Lesson 2

Relative Frequency Tables

2.1: Notice and Wonder: Teacher Degrees (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that two-way tables can be used to think about relative frequency, which will be useful when students create relative frequency tables in a later activity. While students may notice and wonder many things about these images, thinking about the values in the two-way tables relative to the totals are the important discussion points.

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 two-way 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

Several adults in a school building were asked about their highest degree completed and whether they were a teacher.

What do you notice? What do you wonder? 

  teacher not a teacher
associate degree 4% 16%
bachelor’s degree 52% 64%
master’s degree or higher 44% 20%

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 you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

It is not important to answer the questions that students are wondering about at this time. Many of their questions will be addressed later in the lesson. If the concept of relative frequency does not come up during the conversation, ask students, “What does the 44% represent?” (44% of the teachers interviewed have a master’s degree or higher.)

Display the actual values used in the relative frequency table in the task for all to see. 

  teacher not a teacher
associate degree 2 4
bachelor’s degree 26 16
master’s degree or higher 22 5

Point out that, although there are more teachers with bachelor’s degrees than non-teachers with bachelor’s degrees, the percentage is lower. This is because there are more teachers who responded to this survey overall, and the table shows the relative percentage of adults for each group. That is, the 26 teachers with bachelor’s degrees are considered among the 50 teachers while the 16 non-teachers with bachelor’s degrees are only considered among the 25 non-teachers.

Ask students:

  • “What is the categorical variable for the categories that label the rows?” (It is the type of degree earned. It likely resulted from a survey question like: “What is the highest degree that you hold?”)
  • “What is the categorical variable for ‘teacher’ and ‘not a teacher’?” (It is the person’s occupation.)

2.2: City Cat, Country Cat (15 minutes)

Activity

The mathematical purpose of this activity is for students to gain an understanding of the relationship between a two-way frequency table and a two-way relative frequency table. Students have the same information in a two-way table and a segmented bar graph, then are asked to interpret the information given in context. In some cases, it makes sense to view the raw data in the table, and in other cases, it can make sense to understand the data as percentages of various larger groups. Students examine the meaning of data when they are presented as a percentage of the entire group or either of the 2 subgroups.

Launch

Arrange students in groups of 2.

Help orient students by having them notice:

  • The sum of all 4 percentages in the second table is 100%.
  • For each of the next two tables, ask students to find percentages that sum to 100%. (The third table has columns that sum to 100% and the fourth table has rows that sum to 100%.)
Action and Expression: Internalize Executive Functions. To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, instead of presenting tables and graphs all at once, provide 4 different pieces. Ask students to place the table with the actual values at the top of their workspace, and line up the other tables and graphs in a new row below. This allows students to see all of the information while reading and answering the questions without flipping back and forth. Encourage students to move or cover tables to focus on one table at a time.
Supports accessibility for: Memory; Organization

Student Facing

200 people were asked if they prefer dogs or cats, and whether they live in a rural or urban setting.

The actual values collected from the survey are in the first table.

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

The next table shows what percentage of the 200 total people included are represented by each combination of categories. The segmented bar graph represents the same information graphically.

  urban rural
cat 27% 21%
dog 40% 12%
A segmented bar graph. Vertical scale from 0 to 100 by 25’s. 0 to 27 is urban and cat. 27 to 65 is urban and dog. 65 to 88 is rural and cat. 88 to 100 is rural and dog.

The next table shows the percentage of each column that had a certain pet preference in a column relative frequency table. The segmented bar graph represents the same information graphically.

  urban rural
cat 40% 64%
dog 60% 36%
Graph with 2 segmented bars. Vertical scale from 0 to 100 by 25’s. Bars labeled urban, and rural. For urban bar, 0 to 40 is cat. 40 to 100 is dog. For rural bar, 0 to 63 is cat. 63 to 100 is dog.

The last table shows the percentage of each row that live in a certain area in a row relative frequency table. The segmented bar graph represents the same information graphically. 

  urban rural
cat 56% 44%
dog 77% 23%
Graph with 2 segmented bars. Vertical scale from 0 to 100 by 25’s. Bars labeled cat, and dog. For cat bar, 0 to 60 is urban. 60 to 100 is rural. For dog bar, 0 to 77 is urban. 77 to 100 is rural.
  1. For each relative frequency table, select a percentage and explain how numbers from the original table were used to get the percentage.
  2. What percentage of those surveyed live in an urban area and prefer dogs?
  3. Among the people surveyed who prefer dogs, what percentage of them live in an urban setting?
  4. What percentage of people surveyed who live in an urban setting prefer dogs?
  5. How many of the people responded that they prefer dogs and live in an urban setting?
  6. Among the people surveyed, are there more people who prefer dogs or cats?
  7. Your pet food company has access to a billboard in a rural setting. Would you recommend advertising dog food or cat food on this billboard? Which table did you use to make this decision? Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

Students may not understand the difference between what each relative frequency table represents. You can have a whole-class discussion about the data presented before students begin the task. You can also individually ask students questions concerning how the different representations relate to one another. For example: ”Which percentage in the relative frequency table corresponds to a specific portion of the bar graph?” Ensure students understand that each table uses a different total to get the percentages. Ask them, “Which tables are the questions referring to when it says, ‘Among the people surveyed’?”

