Stem-and-Leaf Plots
Objective To use stem-and-leaf plots for organizing and
analyzing data.
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Key Concepts and Skills
Rounding to Estimate Products
• Construct a stem-and-leaf plot from
unsorted data. Math Journal 1, p. 12
Students practice rounding numbers
and estimating products.
[Data and Chance Goal 1]
• Compare different graphical representations
of the same data. [Data and Chance Goal 2]
• Identify landmarks of data (maximum,
minimum, range, median, mode) displayed
in stem-and-leaf plots. [Data and Chance Goal 2]
Key Activities
Students review the basics for stem-and-leaf
plots. They utilize double stems to organize
and display larger sets of data. Students
determine the median, mode, and range
from constructed stem-and-leaf plots.
Math Boxes 1 3
Math Journal 1, p. 13
Geometry Template
Students practice and maintain skills
through Math Box problems.
Study Link 1 3
Math Masters, p. 10
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Informing Instruction See page 30.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Reviewing Stem-and-Leaf Plots
Math Masters, p. 11
measuring tape
Students review how to read a stem-and-leaf
plot.
ENRICHMENT
Reading and Constructing Back-to-Back
Stem-and-Leaf Plots
Math Masters, p. 12
Students use a back-to-back stem-and-leaf
plot to display and compare two sets of data.
EXTRA PRACTICE
5-Minute Math
5-Minute Math™, pp. 116 and 198
Students find and consider landmarks
of data sets.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 11. [Data and Chance Goal 2]
Key Vocabulary
minimum maximum range mode median stem-and-leaf plot stem leaf double-stem plot
Materials
Math Journal 1, pp. 9–11
Study Link 12
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 162–165
Lesson 1 3
27
Mathematical Practices
SMP1, SMP2, SMP4, SMP6, SMP7
Content Standards
Getting Started
6.SP.1, 6.SP.2, 6.SP.3, 6.SP.5a, 6.SP.5c
Bold SMP = Guiding Questions at everydaymathonline.com
Mental Math
and Reflexes
Bold = Focus of lesson
Math Message
Complete Problems 1–3 on journal page 9. Be prepared to discuss
your answers.
Students record dictated numbers on their
slates or dry-erase boards and then
round them to a specified place.
Study Link 1 2 Follow-Up
Round to the nearest hundred.
936; 575; 5,231; 67,351
900; 600; 5,200; 67,400
Briefly review answers. For Problem 3, ask students to explain how they
displayed the two modes on their line plot. If you assigned the Practice
problems, establish a routine for checking these problems.
Round to the nearest thousand.
19,067; 33,733; 483,755; 96,502
19,000; 34,000; 484,000; 97,000
NOTE Some students may benefit from doing the Readiness activity before
beginning Part 1 of the lesson. See Part 3 for details.
Math Test Scores
Stems
(100s and 10s)
Leaves
(1s)
1 Teaching the Lesson
5
5 6 6 6
6
1 3 3 4 8
7
1 5
8
1 4 4 4 4 7 7 8 8
9
1 5 8 9 9
10
▶ Math Message Follow-Up
(Math Journal 1, p. 9)
WHOLE-CLASS
DISCUSSION
SOLVING
Consider having students discuss their answers to Problem 2 of
the Math Message in small groups. Then bring the class together
to share the similarities and differences among Graphs a–d. Expect
statements such as the following:
0 0
Cross out one data value from each end of the
plot until only one (or two) remains.
The number of data points on each graph is the same.
The range of the numbers on each graph is the same.
Student Page
Date
Each graph has the same title.
Time
LESSON
13
The stem-and-leaf plot clearly displays individual data values.
The other graphs are not as specific; they show general trends.
Comparing Graphical Representations
Math Message
Math Test Scores
Stems Leaves
(100s and 10s)
(1s)
5
5 6 6 6
6
1 3 3 4 8
7
1 5
8
1 4 4 4 4 7 7 8 8
9
1 5 8 9 9
10
c.
50–59
60–69
70–79
0 0
80–89
90–99 100–109
Scores
Number of Students
////
////\
//
////\ ////
////\
//
50–59
60–69
70–79
80–89
90–99
100
Math Test Scores
d.
