Median and Mean
Objective To calculate and compare the median and mean
of a data set.
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Math Boxes 1 4
• Calculate and compare the median and
mean of a data set. Math Journal 1, p. 16
metric ruler
Students practice and maintain skills
through Math Box problems.
[Data and Chance Goal 2]
• Examine how the median and the mean
change as data change. [Data and Chance Goal 2]
• Find the mode of a data set. [Data and Chance Goal 2]
Study Link 1 4
Math Masters, p. 13
Students practice and maintain skills
through Study Link activities.
• Determine which landmark best represents
a set of data—mean, median, or mode. [Data and Chance Goal 2]
Key Activities
Students find and compare the median and
the mean of various data sets.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters,
page 404). [Data and Chance Goal 2]
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Defining the Mean
Math Masters, p. 14
unit cubes (25 per student)
Students use manipulatives to represent the
mean and write number sentences that
model how to find the mean.
ENRICHMENT
Using Technology to Find the Median
and Mean
Student Reference Book
computer spreadsheet/graphing software
Students use computer software to find the
median and mean of a data set.
EXTRA PRACTICE
Solving Custom-Made Math Boxes
Math Masters, p. 405
Students complete teacher-generated
Math Boxes.
Materials
Math Journal 1, pp. 14 and 15
Study Link 13
Math Masters, p. 404
calculator
Advance Preparation
Consider having students use computer software for the optional Enrichment activity in Part 3.
Teacher’s Reference Manual, Grades 4–6 pp. 168, 169
Lesson 1 4
33
Mathematical Practices
SMP2, SMP3, SMP4, SMP6
Content Standards
Getting Started
6.SP.2, 6.SP.3, 6.SP.5c, 6.SP.5d
Bold SMP = Guiding Questions at everydaymathonline.com
Bold = Focus of lesson
Math Message
Mental Math and Reflexes
Students solve mental math problems such as the following:
Add 25 and 70. Take away 15. Triple the result. 240
Add 250 and 750. Divide by 10. Subtract 50. 50
Add 35 and 65. Multiply by 5. Double the result. 1,000
Complete Problem 1 on
journal page 14.
Study Link 1 3
Follow-Up
Briefly review answers.
NOTE Some students may benefit from
doing the Readiness activity before beginning
Part 1 of the lesson. See Part 3 for details.
1 Teaching the Lesson
▶ Math Message Follow-Up
(Math Journal 1, p. 14)
Remind students that mean, median, and mode are all landmarks
that are considered measures of the center of a data set. Ask
them to share their thoughts about which of these landmarks best
represents a typical annual salary at the company. Sample answer:
Both the mean and mode are greater than $90,000. Because 9 of
the 12 employees earn less than $90,000, the mean and mode do
not seem to be good representations of the salaries. Since 7 of the
12 salaries are between $40,000 and $63,000, the median salary
of $54,000 best represents the typical annual salary. To support
English language learners, discuss the meaning of typical as well
as annual salary.
Student Page
Time
LESSON
Comparing the Median and Mean
14
Math Message
A small office supply company has 12 employees.
Their yearly salaries appear in the table, as
well as the mean, median, and mode salaries.
1.
Does the mean, the median, or the mode best
represent the typical salary at the company?
Explain.
Sample answer: The mean
and mode salaries are greater
than $90,000. Because 9 of
the 12 salaries are less than
$65,000, the median salary
seems the best representation
of a typical salary at Fancy
Font.
Salaries at Fancy Font
Office Supplies
Job
Annual Salary
President
$275,000
Vice-President: Marketing
$185,000
Vice-President: Sales
$185,000
Marketing Manager
$62,500
Product Manager
$59,000
Sales Manager
$55,500
Promotions Manager
$52,500
Salesperson 1
$45,000
Salesperson 2
$43,500
Salesperson 3
$43,000
Administrative Assistant 1
$39,500
Administrative Assistant 2
$36,000
Mean salary
$90,125
Median salary
$54,000
Mode salary
Discuss how the mean and the median change as data change.
Pose the following situation: Suppose the salary of the company
president increased to $350,000. How would this affect the mean
salary? The mean salary would increase. How would the increase
affect the median salary? The median salary would remain the
same. It is less affected by the maximum and minimum salaries
than the mean salary is. Explore other situations that would affect
the mean and median salaries.
Circulate and assist as students complete journal page 14.
Bring the class together to share answers.
$185,000
Median and Mean
Find the median and mean for each of the following sets of numbers.
2.
6, 9, 10, 15
a.
median
3.
0.50, 0.75, 1, 1.25, 0.80
a.
median
4.
123, 56, 92, 90, 88
a.
median
9.5
0.80
90
b.
mean
b.
mean
b.
mean
10
0.86
89.8
Math Journal 1, p. 14
EM3cuG6MJ1_U01_1-44.indd 14
34
Unit 1
ELL
Review the salary data in the table. Ask students to explain how
the mean salary was calculated. Sample answer: The annual
salaries were added together and the sum was divided by the total
number of employees.
