Rotations and Angles
Objectives To review rotations; and to guide students as they
make and use a full-circle protractor.
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Key Concepts and Skills
Making a Bar Graph
• Use multiples of 30. Math Journal 1, p. 154
Student Reference Book, p. 301
Students make a bar graph showing
percent of population (ages 0–14) for
Region 2 countries.
[Number and Numeration Goal 3]
• Form angles of a given measure. [Measurement and Reference Frames Goal 1]
• Describe right angles. [Geometry Goal 1]
• Rotate objects a given number of degrees. [Geometry Goal 3]
• Investigate the relationship between
rotations and degrees. [Geometry Goal 3]
Key Activities
Students review clockwise rotations. They
make a full-circle protractor by measuring
rotations in degrees; then they use the
protractor to form angles of given measures.
Students solve problems that involve
measuring elapsed time in degrees.
Ongoing Assessment:
Informing Instruction See page 427.
Key Vocabulary
rotation turn clockwise degree right angle
Ongoing Assessment:
Recognizing Student Achievement
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Matching Alternate Time Displays
Math Masters, pp. 186–188
scissors
Students match alternate ways of
naming time.
ENRICHMENT
Use journal page 154. Measuring Elapsed Time in Degrees
[Data and Chance Goal 1]
Math Journal 1, pp. 152 and 153
Math Masters, p. 189
Students determine elapsed time for 1°
increments on a clock face.
Solving Elapsed-Time Problems
Math Journal 1, pp. 154A and 154B
demonstration clock (optional) calculator (optional)
Students solve elapsed-time
problems.
Math Boxes 6 5
Math Journal 1, p. 151
Students practice and maintain skills
through Math Box problems.
Study Link 6 5
Math Masters, p. 185
Students practice and maintain skills
through Study Link activities.
EXTRA PRACTICE
Playing Robot
Students practice making rotations of a
given size.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 140
Students add the term degree to their Math
Word Banks.
Materials
Math Journal 1, pp. 152 and 153
Study Link 64
drinking straws demonstration clock
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 178–180
Lesson 6 5
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Getting Started
Mental Math and Reflexes
Have students imagine standing in the center of a clock with their right hand extended as the minute hand. Ask them to rotate
their bodies to make turns such as the following:
1
_
2
1
_
4
turn clockwise
turn clockwise
full turn clockwise
90° turn clockwise
180° turn clockwise
360° turn clockwise
3
_
4
3
_
4
1
_
2
turn counterclockwise
turn clockwise
turn counterclockwise
Math Message
Study Link 6 4 Follow-Up
How many minutes does it take the minute hand to
move through a full turn on the face of a clock?
60 min A _12 turn? 30 min A _14 turn? 15 min
Have students discuss how they handled the
remainders in Problems 1 and 2.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
The Math Message reminds students who used Third Grade
Everyday Mathematics of previous experiences with rotations.
Review answers using a clock with an hour and minute hand or a
demonstration clock to model movements of the minute hand.
Pose additional problems: A _34 turn? 45 min A _16 turn? 10 min
A _13 turn? 20 min A _23 turn? 40 min
For a _34 turn—Since there are 3 five-minute intervals in _14 of
a turn (3 = _14 of 12), there are three times as many in _34 of a
turn, or 9 five-minute intervals. Therefore, it takes the minute
hand 45 minutes to move through _34 of a turn.
Student Page
Date
LESSON
6 5
䉬
Time
Making a Full-Circle Protractor
There are 360 marks around the circle. They divide the edge of the circle into
360 small spaces. Twelve of the marks are longer than the rest. They are in the same
positions as the 12 numbers around a clock face. Your teacher will tell you how to label
the 12 large marks on the circle.
330°
360°/0°
30°
300°
60°
11 12 1
10
270°
2
9
4
7
6
5
240°
210°
120°
180°
Math Journal 1, p. 152
426
For a _16 turn—Since there are 12 five-minute intervals in 1 full
turn of the minute hand, there are 2 five-minute intervals in
1
_
of a turn. Therefore, it takes the minute hand 10 minutes to
6
move through a _16 turn.
