 # Lesson 3.4 Using a Protractor

```Objectives
To review types of angles, geometric figures, and
the use of the Geometry Template to measure and draw angles.
1
materials
Teaching the Lesson
Key Activities
Students write definitions for acute and obtuse angles, and review other types
of angles. They explore the Geometry Template, measure angles using the
half-circle and full-circle protractors, and draw angles with the half-circle protractor.
Key Concepts and Skills
вЂў Define and classify angles according to their measures.
[Geometry Goal 1]
вЂў Use a full-circle and a half-circle protractor to measure and draw angles.
[Measurement and Reference Frames Goal 1]
вЂў Explore angle types and relationships.
Щ— Math Journal 1, pp. 68 and 69
Щ— Student Reference Book,
pp. 162 and 163
б­њ
Щ— Teaching Masters (Math Masters,
pp. 78 and 79; optional)
Щ— Teaching Aid Masters (Math Masters,
pp. 414 and 419; optional)
Щ— Geometry Template
[Measurement and Reference Frames Goal 1]
Key Vocabulary
acute angle вЂў obtuse angle вЂў right angle вЂў straight angle вЂў reflex angle вЂў
Geometry Template вЂў arc
Ongoing Assessment: Informing Instruction See pages 173 and 174.
Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip.
[Measurement and Reference Frames Goal 1]
2
materials
Ongoing Learning & Practice
Students find the landmarks of data represented by a bar graph.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Informing Instruction See page 175.
3
materials
Differentiation Options
Students review the
vocabulary of angles
and their relationships.
ENRICHMENT
Students use their
knowledge of angle
measures to solve a
baseball challenge
problem.
ELL SUPPORT
Students define and
illustrate angle terms.
Advance Preparation If the Geometry Templates were not distributed earlier, decide how you
will manage them before beginning the lesson. Experienced Everyday Mathematics teachers
suggest writing student ID numbers on the templates with a permanent marker before
distributing them.
170
Unit 3 Geometry Explorations and the American Tour
Щ— Math Journal 1, pp. 70 and 71
Щ— Student Reference Book, p. 121
Щ— Study Link Master (Math Masters, p. 80)
Щ— Student Reference Book, p. 141
Щ— Differentiation Handbook
Щ— Teaching Masters (Math Masters,
pp. 81 and 82)
Щ— Geometry Template
Technology
Assessment Management System
Exit Slip
See the iTLG.
Getting Started
Mental Math and Reflexes
Math Message
Students stand, facing the same direction, and follow
directions related to angle measures or fractions of a
turn. Direct students to focus on the degree equivalents for
quarter (90В°) and half turns (180В°).
Use only the information given on journal page 68
to complete Problems 1 and 2.
б­њ
вЂў Rotate 180В° to the right.
вЂў Make a half turn to the left.
вЂў Make a quarter turn to the right.
вЂў Make a 90В° turn to the left.
вЂў Turn 360В° to the left.
Allow students five minutes to compare their
strategies for estimating and finding the angle measures.
Record strategies on the Class Data Pad for display.
1 Teaching the Lesson
б­¤ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 1, p. 68)
Survey the class for their definitions of acute angle, An angle
whose measure is greater than 0В° and less than 90 В° and obtuse
angle. An angle whose measure is greater than 90В° and less than
180В° For each type of angle, compare the similarities and
differences of studentsвЂ™ definitions. Ask students to agree on a
common definition for each type of angle and record these
definitions on the Class Data Pad. Emphasize the importance of
definitions in mathematics. For example, discuss why narrow and
wide are not specific enough to be acceptable definitions for acute
and obtuse. Ask whether an angle can be both acute and obtuse
and why or why not.
NOTE Working with mathematical definitions helps students build logical
thinking skills that are critical to success in higher mathematics. Definitions are
closely related to classification schemes. The types of angles defined in this
lesson, for example, make up a classification scheme for all angles. Every angle
with a measure greater than 0В° is either an acute angle, right angle, obtuse
angle, straight angle, or reflex angle.
Consider assigning groups to combine their individual definitions into a
group proposal for the class definition.
A U D I T O R Y
б­њ
K I N E S T H E T I C
б­њ
T A C T I L E
б­њ
V I S U A L
Draw a right, a straight, and a reflex angle on the Class Data
Pad. Ask students to name these angles. A 90В° angle is called a
right angle; a 180В° angle is called a straight angle; an angle whose
measure is greater than 180В° and less than 360В° is called a reflex
angle. Draw a small square in the corner of the right angle and
explain that the symbol represents the square corner formed by
perpendicular lines. The symbol indicates that the measure of the
angle is 90В°.
Lesson 3 4
б­њ
171
Ask: Why do you think the 180В° angle is called a straight angle?
Sample answer: Because the two sides of the angle form a straight
line Assign volunteers to label the angles on the Class Data Pad.
Ask partners to demonstrate angle types (right, straight, acute,
and obtuse) using their arms as physical models. Sample answers:
think of the elbow as the vertex of the angle and bend one arm to
form a 90В° angle, an angle smaller than 90В°, an angle between 90В°
and 180В°, and a 180В° angle. Students should name each modeled
angle in turn. (See margin.)
Ask partners to illustrate each of the following terms using their
arms: parallel, perpendicular, and intersecting. Assign volunteers
review the concepts and objects from this lesson listed on the
Class Data Pad with a focus on the vocabulary.
