State Names

• Which letter will you use most? Which letter will you use least?
• Will you use every letter of the alphabet? Are there any letters that you will not use at all?
• Which state name has the most letters?

Allow students to speculate answers to each of these questions, and have them justify their guesses.

Display the names of all 50 states. (You can display the names as an alphabetical list, or you can simply display a map of the United States that shows the state names.) Ask, "Could you answer those questions just by looking at all of the names like this?" The point in asking this question is to make students realize that the data needs to be organized in a better way.

Then, have students use the State Names activity sheet to identify the frequency of each letter.

State Names Activity Sheet

Circulate as students work, and observe their process. If students are not using a systematic approach, ask questions such as, "How will your method guarantee that each letter is counted exactly once?" To check student work, the following are the frequencies of each letter when all 50 state names are written:

As an alternative, students can use the letter frequency tools at the sites listed below to determine how many times each letter appears:

• Letter Frequencies – The Black Chamber
• Frequency Analysis Tool by Dr. Ahmed Desoky
• Decrypting Text – Richard Knights

However, allowing students to determine the frequency by hand is valuable for two reasons. First, it gives students practice using a systematic process for organizing information. Second, and more importantly, the tick marks used to keep track of letter frequency will form a representation when the tally is complete; the number of tick marks indicates how often each letter occurs, and the amount of space required to record all of the tick marks gives a visual representation of the relative frequency.

Once the frequency analysis is completed, explain to students that the data can be represented in various ways.

Your students can create a bar graph of their data using the Bar Grapher Tool. Students can enter the data they collected into the data box.

Alternatively, depending on the availability of technology and the amount of time you want students to spend entering data, you can copy-and-paste the frequency data below into the Bar Grapher Tool, and display it using a projection device.

61, A
2, B
12, C
11, D
28, E
2, F
8, G
15, H
44, I
1, J
10, K
15, L
14, M
43, N
36, O
4, P
0, Q
22, R
32, S
19, T
8, U
5, V
11, W
2, X
6, Y
1, Z

To display the data in the Bar Grapher, highlight the data above, right-click, and choose "Copy". In the Bar Grapher, press the Clear Data button. Click anywhere inside the data box to activate the cursor, and remove the words "(Enter Text Here)". Then right-click in the data box, and choose "Paste" to place the data. Finally, press Graph Data to display a bar graph. (To get the data to display, you may need to change the maximum and minimum values.)

A correct display of the data will appear as follows:

The benefit of using the Bar Grapher Tool is that it minimizes the possibility of human error. Students will not be overwhelmed by the mechanics of constructing a bar graph; instead, they will be able to see how a bar graph organizes the data, and they can interpret the data once it is in the proper form. In addition, the data can be easily manipulated. For instance, if students wish to create a bar graph of the letter frequencies in the state postal codes, they would just need to change the values associated with each letter and hit Graph Data.

Once the bar graph is created, ask students, "Think about the questions that I asked you earlier. Which questions could you answer easily by seeing the data in this form?" Students should realize that the letters used most and least often are easy to identify, by the heights of the bars.

Explain to students that another way to represent data is using a stem-and-leaf plot. This type of graph divides each piece of data into a stem and a leaf. With a two-digit number, the tens digit is the stem, and the units digit is the leaf; for instance, the stem of 36 is 3, and the leaf is 6. (For larger numbers, the stems and leafs may change. In fact, it is unusual to use the units digit as the leaf if the range of the numbers is more than 100.) All numbers with the same stem are then grouped together.

Work with the class to create a stem-and-leaf plot. One way to do this is to assign each letter A–Z to a different student. Each student is responsible for indicating how her number would then be transferred to the stem-and-leaf plot. To model the process, you might assign a letter to yourself; or, to allow students to help one another, you could assign several letters to each group of students.

The data from the frequency analysis above would be represented in the stem-and-leaf plot shown below:

     0 | 0 1 2 2 2 4 5 6 8 8
     1 | 0 1 1 2 4 5 5 9
     2 | 2 8                       Key: 3 | 6 means 36
     3 | 2 6
     4 | 3 4
     5 | 
     6 | 1

Again ask the students, "Think about the questions that I asked you earlier. Which questions could you answer easily by seeing the data in a stem-and-leaf plot?" Students should realize that the letters used most and least often are easy to identify. The letter associated with the highest number (61) is A, and the letter associated with the lowest number (0) is Q.

For older students, the data can also be represented as a box-and-whisker plot. Work with the class to identify the five-number summary of the data set. A box-and-whisker plot is a visual representation of these results.

To help your students best understand how the five-number summary is converted to a box-and-whisker plot, copy-and-paste the following into the Box Plotter Tool.

61
2
12
11
28
2
8
15
44
1
10
15
14
43
36
4
0
22
32
19
8
5
11
2
6
1

To display the data in the Box Plotter, highlight the data above, right-click, and choose "Copy". In the Box Plotter, select "My Data" as the data set. In the data box, remove all of the data that currently appears. Then, right-click in the data-box, and choose "Paste" to place the data. Finally, press Update Boxplot to display the data in a box-and-whisker plot.

Return to the questions posed at the beginning of the lesson, and let students use their graphs to answer them:

• Which letter is used most often? [A]
• Are there any letters that are not used at all? [Q]
• What state name contains the most letters? [North Carolina, South Carolina, and Massachusetts all contain 13 letters. The state name with the most different letters is New Hampshire, with 11 different letters.]

Note that students will not be able to answer the third question from the representations of data that were created during the lesson. This can lead to a nice discussion about how data is used, and what data is necessary to answer different questions. To answer the question about the state with the most letters, students can use their results from the second assessment option, below.