Interactive whiteboard-ready
ePresentations are available at
www.everydaymathonline.com to
help you teach the lesson.
Date
WHOLE-CLASS
DISCUSSION
1/11/11 5:29 PM
Collection, Display, and Interpretation of Data
Adjusting the Activity
Ask students to explain how they could use the mean salary to find the
sum of the salaries.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
▶ Comparing the Median and
V I S U A L
WHOLE-CLASS
DISCUSSION
Mean of a Data Set
Relate the following situation to students.
Imagine that it is the end of the grading period. You have received
the following test scores: 90, 88, 100, and 82. You have one more
test to take before the end of the grading period.
Pose the following questions:
●
If your teacher bases your grade on the median of your test
scores, will you be motivated to study for the last test? Sample
answer: No. There would not be much incentive to study. No
matter how low my last test score is, my median score will not
be lower than 88.
●
If your teacher bases your grade on the mean score, will you be
motivated to study for the last test? Sample answer: Yes. I will
be motivated to study. The better test score I make, the higher
my final grade will be.
●
Suppose you score 20 out of 100 on your final test. Will the
median or the mean give a better picture of your overall
performance? Sample answer: The median (88) seems to
give a better overall picture. The mean will be 76, which is
6 points lower than the lowest of my other test scores.
●
Suppose you score 95 out of 100 on your final test. Will the
median or the mean give a better picture of your overall
performance? Sample answer: Both will be about 90.
●
Most teachers determine final grades by calculating the mean
of the scores for each student. Why do you suppose they use the
mean? Sample answer: One or more low scores can significantly
lower the mean of the scores. Using the mean gives students
incentive to get high scores on all tests.
●
Can you think of a situation outside of school in which the
mode of the data would be most useful? Sample answers:
A small grocery store that can stock only one brand of peanut
butter would want to stock only the most requested brand.
A political candidate often wants to know the most popular
public opinion concerning an issue.
Lesson 1 4
35
Student Page
Date
▶ Finding Landmarks
Time
LESSON
Data Landmarks
14
䉬
The 10 most successful coaches in the history of the
National Football League (NFL) are listed in the table
at the right, along with the number of games won
through the end of the 2002 season.
Most Successful NFL Coaches
Find the following landmarks for the data set
displayed in the table.
1.
median
2.
maximum
3.
minimum
4.
mean
5.
mode
6.
range
203.5
347
155
226.3
none
192
Coach
Games Won
Don Shula
347
George Halas
324
Tom Landry
270
Curly Lambeau
229
Chuck Noll
209
Dan Reeves
198
Chuck Knox
193
Paul Brown
170
Bud Grant
168
Steve Owen
155
10, 14, 12, 11, 12 12
10, 14, 12, 11, 12, 11 11.5
Try This
Denzel’s first three test scores in math were 90, 100, and 90.
a.
(Math Journal 1, p. 15)
Ask students to find the median of the following data sets:
136 137
7.
PARTNER
ACTIVITY
Draw their attention to the number of values in the second data
set. Because the number of values is even, the median is the
number halfway between the two middle values, or the mean
of the two middle values.
Circulate and assist as students complete the journal page.
What must Denzel score on his fourth test to keep his
mean score at 90 or higher?
80 or higher
b.
What must Denzel score on his fourth test to keep his
median test score at 90 or higher?
Ongoing Assessment:
Recognizing Student Achievement
Any score, even 0, will keep the median at
90 or higher.
Exit Slip
Use an Exit Slip (Math Masters, p. 404) to assess students’ understanding of
median and mean. Students are making adequate progress if they can
acknowledge that the median is less affected by outliers than the mean is.
Math Journal 1, p. 15
Pose the following informal assessment item to students: In 2004, baseball player
Derek Jeter earned about $25 million in salary, bonuses, and endorsements. If
you were to report the typical annual earnings for baseball players, would it be
more accurate to report the mean or median earnings? Why?
[Data and Chance Goal 2]
NOTE To help students put Derek Jeter’s salary into context, tell them that the
salaries of baseball players in 2004 ranged from $300,000 to $21,000,000.
2 Ongoing Learning & Practice
Student Page
Date
Math Boxes
14
䉬
1.
▶ Math Boxes 1 4
Time
LESSON
(Math Journal 1, p. 16)
Measure the line segment below to the
nearest centimeter.
2.
Write a data set that fits the following
description.
Sample
answer:
17, 20, 28, 32,
41, 41, 62
cm
Measure the line segment below to the
nearest millimeter.
b.
44
3.
Subtract.
a.
1,000 ⫺ 25 ⫽
136
mm
975
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lessons 1-2 and 1-5a. The skills in
Problems 4 and 5 are prerequisites for Unit 2.
