Lesson 2

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.

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.

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. 

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.)

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.

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%.)

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’?”

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.)

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).

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.

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.

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.)