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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Probability and Statistics
Activity:
What’s in a Name?
TEKS:
(8.12) Probability and statistics. The student uses statistical procedures
to describe data.
The student is expected to:
(A) select the appropriate measure of central tendency or range to
describe a set of data and justify the choice for a particular
situation;
(C) select and use an appropriate representation for presenting and
displaying relationships among collected data, including line
plots, line graphs, stem and leaf plots, circle graphs, bar graphs,
box and whisker plots, histograms, and Venn diagrams, with
and without the use of technology.
(8.16) Underlying processes and mathematical tools. The student
uses logical reasoning to make conjectures and verify conclusions.
The student is expected to:
(A) make conjectures from patterns or sets of examples and
nonexamples
Overview:
Students will construct box-and-whisker plots to analyze and compare
data sets. The teacher will pose the question: Do people with short first
names have shorter last names than people who have long first names?
To investigate this question, students first collect data about the length of
the first names. They will line up in order from the shortest to the longest
first name. The class will create a human box-and-whisker plot and locate
the median, upper and lower quartiles, upper and lower extremes, and
outliers. Adding machine tape is wrapped around the students to create
the box portion of the plot. Yarn is used to create the whiskers. The
median divides the students into two groups, the students with shorter first
names and the students with longer first names. For each group, a boxand-whisker plot for the length of the last names will be created. These
two box-and-whisker plots will be compared and analyzed. Based on
these comparisons, students will reach their conclusions and be prepared
to explain their reasoning.
Materials:
Index cards
One roll of adding machine tape
Scissors
Colored yarn
Graphing calculator (optional)
Labels for Upper Extreme, Lower Extreme, Upper Quartile, Lower
Quartile, Median, Outlier (2)
How Old Where They? (1 copy for each student)
Probability and Statistics
What’s in a Name
Grade 8
Page 1
Mathematics TEKS Refinement 2006 – 6-8
Grouping:
Large group and small group
Time:
Two 45-minute class periods
Tarleton State University
Lesson:
1.
Procedures
Pose the following question to the class:
Notes
Do people with short first names have
shorter last names than people with long first
names?
2.
Ask students to vote by a show of hands on
their opinion:
Yes, people with short first names have
shorter last names than people with long first
names.
No, people with short first names don’t have
short last names or the length of the first
name makes no difference in the length of
the last name.
3.
Ask students how these questions might be
investigated. Hopefully students will suggest
collecting data about the first and last names
of a group of people, organizing the data,
analyzing the data, and drawing a
conclusion.
How might we use the names of
the students in this class to
investigate this question?
4.
Have students record the number of letters in
their first name on an index card. Tell the
students that this data will be used to
organize them into a human box and whisker
plot.
For this class, how could we
determine which first names are
short and which are long?
Have students line up in order from those
with the least number of letters in their first
name to those with the most letters in their
first name.
Sample class data are provided,
and construction of the box and
whisker plot is modeled in the
supplementary materials.
Ask the students how we could find the
median number of letters in the first names
for this group.
Probability and Statistics
What’s in a Name
Grade 8
Page 2
Mathematics TEKS Refinement 2006 – 6-8
Procedures
Have students count off starting at both ends
of the line until they reach the student who is
in the middle. The number of letters in the
student’s first name represents the median
for this group. Give this student a label
which says “median.”
Those students who are above the median
form the group with longer first names, and
those below the median form the group with
shorter first names.
5.
Discuss each of the parts of the box-andwhisker plot with the students as it is being
created.
Give the students who represent the upper
quartile and lower quartile a sign which
labels them as such. Give the student with
the shortest name a label that says lower
extreme and the student with the longest
number a label that says upper extreme.
To create the box portion of the box-andwhisker plot, wrap the students from the
lower quartile to the upper quartile in adding
machine tape.
Use the yarn to create the whiskers. One
piece of yarn should be stretched from the
lower quartile to the lower extreme and
another from the upper quartile to the upper
extreme. Check out any outliers (see
Notes).
Draw and label a box-and-whisker plot on
the board so that students have a concrete
representation of the human plot they form.
Tarleton State University
Notes
Order the data. Then the median
of the data is the middle number if
there is an odd number of data
points or the average of the two
middle numbers if there is an even
number of data points.
