Objectives
To introduce rates; and to provide practice
collecting and comparing rate data.
1
materials
Teaching the Lesson
Key Activities
Students collect data on how many times classmates blink in one minute. They compare
median blinking rates for students at rest and for students who are reading. Students list
examples of rates.
Key Concepts and Skills
• Describe examples of rates. [Operations and Computation Goal 7]
• Collect and organize data to create a table.
ⵧ Math Journal 2, p. 309
ⵧ Student Reference Book, pp. 271
and 299
ⵧ Teaching Aid Master (Math Masters,
p. 423)
ⵧ timer or clock with a second hand
ⵧ index cards
ⵧ slate
[Data and Chance Goal 1]
• Find the median and mean of a data set. [Data and Chance Goal 2]
• Use data to draw conclusions and make predictions.
[Data and Chance Goal 2]
ⵧ calculator (optional)
See Advance Preparation
• Write a number sentence with parentheses.
[Patterns, Functions, and Algebra Goal 3]
Key Vocabulary
rate • per
2
materials
Ongoing Learning & Practice
Students fill in missing fractions and decimals on number lines.
Students practice and maintain skills through Math Boxes and Study Link activities.
ⵧ Math Journal 2, pp. 310 and 311
ⵧ Study Link Master (Math Masters,
p. 339)
Ongoing Assessment: Recognizing Student Achievement Use journal page 311.
[Number and Numeration Goal 6]
3
materials
Differentiation Options
READINESS
Students analyze the
median and mean of a
data set.
ENRICHMENT
Students create a
side-by-side (double)
bar graph to display
eye-blinking rates.
ELL SUPPORT
Students create a Rates All
Around Museum.
ⵧ Teaching Master (Math Masters,
p. 340)
ⵧ Teaching Aid Master (Math Masters,
p. 403)
ⵧ chart paper; colored pencils or
crayons
See Advance Preparation
Additional Information
Advance Preparation For Part 1, read the description of the eye-blinking experiment before
class and decide how best to conduct it. For the optional ELL Support activity in Part 3, post
chart paper for a Rates All Around Museum.
908
Unit 12 Rates
Technology
Assessment Management System
Math Boxes, Problems 4a–4c
See the iTLG.
Getting Started
Mental Math and Reflexes
Pose rate problems. Have copies of the multiplication/division diagram (Math Masters, page 423) available
for student use. Suggestions:
There are 9 stickers per sheet.
How many stickers on
5 sheets? 45 stickers
7 sheets? 63 stickers
9 sheets? 81 stickers
There are 40 books per shelf.
How many books on
10 shelves? 400 books
30 shelves? 1,200 books
50 shelves? 2,000 books
Pencils cost $1.20 per box.
What is the cost of
4 boxes? $4.80
6 boxes? $7.20
11 boxes? $13.20
Math Message
Find the median for each set of numbers.
a. 4, 9, 3, 12, 15, 9, 7
b. 2, 10, 6, 9
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
The median of a set of numbers is the middle number when the
numbers are listed in order from smallest to largest or from
largest to smallest. Nine is the median of the first set of numbers
because in the ordered list 3, 4, 7, 9, 9, 12, 15, the number 9 is
the middle number.
If there is an even number of numbers in the set, the median is
the mean (average) of the two middle numbers. In the ordered list
2, 6, 9, 10, there are two middle numbers. The median is
1
(6 9) / 2, or 72.
Adjusting the Activity
Ask students to find the mean of each data set and write a number
model with parentheses to describe their strategy.
䉯 (3 4 7 9 9 12 15) / 7 8.4286
䉯 (2 6 9 10) / 4 6.75
Students should note that the median and the mean for a set of data usually are
not equal, but the two are often close.
A U D I T O R Y
䉬
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
Lesson 12 1
䉬
909
Student Page
Date
䉴 Collecting Eye-Blinking Data
Time
LESSON
12 1
䉬
Rates
47
Answers times in one minute.
vary.
While reading, a typical student in my class blinks Answers times in one minute.
vary.
1. While at rest, a typical student in my class blinks
2.
Take half the class aside, outside the hearing range of the other
half. Tell these students that they are going to collect data on
their classmates’ eye-blinking rates, but they must do so secretly.
Explain the procedure they are to follow:
3. In Problems 1 and 2, what is meant by the phrase a typical student?
Sample answer: one that blinks about the
same number of times as most others
4. Calculate the mean for each set of data.
a. At rest:
䉯 Each student in the data-collecting group—Group A—is paired
with a student in the other group—Group B. Partners sit across
from each other and, at your signal, they look at each other.
blinks per minute
b. While reading:
WHOLE-CLASS
ACTIVITY
blinks per minute
5. List as many examples of rates as you can.
6. Find at least 2 examples of rates in your Student Reference Book.
(Hint: Look at pages 271 and 299.)