For a _23 turn—Since there are 4 five-minute intervals in _13 of a
turn (4 = _13 of 12), there are twice as many in _23 of a turn, or
8 five-minute intervals. Therefore, it takes the minute hand
40 minutes to move through _23 of a turn.
90°
3
8
92
150°
Investigating Rotations
and Degree Measures
WHOLE-CLASS
ACTIVITY
ELL
(Math Journal 1, p. 152)
Tell the class that in this lesson they will investigate the markings
on a full-circle protractor and compare them to familiar markings
on an analog clock.
Unit 6 Division; Map Reference Frames; Measures of Angles
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Discuss the marked circle on journal page 152.
There are three different lengths of marks.
The shortest marks divide the circle into 360 small spaces.
360 ∗ 1° = 360°
The longest marks are in the same positions as the 12 numbers
around a clock face. These 12 long marks divide the circle into
12 spaces. 12 ∗ 30° = 360°
The middle-size marks divide the circle into 72 spaces.
72 ∗ 5° = 360°
Ask students to write 0° beneath the large mark at the 12 o’clock
position on the circle.
Ask students to fold a straw in half. Show them how to place it on
the circle on journal page 152. The bend of the straw should touch
the center of the circle, and both halves of the straw should point
to the 0-degree mark.
NOTE Think of an angle as “in motion”
opening from 0° to the desired angle. For
example, to measure a 15° angle, start with a
0° angle and open the angle to 1°, 2°, and so
on until 15° is reached. Thinking this way can
help students realize that counting the spaces
in between the marks of the protractor is
more accurate than counting the marks.
Keeping one part of the straw pointing to the 0-degree mark, move
the other half of the straw clockwise to the first large mark, or
1
_
of a turn.
12
Links to the Future
Students will discuss counterclockwise rotations in Lesson 6-6.
The straw-halves form an angle. Remind students that angles are
measured in degrees and that the degree symbol (◦) is often used
in place of the word degree. To support English language learners,
write degree on the board and explain that this word has different
meanings when it is used to measure angles and temperature.
30°
Now show students how to measure the straw angle they just
made: To measure the angle, count the number of small spaces
created by the shortest marks. (See note in margin.) 30 spaces, so
the angle measures 30 degrees, or 30◦ Ask students to write 30◦
at the first large mark on the circle.
Tell students to move the straw-half back to its original position
and then repeat the routine for a _14 turn. (See margin.)
● What is the measure of the angle? 90◦ Ask students to write
90◦ at the _1 -turn mark on the circle. (See margin.)
4
0°
30°
Ongoing Assessment: Informing Instruction
Watch for the different strategies that students use to determine that the result of
a
1
_
4
90°
turn of the straw is an angle that measures 90°.
Count the 90 spaces along the circle between the sides of the straw.
Recognize that the angle is 3 times as large as the first angle, and multiply
by 3 to get 90°.
Recognize that the angle is a right angle, and right angles measure 90°.
Lesson 6 5
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Student Page
Date
Time
LESSON
65
●
Clock Angles
Use the clock below and the full-circle protractor on journal page 152 to help you
answer the questions.
92 141
How many minutes and how many degrees does the minute hand move
1.
a.
from 3:00 to 4:00?
b.
from 7:00 to 7:45?
c.
from 8:15 to 8:45?
d.
from 6:30 to 6:50?
e.
from 5:15 to 5:30?
f.
60
45
30
20
15
10
5
1
from 1:00 to 1:10?
g.
from 12:00 to 12:05?
h.
from 5:00 to 5:01?
360
270
180
120
90
60
30
6
minutes
minutes
minutes
minutes
minutes
minutes
minutes
minutes
°
°
12
11
10
°
°
2
3
8
°
Now tell students to move the straw-half back to its original
position, and then repeat the procedure for a full turn.
