## Numerical Data

3-5
2

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.

Suggest to the class that they figure out a way to determine the typical height of students in the class. Students may have several suggestions: Find the average (the mean--add all the heights, then divide the sum of heights by the number of students), make a graph, or line up and figure out the heights.

Suggest that students line up according to height. As much as possible, have them take a look at themselves (the shape of the data). Are there many students of similar heights? Are students of similar heights usually toward the ends of the line or in the middle of the line? After this discussion, have the tallest and shortest students leave the line and sit down. Then, the second tallest/shortest students, etc. Go through this process very methodically, so that in the end you will have either one or two students left. Measure the heights of the remaining student(s). If there is only one student, that measurement will represent the middle or typical height, the median, in the classroom. If there are two students left, measure them and ask students how to determine what the middle height is. Probably they will suggest find the average of the two students, which is exactly what is needed. Remind students that earlier in the activity they noticed that most students of similar heights were standing near the middle of the line. Explain that they have just found the median height of students in their fourth grade class or the middle of the data set.

### Activity: Students gather, represent, and analyze numerical data

Explain to students that for this part of the lesson, they will study the following three questions, but that initially they will concentrate on the first one:

• What is the typical height of students in this class?
• What is the typical length of the feet of students in this class?
• What differences and similarities do we notice between these two data sets?

Have a discussion with students about what they will need to do in order to answer these three questions. Some may suggest that the class already knows the answer to the first question. Others may point out that the class will need more information about the first question in order to answer the third question. Eventually, students will probably realize that they need to measure their heights and foot lengths.

In order to collect accurate data, it is important that students discuss the process of measurement. Encourage them to come up with questions that need to be addressed before they collect the measurement data. Questions such as the following may occur to them:

• When we measure height, should we take our shoes off? Should we stand against the wall? What if someone's hair sticks up; should we measure that?
• When we measure people's feet, should they have their shoes on? Where exactly on the foot should we measure? From the heel to the big toe? From the heel to the little toe?
• Should we use centimeters and meters or inches and feet?
• Can we measure heights in meters [feet] and then change to centimeters [inches] when we're finished?

Once these questions have been answered to the students' satisfaction, set up the measuring activity. First, concentrate on the height data. Students might work in groups of two or three to measure one another and to check the measurements. When students have all been measured, have them line up according to their heights as actually measured in class. How does this line compare with the line formed in the Launch activity? Do the measurements seem reasonable? Are students in about the same places they were during the first part of the lesson? Or is someone in a very different position this time? When students look at this line based on actual measures, are they really lined up according to height? Or would it be good to measure some or all students over again more carefully? If some measures are obviously incorrect, ask students to speculate about the source of the errors. Did they hold the measuring instruments perpendicular to the floor? Since the meter stick (or yard stick) was probably not long enough to measure the whole person, did students mark the end of the first measurement carefully before moving the measuring tool to continue the measurement? Did they begin measuring from the proper point on the measuring tool?

When measurements have been completed, give students post-it notes and ask them to record their names and measurements on them. Students should then organize this data on a graph. Spend some time discussing which measurement numbers should be placed on the horizontal axis of the graph. Help students to reason about the desirability of including values that are within the range of the data set even though they are not actually in the data set. (For example, if students are measuring in inches, and the data are 49, 52, 54, 54, 54, . . ., it is important to include 50, 51, and 53 on the horizontal axis so that gaps in the data will be evident.) Then, students can copy the class data onto their own grid paper.

Once the data set has been organized ask students to make some observations about the shape of the data. Are there places where the data are concentrated or clumped, values for which there are no data, or data points that have unusual values? What is the range of the data? Between what values (measures) do most of the data fall in each set? Do these values (measures) seem to be consistent with the class's observations about the human graph formed earlier in the lesson? Students can then determine the median and mode,then relate these figures to the human graph.

### Activity: Students compare data sets with similar shapes

Continue the lesson by telling students they will focus on the second and third questions being studied:

• What is the typical height of students in this class?
• What is the typical length of the feet of students in this class?
• What differences and similarities do we notice between these two data sets?

Follow the procedure described in previously, for measuring class members' feet, but use a different color of post-it notes when students are ready to record the data. Students should analyze the data as suggested above. Next they should compare the two data sets. Are the measurements similar? Are the shapes of both graphs similar or different? In what ways? What accounts for the similarities/differences? Are the "ups and downs," modes, and outliers in similar places on both graphs?

Students can also make observations about the proportional relationships within and between these data sets, just as they did with categorical data. What is the proportional relationship between the median height and the median foot length? (For example, "the median height is roughly seven times the length of the median foot.") When students look at the proportional relationship between individuals' heights and feet, is it similar? If students were characters in "Alice in Wonderland" and drank a potion that made them larger (or smaller), what would they expect their height and foot length graphs to look like?

Activity: Comparing Data Sets with Different Shapes

For this part of the lesson, students will probe the following questions:

• How far have we traveled from home?
• How far have our parents traveled from home?

Before gathering information, students will need to make the definitions and process clear. Questions that may arise include:

• What does home mean--the home I live in now, or the home I used to live in? What if I went to London before I lived here? Do I count the miles to London from here or from my other city?
• What if I haven't traveled very far?
• How should we figure out how many miles are between cities? Measure on a map? Use a chart?
• Should people write down the cities of their homes and the cities of their destinations so we can check to see that the number of miles is reasonable? Or should we assume the mileage will be correct?

Use an online map tool to find distances between cites in the US, Canada and Mexico. Simply type in the two cities and get directions and distances.

Students should collect, record, organize, and analyze the data as they did with previous data questions. When collecting and recording information, students might find it helpful to use one color for themselves and one color for parents. The main difference in this discussion will be that the data sets for this survey will probably not have similar shapes. Students should make observations about these differences. Where are most of the data located in each data set? What about the "ups and downs" of the data. Are the data more spread out in one situation? What about the mode in each case? Are there any outliers?

Students might consider organizing the data differently by combining all of the data from both sets into one set. What suggestions do students have for organizing such a graph? What observations become clearer to students when they examine this newly formed graph?

In conclusion, students should make and explain several mathematical observations about the data sets.

In some of the discussions on numerical data, the concept of mean may arise. Generally students have learned that to find the mean or average, they add up a series of numbers, then divide the sum by the number of numbers they added. However, often students do not have an understanding of what this mean actually signifies. A complex topic appropriate for fifth and sixth graders, the mean is not emphasized in this lesson.

• Yardsticks or meter sticks
• Grid Paper
• Computers with Internet access

Assessment Option

Collect student graphs and written responses to any key questions. This data can be used for assessment. In addition, students will need access to their graphs in the next lesson.

Extension

Move on to the last lesson, Comparing Categorical and Numerical Data.

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 activity?
• Were students able to explain their reasoning in a clear and logical manner?
• What additional extensions/experiences would be appropriate?

### Categorical Data

3-5
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.

### 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:

• Pose and refine questions that can be addressed with numerical data.
• Consider aspects of data collection such as how to obtain measurements and record the data they collect.
• Represent, then 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.
• Measure with standard units and carry out simple unit conversions, such as from centimeters to meters or feet to inches.

### NCTM Standards and Expectations

• Carry out simple unit conversions, such as from centimeters to meters, within a system of measurement.
• Design investigations to address a question and consider how data-collection methods affect the nature of the data set.
• Collect data using observations, surveys, and experiments.
• Represent data using tables and graphs such as line plots, bar graphs, and line graphs.
• Select and apply appropriate standard units and tools to measure length, area, volume, weight, time,