Say to students, "Spell the names of all 50 states." Before they get
too far in writing all the state names, ask the following questions as
an introduction to this lesson:
- 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:
Letter | Frequency |
| Letter | Frequency |
A | 61 |
| N | 43 |
B | 2 |
| O | 36 |
C | 12 |
| P | 4 |
D | 11 |
| Q | 0 |
E | 28 |
| R | 22 |
F | 2 |
| S | 32 |
G | 8 |
| T | 19 |
H | 15 |
| U | 8 |
I | 44 |
| V | 5 |
J | 1 |
| W | 11 |
K | 10 |
| X | 2 |
L | 15 |
| Y | 6 |
M | 14 |
| Z | 1 |
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.
Bar Grapher Tool
Have students 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.
Five-Number Summary
The five-number summary consists of the upper and lower extremes, the median, and the upper and lower quartile.
- The upper and lower extremes are the greatest and least numbers that occur in the set.
- The median is the middle term when the data is arranged from least to greatest.
- The upper and lower quartiles are the median of the upper and
lower halves of the data, respectively. Note that there are various
methods used for determining the upper and lower quartiles of a set of
data; refer to your local curriculum for the method that should be used
with your students. If there is no recommended method for your district
or state, then you can use the following process for identifying the
upper and lower quartiles:
- Arrange the data in order from least to greatest, and identify the median.
- Identify the middle term of each half of the data on either side of the median. These values are the upper and lower extremes.
Example: Consider the set {1, 3, 4, 5, 6, 7, 9}.
- The lower extreme is 1.
- The lower half is {1, 3, 4}, and the middle term of that half is 3. Therefore, the lower quartile is 3.
- The median is the middle term, 5.
- The upper half is {6, 7, 9}, and the middle term of that half is 7. Therefore, the upper quartile is 7.
- The upper extreme is 9.
Box Plotter Tool
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
names with the most different letters are New Hampshire and South Carolina, each 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.