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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Probability and Statistics
Activity:
Your Average Joe
TEKS:
(6.10) Probability and statistics. The student uses statistical
representations to analyze data.
The student is expected to:
(B) identify mean (using concrete objects and pictorial models),
median, mode, and range of a set of data;
(6.11) Underlying processes and mathematical tools. The student
applies Grade 6 mathematics to solve problems connected to everyday
experiences, investigations in other disciplines, and activities in and
outside of school
The student is expected to:
(D) select tools such as real objects, manipulatives, paper/pencil,
and technology or techniques such as mental math, estimation,
and number sense to solve problems.
Overview:
In this activity, students will answer a question about the average number
of letters in a first name. Using linking cubes, students will physically
model the mean, median, and mode of the number of letters in their first
names. Definitions will be formulated for these terms by referring to the
physical activities used to determine their value. Example: to find the
median, we first arranged ourselves in order from the shortest first name
to the longest first name. Finally, students will be given a problem to solve
in pairs, and then solution strategies will be shared with the class.
Materials:
Linking cubes
Grid paper
Transparencies 1-4
Handout 1
Calculator
Grouping:
Large group and pairs of students
Time:
Two 45-minute class periods
Lesson:
1.
Procedures
(5 minutes) Discuss the meaning of the
word “average” and how it is used in
statistics. Perhaps have some newspaper or
magazine articles that show different uses of
“average.”
Probability and Statistics
Your Average Joe
Notes
Possible question:
What do you think of when you
hear the word average?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
2.
Procedures
(10 minutes) Ask students what they think
of when they hear the expression “he’s just
your average Joe.”
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Notes
Possible question:
What do you think of when you
hear the phrase “he’s just your
average Joe?”
Place Transparency 1 on the overhead.
Give students time to use linking cubes to
make a tower with the number of cubes that
represents the number of letters in their first
name. (If there is an even number of
students in the group, the teacher should
participate.)
3.
4.
5.
(10 minutes) Ask students to hold up their
towers. Remark that the data is random and
difficult to analyze. Suggest that arranging
the data in some way may help.
Possible questions:
Have the students with a first name
containing the smallest number of letters
stand on one side of the room with their
name tower and the students with the largest
number of letters stand on the opposite side
of the room with their tower. Ask the other
students to come up with their name towers
and line up between the least and greatest
numbers in order according to the number of
cubes in their towers. If there are students
with the same number of letters in their first
name, have them stand side by side.
Identify the smallest and largest number of
letters and discuss the spread of the data.
Who has the least number of
letters in their name? Who has the
greatest number of letters in their
name? What does this tell us
about the data?
(5 minutes) Have students count off from
both ends of the line simultaneously. When
you reach the student in the center, the
number of cubes in their tower is the median
number of letters in the first names for this
group. Remove one student from the line
and talk about finding the median when two
numbers are in the middle.
Possible question:
(5 minutes) Ask students with the same
Possible questions:
Probability and Statistics
Your Average Joe
How might we arrange the data to
make it easier to analyze?
Remind students that by looking at
the greatest and least numbers in
this or any set of data we can
determine how much the data
varies. The range of the set of
data is the difference between the
greatest and least numbers in the
set and is one way to express the
spread of the data. If the range is
a small number, the data are close
together.
How would we find the median of
the data if there was not a number
in the middle?
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
6.
7.
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Procedures
number of letters in their name to line up
behind one another. Identify the longest line.
The number of cubes that each person in the
longest line has represents the mode of the
number of letters used most often in first
names for this group of people.
Notes
Which number of letters occurs
most frequently in this class?
How do you think this would
compare to other classes of 6th
graders?
(10 minutes) To model finding the mean,
have participants either give away or take
cubes until all participants have towers of the
same length. Hold extra cubes aside and
discuss what to do with them. Discuss the
mean of the group by looking at their even
cube towers. Have students estimate the
mean. This is particularly important if there
are extra cubes.
Possible questions:
How might we use the cubes to
find the mean number of letters in
our names? What do we do with
any extra cubes?
(10 to 15 minutes) Have the students return
to their seats. Place Transparency 2 on the
overhead and define median, mode, and
mean by referring to the physical activities.
Possible questions:
What did we do first to find the
median? After we were arranged
in order, how did we find the
median?
Calculate the mean using a calculator and
compare to the earlier estimate.
Talk about the fact that we could
have taken all the towers apart,
put the cubes in a pile, and let
each person take cubes from the
pile until everyone had the same
number of cubes. This is more
like the “add and divide” method of
finding the mean with which most
students are familiar.
How did we determine the mode?
How did we determine the mean?
How does the mean we found
using the cubes compare to the
mean we computed with the
calculator?
8.
(20 to 30 minutes) Pose another problem
such as the one on Transparency 3 for
students to model.
Ask students to work in pairs, use the linking
cubes to model their thinking, and draw a
diagram or sketch to explain their work.
Provide grid paper for the drawings.
Probability and Statistics
Your Average Joe
Students may approach this
problem in several different ways:
(1) Some may build the five
towers and then level them off by
moving cubes one-by-one from
the taller to the shorter towers.
(2) Some may make one long
tower and then break it into five
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Procedures
Notes
Give several pairs of students the
equal towers.
opportunity to share their work with the class. (3) Others may break the towers
into a large pile and rebuild them
into five equal towers.
Experience with leveling off towers
of cubes, describing their
methods, and listening to others’
methods helps students to
develop a strong visual model for
the meaning of average or mean.
Questions to ask during sharing:
How is your solution different from
this one?
How is your solution the same as
this one?
9.
Student Reflection: How will this activity
help you remember how to determine the
median, mode and mean for a set of data?
Homework:
Handout 1 can be used as homework.
Extensions:
See Transparency 4.
Probability and Statistics
Your Average Joe
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Your Average Joe
You’ve heard the expression “your average Joe.” Is Joe
really average? The name Joe only contains three letters.
Does the average name contain three letters? How many
letters do you think an average name might have?
Let’s look at the names of the people in this class and see
what we can determine about the average name. Count
the number of letters in your first name. Use the linking
cubes to make a tower with this number of cubes.
Transparency 1
Probability and Statistics
Your Average Joe
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Measures of Central Tendency
Median
Mode
Mean
Transparency 2
Probability and Statistics
Your Average Joe
Grade 6
Page 6
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Missing from Math Class
The table shows the number of students absent from a
mathematics class each day last week. What was the
mean number of students absent per day last week?
Day of
Number of
the Week Students Absent
Monday
2
Tuesday
6
Wednesday
4
Thursday
2
Friday
1
Find the median and mode for the number of days absent.
How do these measures of the data compare to the
mean? Explain why this happens.
Transparency 3
Probability and Statistics
Your Average Joe
Grade 6
Page 7
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Finding the Mean, Median, and Mode
Use your own paper to record your work.
1.
In the last four softball games, Michelle scored 5, 3, 2, and 6 runs.
a. Use concrete objects to model the mean and draw of diagram of your thinking.
b. What is the median number of runs and how does to compare to the mean?
c. Does this data set have a mode? Explain.
2.
Dave helped his dad with his landscaping business for 5 days. He earned $9, $11, $14,
$15, and $11.
a. What was the mean amount he earned each day? Explain how you determined the
answer.
b. What is the median amount he earned?
c. Does this data set have a mode? Explain.
3.
Jacob scored an average (mean) of 18 points per game during the first 5 games of the
season. Before the 6th game he was injured and didn’t get to play. What was his 6-game
average? Explain how you determined your answer.
4.
You want to convince your parents to raise your allowance. You ask several friends how
much allowance they get each week. Their responses are recorded in the table.
Name
Andrew
Collin
Grayson
Reid
Sally
Weekly Allowance
$9.00
$7.00
$8.00
$50.00
$8.00
a. What is the mean of their weekly allowances? Explain how you determined your
answer.
b. Does the mean describe the typical allowance for this group of friends? Explain.
c. Does the median describe the typical allowance for this group? Explain
5.
Yolanda has an average of 84 points on 3 assignments. She wants to bring up her
average to 88 points. What must she score on her 4th assignment? Use a diagram, sketch,
and/or words to explain your answer.
Handout 1
Probability and Statistics
Your Average Joe
Grade 6
Page 8
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Think About It!
For a group of 7 students, the number of letters in their
last names is 5, 5, 8, 6, 4, and 10. As you can see, one
piece of data is missing, but we know that the median is 6.
Can we determine the missing number? Why or why not?
Suppose we know the mean is 6, can we determine the
missing number? Why or why not?
