Students may be confused as to how to interpret the table. Encourage students to use the table to look for the specific numbers mentioned in the questions. Then students can describe the larger group as well as the subgroup being considered in each respective question.

The mathematical purpose of this activity is to look for associations in categorical data. Since the subgroups represented by rows or columns are not always equally represented in the data, associations are often best recognized by considering row or column frequencies and comparing across the categories. Students should determine that there is likely not an association between the categories if the percentages are similar, or that there is evidence of an association between the variables when the percentages are very different.

Monitor for students who use the term “association” in their explanations. Identify students who struggle with knowing whether to compare frequencies in the columns or the rows and students who do not know how to find relative frequencies.

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

Students may not know how to determine if they notice an association between the variables. Ask students to describe whether coral with low nitrate concentration tends to be healthy or unhealthy, and then compare that to a description of what a high-nitrate coral might have. Then ask students whether they think it matters whether the concentration is high or low.

The purpose of this discussion is for students to gain a better understanding of when there is an association in the data. Discuss how calculating the relative frequency for each row or each column can help one get a better sense of the data.

Here are some questions for discussion:

• “What does it mean for two categorical variables to have an association?” (It means that the variables are statistically related to each other.)
• “For the data in Jada’s survey, the same number of people preferred sneakers without laces in the two youngest age groups. Why doesn’t this matter when deciding if there is an association between age group and shoe preference?” (It does not matter because we are looking for the relative frequency. Sneakers without laces represents the most frequent condition for the youngest age group and the least frequent condition for the second-youngest age group.)

It may be worth noting that silicon dioxide is the main chemical component in sand, so it should make sense that it has little impact on coral health.

If time permits, you may want to discuss associations in Jada’s data using different two-way relative frequency tables.

The first table shows row relative frequencies. It can help show that most of the younger kids (ages 4–10) prefer sneakers without laces, while many of the young adults (ages 18–24) prefer shoes that are not sneakers. This indicates that shoe preference is likely associated with age group. 

In the next table, column relative frequencies are shown. It can help show that among people who prefer shoes that are not sneakers, a very small percentage are younger kids. Conversely, among people who prefer sneakers without laces, about a third are younger kids. Since there are such large differences in the percentages, there is likely an association between age group and shoe preference. 

The mathematical purpose of this activity is for students to understand what tables might look like when two categories are possibly associated and when they are not. Students invent their own pair of variables that they expect to be associated and another pair that they expect not to be associated. Then, students prepare a two-way table of invented data that shows (or does not show) evidence of an association. Then, students create a display of their information for the class to see.

Notice groups that create displays that communicate their mathematical thinking clearly, contain an error that would be instructive to discuss, or organize the information in a way that is useful for all to see.

For students struggling to think of examples, these are some suggestions:

Associated categories:

• Age groups and price of a movie ticket
• Region of the country (North, West, or South) and whether a person enjoys skiing
• Day of the week and wake-up time (before 7 a.m. or after 7 a.m.)

Unassociated categories:

• Eye color and whether or not a person sings in the chorus
• Saltwater or freshwater fish tank and whether or not there are decorations in the tank
• Favorite sport and whether or not a person eats broccoli

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

Students may struggle with knowing what numbers to use in their made-up data set. As students create their two-way tables, ensure that they keep the association in mind. Encourage students to fill in the totals for the rows (or columns) first and then adjust the numbers in the cells to show a clear difference between the rows (or columns).

The goal of the discussion is to make sure students understand when there is a possible association and when there is no association in categorical data.

Invite each group to present their chosen variable pairs along with the display they created. After each group presents, discuss how the group created the data for each two-way table and how others can recognize if there is an association present or not. If time permits, ask questions such as:

• “Was one table easier to create than the other? Why?” (The data with no association was more difficult to create because I had to think about getting the relative frequencies similar.)
• “What insight into two-way tables, relative frequencies, or associations in categorical data did you gain as a result of doing this task?” (First, this really got me thinking about categorical data because I kept thinking about numerical data and that did not work. Second, it really helped me understand why we needed relative frequencies and not just the frequencies from the two-way table. The relative frequencies allowed me to compare data in the rows to know whether or not they were associated.)