Activity Synthesis

The purpose of this discussion is to make sure students understand how to interpret information in a relative frequency table and to prime student thinking about associations between categorical variables.

After selecting a student to share a response, select another student to share which table or graph can be used to answer the question, then select a third student to explain the connection between the question and the table or graph.

Ask students:

  • “What information can be understood from each type of relative frequency table?” (The first table can explain how each combination of categories fits into the whole group. The second table divides the entire group by their location and shows the pet preference in those locations. The third table divides the entire group by pet preference and shows the location of people for each pet preference.)
  • “Do you notice any interesting patterns in the data? Explain your reasoning.” (It is okay if students struggle with this question at this time. The focus of the next lesson is on associations between categorical variables. Students may notice that dog lovers tend to live in urban settings, since 77% of dog lovers are urban.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. As students share their ideas, press for details by requesting students to challenge an idea, elaborate on an idea, or give an example. Revoice student responses to demonstrate mathematical language use by applying appropriate language. For instance, revoice the statement: “The percentage of people living in a rural setting who love cats,” as “The percentage of cat lovers who are rural.” Call students’ attention to any words or phrases that helped clarify the original statement. If needed, practice phrases or words through choral response. This provides more students with an opportunity to produce language as they share their ideas about data represented in relative frequency tables.
Design Principle(s): Support sense-making

2.3: Analyzing a Study With Two Treatments (15 minutes)

Activity

The mathematical purpose of this activity is for students to create and interpret different relative frequency tables in context. Identify students who record the totals for the columns and rows on the first table.

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

Launch

Arrange students in groups of two. Read the directions for the activity to the class. Explain to students that a placebo pill is a pill that looks just like the one that has the vitamin C, but does not actually have any vitamin C in it. In this study, the person taking it does not know which pill is which. Allow students five minutes to work through the problems, then have a whole-group discussion.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, as students answer the questions, demonstrate identifying and isolating the categorical value referred to in the question by circling the column or row pertaining to the value in a different color.
Supports accessibility for: Visual-spatial processing

Student Facing

In an experiment to test the effectiveness of vitamin C on the length of colds, two groups of people with colds are given a pill to take once a day. The pill for one of the groups contains 1,000 mg of vitamin C, while the other group takes a placebo pill. The researchers record the results in a table.

  group A group B
cold lasts less than a week 16 27
cold lasts a week or more 17 53
  1. First, the researchers want to know what percentage (to the nearest whole percent) of people are in each combination of categories. Fourteen percent of all the participants had a cold that lasted less than a week and were in group A. What percentage of all the participants had a cold that lasted less than a week and were in group B? Complete the rest of the relative frequency table with the corresponding percentages. 
     

      group A group B
    cold lasts less than a week 14% (\(\frac{16}{16+27+17+53} \approx 0.14\))  
    cold lasts a week or more    
  2. Next, the researchers notice that, among participants who had colds that lasted less than a week, 37% were in group A. Among participants who had colds that lasted a week or more, what percentage were in group B? Complete the table with the corresponding percentages.
     

      group A group B
    cold lasts less than a week 37% (\(\frac{16}{16+27} \approx 0.37\))  
    cold lasts a week or more    
  3. Finally, the researchers notice that, among the participants in group A, 48% had colds that lasted less than one week. Among the participants in group B, how many had colds that lasted a week or more? Complete the table with the corresponding percentages. 
     

      group A group B
    cold lasts less than a week 48% (\(\frac{16}{16+17} \approx 0.48\))  
    cold lasts a week or more    
  4. To understand the results, the researchers want to know: Among people whose colds lasted less than a week, what percentage are in group B? Explain your reasoning. 
  5. If the researchers believe that vitamin C has a small effect on the length of a cold, which group most likely got the pills containing vitamin C? Explain your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Are you ready for more?

A teacher surveyed a group of 25 8th graders and a group of 20 12th graders who indicated they knew a computer programming language. Python and Scratch are programming languages. The results from the 8t-grade survey are displayed in the two-way table.