1
9
°
●
4
7
6
5
°
Try This
a.
30
15
in 1 hour?
1
_
b. in 2 hour?
c.
°
°
5
in 10 minutes?
Forming Angles of
°
Explain how you solved Problem 2c.
3.
(Math Journal 1, p. 152)
Math Journal 1, p. 153
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in Degrees
Have students describe a strategy
for labeling the remaining large tick marks
without using the folded straw. Sample
answer: Count by 30s; each large tick mark
corresponds to a multiple of 30.
KINESTHETIC
TACTILE
LESSON
65
2 Ongoing Learning & Practice
Time
Population Bar Graph
The table below shows the percent of
the population (number of people out
of 100) who are 14 years old or
younger in the Region 2 countries.
Percent of Population
Ages 0–14
Country
France
19
Greece
15
Hungary
1.
Making a Bar Graph
76 301
16
Iceland
23
Italy
14
Netherlands
18
Norway
20
Poland
18
Spain
15
United Kingdom
19
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 154; Student Reference Book, p. 301)
Social Studies Link Students make a bar graph to show
percent of population (ages 0–14 years) for Region 2
countries. Direct students to page 301 of the Student Reference
Book for additional data.
Make a bar graph to display the information given in the table above.
PROBLEM
PR
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SOLVING
SO
S
OL
O
LV
VIN
IIN
NG
Students solve problems and share solution strategies about the
number of degrees the minute and hour hands of a clock move in
a given amount of time.
Student Page
Date
PARTNER
ACTIVITY
(Math Journal 1, pp. 152 and 153)
VISUAL
Direct students to use their bent straws to form angles of various
degree measures. For example, Show me a 120° angle; a 45° angle;
a 77° angle.
Measuring Elapsed Time
Adjusting the Activity
WHOLE-CLASS
ACTIVITY
Given Measures
Sample answer: I know that the hour hand
moves 30° in 1 hour. There are 6 groups of
10 minutes in 1 hour (60 minutes). So there
are 6 groups of 5° in 30°, or 30°/ 6 = 5°.
AUDITORY
What is the measure of the angle? 360°
Ask students to write 360° at the full-turn mark on the circle,
right next to the 0° mark. Have students label the rest of the large
tick marks on the circle until all 12 marks have been labeled.
°
How many degrees does the hour hand move
2.
What is another name for a 90-degree angle? A right angle
To support English language learners, review the different
meanings of right discussed in Unit 1—right answer, right
hand, right angle.
Percent of Population Ages 0–14
25
Percent
20
15
Ongoing Assessment:
Recognizing Student Achievement
10
5
Journal
page 154
ain
om
Sp
gd
Kin
d
ay
nd
la
w
or
Po
Use journal page 154 to assess students’ ability to create a bar graph. Students
are making adequate progress if they can draw the bars at the appropriate
height on the graph. Some students may be able to provide a title and label
each axis.
[Data and Chance Goal 1]
U
nit
e
et
h
N
N
Ita
ly
la
nd
s
er
ela
nd
Ic
e
ry
H
un
ga
ec
re
G
Fr
an
ce
0
Country in Region 2
2.
Why might it be important to know what percent of the population of a country
is 0 through 14 years of age?
Sample answer: A government will know how much money
to give schools for grades 8 and below.
154
Math Journal 1, p. 154
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Unit 6 Division; Map Reference Frames; Measures of Angles
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Student Page
Solving Elapsed-Time
Date
INDEPENDENT
ACTIVITY
Problems
Time
LESSON
Elapsed Time
65
Record the times on each pair of clocks. Then record the elapsed time.
Start
1.
