the values on the top of the table.

Roll your number cube. Record the result of the roll. For example, if partner A rolls a 3 and partner B rolls a 5, then record 3,5. Repeat this process until your teacher tells you to stop.

Use the table to summarize the results. For example, if 6,6 appears on your list a total of five times, write a 5 in the bottom right cell of the table.

  1 2 3 4 5 6
1            
2            
3            
4            
5            
6            
  1. Do the values in the table match your expectation? Explain your reasoning.
  2. Based on the table, how many times did partner A roll a 5?
  3. How many times did you both roll the same number?
  4. What percentage of the rolls resulted in the same number from both partners?
  5. What percentage of the rolls resulted in partner A rolling a 3 and partner B rolling a 6?
  6. Based on the table, estimate the probability that partner A will roll a 2 and partner B will roll a 4. Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The purpose of this discussion is to make the connection between a two-way table, relative frequencies, and estimated probabilities. Here are some questions for discussion.

  • “How would you draw a similar table showing the estimated probabilities for each cell in the table?” (Divide each cell by the total number of rolls.)
  • “Which roll had the highest estimated probability?” (My group rolled 60 times and 3,4 showed up 7 times.)
  • “Which roll had the lowest estimated probability?” (The lowest estimated probability for my groups was 2,2 and 6,1. Each of these only appeared once. The estimated probability for each was \(\frac{1}{70}\) since my group rolled 70 times.)
  • “How are relative frequencies and estimated probability related?” (The relative frequency for the total number rolls can be used to estimate the probability.)
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making
Action and Expression: Develop Expression and Communication. Provide options for communicating understanding. Give students a copy of the questions from the discussion to preview before the class discussion. Allow students to provide their answers in written form and provide sentence frames if needed. 
Supports accessibility for: Language; Organization

4.3: Traveling Methods (10 minutes)

Activity

The mathematical purpose of this activity is to use a two-way table to estimate probabilities for some events. Monitor for students who struggle with the questions in which a person is selected from only a subgroup of the population.

Launch

Help students to imagine someone going to the office in Copenhagen and selecting someone at random while there. Ask them, “How many people did they have to choose from? How many of those people rode a bike to work that day?”

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select either question 1 or question 2 to complete. 
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

  1. A company has an office in Austin, Texas, and an office in Copenhagen, Denmark. The company wants to know how employees get to work, so they take a survey of all the employees and summarize the results in a table. 
     

    walk car public transit bike total
    Austin 63 376 125 63 626
    Copenhagen 48 67 95 267 477
    total 111 443 220 330 1,103
    1. If an employee is selected at random, what is the probability that they work in Austin and drive a car to work?

    2. If an employee is selected at random, what is the probability that they work in Copenhagen and ride a bike to work?

    3. If an employee is selected at random, what is the probability that they take public transit to work?

    4. If an employee from Copenhagen is selected at random, what is the probability that they ride a bike to work?

    5. If an employee who takes public transit to work is selected at random, what is the probability they work in Austin?

    6. How are the last two questions different from the first three?

  2. A school district is interested in how students get to school, so they survey their high school students to see how they get to school and separate the numbers by grade level. The results of the survey are summarized in the table. 
     

    car bus other method total
    grade 9 1,141 3,196 228 4,565
    grade 10 1,126 1,770 322 3,218
    grade 11 1,732 799 133 2,664
    grade 12 1,676 447 111 2,234
    total 5,675 6,212 794 12,681
    1. If a high school student is selected at random, what is the probability they are in grade 9 and ride the bus to school?

    2. If a high school student is selected at random, what is the probability that they are in grade 12?

    3. If a high school student is selected at random, what is the probability that they take a car to school?

    4. If a grade 10 student is selected at random, what is the probability that they ride a bus to school?

    5. If a grade 12 student is selected at random, what is the probability that they ride a bus to school?

    6. If a student who rides the bus to school is selected at random, what is the probability that they are in grade 9?

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

Clare surveyed 40 students at her school as part of a psychology project. Here are the two questions she asked:

  • Do you like to swim? (Yes or No)
  • What is your favorite season? (Winter, Spring, Summer, or Fall)

Here are the results of her survey.

likes to swim

does not like to swim

winter

5

4

spring

8

3

summer

11

1

fall

4

4

  1. Create a relative frequency table for Clare’s data.
  2. If a student that took Clare’s survey is selected at random, what is the probability that they said they like to swim and their favorite season is summer?
  3. Create two of your own survey questions or use Clare’s questions to survey 20 or more people. Record your survey questions and display the results in a table.
  4. Create a relative frequency table for your own data. What is the most common response to your survey? What is the least common response to your survey?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Some students may struggle finding the probability for the questions in which a person is selected from only a subgroup of the population. Prompt students to look at the table of the students in a school district and ask, "How many students are in grade 9?" (4,565) and then ask, "If a grade 9 student is selected at random, what is the probability that the student takes a car to school?" (\(\frac{1,141}{4,565}\)) . Emphasize that the denominator in the probability is the total number of students in grade 9 (4,565) and the numerator is the number of students in grade 9 who take a car to school (1,141).

Activity Synthesis

The goal of this discussion is to make sure students understand how to estimate probabilities using information in a two-way table.

Here are some questions for discussion.

  • “How do you use a two-way table to estimate probabilities?” (You look at the cell determined by the question being asked and divide it by the appropriate total).
  • “How do you know when to use the total for the whole table, or a total for a row or column to estimate the probability?” (You really have to look at how the question is worded. For example, for the second table, if it is a high school student selected at random then you would use the total for the whole table, but if is a student from one grade being selected at random the you would just use the total for that grade.)
  • “Did any of the probability questions stand out to you as more difficult? Explain your reasoning.” (I found that the questions that used row or column totals were more difficult because I could not just use the total for the table by default. I really had to think about what the question was asking for.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each idea 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

The table shows the responses to the survey question, "Do you like homework?" for students taking either statistics or calculus.

yes no total
statistics 7 13 20
calculus 12 18 30
total 19 31 50

Here are some questions for discussion.

  • “If a student is selected at random, what is the probability that they said yes to the survey question and are taking calculus?” (\(\frac{12}{50}\))
  • “If a student taking statistics is selected at random, what is the probability that they said yes to the survey question?” (\(\frac{18}{30}\))
  • “What is the probability that a student selected at random answered yes to the survey question?” (\(\frac{19}{50}\))
  • “If a student who answered yes to the survey question is selected at random, what is the probability that they are taking calculus?” (\(\frac{12}{19}\))

4.4: Cool-down - School Attendance (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

Tables provide a useful structure for organizing data. When several responses have been collected about some categorical variables, the data can be organized into a frequency table. The table can be used to calculate relative frequencies, which can be interpreted as probabilities.

For example, 243 participants in a survey responded to questions about their favorite season and whether they like wearing pants or shorts better. The results are summarized in the table.

pants shorts
winter 21 16
spring 43 20
summer 18 56
autumn 40 29

This table can be turned into a relative frequency table by dividing each of the values in the cells by the total number of participants.

pants shorts
winter 0.09 0.07
spring 0.18 0.08
summer 0.07 0.23
autumn 0.16 0.12

If a person is randomly selected from among these 243 participants, we can see that the probability that the chosen person’s favorite season is spring and likes shorts better than pants is 0.08. We can also use the fact that there were 63 people who listed spring as their favorite season (\(43+20\)), so the probability that a randomly selected person from this group likes spring best is around 0.26 (\(\frac{63}{243} \approx 0.26\)).