X
X
XX
50
55
X
X XX
60
65
X
X
70
X
75
X
80
X
X
X
X
85
XX
XX
X
90
X
XX
XX X
95 100
Use pages 134, 135, and 138 of the Student Reference Book to identify each of the data
representations (a–d) above.
a.
c.
2.
Math Test Scores
10
9
8
7
6
5
4
3
2
1
0
Math Test Scores
Scores
1.
b.
Number of Students
a.
Stem-and-leaf plot
Tally chart (of grouped data)
b.
d.
Bar graph
Line plot
Ask: What is a statistical question that can be asked about the
data shown in the graphs? Sample answer: What is the range
of test scores for Mr. Smith’s 6th grade class? How do you know
the question is a statistical question? Sample answer: In order
to answer the question, I have to analyze the data shown in the
graphs.
Ask students to read the leaves along one of the stems. Sample
answer: 61, 63, 63, 64, 68 Ask: How many test scores are greater
than 95? 5 How many scores are less than 75? 10 How many total
test scores are shown? 27
Explain how these data representations are alike and how they are different.
Each graph displays the same set of data. Only the stem-andleaf plot and line plot show individual values. The stem-and-leaf
plot shows the individual values more clearly.
3.
Which graphical representation helps you identify the range, median, and mode most easily?
Explain your choice. Sample answer:
The stem-and-leaf plot displays individual values so it is easy
to determine data landmarks.
Math Journal 1, p. 9
EM3cuG6MJ1_U01_1-44.indd 9
28
1/11/11 5:29 PM
Unit 1
Collection, Display, and Interpretation of Data
Student Page
Expect that most students will be able to identify the minimum
55, maximum 100, range 45, and mode 84. Students will see
that the procedure for finding a median is the same for any data
set that is displayed in numerical order from smallest to largest or
vice versa. In each case, students cross out or count one data value
from each end of the data set until one value (or two) remains. (See
margin of page 28.) There are 27 numbers, so once the numbers
are put in order from smallest to largest, the middle value is the
14th number because there are 13 numbers above it and 13 below
it. The median is 84. Remind students that the median is one
measure of the center of a data set. The range is a measure of the
spread, or variability, of a data set.
Date
Time
LESSON
Old Faithful Erupts
13
䉬
A geyser is a natural fountain of water and steam that erupts from the ground. Old
Faithful is perhaps the most studied geyser of Yellowstone Park. Its eruptions have
been recorded since its discovery in 1870. Mathematicians have examined the
relationship between the time in minutes an eruption lasts, which is called the
duration, and the time to the next eruption, which is called the interval.
135 136
Duration data appear in the table below.
Duration of Old Faithful Eruptions (in min)
(Number of Observations: 48)
4.9
1.7
2.3
3.5
2.3
3.9
4.3
2.5
3.4
4.8
4.1
1.9
4.6
4.1
2.9
3.7
3.4
1.7
1.7
3.3
4.0
4.6
3.1
2.9
4.1
4.6
2.0
3.5
4.2
4.7
1.8
4.0
1.8
1.9
2.3
2.0
4.5
3.7
3.9
3.9
1.9
4.3
3.2
4.7
3.5
2.0
1.8
4.5
A stem-and-leaf plot is a useful way to find landmarks when there are many data
values in random order. The stem-and-leaf plot of the eruption data appears below.
Duration of Old Faithful Eruptions
(Number of Observations: 48)
Stems
(ones)
Leaves
(tenths)
NOTE Most students understand that the mode of a data set is the value that
1
7 7 7 8 8 8 9 9 9
2
0 0 0 3 3 3 5 9 9
appears most often. However, students may not realize that a mode does not
always exist or that a mode does not have to be unique.
3
1 2 3 4 4 5 5 5 7 7 9 9 9
4
0 0 1 1 1 2 3 3 5 5 6 6 6 7 7 8 9
The data in the plot are ordered, making it easier to determine data landmarks.
Using the stem-and-leaf plot above, find the minimum, maximum, and range of
the duration data.
▶ Constructing
a.
WHOLE-CLASS
ACTIVITY
Double-Stem Plots
(Math Journal 1, pp. 10 and 11)
minimum
1.7
b.
maximum
4.9
c.
range
3.2
Math Journal 1, p. 10
A stem-and-leaf plot is a useful way to organize data that are in
a random order. Decide what unit to use for the stems and what
unit to use for the leaves. The Old Faithful eruption data on
journal page 10 involve a number of whole minutes and tenths of a
minute. A duration of 4.9 minutes is split as 4 | 9 in the stem-andleaf plot. The digits to the left of the vertical line form the stem.