(Math Journal 1, pp. 154A and 154B)
12
11
10
End
1
2
9
3
8
6
12
11
10
2
6
Start
12
11
10
11
10
2
9
3
8
6
5
11
10
2
3
8
6
INDEPENDENT
ACTIVITY
5
7
5
6
End
11
10
1
2
1
2
3
8
4
6
12
9
3
7
4
7
5
6
5
11:32
6:28
6 hours, 56 minutes
Record how much time has passed between the start time and the end time.
5.
Start 11:00 A.M.
End 4:30 P.M.
Elapsed time:
5 hours, 30 minutes
6.
Start 2:20 P.M.
End 6:35 P.M.
Elapsed time:
4 hours, 15 minutes
7.
Start 9:12 A.M.
End 11:43 P.M.
Elapsed time:
2 hours, 31 minutes
(Math Journal 1, p. 151)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-7. The skill in Problem 6
previews Unit 7 content.
12
8
4:02
1:05
2 hours, 57 minutes
Math Boxes 6 5
3
4
5
9
4
7
2
8
Start
4.
1
9
4
7
12
6
1
9
2:35
3:15
40 minutes
End
1
2
4
7
12
11
10
3
5
1:00
8:00
7 hours
3.
End
1
8
4
7
12
9
3
8
5
Start
2.
1
9
4
7
Students solve elapsed-time problems. Students may use the
demonstration clock, calculators, or anything else that may help.
Some students may find it helpful to use an open number line to
illustrate the strategy of counting up in hours and minutes. See
Lesson 3-6 for more information.
11
10
Math Journal 1, p. 154A
Study Link 6 5
INDEPENDENT
ACTIVITY
EM3MJ1_G4_U06_137-169.indd 154A
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(Math Masters, p. 185)
Home Connection Students follow directions, given as
fractions of turns and distances, to trace a path on a
coordinate grid. When reviewing answers, point out that
the length of each horizontal line segment equals the difference of
the x-coordinates and that the length of each vertical line segment
equals the difference of the y-coordinates.
Student Page
Student Page
Date
Date
Time
LESSON
Math Boxes
65
Time
LESSON
Elapsed Time
65
continued
Read the time on each clock. What time will it be in 50 minutes?
1.
Insert parentheses to make each number
sentence true.
(15 + 5)∗ 6 = 120
b. 7 +(9 ∗ 2)= 25
c. 77 =(1 + 6)∗(6 + 5)
2.
Draw a line segment that is 2 inches
long. Mark and label the following inch
measurements on the line segment:
a.
3.
8.
11
10
3
1 _
1
_
_
2 , 4 , 1, 1 2 , and 2
9.
2
9
150
4.
1 1
1
2
2
4,092
7
8:50
11.
908 4 = d
$227
14.
908 4 = 227
11:00 A.M.
12.
11
10
12
2
22
23
5
9:53 P.M.
11:13 P.M.
6
11
10
12
11
10
2
1
2
3
8
4
6
12
9
3
7
5
16.
1
4
7
5
6
5
18 19
7:15
1
_
6. a. Shade 2 of
the square.
Complete.
e.
13.
8
4
7
d.
6:45 P.M.
9
3
8
c.
6
8:42
8:05 P.M.
15.
1
9
b.
3
7
Read the time on each clock. What time was it 30 minutes ago?
Summary number model:
900 cm
1,500 cm = 15 m
350 cm = 3.5 m
4 m 58
458 cm =
3.2 m = 320 cm
2
4
5
4:15
12:20 P.M.
Number model with unknown:
a.
6
1
8
4
5
12
9
3
For each time, record what time it will be in 1 hour and 20 minutes.
How much money will each team receive?
5.
11
10
2
128
Multiply with a paper-and-pencil algorithm.
66 ∗ 62 =
10.
1
8
4
6
12
9
3
7
1 3
2 4
11
10
1
8
The Sports Boosters raised $908 at their
annual chili supper. Four athletic teams
will share the money equally.
Answer:
12
10:50
4:39
9m=
Sample
answers:
b.
cm
129
For each time, record what time it was 2 hours and 15 minutes ago.