The upper quartile is the middle
or median of the upper half of the
data, and the lower quartile is the
median of the lower half of the
data.
The highest value in the data set
is called the upper extreme, and
the lowest value in the data set is
called the lower extreme.
Values which are widely
separated from the rest of the data
are called outliers. An outlier is
any value more than 1.5 inter
quartile ranges above the upper
quartile and/or more than 1.5 inter
quartile ranges below the lower
quartile. The inter quartile range
is the difference between the
upper quartile and the lower
quartile.
Questions to ask: How many
students are represented by each
part of the box plot? Look at the
whisker from the lower extreme to
the lower quartile – about what
percent of the class is represented
by it?
Answer the same question about
the “box” and the two sections of
Probability and Statistics
What’s in a Name
Grade 8
Page 3
Mathematics TEKS Refinement 2006 – 6-8
Procedures
Tarleton State University
Notes
the “box” formed when the median
is drawn and the whisker drawn
from the upper quartile to the
upper extreme.
It is important for students to note
that each of these parts
represents 25% of the group.
Remind students that the big questions to
answer are for this class:
How many letters are in short first name?
How many letters are in a long first name?
Questions to ask: According to
the data about our first names,
how many letters are in a short
first name? How many letters are
in a long first name?
See the step-by-step instructions on creating
a box-and-whisker plot for a sample class
roster.
Steps to follow when constructing
a box-and-whisker plot:
1. Write the data in numerical
order.
2. Identify the five values you will
use to construct the plot:
• median
• lower extreme
• upper extreme
• lower quartile
• upper quartiles
3. Draw a number line which will
accommodate the range of the
data set.
4. Plot the five identified values
from the data set below the
number line.
5. Draw a box from the lower
quartile to the upper quartile.
6. Draw a vertical line for the
median.
7. Draw the whiskers from the box
to the extremes of the data set.
Probability and Statistics
What’s in a Name
Grade 8
Page 4
Mathematics TEKS Refinement 2006 – 6-8
6.
7.
Procedures
Collect the data about the number of letters
in the last names. The data to be collected
should be based on the two groupings so
that students are able to compare the data
(number of letters in the last name) of
students with short first names with the data
of students with long first names.
Tarleton State University
Notes
Write “Short Names” and “Long
Names” on the board and have
students list the number of letters
in their last name under the label
that describes their first name.
When the last name data has been recorded, Option: Students could use a
have students work in small groups to
graphing calculator to create the
construct two box- and-whisker plots – one
box-and-whisker plots.
for the number of letters in the last names for
students with short first names and a second
one for the number of letters in the last
names for students with long first names.
Their task is to analyze the data to answer
the original question: Do people with short
first names have shorter last names than
people who have long first names?
Have students prepare a poster of their work
which includes the box-and-whisker plot,
their conclusion, and a justification for their
conclusion.
8.
Display the posters, and ask the groups to
take turns presenting their work.
9.
Student Reflection: How was the box-andwhisker plot helpful in comparing the length
of last names? Were there questions about
the length of last names that you couldn’t
answer from the box plot?
Homework:
How Old Were They?
Extensions:
Make a scatterplot using the ordered pairs (number of letters in first
name, number of letters in last name) for the students in this class.
Can you draw any conclusions or make any predictions based on this
data?
Compare this graph to the box-and-whisker plots. What can you tell
about the data from the scatterplot that you can’t tell from the box-and-
Probability and Statistics
What’s in a Name
Grade 8
Page 5
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
whisker plot? What can you tell about the data from the box-andwhisker plot that you can’t tell from the scatterplot?
Resources:
Eric W. Weisstein. "Box-and-Whisker Plot." From MathWorld--A
Wolfram Web Resource. http://mathworld.wolfram.com/Box-andWhiskerPlot.html
http://www.purplemath.com/modules/boxwhisk.htm
Probability and Statistics
What’s in a Name
Grade 8
Page 6
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
What’s in a Name?
