The Mean and the Median
Objectives To introduce the mean of a set of data; and to review
the
t median of a set of data.
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Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Interpreting a Pictograph
• Order whole numbers. Math Journal 2, pp. 255A and 255B
Student Reference Book, pp. 88
and 89
Children use data from a pictograph
to answer questions. Then they create
a bar graph to show the same data.
[Number and Numeration Goal 6]
• Use data to complete a bar graph. [Data and Chance Goal 1]
• Find the median and mean of data sets. [Data and Chance Goal 2]
• Use graphs to ask and answer questions. [Data and Chance Goal 2]
Key Activities
Children make bar graphs for given sets
of data. They model the bar graphs with
pennies and then rearrange the pennies to
determine the mean (average) for each data
set. They compare the mean and median of
data sets.
Math Boxes 10 6
Math Journal 2, p. 255
Children practice and maintain skills
through Math Box problems.
Home Link 10 6
Math Masters, p. 340
Children practice and maintain skills
through Home Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Graphing Dice Rolls
Math Masters, p. 339
per partnership: 2 dice
Children make a bar graph to record
dice-roll results.
ENRICHMENT
Making a Data Set
Math Masters, pp. 341 and 342
per partnership: counters
Children determine a set of data based
on given landmarks and graph the data.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 132
Children add the term mean to their
Math Word Banks.
Use journal page 254. [Data and Chance Goal 1]
Ongoing Assessment:
Informing Instruction See page 838.
Key Vocabulary
mean average median
Materials
Math Journal 2, pp. 253 and 254
Student Reference Book, pp. 80 and 83–85
Home Link 10 5
ruler or straightedge tool-kit pennies or
counters (30 per child) stick-on notes
(optional)
Advance Preparation
Teacher’s Reference Manual, Grades 1– 3 pp. 124–126
Lesson 10 6
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Getting Started
Mental Math and Reflexes
Math Message
Pose fact and fact extension problems like the
following. Have children record the facts on slates and
share their strategies for solving the fact extensions.
6 × 9 = 54
60 × 70 = 4,200
7 × 8 = 56
80 × 80 = 6,400
Make a bar graph of the data in the table on journal
page 253.
9 × 4 = 36
3 × 90 = 270
70 × 7 = 490
80 × 6 = 480
40
× 90 = 3,600
Home Link 10 5 Follow-Up
Have partners share answers.
1 Teaching the Lesson
Math Message Follow-Up
NOTE Everyday Mathematics does not
draw a distinction between bar graphs and
histograms. For a discussion on how some
people contrast them, see section 12.2.3,
Organizing and Displaying Data in the
Teacher’s Reference Manual.
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 253)
Check that children have completed the bar graph. Ask: What does
the height of each bar represent? The number of children in that
family
Finding the Mean
Number of Children
(Math Journal 2, p. 253; Student Reference Book,
pp. 83–85)
Student Page
Date
Time
LESSON
10 6
A Mean, or Average, Number of Children
Activity 1 Make a bar graph of the data in the table.
Family Sizes
8
7
Number of Children
6
5
Family
Number of
Children
Kugel
5
Abuka
1
Lauer
2
Miller
7
Ellis
1
Bosnak
2
WHOLE-CLASS
ACTIVITY
ELL
PROBLEM
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Ask whether anyone can remember how the class found an
average class shoe length in an earlier unit. (In Lesson 3-1,
16 children lined up along a paper tape, each placing one foot
heel-to-toe on the tape. After cutting off the leftover piece of tape,
children folded the tape into 16 equal parts. Each part
represented an average class shoe length.) Explain to the class
that they are going to find the average number of children per
family shown in the table using another method. Lead them in the
following routine:
Activity 2
4
(to be done later)
Use the table above.
List the number of
children in order.
3
2
1
0
Kugel Abuka Lauer Miller
Families
Ellis
The mean, or average, number of
children in the six families in the table is
Bosnak
3
.
1. Cover each shaded cell in the bar graph with a penny (or
other counter). What does each penny represent? A child
1
1
2
2
5
7
2. Level off the pennies; that is, rearrange the pennies so each
column in the graph has the same number of pennies.
3. With a straightedge, draw a horizontal line just above the
top row of pennies.
The median number
of children in the six
families in the table is
2
4. Remove the pennies.
.
Math Journal 2, p. 253
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Unit 10 Measurement and Data
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Student Page
The horizontal line represents the mean, or average, of this set
of data. In this example, the mean number of children per family
is 3. To support English language learners, discuss the social and
mathematical meanings of the words mean and average.
Emphasize that in this mathematical context, mean and average
describe the same thing.
Date
10 6
How could you find the place to draw the horizontal line
without using pennies? By trial and error, draw a horizontal
line so that the number of shaded squares above the line is the
same as the number of unshaded squares below the line.