11. 10:15 A.M.
2
Shade _
3 of
the square.
8:00 A.M.
2:05 P.M.
11:50 A.M.
13.
1:12 A.M.
10:57 P.M.
44
Math Journal 1, p. 154B
Math Journal 1, p. 151
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Lesson 6 5
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Study Link Master
Name
Date
STUDY LINK
3 Differentiation Options
Time
Treasure Hunt
65
Marge and her friends are playing Treasure Hunt. Help them find the treasure.
Follow the directions. Draw the path from the oak tree to the treasure. Mark the
spot where the treasure is buried.
1.
Start at the dot under the oak tree; face north. Walk 4 steps.
2.
Make a quarter turn, clockwise. Walk 5 steps.
3.
Face south. Walk 2 steps.
5.
1
Face east. Walk 2_2 steps.
3
Make a _4 turn, clockwise. Walk 5 steps.
6.
3
1
Make a _4 turn, clockwise. Walk 6 _2 steps.
7.
Make an X to mark the spot where you end.
4.
107
Matching Alternate Time Displays
To explore alternate ways of naming time, have students
match cards that indicate the same time in analog, digital,
and word forms.
9
8
7
N
W
6
E
S
5
1 step
4
ENRICHMENT
3
Measuring Elapsed Time
2
Oak Tree
1
0
0
1
2
3
4
5
6
7
8
9
29 R1
86 R1 = 603 7
88 ÷ 3 =
9.
11.
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To further investigate the relationship between elapsed time and
angle measures, have students use the full-circle protractor on
journal page 152 to determine how long it takes the minute and
the hour hands to move 1 degree.
EXTRA PRACTICE
Playing Robot
12
1
Quarterpast
5 o’clock
2
9
3
8
4
7
6
5–15 Min
(Math Journal 1, pp. 152 and 153; Math Masters, p. 189)
11 R5 = 71 ÷ 6
934 5 = 186 R4
Math Masters, p. 185
11
10
INDEPENDENT
ACTIVITY
in Degrees
10
Practice
10.
5–15 Min
(Math Masters, pp. 186–188)
10
8.
PARTNER
ACTIVITY
READINESS
5
Alternate ways of naming time from Math Masters,
pages 186 –188
PARTNER
ACTIVITY
5–15 Min
To practice rotations expressed as both fractions of turns and
degree measures, have students play Robot. One partner is the
“Controller” and the other is the “Robot.” The Controller picks a
destination. The Controller gives the Robot directions for the
amount of each turn and the number of steps to take until the
Robot reaches the destination. The amount of each turn may be
given as a fraction of a full turn or as a degree measure.
Example: “Make a half-turn clockwise and go forward 5 steps.
Now turn clockwise a quarter-turn (90 degrees), and go back
3 steps.”
ELL SUPPORT
Building a Math Word Bank
Name
Date
LESSON
䉬
Use the full-circle protractor and the clock from journal pages 152
and 153 to help you solve the problems below.
Sample explanations:
1.
How long does it take the hour hand to move 1°?
Explain.
141 142
2 minutes
The hour hand takes 60 minutes to move 30ⴗ, so it
takes 10 minutes to move 5ⴗ and 2 minutes to move 1ⴗ.
2.
How long does it take the minute hand to move 1°?
Explain.
5–15 Min
(Differentiation Handbook, p. 140)
Time
Clock Angle Challenge
65
PARTNER
ACTIVITY
10 seconds
To provide language support for angle rotations, have students
use the Word Bank template found on Differentiation Handbook,
page 140. Ask students to write the term degree, draw a picture
or give an example to represent the term, and write other related
words. See the Differentiation Handbook for more information.
It takes the minute hand 60 minutes to move 360ⴗ,
so it moves 6ⴗ every minute (or 60 seconds). Dividing
60 by 6, I get that it moves 1ⴗ every 10 seconds.
Math Masters, page 189
430
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