Sample Class Roster
First Name
Kyle
Alissa
Ryan
Stephanie
Kathy
Logan
Sanjuanita
Andrew
Arlene
Angela
Heather
Layne
Lee
Bob
Jeremy
Collin
Lisa
Greyson
Rafaeal
Rosemary
Juan
Larry
Joe
Tricia
Devin
Jeff
Macey
Jonathan
Brenda
Susan
Michael
Probability and Statistics
What’s in a Name
Last Name
Alexander
Arendall
Belford
Box
Ceeley
Contreras
Daroza
Dominguez
Gonzalez
Green
Hail
Jackson
Komel
Littleton
MacMillan
McMurray
Night
Noske
Ortega
Orvera
Padula
Pizzini
Renfro
Rothenbury
Smith
Suhr
Teran
Thill
Tindle
Williams
Wusterhausen
Grade 8
Page 7
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Steps to follow when constructing a box-and-whisker plot:
(The example below uses first names from the sample class roster. The objective is to find out
how many letters are in a short first name and a long first name for this class.)
1. Write the data in numerical order.
The numbers of letters in the first names of the students on the sample roster are listed in order
from least to greatest:
3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10
2. Identify the five values you will use to construct the plot: median, lower and upper
extremes, the lower and upper quartiles, and the outliers (if there are any).
The median of the data is the middle number if there are an odd number of data points
or the average of the two middle numbers if there is any even number of data points.
This class has 31 students so the median is the data point that is in the 16th position in
the ordered data set. The median is 6 letters in the first name.
The upper quartile is the middle or median of the upper half of the data and the lower
quartile is the median of the lower half of the data.
For this set of data, the upper quartile is 7 and the lower quartile is 4.
The highest value in the data set is called the upper extreme (for this set of data, 10)
and the lowest value in the data set is called the lower extreme (for this set of data, 3).
Values which are widely separated from the rest of the data are called outliers. An
outlier is any value more than 1.5 inter quartile ranges above the upper quartile and/or
more than 1.5 inter quartile ranges below the lower quartile. The inter quartile range is
the difference between the upper quartile and the lower quartile. For this data, the inter
quartile range is 7 – 4 or 3. 1.5 times 3 is 4.5. An outlier above the upper quartile for
this data would be 4.5 added to 7 which is 11.5. Since 10 is the greatest number of
letters, there are no outliers on the upper end of the data. There are no outliers on the
lower end of the data as well
3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10
Lower Extreme
Lower quartile
median
Upper quartile
Upper Extreme
.
Probability and Statistics
What’s in a Name
Grade 8
Page 8
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
3. Draw a number line which will accommodate the range of the data set.
3
4
5
6
7
8
9
10
4. Plot the five identified values from the data set below the number line.
See below LE is the lower extreme, 3; LQ is the lower quartile, 4; M is median, 6, UQ is upper
quartile, 7; UE is upper extreme, 10.
3
•
LE
4
•
LQ
5
6
•
M
7
•
UQ
8
9
10
•
UE
5. Draw a box from the lower quartile to the upper quartile.
3•
4•
5
6•
•7
8
9
10
•
9
10
•
6. Draw a vertical line for the median.
3
•
4
•
5
6
•
7
•
8
7. Draw the whiskers from the box to the extremes of the data set.
3
4
5
Probability and Statistics
What’s in a Name
6
7
8
9
10
Grade 8
Page 9
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
NOTES: After the box-and-whisker plot is constructed, there are many interesting
questions to explore. Examples:
How many people are represented by the shorter whisker? the longer whisker?
How many people are represented by the box drawn between the lower and upper
quartile?
How many people are represented by the boxes inside the larger box?
Why are the whiskers different lengths? What does that tell us about the first names in
this class?
Why is the larger box divided into two sections that are different sizes? What does that
tell us about the first names in this class?
Are there any outliers in this data?
Remember that our objective was to divide the class into two groups according to the
number of letters in the first names. One group would have short first names and the
other long first names. Then we can compare the last names for these two groups to
answer the original question: Do people with short first names have shorter last names
than people who have long first names?
For the sample class roster, short names had 3 to 5 letters, and long names had 6 to 10
letters. For each group, a box-and-whisker plot for the length of the last names will be
created. These two box-and-whisker plots will be compared and analyzed.
Number of letters of last names of students with
short first names
Number of letters of last names of students with
long first names
Probability and Statistics
What’s in a Name
Grade 8
Page 10
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
On the previous screen shot, the top plot represents the number of letters in the last
names of students with short first names and the bottom plot represents the number of
letters in the last names of students with long first names.
Possible questions to ask:
How many letters are in the shortest last name of the students with short first names?