Another method of finding the mean is to model each family by
drawing a medium-size circle for each family and putting in the
required number of pennies or counters to represent the number of
children. The counters are then redistributed among the families
(without adding or subtracting any) so each family has the same
number.
Ostrich Clutches
10
Clutch
9
8
7
Number of Eggs
●
Are the number of cells the same? Yes. Each time a penny was
moved, a shaded cell was matched with an unshaded cell.
A Mean, or Average, Number of Eggs
Activity 1 Make a bar graph of the data in the table.
Ask children to count the number of shaded cells above the line
and the number of unshaded cells below the line.
●
Time
LESSON
Number
of Eggs
a
6
b
10
c
4
d
2
e
8
6
The mean, or average,
number of eggs in the
6 .
five clutches is
5
Activity 2
(to be done later)
List the number of eggs
in the clutches in order.
4
2
3
4
2
6
8
1
10
0
a
b
c
d
Ostrich Clutches
e
The median is
eggs per clutch.
6
Math Journal 2, p. 254
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Have children read the essay on The Mean (Average) on pages
83–85 in the Student Reference Book.
NOTE Although the mean for this set of
Finding the Mean of
INDEPENDENT
ACTIVITY
Ostrich Egg Clutches
PROBLEM
PR
PRO
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SOLVING
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data is a whole number, the mean for other
data sets will likely be between two whole
numbers. A more efficient way to find the
mean will be introduced in Lesson 10-7.
(Math Journal 2, p. 254)
Children make bar graphs of the data set about ostrich egg
clutches. They use one of the methods from the previous activity to
find the mean number of eggs.
Student Page
Date
Time
LESSON
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 254
Use journal page 254 to assess children’s ability to complete a bar graph.
Children are making adequate progress if they successfully complete the bar
graph. Some children may be able to find the mean number of eggs.
10 6
Interpreting a Pictograph
Big Bend National Park is located in southwestern Texas. It contains
801,163 acres of protected wilderness. About 350,000 people visit the
park each year.
The pictograph below shows attendance at Big Bend National Park
for one week in August. Use the data to answer the questions below.
Visitors to Big Bend National Park
KEY:
Day 1
= 50 people
Day 2
[Data and Chance Goal 1]
Day 3
Day 4
Day 5
Day 6
Links to the Future
The activities in this lesson are an early exposure to finding the mean of a set of
data. Finding the mean of a data set is a Grade 5 Goal.
Day 7
1. How many people visited the park on Day 4?
2. How many people visited the park on Day 2?
3. How many people visited the park on Day 5?
500
350
475
people
people
people
4. How many more people came to Big Bend National Park
on Day 7 than Day 6?
175
more people
5. How many fewer people came on Day 1 than Day 3?
125
fewer people
Math Journal 2, p. 255A
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Lesson 10 6
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Student Page
Date
Finding the Median
Time
LESSON
10 6
Making a Bar Graph from a Pictograph
Use the information shown in the pictograph to make a bar graph.
Remember to add labels and a title to your graph.
KEY:
INDEPENDENT
ACTIVITY
of Sets of Data
Visitors to Big Bend National Park
Day 1
PROBLEM
PRO
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SOLVING
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(Math Journal 2, pp. 253 and 254; Student Reference Book, p. 80)
= 50 people
Day 2
Day 3
Day 4
Have children complete Activity 2 on journal pages 253 and 254.
Children find the median number of children per family and the
median number of eggs per clutch. If necessary, have them read
page 80 in the Student Reference Book to review how to find the
median, or middle value, of a set of data.
Day 5
Day 6
Sample answer: Number of Visitors
Day 7
Visitors to Big Bend National Park
550
500
450
400
350
300
250
200
150
100
50
0
Ongoing Assessment: Informing Instruction
Watch for children who have difficulty finding the median. They can write the
number of eggs in each nest on a small slip of paper or a stick-on note. Have
them put the numbers in order and then find the middle number(s).
1
2
3
4
5
6
7
Sample answer: Days of Visits
255B
Math Journal 2, p. 255B
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2 Ongoing Learning & Practice
Interpreting a Pictograph and
PARTNER
ACTIVITY
Creating a Bar Graph
(Math Journal 2, pp. 255A and 255B; Student Reference Book,
pp. 88 and 89)
Student Page
Date
Time
LESSON
10 6
1.
Math Boxes
Measure each side of the triangle
to the nearest centimeter.
4
2.
cm
There are 5 blocks in a bag.
2 blocks are red, 2 blocks are blue,
and 1 block is green. What are the
chances of pulling out a red block?
