## Categorical Data

3-5
3

In the first lesson of this unit, students formulate and refine questions that can be addressed with categorical data. They consider aspects of data collection such as how to word questions and how to record the data they collect. Finally they represent and analyze the data in order to answer the question posed.

To begin the lesson, tell students you are interested in knowing the eye colors of the students in the classroom. Explain that you want to know the answer to this question:

• What colors are the eyes of the students in this class?

How can you find that out? Students may suggest that you count, that you ask all the students whose eyes are the same color to stand together in one part of the room, or that you make a graph. Listen carefully to their suggestions in order to learn what the students know about data collection and analysis. After the discussion, give students colored paper matching their eye colors and have them hold the colored paper in front of them as they form a human bar graph on a line in the room. Record the number of students in each category.

• Ask the students to think about the relationships of the counts to one another. How many more brown are there than green? Or alternately, how many fewer green than brown? What fractional part of the class has blue eyes? How many times as many blue are there as green? Green is what fractional part of blue?
• Have the students point out these mathematical relationships in their human graph. If there are 7 more brown than green, show that on the graph. Move the emphasis toward multiplicative reasoning by asking students to point out on the human graph that there are, for example, twice as many blue as green, or half as many green as blue.

Explain to students that this data can be represented in a circle graph as well as a bar graph. Ask them to stay with students who have the same eye color, to hold their colored paper in front of them, and to move into a circle. Ask a teacher's aide to stand in the center of the circle. Give a string to any student who is standing at the point where one eye color ends. Use these strings to form the radii (from the outer rim to the teacher's aide in the center) bounding the eye-color sectors of the circle. Which colors take up the most room in the circle? The least? Are the fractional relationships the same as in the bar graph? Be specific: Is green still half as much as blue, etc.?

Activity: Students formulate questions, gather and record data

When students are seated, tell them they have been working with categorical data and that they have represented the data in two ways — in a bar graph and in a circle graph. Explain that usually questions about categorical data (the kind of data they were obtaining in the eye-color activity) are answered with words. Give students some examples and ask them to think of additional ones:

• What color are the eyes of students in this classroom?
• What is your favorite dessert?
• Whom do you want to be president?

Then tell students they are going to conduct a survey that will give them a chance to think about categorical data. Remind them of the question and explain the importance of phrasing the question carefully. Ask them to speculate about whether responses in all classrooms of the world would be similar to their responses, and point out the importance of determining who will be asked the survey question.

Consider the following issues:

• What exact information do you want your classmates to give you?
• Do any words need to be defined more clearly?
• Do any questions need to be reworded?
• If the class wanted to group some of these responses into additional categories, would it help to ask a few more questions?
• Should we have everyone write down their answers, or should they just tell us the answers?
• Should we keep track of the names of people we survey?

Students can work in groups of two or three to select their survey question. Using the Survey Form, each group can collect its data during an appropriate period of time.

 Survey Form

Alternatively, students can collect the data on individual index cards, which are set up as follows:

Student Survey Form

 Name Question Student Response

Activity: Students represent and analyze data

Now that students have collected the data, they need to think about organizing a bar graph of the raw data (names of books, leisure activities, or TV shows). In graphing this raw data, students are graphing categorical data, just as they have frequently done in primary grades. Usually in these earlier grades, students identified the mode and were able to pick out their own data on the graph. In addition, students need to organize the data in a variety of ways, analyze the data mathematically, and compare data sets.

Students should take a close look at the data to see if they can think of other ways of organizing it on bar graphs. For example, students might order the favorite books information according to fiction/nonfiction; if most responses are fiction, students might consider fictional subcategories. For free time activities, students can organize information through the alone/with other, or indoor/outdoor categories. Other categories may occur to students as they work with the data.

Students can create bar grpahs of their survey results by using the NCTM Bar Grapher tool.

 Bar Grapher

Remind the students of the mathematical relationships they discovered when they analyzed the human graphs on eye color. Ask that they make similar observations based on their current representations of data. Using the total number of responses as the whole, they should count the number of responses in each category. Then, they should record that number in relation to the whole. For example, 15 out of 23 students chose fiction; 8 out of 23 chose non-fiction. Then, give students time to come up with further observations. Be sure to give them time to generate ideas. If eventually students don't come up with observations, you might offer some conjectures (true or false) and ask the students to use their reasoning to refute or support them. The conversation about the data will, of course, depend on the data collected by the students.

In conclusion, students should refer to the original survey question. How do they think the question should be answered? Each student might select one mathematical observation based on the data and write a thorough explanation of it (as in the conversations above). Students should also discuss any questions which the data suggest should be pursued if the survey project were to be continued.

• Half-sheets of colored paper to match the eye colors of students (most likely brown, green, blue, and gray)
• Five or six lengths of string, each about 6-10 yards long
• Bar Grapher
• Survey Form

Assessments

1. Students should use multiple representations to explain and justify their thinking and their strategies in approaching mathematical tasks.
2. Students can be given the following categorical data set and asked to make three mathematical observations which they justify. A scoring guide for assessing students' responses is provided.
Favorite Dogs Survey Results

3. In the third lesson of this unit, students will summarize the differences between representing categorical and numerical data. Therefore, it is important to keep the graphs created in this lesson so that students may refer to them in that discussion. If the graphs are too large, you may want to record the information on paper in order to preserve the data.

Extensions

1. Students who have worked with percentage might enjoy using the class survey question with their parents/caretakers. With this data students can create a variety of bar graphs:
• A data set of students' responses and a data set of adults' responses
• A data set of girls' responses and a data set of women's responses
• A data set of boys' responses and a data set of men's responses
• A data set of women's responses and a data set of men's responses
• A data set of girls' responses and a data set of boys' responses.

Students should attempt to make some general comparisons between the two data sets, but will probably have difficulty making specific comparisons because of the different quantities in the two data sets. (The parent data set may be a near multiple of the class data set, and if so, that relationship may be helpful in making comparisons. However, in many cases, this proportional relationship may not exist.)

2. In this lesson, students will be asked to formulate data collection questions, develop simple recording instruments for data collection, represent the data, categorize the data, analyze the data, and answer the original question. Since many of these skills are important in other subject areas as well as mathematics (science, social studies, and language arts, for example), sections of the lesson can be integrated into other subject areas.

Questions for Students

Refer to the instructional plan

Teacher Reflection

• Did students achieve the objectives for this lesson? What evidence supports this claim? What changes should I make to create a more effective lesson?
• What additional experiences do students need to be successful with this activity?
• What additional experiences do students need before moving to the next lesson?
• Were students able to explain their reasoning in a clear and logical manner?
• What additional extensions would be appropriate?

### Numerical Data

3-5
In the second lesson of this unit, students pose and refine questions that can be addressed with numerical data. They consider aspects of data collection such as how to obtain measurements and record the data they collect. They represent and analyze the ordered numerical data by describing the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed. In collecting data, students measure with standard units and carry out simple unit conversions, such as from centimeters to meters or feet to inches.

### Comparing Categorical and Numerical Data

3-5
In the final lesson of this unit, students recognize differences in representing and analyzing categorical and numerical data. Students also identify examples of each type of data.

### Learning Objectives

Students will:

• formulate and refine questions that can be addressed with categorical data
• consider aspects of data collection such as how to word questions and how to record the data they collect
• represent and analyze the data in order to answer the question posed

### Common Core State Standards – Mathematics

Grade 3, Measurement & Data

• CCSS.Math.Content.3.MD.B.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ''how many more'' and ''how many less'' problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.