Transparency 4
Probability and Statistics
Your Average Joe
Grade 6
Page 9
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
Finding the Mean, Median and Mode
Possible Solutions
Use your own paper to record your work.
1. In the last four softball games, Michelle scored 5, 3, 2, and 6 runs.
a. Use concrete objects to model the mean and draw of diagram of your thinking.
Possible solution:
Move one cube from the first tower to the second, move two cubes from the last
tower to the third tower. The towers are level at four cubes in each, so the mean
number of runs is four.
b. What is the median number of runs and how does it compare to the mean?
First I put the runs in order. (2, 3, 5, 6) Because there are 4 numbers, there is no
middle number. The median is the average of the two middle scores, 3 and 5.
The median is 4 which is the same as the mean.
c. Does this data set have a mode? Explain.
No, there is no mode because each number of runs occurs only once.
2. Dave helped his dad with his landscaping business for 5 days. He earned $9, $11,
$14, $15, and $11.
a. What was the mean amount he earned each day? Explain how you determined
the answer.
Possible Solution Method:
Since he earned the most on the fourth
day, I started there and shared some of
that money with day 1 and day 5. Then I
moved some of the money from day 3 to
day 1 and day 2. Once the cubes were
level, I could see that the mean amount
he earned each day was $12.
Probability and Statistics
Your Average Joe
Grade 6
Page 10
Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
b. What is the median amount he earned and does it compare to the mean?
First I put the amounts of money in order. (9, 11, 11, 14, 15) Because there are
5 numbers, the median is the one in the middle or 11. The median is $11 which is
less than the mean.
c. Does this data set have a mode? Explain.
Yes. The mode is $11 because it occurs most often in the list of the amounts he
earned.
3. Jacob scored an average (mean) of 18 points per game during the first 5 games of
the season. Before the 6th game, he was injured and didn’t get to play. What was his
6-game average? Explain how you determined your answer.
If he scored an average of 18 points for 5 games, that is a total of 90 points. If he
didn’t score during the 6th game, the 90 points now must be divided by 6. So, his 6game average is 15 points.
4. You want to convince your parents to raise your allowance. You ask several friends
how much allowance they get each week. Their responses are recorded in the
table.
Name
Andrew
Collin
Grayson
Reid
Sally
Weekly Allowance
$9.00
$7.00
$8.00
$50.00
$8.00
a. What is the mean of their weekly allowances? Explain how you determined your
answer.
I added the allowances together and divided by 5. The mean is $16.00.
b. Does the mean describe the typical allowance for this group of friends? Explain.
No, because the mean is $16 which is not close to any of the weekly allowances.
c. Does the median describe the typical allowance for this group? Explain
Yes, because the median is $8 which is close to 4 of the 5 allowances.
Probability and Statistics
Your Average Joe
Grade 6
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Mathematics TEKS Refinement 2006 – 6-8
Tarleton State University
5. Yolanda has an average of 84 points on 3 assignments. She wants to bring up her
average to 88 points. What must she score on her 4th assignment? Use a diagram,
sketch, and/or words to explain your answer. Possible Solution Method:
The first 3 towers drawn
with a solid black line
represent the 84 points
scored on the first three
assignments. She needs to
score 4 more points to raise
each of these assignments to
The fourth tower
represents a
score of 88 on
the fourth
assignment.
She needs to score 88 points plus 12 more
points to raise each of the first three
assignments to 88. That means Yolanda has
to score 100 on the fourth assignment to
bring up her average to 88 points. She
better start studying!
100
88
84
Probability and Statistics
Your Average Joe
Grade 6
Page 12
Possible Solutions
Think About It!
Tarleton State University
Probability and Statistics
Your Average Joe
The missing number is 4, so the person missing has 4 letters in his or her first name.
7 X 6 = 42
42 – 5 – 5 – 8 – 6 – 4 – 10 = 4
Grade 6
Page 13
Yes. Since we know the mean of the numbers is 6 we can find the sum of the numbers. Then we can
subtract the numbers we know it find the missing number.
Suppose we know the mean is 6, can we determine the missing number? Why or
why not?
No. All we know is that the missing number must be greater than or equal to 6 in order for 6 to be in the
middle.
Can we determine the missing number? Why or why not?
For a group of 7 students, the number of letters in their last names is 5, 5, 8, 6, 4,
and 10. As you can see, once piece of data is missing, but we know that the
median is 6.
Mathematics TEKS Refinement 2006 – K-5