## Alphabet Soup

3-5
1

In this lesson, students construct box-and-whisker plots. Students use the box-and-whisker plots to identify the mean, mode, median, and range of the data set. Representation is the major focus of this lesson.

Give each student a small handful of pasta letters of the type used in alphabet soup. Ask the students to sort the letters, write their first and last names using the letters, and then glue the alphabet letters found in their name onto an index card. (Optional opening: Ask the students to write their first and last names on an index card, then count the number of letters altogether.) Have the students write the total on the other side of the card. (In this lesson, the students will learn about a different way to graph data, the box-and-whisker plot (or box plot). This graph clearly displays the endpoints, range, and median of quantitative data. Its construction begins with ordering the data.]

Now help the students form a line in which they order themselves from greatest to least according to the number on their index card. [If more than one student has the same number, the students should stand side by side.] Ask them to face front. Now give the student with the smallest number a card on which you have written "Minimum." Now give the student with the highest number a card on which you have written "Maximum." Ask the students to find the range of the data. [To find the range, subtract the minimum from the maximum.] Record the range on the board.

Next, have the students determine whether any value occurs more times than all others. Identify that value as the mode, and record it on the board. Next, ask the students at the two ends of the line to say "one" at the same time, then the students next to them to say "two." Continue counting off in this fashion until the middle of the line is reached. [If there is an odd number of students, one student will be at the middle; if there is an even number, two students will be there. If there is one student, the number he or she holds is the median. If there are two students, the arithmetic average of their numbers is the median. If this happens, you may need to work out the problem on the board.] Ask the students what this "middle" number is called. Write the median on the board under the mean, and label it. Provide the student(s) who represent the median with a card on which you have written "Median." Tell the students that the halfway mark is called the 50th percentile, just as a half-dollar represents 50 cents.

Now have the students on either side of the median find the median of just their side. Provide a card that says "75th Percentile" to the center student on the higher end and a card that says "25th Percentile" to the center student on the lower end. [As this terminology may be new to students, you may wish to explain that the 25th percentile is that point greater than 25 percent of the score. In the money analogy, it is like a quarter. Similarly, the 75th percentile is the point greater than 75 percent of the scores, and, in the money analogy, is like 75 cents.]

Give the student at the 75th percentile place one end of a long piece of yarn to hold in his or her right hand. Then, holding the yarn, walk to the student who holds the 25th percentile card and place yarn in his or her right hand. Walk in front of that student and place the yarn in his or her left hand as well. Then, carrying the yarn, walk back to the student holding the 75th percentile card and put the other end of the yarn in his or her left hand to complete the loop. Now have those students hold out their arms, so that a yarn "box" is formed. Explain that they have made a human box-and-whisker plot.

Allow students, a few at a time, to leave the line and stand where they can see the box. Call on a volunteer to draw the figure on the board. Then collect the yarn and the cards and ask the students to take their seats and copy the plot, naming the high and low scores and the median. Encourage them to use color to show the various parts of the plot.

Now tell the students to use their calculators to find the mean. When they have found it, enter the mean under the median on the board. Now ask them how they could find the mode. When they have suggested a way and found the mode, have them add it to the list of measures of center. Identify these statistics as measures of center or central tendency. [The mode, which cannot be determined from a box plot, is the data point that occurs most often. The mean is the arithmetic average. The median is the halfway point in the ordered data, one-half the observations are above it and one-half are below it. These three statistics are called measures of central tendency or averages.] Ask the students what they notice about the averages and which one best describes the "average" length of names in the class. [The averages are probably not the same. The median is the best average in this case.]

Go to the National Library of Virtual Manipulative's Box Plot. Call on one or more students to collect the index cards and enter the data on the Web site. When it is entered, generate the box plot. You may wish to line the index cards up on the board tray so the data is visible to all the students. If you do so, you could indicate the low and high values, the median, the mean, and the mode with the labeled index cards used in the human box-and-whisker plot.