Sample answers: There are 555 televisions per
1,000 people in Spain. There are 757 cars per
1,000 people in the United States.
309
Math Journal 2, p. 309
䉯 While looking at each other, students in Group A count the
number of times their partners in Group B blink in one minute.
At the end of one minute, you give the signal to stop. Students
in Group A secretly write the number of times their partner
blinked in one minute.
䉯 Next, instruct the students in Group B to open a book. At
your signal, the students in Group B start reading, while the
students in Group A again count their partners’ number
of blinks. Again, you give the signal to stop at the end of
one minute.
䉯 The students in Group A then write the number of times their
partners blinked while reading for one minute.
Tell students in Group A that they will follow this procedure so
that their partners remain unaware of what is taking place.
Otherwise, they might blink unnaturally.
Bring the class together and conduct the experiment.
䉴 Comparing Eye-Blinking
WHOLE-CLASS
ACTIVITY
Rates
(Math Journal 2, p. 309)
Number of Blinks in 1 Minute
At Rest
Reading
14
10
18
1
2
2
Table for recording blinking rates (sample data
provided)
Ask one of the students in Group A to describe the experiment.
Have students in Group B speculate about whether a person
blinks more often while reading or at rest, or whether the number
of blinks remains about the same.
Take a vote and ask students to discuss why they voted the way
they did. Make a table on the board and record each student’s
blinking rates on a separate line. (See margin.)
Ask partners to find the median for each set of data, record them
in Problems 1 and 2 in their journals, and describe what is meant
in Problem 3 by the phrase “a typical student.”
ELL
Adjusting the Activity
Have students record the numbers in each column of the table on
separate index cards. They can then order the index cards to find the middle
value for each data group.
A U D I T O R Y
910
Unit 12 Rates
䉬
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
Bring the class together to discuss the results. Ask:
●
Why might a person’s blinking rate vary, depending on the
activity? Sample answer: A person may concentrate more and
blink less when reading.
●
What might be some other factors that can affect blinking
rates? Sample answers: Brightness of light; how tired a person
is; whether a person wears glasses or contact lenses; whether a
person is interested or bored
●
Based on the data, can you make a prediction about a person’s
blinking rate while exercising? Sample answer: When a person
concentrates on moving certain muscles, he or she may blink
less. However, if a person is exercising outdoors, the wind and
sunlight may make the person blink more.
䉴 Listing Examples of Rates
PARTNER
ACTIVITY
(Math Journal 2, p. 309; Student Reference Book, pp. 271 and 299)
The number of times a person blinks in one minute is an example
of a rate. A rate tells how many there are of one thing for a
certain number of another thing. Rates often contain the word
per, which means “for each,” such as in 15 blinks per minute or
55 miles per hour. A rate can be written with a slash to represent
the word per, as in $2.25/lb.
Ask students to list other examples of rates in Problem 5 in their
journals. You might suggest a few categories:
䉯 Food: calories per serving
䉯 Packaging: paper clips per box
䉯 Price: dollars per pound
䉯 Transportation: miles per hour
䉯 Sports: minutes per half in basketball
Social Studies Link In Problem 6, have students record at
least two examples of rates from pages 271 and 299 of the
Student Reference Book World Tour section. Bring students
together to share their examples. Throughout the unit, encourage
the class to find other examples of rates and display them in a
Rates All Around Museum. See the optional ELL Support activity
in Part 3.
Lesson 12 1
䉬
911
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Counting with Fractions and Decimals
12 1
䉬
Fill in the missing fractions on the number lines below.
1.
2
3
1
3
0
1
䉴 Identifying Fractions and
2.
1
6
0
2
6
3
6
4
6
5
6
INDEPENDENT
ACTIVITY
Decimals on Number Lines
1
(Math Journal 2, p. 310)
3.
1
8
0
2
8
3
8
4
8
5
8
6
8
7
8
1
Students fill in the missing fractions and decimals on
number lines.
Fill in the missing decimals on the number lines below.
4.
2
2.5
3
4
3.5
䉴 Math Boxes 12 1
Try This
INDEPENDENT
ACTIVITY
䉬
5.
7
7.25
7.50
8
7.75
(Math Journal 2, p. 311)
6.
–2
1.5
1
0.5
0
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 12-3.
1
0.5
Math Journal 2, p. 310
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problems
4a–4c
夹
Use Math Boxes, Problems 4a–4c to assess students’ ability to compare
integers between 100 and –100. Students are making adequate progress if they
insert , , or to make true number sentences. Some students may be able
to solve Problems 4d and 4e, which involve the comparison of rational numbers.
[Number and Numeration Goal 6]
䉴 Study Link 12 1
䉬
INDEPENDENT
ACTIVITY
(Math Masters, p. 339)
Home Connection Students look for examples of rates in
newspapers, in magazines, and on labels and bring them
to class.
Student Page
Date
Time
LESSON
12 1
䉬
Math Boxes
2. Calculate the volume.
1.
6 in.
a. Pick a face of the cube. How many
other faces are parallel to it?
1
8 in.
face(s)
5 in.
b. Pick an edge of the cube. How many
other edges are parallel to it?
3
edge(s)
94 101
3. Write A, P, or V to tell whether you would
need to find the area, perimeter, or volume
in each situation.
Number model:
Volume around a circular track
P
b. Buying tile for a bathroom floor
sentence.
c. Filling a pool with water
b.
A
V
c.
d.
e.
131 133
137
5. For which number is 8 a factor? Fill in
in3
3
43 21
68 100
0.78
13
13
15
夹
䉴 Finding the Median and
INDEPENDENT
ACTIVITY
5–15 Min
Mean of a Data Set
3
4
1
2
(Math Masters, p. 340)
60
hundredth.
A
253
a. 12.368
B
94
b. 234.989
C
120
c. 1.225
D
884
d. 12.304
e. 0.550
7
12.37
234.99
1.23
12.30
0.55
To provide experience finding the median and calculating the
mean of a data set, have students complete Math Masters,
page 340.
182 183
311
Math Journal 2, p. 311
Unit 12 Rates
READINESS
6. Round each number to the nearest
the circle next to the best answer.
912
3 Differentiation Options
138
4. Insert , , or to make a true number
a.
a. Finding the distance
(5 ⴱ 8) ⴱ 6 240
240
Study Link Master
PARTNER
ACTIVITY
ENRICHMENT
䉴 Creating Side-by-Side
Name
䉬
15–30 Min
Time
Examples of Rates
12 1
1.
Bar Graphs
Date
STUDY LINK
47
Look for examples of rates in newspapers, in magazines, and on labels.
Study the two examples below, and then list some of the examples you find.
If possible, bring your samples to class.
Label on a can of corn
1
says “Servings Per Container 3 2 ”
Example:
(Math Masters, p. 403)
Serving Size 110 g
Servings Per Container 3 1/2
Amount Per Serving
Lightbulbs come in packages of 4 bulbs.
The package doesn’t say so, but there are always
4 bulbs in each package.
Example:
To apply students’ ability to organize and compare data,
have them create a side-by-side (double) bar graph to
display the eye-blinking rates for students at rest and
while reading. Remind students to choose a reasonable title and
labels for the graph and to include a key for the color-coded bars.
For example:
Example:
Example:
Example:
Eye-Blink Rates
14
13
Number of Students
12
11
Practice
10
4
2. 5
9
8
1
5
9
,
9
4.
3
5
or 1
3.
1
9
8
9
1
5. 3
3
6
1
8
5
6
7
8
3
4
7
6
Math Masters, p. 339
5
4
3
2
1
0
6–10
0–5
11–15
16–20
21–25
26+
Number of Eye Blinks in 1 Minute
Key
at rest
reading
When students have completed their graphs, ask:
●
Why might it be useful to show data in a side-by-side graph
like this rather than two individual bar graphs? Sample
answer: When the two bars are side by side, you can
immediately compare the rates without looking at the numbers.
●
How did you determine the scale to use for the vertical and
horizontal axes? Sample answer: I decided to clump the
number of eye blinks into categories of 0–5, 6–10, 11–15,
16–20, and 26 on the horizontal axis, because the chart
would be too crowded with the individual data.
ELL SUPPORT
䉴 Creating a Rates All
Teaching Master
Name
LESSON
12 1
䉬
85
73 75
1.
What is Anthony’s median score?
2.
What score must Anthony get on his
next test to maintain his median score?
85 % Explain your answer.
Sample answer: Any other score would
result in a change in median.
3.
Anthony would like to raise his mean score to 90% or
higher. If he takes one more spelling test, can he do it?
Around Museum
To provide language support for rates, have students create a
Rates All Around Museum. Ask them to read the numbers and
describe some of the ways that rates are used in the museum;
for example, what the numbers mean or the units attached to
the rates.
Time
Anthony’s first 4 test scores for his weekly 20-word spelling tests were
80%, 90%, 100%, and 75%.
SMALL-GROUP
ACTIVITY
15–30 Min
Date
Median and Mean
%
No Explain your answer.
Even if he makes 100% on the next test, his
average will be (80 90 100 75 100) / 5 89%.
Name
LESSON
12 1
䉬
Date
Time
Median and Mean
Anthony’s first 4 test scores for his weekly 20-word spelling tests were
80%, 90%, 100%, and 75%.
1.
What is Anthony’s median score?
2.
What score must Anthony get on his
next test to maintain his median score?
3.
73 75
%
% Explain your answer.
Anthony would like to raise his mean score to 90% or
higher. If he takes one more spelling test, can he do it?
Explain your answer.
Math Masters, p. 340
Lesson 12 1
䉬
913