There are 7 numbers in the data set.
The minimum is 17.
The range is 45.
The median is 32.
The mode is 41.
a.
3
b.
INDEPENDENT
ACTIVITY
2,037 ⫺ 294 ⫽
1,743
c.
996
▶ Study Link 1 4
⫽ 7,214 ⫺ 6,218
INDEPENDENT
ACTIVITY
(Math Masters, p. 13)
Home Connection Students construct stem-and-leaf plots
and practice finding the mean and median of data sets.
15–17
4.
Find the quotient.
5.
17冄4
苶5
苶9
苶
Complete.
a.
b.
c.
d.
e.
459 ⫼ 17 ⫽
27
22–24
60 ⫽ 48,000
2,400 ⫽ 60 ⴱ 40
30
1,500 ⫽ 50 ⴱ
700
630,000 ⫽ 900 ⴱ
90 ⴱ 300 ⫽ 27,000
800 ⴱ
18
Math Journal 1, p. 16
36
Unit 1
Collection, Display, and Interpretation of Data
Study Link Master
Name
3 Differentiation Options
Date
STUDY LINK
14
䉬
Time
Median and Mean
Mia’s quiz scores are 75, 70, 75, 85, 75, 85, 80, 95, and 80.
135–137
Nico’s quiz scores are 55, 85, 95, 100, 75, 75, 65, 95, and 75.
READINESS
▶ Defining the Mean
PARTNER
ACTIVITY
Find each student’s mean score. Mia
2.
Make a stem-and-leaf plot for each student’s scores.
a.
Mia’s Quiz Scores
Stems
(100s and 10s)
5–15 Min
b.
Nico
80
Nico’s Quiz Scores
Leaves
(1s)
Stems
(100s and 10s)
7 0555
8 0055
9 5
(Math Masters, p. 14)
To provide experience with modeling and finding the mean,
have students use unit cubes to represent data values. They
redistribute, or even out, the unit cubes to find the mean value.
Students then write a number sentence to model how they
redistributed the unit cubes to help determine the mean of the
data set.
80
1.
Leaves
(1s)
5
6
7
8
9
10
80
5
5
555
5
55
0
75
3.
Find each student’s median score. Mia
4.
What is the range of scores for each student? Mia
Nico
5.
Which landmark, mean or median, is the better indicator of each student’s
overall performance? Explain.
25
Nico
45
Because Mia’s mean and median scores are the
same (80), either landmark is a good indicator for
her. Nico’s median score is the better indicator of
his performance. His mean is not as good an
indicator because of the range of his scores.
ENRICHMENT
▶ Using Technology to Find the
SMALL-GROUP
ACTIVITY
Practice
6.
$4.57 ⫹ $1.25 ⫽
8.
$19.99 ⫺ $5.75 ⫽
$5.82
$14.24
7.
$14.49 ⫹ $15.78 ⫽
9.
$39.25 ⫺ $18.75 ⫽
$30.27
$20.50
15–30 Min
Math Masters, p. 13
Median and Mean
Assign each group one of the data sets provided in the Data and
Probability section of the Student Reference Book. Students enter
the data into a spreadsheet or graphing program to sort the values
from least to greatest to determine the median. They should then
enter a formula to calculate the mean of the same data set.
EXTRA PRACTICE
▶ Solving Custom-Made
INDEPENDENT
ACTIVITY
5–15 Min
Math Boxes
(Math Masters, p. 405)
To provide extra practice, use Math Masters, page 405 to
generate Math Box questions that focus on a particular concept
or skill for which students need extra practice.
Teaching Master
Name
LESSON
14
䉬
Date
Defining the Mean
The table at the right shows the number of students
absent from gym class during the week.
1.
Time
Place unit cubes on the line below to show
the number of absent students for each day.
Monday
Tuesday
Day
Students Absent
Monday
6
Tuesday
2
Wednesday
5
Thursday
4
Friday
8
Wednesday Thursday
Friday
If you redistribute, or even out, the number of absent students so the number is the same
for each day, you are finding the mean. The mean is a useful landmark when there are not
one or two numbers that are far away from the rest of the data values (outliers).
2.
Move the cubes on the line plot so that each day has the same number.
After you’ve evened out the cubes, how many does each day have?
5
You can use a number sentence to model how you evened out the cubes. You started
with 6 ⫹ 2 ⫹ 5 ⫹ 4 ⫹ 8 ⫽ 25 cubes. Then you redistributed the cubes so that the total
number of cubes (25) was the same for each of the 5 days, or 25 ⫼ 5 ⫽ 5.
3.
Use the cubes to find the mean of the following number of absent students.
Monday: 5; Tuesday: 0; Wednesday: 6; Thursday: 2; Friday: 7
Then write a number sentence to model what you did.
20 ⫼ 5 ⫽ 4
Math Masters, p. 14
Lesson 1 4
37