If time allows, ask the students how the box-and-whisker plot would change if the length of the teacher's name were included in the data set. Finally, ask the students to add the measures of central tendency to their copy of the box plot so they can have a record for their files.

• Crayons
• Looseleaf or Copy Paper
• Index cards
• Yarn
• Dried pasta in the shape of alphabet letters
• Calculators
• Glue
• Class Notes Recording Sheet

Assessment Options

1. At this stage of the unit, it is important to know whether students can do the following:
• Construct and read a box-and-whisker plot
• Identify the mean, median, mode, and range in a set of data

Extensions

1. Students may also wish to use the NCTM Box Plotter to create their box-and-whisker plots. Compare these box-and-whisker plots to those created using the other tool.
Box Plotter
2. Move on to the last lesson, Glyphs for All Reasons.

Questions for Students

1. What type of graph did we make today?

[Box-and-whisker plot, or box plot]

2. What length of name was most common in our class? What name is given to this measure of central tendency?

[The name will depend upon the data collected. This measure of central tendency is known as the mode.]

3. What was the shortest name in the class? The longest? How did we show these values on the box plot? What was the difference between these numbers? What do we call that difference?

[These will depend upon the data collected. The range is what we call that difference.]

4. What were the mean and median of the data set? What does each term mean? How did we find the mean? The median?

[The mean and median will depend upon the data collected. To find the mean, add up all of the data, and divide by the number of people (in this case.) To find the median, order all of the data from least to greatest, and find the middle number. (If there are two "middle" numbers, find the average of those two numbers to find the median.)]

5. Suppose a new student named Michael Burton came into the class. How would that change the plot that we made? (Repeat with other scenarios.)

[The answer will depend upon the data collected.]

6. Suppose (student name) moved away. How would that change the graph we made? (Repeat with other scenarios.)

[The answer will depend upon the data collected.]

7. How many students in the class had names longer than the name at the 75th percentile? How many students had names shorter than the length of the name at the 25th percentile?

[The answers will depend upon the data collected.]

8. Suppose the median is like a half-dollar. What amount is the 25th percentile like? What does 25th percentile mean?

[25 cents or a quarter; 25 percent of the class is accounted for when we get to this piece of data.]

9. How were these two points shown on the plot?

[They form the ends of the box.]

Teacher Reflection

• Which students were able to copy the box-and-whisker plot with minimum supervision?
• Which students easily found the range and mode? The median? The mean?
• Which students could compare the measures of central tendency with understanding?
• How can I extend this instructional experience? What will I do differently the next time that I teach this lesson?

### Tally Time

3-5
Students tally data about food preferences and learn the convention of displaying a set of five tallies. Students also answer pose and answer questions about the data.

### Can You Picture It?

3-5
This lesson builds on the experiences of the previous lesson. Students collect data about favorite vegetables and record the data in a pictograph and interpret this representation. They also create and use legends for the pictograph.

### Healthy Eating

3-5
Students collect data about classmates' healthy food knowledge. They create bar graphs, pose and answer questions about the data by looking at the graphs, and find the range and mode.

3-5
Students make human bar graphs and circle graphs, then draw them on paper and use a Web site to generate them. Posing and answering questions using the graphs will give the students an opportunity to apply their problem-solving and communication skills. They will also find the mode for a set of data.

### Let's Compare

3-5
Students collect numerical data, generate graphs, and compare two data sets. They also find the mean, mode, median, and range of the data sets. Students communicate with each other and the teacher and practice their problem-solving skills.

### Glyphs for All Reasons

3-5
Students learn a powerful way to display data, the glyph. Representation, communication, and problem solving are important parts of this lesson.

### Learning Objectives

Students will:

• Construct and read box-and-whisker plots.
• Identify the mean, median, mode, and range for a given set of data.

### NCTM Standards and Expectations

• Represent data using tables and graphs such as line plots, bar graphs, and line graphs.
• Describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed.