• Evaluate student engagement in the applet exploration (Mean and Median) by looking for repeating rounds of data entries with different data. Require students to explain their understanding of the two measures.
• Use the Exit Ticket (M-6-3-1_Exit Ticket.docx and M-6-3-1_Exit Ticket KEY.docx) to evaluate the accuracy of each student’s representation of the box-and-whisker plot.

Distribute the Measures of Center handout (M-6-3-1_Measures of Center.docx and M-6-3-1_Measures of Center KEY.docx). Together with students, review and/or discuss each different measure (for now, exclude the upper and lower quartiles) and have them fill out the corresponding “Definition,” “How to Calculate,” and “Example” columns in the table (leave the “When to Report” column blank for now.)

“As you can see, there are many different measures we can use to describe a set of data. Similarly, we have many different ways of representing or organizing the data in the first place. Today we will discuss two common representations, called stem-and-leaf plots and box-and-whisker plots.”

Distribute the Two Representations handout (M-6-3-1_Two Representations.docx). Go through the following information as students refer to the examples.

• The stem-and-leaf-plot is a type of data display that presents numerical data between
  1 and 99 by separating each number into its tens-digit stem and its ones-digit leaf. Then the data are displayed in the ascending order of the tens place.
  • In the example below, 58 appears alone on the first line because there are no other data values between 50 and 59. The data values 61, 63, 65, and 69 appear on the second line as 6 representing one tens place and 1, 3, 5, and 9 representing four data values with four corresponding ones place values.

Example Stem-and-Leaf Plot

Thirteen data values used: 71, 87, 58, 76, 78, 63, 69, 71, 72, 92, 61, 65, 83

• The box-and-whisker plot is a type of data display that graphically illustrates the “five-number summary” of a set of data. These five numbers are: minimum value, lower quartile, median, upper quartile, and maximum value. The plot is placed on a number line showing the full range of the sample. A line is drawn at the median value and a “box” is created from the lower quartile to the median, and similarly from the median to the upper quartile. “Whiskers” extend from the left and right sides of the “box,” reaching the sample’s minimum and maximum values. Emphasize that data values must be arranged in order from least to greatest before finding the five pieces of the box plot.

Example Box-and-Whisker Plot

Thirteen data values used: 71, 87, 58, 76, 78, 63, 69, 71, 72, 92, 61, 65, 83

Now that students are familiar with lower and upper quartiles, instruct them to complete the corresponding “Definition,” “How to Calculate,” and “Example” columns of the Measures of Center handout.

Activity 1: Human Stem-and-Leaf and Box-and-Whisker Plot

Find a data set that would be of interest to your class and has ample values for the total number of students in your class; write each piece of data on a separate index card. (Note: The Internet has many ready-to-view databases. Refer to the following sources: TIMSS data, http://nces.ed.gov/timss/; U.S. Census Bureau, http://www.census.gov; and U.S. Department of Agriculture, http://www.ers.usda.gov/Data/.) Distribute an index card to each student and show the data on the board or on an overhead projector.

“Look at the data values in the set. Each data value is written on one of these index cards. We will be creating a human stem-and-leaf plot and box-and-whisker plot with this data. This activity will help us compare and contrast stem-and-leaf plots with box-and-whisker plots. It will also help us learn to extract specific values from these representations.”

Activity 1 Procedures:

1. Give each student an index card with a data value.
   1. Tell students to position themselves from least to greatest, according to their data value.
   2. Once students are arranged from least to greatest, have them determine what stems they will need for a stem-and-leaf plot. Write the number of each stem on a piece of paper, and position them vertically on the floor (mimicking the stem of a stem-and-leaf plot).
   3. Instruct students to move to their appropriate location on the stem-and-leaf plot.
   4. Once students are arranged in the stem-and-leaf plot, have them determine the mode and mean. (Promote discussion about strategies to find the mode and mean from a stem-and-leaf plot.)
   5. Next, have students determine the range, minimum, maximum, and median values. (Promote discussion about strategies to find the range, minimum, maximum, and median values from a stem-and-leaf plot.)
   6. Separate the groups at the median and have students determine the lower and upper quartile values.
   7. Students should now have identified the full five-number summary. Write each value from the five-number summary on a piece of paper. Use tape to create a large horizontal number line on the floor.
   8. Give students the pieces of paper showing the five-number summary values and the tape. Working together, they must complete the box-and-whisker plot by using tape to outline the boxes and whiskers and appropriately placing the five-number summary values.
   9. Repeat with a different data set if students need additional practice.

Investigating the Mean and Median

Use the same data set presented in Activity 1. Students will examine the effects of new values, including extreme values, on the mean and median of this data set. Students will use the table feature of a graphing calculator to calculate new mean and median values and, by using the Mean and Median applet from NCTM (see Related Resources), they will visually examine the effects of the new values.

Direct students with the following questions:

“What happens to the median when the same number of higher and lower data values are added?” (The median does not change if the same number of lower and higher data values are included.)

“What kind of data will make the mean increase or decrease?” (Extremely low or extremely high data values)

“When do you think it is better to represent a data set using the median? When do you think it is better to represent a data set using the mean?” (If there are major outliers, the median may provide a better representation. If there are a skewed number of repeated values, the mean may be better.)

Activity 2: Comparing Measures of Central Tendencies

Students often do not understand the fundamental differences between mean and median. Furthermore, they do not realize when it is more appropriate to report the mode of a data set. Thus, they have difficulty deciding which measure of center is most appropriate in different real-world situations.

The purpose of this activity is to prompt students to engage in self-reflection regarding the conceptual basis for each descriptive statistic. (For example: In what situations do I want to report the median? In what situations do I want to report the mean? In what situations will the mode serve me better?)

To prepare for the activity, engage students in full-class or small-group discussions to compare/contrast mean, median, and mode. Ask students to share ideas. After these discussions, help students synthesize this information and instruct them to complete the “When to Report” columns in the table from the Measures of Center handout.

Students will examine various data sets and real-world situations and determine which measure of center is most appropriate to report.

Divide students into groups of four or five. Distribute the Data Sets worksheet (M-6-3-1_Data Sets.docx and M-6-3-1_Data Sets KEY.docx). Notice the worksheet lists five different data sets and five real-world scenarios. For an optimal discussion, create five or six worksheets, similar to the one in the Resources folder so students have ample examples to explore and discuss.

Within each group, ask students to discuss situations and determine the most appropriate measure of center for each. At the close of the activity, ask a representative from each group to explain the group’s rationale for choices in each situation.

Extension:

• Routine: Grouping, open-ended discussion, exploration, connection to real-world scenarios, and self-reflection each promote engagement and learning within the lesson. The inclusion of a variety of representations and models aims at reaching all students with varying learning styles. If students have difficulty with parts of the lesson, they can be partnered with another student. If necessary, provide more activities and practice problems related to stem-and-leaf and box-and-whisker plot exploration, examination of components, and extraction and application of numbers to real-world settings.
• Expansion: Have students create another kinesthetic activity that involves the whole class in the learning of measures of center and spread. Ingenuity should be encouraged. For example, students could tweak the previous activity by using linking cubes to represent values and then explore measures of central tendency, the need for ranking when finding median, and the effect of change in data values on center and spread via movement of cubes.
• Technology: This lesson involves use of virtual applets and a graphing calculator for demonstrative and exploratory purposes.