How do you know?
How many letters are in the longest last name of the students with short first names?
How do you know?
How many letters are in the shortest last name of the students with long first names?
How do you know?
How many letters are in the longest last name of the students with long first names?
How do you know?
What is the median length of a last name for each group? How do you know?
Write three statements comparing the two sets of students that can be supported by
analyzing these plots.
Probability and Statistics
What’s in a Name
Grade 8
Page 11
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
How Old Were They?
The following box-and-whisker plot shows the ages of the 17 Presidents of the United
States elected in the twentieth-century at their inaugurations.
40
50
60
70
80
1.
The youngest president was John F. Kennedy. How old was he at his
inauguration?
2.
The oldest president at inauguration was Ronald Reagan. How old was he at his
inauguration?
3.
What was the median age at inauguration?
4.
What percentage of the presidents were over 60 (the upper quartile)?
5.
What percentage of the presidents were under 51 (the lower quartile)?
The following table gives the ages of the 21 Vice-presidents of the United States at the
time they took office:
Vice-president
Age when he took office
Charles Fairbanks
52
James Sherman
53
Thomas Marshall
58
Calvin Coolidge
48
Charles Dawes
59
Charles Curtis
68
John Garner
64
Henry Wallace
52
Harry Truman
60
Alben Barkley
71
Richard Nixon
40
Lyndon Johnson
52
Hubert Humphrey
53
Spiro Agnew
51
Gerald Ford
59
Nelson Rockafeller
65
Walter Mondale
49
George Bush
56
Dan Quayle
41
Albert Gore
44
Probability and Statistics
What’s in a Name
Grade 8
Page 12
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
6. Make a box-and-whisker plot of the ages of the vice-presidents. Below the same
number line, copy the box plot for the ages of the presidents. Compare the ages of
the twentieth century presidents and vice-presidents. Write at least three things you
learned from analyzing the data in the box plots. Include a statement about how old
a person must be to become president or vice-president, and tell how this compares
to the ages of the people who served in these offices in the twentieth century.
Probability and Statistics
What’s in a Name
Grade 8
Page 13
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
How Old Were They?
Solutions
The following box-and-whisker plot shows the ages of the 17 Presidents of the United
States elected in the twentieth-century at their inaugurations.
40
50
60
70
80
1.
The youngest president was John F. Kennedy. How old was he at his
inauguration? 43 years old
2.
The oldest president at inauguration was Ronald Reagan. How old was he at his
inauguration? 69 years old
3.
What was the median age at inauguration? 55 years old
4.
What percentage of the presidents were over 60 (the upper quartile)? 25%
5.
What percentage of the presidents were under 51 (the lower quartile)? 25%
The following table gives the ages of the 21 Vice-presidents of the United States at the
time they took office:
Vice-president
Age when he took office
Charles Fairbanks
52
James Sherman
53
Thomas Marshall
58
Calvin Coolidge
48
Charles Dawes
59
Charles Curtis
68
John Garner
64
Henry Wallace
52
Harry Truman
60
Alben Barkley
71
Richard Nixon
40
Lyndon Johnson
52
Hubert Humphrey
53
Spiro Agnew
51
Gerald Ford
59
Nelson Rockafeller
65
Walter Mondale
49
George Bush
56
Dan Quayle
41
Albert Gore
44
Probability and Statistics
What’s in a Name
Grade 8
Page 14
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
6. Make a box-and-whisker plot of the ages of the vice-presidents. Below the same
number line copy the box plot for the ages of the presidents. Compare the ages of
the twentieth century presidents and vice-presidents. Write at least three things you
learned from analyzing the data in the box plots. Include a statement about how old
a person must be to serve as president or vice-president.
40
50
60
70
80
Vice-Presidents
Presidents
The youngest Vice-President was 40 years old while the youngest President was 43
years old. The oldest Vice-President was 71 years old while the oldest President
was 69 years old. The median age for the Vice-Presidents was 53 years and for the
Presidents the median was 55 years. The Presidents were slightly older than the
Vice-Presidents since half of them were over 55 years old.
According to Article II of the United States Constitution, you must be 35 years old to
hold the office of President. The Twelfth Amendment makes this apply to the VicePresident as well.
Probability and Statistics
What’s in a Name
Grade 8
Page 15