Two-Way Tables
10 min
Narrative
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:
- Go back to the original statements to answer each question from scratch.
- Draw shapes to represent the students and group them into utensil and paper preferences.
- Create a table to keep track of the information, complete the table, and then answer the questions.
Launch
Allow students 3-5 minutes to work the questions, and then have a whole-class discussion.
Student Task
Several students are surveyed about whether they prefer writing with a pen or a pencil, and they are also asked whether they prefer lined paper or unlined paper. Some of the results are:
- The survey included 100 different students.
- 40 students said they prefer using a pen more than a pencil.
- 45 students said they prefer using unlined paper more than lined paper.
- 10 students said they prefer lined paper and a pen.
- 45 students said they prefer a pencil and lined paper.
For each part, explain or show your reasoning.
- How many students prefer using pencil more than pen?
- How many students prefer using pen and unlined paper?
- How many students prefer using pencil and unlined paper?
Sample Response
- 60. Sample reasoning: Since there are 100 students total and 40 who prefer pen, the rest ( 100−60=40) must prefer pencil.
- 30. Sample reasoning: Since there are 40 students total who prefer pen, and 10 of those prefer lined paper, the rest (40−10=30) must prefer unlined paper.
- 15. Sample reasoning: Since there are 45 students who prefer unlined paper, and I just determined there are 30 who prefer pen and unlined paper, the rest (45−30=15) must prefer pencil and unlined paper.
Activity Synthesis (Teacher Notes)
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:
- Answering the question from scratch
- Drawing shapes for groupings
- Creating a table
Point out that a variable is a characteristic that can take on different values and a categorical variable represents data which can be divided into groups or categories. In this example, writing utensil preference 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.
| prefer pen | prefer pencil | total | |
|---|---|---|---|
| prefer lined paper | 10 | 45 | 55 |
| prefer unlined paper | 30 | 15 | 45 |
| total | 40 | 60 | 100 |
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 a two-way table refer to? How is it seen in the example from the task?” (It refers to 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.”
Standards
- HSS-ID.B.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
10 min
10 min
Knowledge Components
All skills for this lesson
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Two-Way Tables
10 min
Narrative
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:
- Go back to the original statements to answer each question from scratch.
- Draw shapes to represent the students and group them into utensil and paper preferences.
- Create a table to keep track of the information, complete the table, and then answer the questions.
Launch
Allow students 3-5 minutes to work the questions, and then have a whole-class discussion.
Student Task
Several students are surveyed about whether they prefer writing with a pen or a pencil, and they are also asked whether they prefer lined paper or unlined paper. Some of the results are:
- The survey included 100 different students.
- 40 students said they prefer using a pen more than a pencil.
- 45 students said they prefer using unlined paper more than lined paper.
- 10 students said they prefer lined paper and a pen.
- 45 students said they prefer a pencil and lined paper.
For each part, explain or show your reasoning.
- How many students prefer using pencil more than pen?
- How many students prefer using pen and unlined paper?
- How many students prefer using pencil and unlined paper?
Sample Response
- 60. Sample reasoning: Since there are 100 students total and 40 who prefer pen, the rest ( 100−60=40) must prefer pencil.
- 30. Sample reasoning: Since there are 40 students total who prefer pen, and 10 of those prefer lined paper, the rest (40−10=30) must prefer unlined paper.
- 15. Sample reasoning: Since there are 45 students who prefer unlined paper, and I just determined there are 30 who prefer pen and unlined paper, the rest (45−30=15) must prefer pencil and unlined paper.
Activity Synthesis (Teacher Notes)
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:
- Answering the question from scratch
- Drawing shapes for groupings
- Creating a table
Point out that a variable is a characteristic that can take on different values and a categorical variable represents data which can be divided into groups or categories. In this example, writing utensil preference 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.
| prefer pen | prefer pencil | total | |
|---|---|---|---|
| prefer lined paper | 10 | 45 | 55 |
| prefer unlined paper | 30 | 15 | 45 |
| total | 40 | 60 | 100 |
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 a two-way table refer to? How is it seen in the example from the task?” (It refers to 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.”
Standards
- HSS-ID.B.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.
- S-ID.5·Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.