The digits to the right of the vertical line form the leaf. In the case
of 4 | 9, the stem is 4 minutes and the leaf is 0.9 minutes. The
stem is written only once, but the leaves are listed every time
they appear. In this way, individual data values are displayed. This
individual display is an advantage of using a stem-and-leaf plot.
Adjusting the Activity
ELL
To support English language learners, make a connection between a
stem-and-leaf plot and the similarly named parts of a plant or tree.
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
V I S U A L
Ask students to use the stem-and-leaf plot on journal page 10 to
find the maximum, minimum, and range. After students have
recorded these landmarks on the journal page, draw attention to
the many leaves in the plot, particularly for stems 3 and 4. Long
rows of leaves make it cumbersome to find the median. In cases
like the eruption-duration data, in which many leaves fall on a few
stems, students can split the stems to make a double-stem plot.
In a double-stem plot, each stem having leaves that span from 0
to 9 is listed twice. Leaves 0 to 4 go on the upper stem, and leaves
5 to 9 go on the lower stem. (See margin.)
Notice that because the values of leaves for the first stem begin
with 7, the first stem has not been split. The rows of leaves are
shorter, so it is easier to find the median.
Duration of Old Faithful Eruptions
(Number of Observations: 48)
Stems
Leaves
(ones)
(tenths)
1
7 7 7 8 8 8 9 9 9
2
0 0 0 3 3 3
2
5 9 9
3
1 2 3 4 4
3
5 5 5 7 7 9 9 9
4
0 0 1 1 1 2 3 3
4
5 5 6 6 6 7 7 8 9
Lesson 1 3
29
Student Page
Date
Time
LESSON
Circulate and assist as students work on journal page 11. The
table on the page shows the time between eruptions in minutes.
Have students complete the double-stem plot and find the
maximum, minimum, range, median, and mode of the data set.
Stem-and-Leaf Plot: Double Stems
13
䉬
Predicting Old Faithful’s eruptions can be difficult. To predict its next eruption, mathematicians
have studied the length of time between eruptions, which is called the interval.
Interval data appear in the table below.
Interval of Old Faithful Eruptions (in min)
(Number of Observations: 48)
95
60
49
61
75
68
70
86
58
66
88
93
42
91
45
69
81
57
54
67
80
86
67
83
79
48
50
53
81
77
56
86
72
80
76
53
61
72
88
57
53
51
86
81
77
83
78
70
The stem-and-leaf plot of the interval data has been started for you. Complete the plot by filling in
the leaves for each double stem. Remember that for each pair of identical stems, leaves with
values of 0–4 go on the upper stem, and leaves with values of 5–9 go on the lower stem.
Interval of Old Faithful Eruptions
(Number of Observations: 48)
Stems
(tens)
4
4
5
5
6
6
7
7
8
8
9
9
Leaves
(ones)
2
5
0
6
0
6
0
5
0
6
1
5
8
1
7
1
7
0
6
0
6
3
9
3
7
1
7
2
7
1
6
3 3 4
8
8
2
7
1
6
Ongoing Assessment: Informing Instruction
Watch for students who are having difficulty keeping track of data values that
they have recorded in the plot and values that they still need to record.
Encourage students to develop a system of crossing out values in the table once
they’ve been recorded in the plot. Remind students to check that the number of
leaves in their plot is the same as the number of values in the table.
9
8 9
1 3 3
8 8
Journal
Page 11
Problems a–d
Ongoing Assessment:
Recognizing Student Achievement
夹
Use your completed stem-and-leaf plot to find the following landmarks:
a.
minimum
d.
mode
42
86
b.
maximum
e.
median
Math Journal 1, p. 11
95
71
c.
range
53
11
Use journal page 11, Problems a–d to assess students’ ability to find the
minimum, maximum, range, and mode of data displayed in a stem-and-leaf plot.
Students are making adequate progress if they accurately calculate these
landmarks from their constructed plots. Some students may be able to navigate
the double stems to find the median (near the 24th and 25th values).