2
4
4
5
out of
Have children examine the pictograph on journal page 255A. Ask
them to explain what each smiley face means. 50 people Next ask
them to explain what one-half of a smiley face means. 25 people
Have children work independently or with a partner to complete
journal page 255A. When they have finished, children create a bar
graph on journal page 255B to show the same data. Discuss the
scale for the bar graph. Since the pictograph key shows that each
smiley face represents 50 people, the scale for the bar graph could
be in increments of 50.
chances
Math Boxes 10 6
cm
cm
Perimeter = 12 cm
3.
137–139
James built a rectangular prism out
of base-10 blocks. He used 30 cm
cubes to make the base. He put 4
more layers of cubes on top of that.
What is the volume of the prism he
built?
94
4.
3
7
4
quarts
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 10-8. The skill in Problem 6
previews Unit 11 content.
gallons = 12 quarts
1 pint =
150 cubic centimeters
(Math Journal 2, p. 255)
Complete.
1 gallon =
2
cups
pints = 14 cups
1,000milliliters
1 liter =
5
157–159
5.
Molly is playing with 5 toy cars.
1
This is only _
3 of her set of cars.
How many cars are in her complete
set? Fill in the circle
next to the best answer.
A
5
_
3 cars
C 10 cars
B
5 cars
D 15 cars
6.
liters = 5,000 milliliters
1
_
3 green
blue
yellow
green
1
_
6 yellow
24
160 161
Color the circle so that it matches
the description.
1
_
2 blue
Which color would you
expect the spinner to
land on most often?
blue
93
Math Journal 2, p. 255
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INDEPENDENT
ACTIVITY
Writing/Reasoning Have children write an answer to the
following: Choose one of the units of capacity in Problem 4.
Describe the unit of capacity by using one of the personal
references for units of capacity from Lesson 10-5. For example,
describe an object that has about the same measure as a liter (or the
unit of capacity you chose). Sample answer: I have a water bottle
that holds 1 liter of water.
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Home Link 10 6
PROBLEM
PRO
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B
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SOLVING
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INDEPENDENT
ACTIVITY
(Math Masters, p. 340)
Home Link Master
Name
Date
HOME LINK
Mean, or Average, Number of Fish
10 6
Home Connection Children find the mean of a data set.
Time
Family
Note
Many of us learned that to find the mean (average) of a set of numbers, we add all the
numbers and then divide the total by how many numbers we added. In today’s lesson, the
class tried a different method of finding the mean. After your child has completed the page,
ask him or her to explain how this method works. In the next lesson, we will introduce
finding the mean by adding the numbers and dividing to find the answer.
83–85
Please return this Home Link to school tomorrow.
The table below lists
how many goldfish
each child won at
the school fun fair.
3 Differentiation Options
Graphing Dice Rolls
6
Name Number of
Goldfish
PARTNER
ACTIVITY
5–15 Min
Reba
3
Bill
1
Lucy
7
Meg
0
Nate
5
Pat
2
5
Number of Goldfish
READINESS
7
1. Put a penny
3
2
over each
shaded square
in the bar graph.
(Math Masters, p. 339)
4
1
2. Move the pennies
To provide experience with making a bar graph, have
children predict how many times they think they will roll
a 1 and then make a graph to record their results. When
children have completed their graphs, have them discuss their
predictions and results.
so that each
0
column has the
Reba
same number of pennies.
Meg
Lucy
Nate
Names
3. Draw a horizontal line across your graph to show the height
of the pennies when all of the columns are the same height.
Bill
4. The mean (average) number of
goldfish won by children at the fun fair is
3
Pat
.
Math Masters, p. 340
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ENRICHMENT
Making a Data Set
2/28/11 2:17 PM
PARTNER
ACTIVITY
15–30 Min
(Math Masters, pp. 341 and 342)
To apply children’s understanding of landmarks
(maximum, minimum, range, mode, and median), have
them use the given median, range, and mode to display a
possible data set when 5 children share 15 cookies. Children
record their work on Math Masters, page 342 and make a bar
graph of their data set on Math Masters, page 341.
Teaching Master
Name
Date
The mode of the number of cookies is 2.
How many cookies could each child have?
The range of the number of cookies is 5.
The median number of cookies is 2.
Lunch bag D
Lunch bag C
To provide language support for landmarks, have children use the
Math Word Bank template found on Differentiation Handbook,
page 132. Ask children to write the term mean, draw a picture
representing the term, and write other related words. See the
Differentiation Handbook for more information.
Altogether, 5 children have 15 cookies in their lunch bags.
(Differentiation Handbook, p. 132)
Lunch bag B
5–15 Min
79– 81
Lunch bag A
Lunch bag E
Making a Data Set
10 6
Sample answer:
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
Use counters and the drawings of lunch bags below to organize your data. Draw cookies in the lunch
bags to match the description above.
ELL SUPPORT
Time
Graph your results on Math Masters, page 341. Remember to include labels
and a title.
LESSON
Math Masters, p. 342
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Lesson 10 6
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