Lesson 1

The purpose of this activity is not to construct a perfect two-way table, but to motivate the need for an efficient representation. Monitor for students who:

1. go back to the original statements to answer each question from scratch
2. draw shapes to represent the students and group them into utensil and paper preferences
3. create a table to keep track of the information, complete the table, and then answer the questions

Allow students 3 to 5 minutes to work the questions then have a whole group discussion.

The purpose of this discussion is for students to share the different ways they approached this problem and to introduce how to represent data using a two-way table.

Select previously identified students who used different solving methods in the order: 

1. answering the question from scratch
2. drawing shapes for groupings
3. creating a table

Point out that a categorical variable represents data which can be divided into groups or categories. In this example, utensil and paper preference are two categorical variables.

If no students used a table to organize the information and answer the questions, demonstrate how to solve the questions by creating and completing a two-way table. 

Ask students:

• “Why is this table an effective method for solving this question?” (It is effective because it keeps track of all the information we know and don’t know in a way that is organized to show the relationship between each piece of information.)
• “Was using a table the only way to solve the questions?” (No, there are other methods to arrive at a good solution. Reasoning from the original statements, drawing a representation of the situation, or other similar methods are also good ways to approach the problem.)
• “What does the word ‘two’ in two-way table refer to? How is it seen in the example from the task?” (The number of categorical variables, not the number of categories for each variable. In this example, the two variables are writing utensil preference and paper preference.)

Summarize by stating: “A two-way table can be used to organize data from two different categorical variables.”

The mathematical purpose of this activity is for students to interpret data in a two-way table. Students answer questions such as the meaning of particular values or totals for a group using information in a two-way table.

Arrange students in groups of 2. After quiet work time, ask students to compare their responses to their partner’s and justify their reasoning for any differences in the responses.

When asked about a category, students may not include both columns or rows in the total. Ask students to describe what each number in the table represents and which ones might apply to flies with white eyes.

The goal is to make sure students understand how to interpret information in a two-way table. Here are some questions for discussion.

• “What does the 13 in the table represent?” (The number of offspring with white eyes and standard wings.)
• “How many selected fruit fly offspring have red eyes and standard wings?” (45)
• “How many selected fruit fly offspring are represented in the table?” (80)
• “How many selected fruit fly offspring have curled wings?” (22)

Restate, “A two-way table can be used to organize data from two different categorical variables.”

This Info Gap activity gives students an opportunity to determine and request the information needed to understand the relationships among cells in a two-way table.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Keep students in their groups of two. Give students 5 minutes to fill in the two-way tables then have a whole class discussion.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What strategy did you use to find additional values that were not provided on the data card?” (If I knew the row total, I knew the sum of the two values in the same row, so I could subtract one of the values from the total to get the other.)
• “How do the totals along the bottom of the table differ from the totals along the right side of the table?” (Along the bottom, the totals are the sum of the values in the column describing—for example, how many people reported that they floss. Along the right side of the table, the totals represent the sum of the values in each row—for example, how many people reported that they use mouthwash.)
• “How can you describe the value of 8 in the first problem?” (The times Jada ran with headphones off and took under 25 minutes to finish.)

Highlight for students the meaning of the values in the cells and various ways to describe the intersection of two categories.