[Data and Chance Goal 2]
Student Page
Date
LESSON
13
䉬
Time
Rounding to Estimate Products
Round each number to its greatest place value.
700
7,000
2. 6,557; thousands
20,000
3. 22,698; ten-thousands
4. 1,943,007; millions 2,000,000
30
5. 34; tens
6. 956,391; hundred-thousands 1,000,000
1.
261 262
711; hundreds
Rounding is one strategy you can use to estimate products. One way you can estimate a
product is by first rounding each number to its greatest place value and then computing.
Round each number to its greatest place value. Then estimate.
Example: 28 ⴱ 52
7.
62 ⴱ 79
8.
876 ⴱ 82
9.
456 ⴱ 714
10.
5,473 ⴱ 44
11.
3,736 ⴱ 633
12.
4,892 ⴱ 452
13.
3,491 ⴱ 5,347
14.
46,932 ⴱ 72
15.
16,236 ⴱ 284
Rounded factors 30 ⴱ 50
60 ⴱ 80
900 ⴱ 80
Rounded factors
500 ⴱ 700
Rounded factors
40
Rounded factors 5,000 ⴱ
Rounded factors 4,000 ⴱ 600
Rounded factors 5,000 ⴱ 500
Rounded factors 3,000 ⴱ 5,000
70
Rounded factors 50,000 ⴱ
Rounded factors 20,000 ⴱ 300
Rounded factors
Math Journal 1, p. 12
30
Unit 1
Collection, Display, and Interpretation of Data
Estimate 1,500
4,800
72,000
350,000
Estimate
200,000
Estimate
Estimate 2,400,000
Estimate 2,500,000
Estimate 15,000,000
Estimate 3,500,000
Estimate 6,000,000
Estimate
Estimate
Student Page
Date
2 Ongoing Learning & Practice
▶ Rounding to Estimate Products
䉬
1.
2.
Draw line segments having the following lengths.
a.
134 inches
b.
268 inches
c.
12
16
c.
The problems on journal page 12 provide practice rounding
numbers and estimating products.
inch
Add.
a.
▶ Math Boxes 1 3
Math Boxes
13
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 12)
Time
LESSON
4,209 6,385 1,366
10,594
b.
472 38,529 39,001
4 263 1,020 79
13 14
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 13)
3.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 1-1. The skill in Problem 4
previews Unit 2 content.
▶ Study Link 1 3
INDEPENDENT
ACTIVITY
Use your Geometry Template to draw a
regular hexagon. Then divide this figure
into 6 congruent triangles.
4.
Solve.
a.
$7.22 $3.43 $3.79
b.
$9.28 $2.76 $12.04
What kind of triangles are these?
Circle the best answer.
A.
scalene triangles
B.
equilateral triangles
C.
isosceles triangles
D.
right triangles
164 165
31–33
Math Journal 1, p. 13
(Math Masters, p. 10)
Home Connection Students construct a stem-and-leaf plot
for decimal number values.
Study Link Master
Name
Date
STUDY LINK
13
䉬
Time
Stem-and-Leaf Plots
Every day, there are many earthquakes
worldwide. Most are too small for people
to notice. Scientists refer to the size of an
earthquake as its magnitude. Earthquakes
are classified in categories from minor
to great, depending on magnitude.
Class
Magnitude
Great
8.0 or more
Major
7–7.9
Strong
6–6.9
Moderate
5–5.9
Light
4–4.9
Minor
3–3.9
135 136
The table below shows the magnitude of 21 earthquakes that occurred on June 28, 2004.
Magnitude of Earthquakes Occurring June 28, 2004
4.2
5.2
2.8
4.8
3.9
2.0
3.3
4.8
4.5
3.5
2.6
3.4
6.8
3.0
4.7
2.8
4.2
4.1
5.4
5.1
2.2
1.
Construct a stem-and-leaf plot
of the earthquake magnitude data.
Magnitude of Earthquakes
Occurring on June 28, 2004
2.
Use your stem-and-leaf plot to
find the following landmarks.
Stems
(ones)
4.8
2.8, 4.2, 4.8
median 4.1
a.
range
b.
mode(s)
c.
2
3
4
5
6
Leaves
(tenths)
0
0
1
1
8
2
3
2
2
6 8 8
4 5 9
2 5 7 8 8
4
Practice
80
3.