 

I know Python the best

I know Scratch the best

I know a different programming language the best

I have been taught a programming language at school

8

6

1

I have not been taught a programming language at school

1

7

2

The results from the 12th-grade survey are displayed in the two-way table.

 

I know Python the best

I know Scratch the best

I know a different programming language the best

I have been taught a programming language at school

25%

35%

0%

I have not been taught a programming language at school

30%

5%

5%

  1. Which programming language did a majority of 8th graders surveyed know best?

  2. Which programming language did a majority of 12th graders surveyed know best?

  3. How many of 12th graders surveyed reported that they were taught a programming language at school?

  4. What percentage of 8th graders surveyed reported that they knew Python the best and were not taught a programming language at school?

  5. Why is it difficult to decide if 12th graders or 8th graders use Python more with the way the information is given in the tables?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

When asked to complete the relative frequency table with percentages, students may struggle with using the correct totals. Bring to their attention the numbers used in the examples, and ask students where the denominators came from.

Activity Synthesis

The goal of the discussion is to make sure students know how to create and interpret different two-way relative frequency tables from the same two-way frequency table. Ask the students identified as writing down the totals for the two-way table, “Why did you write down the totals for the two-way table?” (I wrote them down because I knew we would need them to calculate relative frequencies.) Talk about overall relative frequency, row relative frequency, and column relative frequency, and the different interpretations of them.

  • “Why is it okay that the second column in the table for number 2 adds up to more than 100%?” (They are percentages of different wholes. It does not make sense to add them together.)
  • “How did you get the percentages for group B in the table for number 3?” (I divided the values for group B in the first table by 80, the total number in group B, then multiplied by 100.)
  • “What questions did the relative frequency tables help the researchers answer? Why are they powerful tools?” (They can help the researchers determine whether vitamin C has an effect on the length of a cold. The relative frequency tables are powerful because there were many more people in group B, so it was hard to compare the groups from the actual frequencies.)

Lesson Synthesis

Lesson Synthesis

The mathematical purpose of this lesson is for students to understand how to create and interpret relative frequency tables. Here are some questions for discussion.

  • “How do you create a relative frequency table?” (There are several different ways to do it. In short, you divide the numbers in a two-way table by a total or totals. The total can be for the entire table, or the totals can be for each row or column, depending on what relationship you are trying to find.)
  • “What are some categorical variables you could use to make a frequency table? Share them with your partner.” (Here are some examples. Categorize bird species based on the dominant color they display on their feathers. Categorize students from two different sports teams and whether or not they are in band. Categorize students by their type of cell phone and what type of social media they use.)

2.4: Cool-down - Writing Sample (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

Converting two-way tables to relative frequency tables can help reveal patterns in paired categorical variables. Relative frequency tables are created by dividing the value in each cell in a two-way table by the total number of responses in the entire table, or the total responses in a row or a column. Depending on what patterns are important, different types of relative frequency tables are used. To examine how individual combinations of the categorical variables relate to the whole group, divide each value in a two-way table by the total number of responses in the entire table to find the relative frequency.

For example, this two-way table displays the condition of a certain textbook and its price for 120 of the books at a college bookstore. 

$10 or less more than \$10 but less than $30 $30 or more
new 3 9 27
used 33 36 12

A two-way relative frequency table is created by dividing each number in the two-way table by 120, because there are 120 values (\(3 + 9 + 27 + 33 + 36 + 12\)) in this data set. The resulting two-way relative frequency table can be represented using fractions or decimals. 

$10 or less more than \$10 but less than $30 $30 or more
new 0.025 0.075 0.225
used 0.275 0.300 0.100

This two-way relative frequency table allows you to see what proportion of the total is represented by each number in the two-way table. The number 33 in the original two-way table represents the number of used books that also sell for $10 or less, which is 27.5% of all the books in the data set. Using this two-way relative frequency table, you can see that there are very few (2.5%) new books that are also inexpensive and that 10% of the books in the bookstore are both expensive and in used condition.

In other situations, it makes sense to examine row or column proportions in a relative frequency table. For example, to convert the original two-way table to a column relative frequency table using column proportions, divide each value by the sum of the column. 

$10 or less more than \$10 but less than $30 $30 or more
new 0.08 0.2 0.692
used 0.917 0.8 0.308

This shows that about 91.7% (\(\frac{33}{3+33} \approx 0.917\)) of the books that are sold for $10 or less are in used condition. Notice that each column of this column relative frequency table reveals the proportions of the books in each price category that are in each condition and the relative frequencies in each column sum to 1. In particular, this shows that most of the inexpensive and moderately priced books are used, and most of the expensive books are new.