6,400 80 5.
3,000,000 6,000 500
4.
121,000 1,100 6.
600,000 12,000 110
50
Math Masters, p. 10
Lesson 1 3
31
Teaching Master
Name
Date
LESSON
Time
3 Differentiation Options
Reviewing Stem-and-Leaf Plots
13
䉬
Students in Mr. Conley’s sixth-grade class measured
how far they could reach and jump. Each student stood
with legs together, feet flat on the floor, and one arm
stretched up as high as possible. Arm reach was then
measured from top fingertip to floor.
1.
arm
reach
READINESS
Using a tape measure, measure your arm reach in inches.
Record this measurement.
in.
▶ Reviewing Stem-and-Leaf Plots
Answers vary.
In the standing jump, each student stood with knees
bent and then jumped forward as far as possible. The
distance was measured from the starting line to the
point closest to where the student’s heels came down.
Stems
(10s)
2.
jump
distance
Plot 2
Unit: inches
Leaves
(1s)
Stems
(10s)
Leaves
(1s)
4
4 6 8
6
1 7
5
0 0 3 3 4 5 6 7 7 8 8 9
7
0 1 2 2 2 3 3 4 5 6 6 6 6 8 9 9
6
0 0 1 3 8 9
8
3 4 7
To provide experience with stem-and-leaf plots, have students
work in pairs to complete Math Masters, page 11 and then
compare their answers. Have students describe the stem-andleaf plot when sharing their results.
Use what you know about your arm reach to figure out which stem-and-leaf
plot represents the class data for the standing jump.
a.
b.
15–30 Min
(Math Masters, p. 11)
Students displayed their results using two different
stem-and-leaf plots.
Plot 1
Unit: inches
PARTNER
ACTIVITY
1
Explain why you think so. Sample answer: The smallest
number in Plot 2 is 61, which is more than
5 feet. Most sixth-grade students cannot
jump that far.
Which plot do you think it is? Plot
Math Masters, p. 11
ENRICHMENT
▶ Reading and Constructing
INDEPENDENT
ACTIVITY
15–30 Min
Back-to-Back
Stem-and-Leaf Plots
(Math Masters, p. 12)
To extend their knowledge of stem-and-leaf plots, have
students study an example of a back-to-back stem-andleaf plot. They then construct a back-to-back stemand-leaf plot that has 2-digit stems.
EXTRA PRACTICE
▶ 5-Minute Math
SMALL-GROUP
ACTIVITY
5–15 Min
To offer more practice finding the range and mean of data sets, see
5-Minute Math, pages 116 and 198.
Teaching Master
Name
Date
LESSON
13
䉬
Time
Back-to-Back Stem-and-Leaf Plots
You can compare two related sets of data in a back-to-back stem-and-leaf plot.
In this type of plot, the stem is written in the center, with one set of leaves to the right
and another set of leaves to the left.
The ages of Wimbledon tennis champions in the women’s and men’s singles from
1993–2003 are shown in the back-to-back stem-and-leaf plot below.
Ages of Wimbledon Tennis
Champions 1993–2003
Women
Leaves
(1s)
Stems
(10s)
7
Men
Leaves
(1s)
1.
How many women champions were in
9
their twenties?
2.
What is the mode age for women?
3.
What is the median age for women?
21, 22 For men?
22
1
4 3 2 2 1 1 0
2
1 2 2 3 4
7 6
2
5 6 7 8 9
3
0
22
25
For men?
Students’ Heights (in centimeters)
4.
Girls’ Heights
Boys’ Heights
162, 126, 134, 145, 127, 134, 143, 159,
147, 169, 164, 171, 163, 171, 154, 157
154, 137, 162, 147, 145, 174, 132, 151,
170, 161, 166, 136, 168, 155, 153, 143
The data table above shows students’
heights in centimeters. Make a
back-to-back stem-and-leaf plot to
display the data. Use the 2-digit
stems provided.
Students’ Heights
Girls
Leaves
(1s)
7
4
75
97
943
1
6
4
3
4
2
1
Stems
(10s)
Boys
Leaves
(1s)
12
13
14
15
16
17
2
3
1
1
0
6
5
3
2
4
7
7
45
68
Math Masters, p. 12
32
Unit 1
Collection, Display, and Interpretation of Data