Demonstrating a right angle
б­¤ Introducing the Geometry
WHOLE-CLASS
ACTIVITY
Template
(Student Reference Book, pp. 162 and 163; Math Masters, p. 419)
Transition to this activity by displaying the Geometry Template,
either by placing the template on the overhead or by using a
transparency of Math Masters, page 419.
Ask students to turn to pages 162 and 163 of the Student
Reference Book. Consider assigning groups one each of the
paragraph topics on page 162. Groups should read their assigned
paragraph(s), identify the important information, and present it to
the class. As groups present their information, make sure that the
following features of the Geometry Template are noted:
б­џ There is an inch ruler along the left edge and a centimeter
ruler along the right edge.
б­џ As part of the geometric figures, there are six pattern-block
shapesвЂ”equilateral triangle, square, trapezoid, two
rhombuses, and regular hexagonвЂ”labeled PB. The regular
pentagon and octagon are not pattern-block shapes.
б­џ Some of the other shapes have sides that are the same length
as the pattern-block shapes.
б­џ The squares and circles of various sizes can be used to draw
figures quickly.
The Geometry Template appears on both
Math Masters, page 419 and Student
Reference Book, page 163.
172
Unit 3 Geometry Explorations and the American Tour
Ask students to find the relationship between the template shapes
and actual pattern blocks. The sides of the template shapes are
half as long as the edges of the pattern blocks.
The triangles on the Geometry Template will be discussed in Lesson 3-6. The
Percent Circle is used to read and make circle (pie) graphs and will be discussed
in Lessons 5-10 and 5-11.
Ask students to use their templates to make tracings that identify
different figures on the Geometry Template that have sides of the
same length. Have volunteers record this list of figures on the
Class Data Pad. The sides of the pentagon, octagon, five
pattern-block shapes, and three sides of the pattern-block
trapezoid are the same length; the fourth side of the trapezoid is
twice as long.
Comparing each figure with the octagon shows
that all regular polygons on the Geometry
Template have sides of the same length. Other
shapes can be compared in the same way.
Remind students that any two figures can be compared by tracing them
so they are adjacent; that is, they share a common side. (See margin.)
A U D I T O R Y
б­њ
K I N E S T H E T I C
б­њ
T A C T I L E
б­њ
NOTE The key to using the half-circle
protractor is knowing which scale to read. The
key to using the full-circle protractor is
knowing that the scale runs clockwise. The 0В°
mark must be aligned with one side of the
angle. Students measure from 0В° in the
clockwise direction to find the answers. One
way to reinforce this habit is for students to
estimate whether the angle is more or less
than 90В° before measuring it.
V I S U A L
Ask students to identify the half-circle protractor and the
full-circle protractor on the Geometry Template. Then ask a
volunteer how they would measure one of the angles from Problem
1 on journal page 68. Ask another volunteer to use the full-circle
protractor to measure one of the angles in Problem 2.
Student Page
Ongoing Assessment: Informing Instruction
Date
Watch for students who find it difficult to align the protractor when the Geometry
Template spans across two pages. Have students try turning the page and/or
aligning a different leg of the angle with the 0В° mark on the protractor.
Time
LESSON
3 4
б­њ
Acute and Obtuse Angles
Math Message
1. Acute Angles
NOT Acute Angles
91В°
90В°
28В°
65В°
б­¤ Measuring Angles
120В°
88В°
PARTNER
ACTIVITY
(Math Journal 1, p. 68)
Sample answer: An acute angle is
an angle whose measure is less than 90В°.
Write a definition for acute angle.
2. Obtuse Angles
NOT Obtuse Angles
91В°
120В°
Partners work on Problem 3 at the bottom of the journal page.
Remind students to estimate the angle measures before they use a
protractor. (See margin note.) Explain that estimating first will
help them confirm whether theyвЂ™re reading the correct scale or
moving in the correct direction on the protractor. Allow five to ten
minutes, then discuss studentsвЂ™ answers. TheresaвЂ™s and DevonвЂ™s
answers are incorrect. Devon interpreted the angle to be a reflex
angle (the angle larger than 180В°). Theresa used the wrong scale
on her half-circle protractor.
170В°
55В°
89В°
angle is an angle whose measure is greater than 90В°.
Write a definition for obtuse angle.
Measuring and Drawing Angles with a Protractor
Sarah used her half-circle protractor to measure the angle at the
right. She said it measures about 35РЉ. Theresa measured it with
her half-circle protractor. Theresa said it measures about 145РЉ.
Devon measured it with his full-circle protractor. Devon said it
E
3. a. Use both your template protractors to measure the angle. Do you agree with
Sarah, Theresa, or Devon?
Sample answer: I agree with Sarah.
Sample answer: The measure of the angle with the
b. Why?
68
Math Journal 1, p. 68
Lesson 3 4
б­њ
173
Student Page
Date
Time
LESSON
Measuring and Drawing Angles with a Protractor
34
б­њ
Use your half-circle protractor. Measure each angle as accurately as you can.
E
Explain that in Everyday Mathematics when students measure an
angle, they will normally measure the smaller angle, not the reflex
angle. When they are supposed to measure the reflex angle, it is
indicated with an arc.
D
A
S
56В° .
T
b. mР„EDS is about 115В° .
88В° .
Use your full-circle protractor to measure each angle.
330В° angle
U
C
E
G
35В° .
121В° .
78В° .
б­¤ Practicing Measuring and
Draw and label the following angles. Use your half-circle protractor.
6. a. Р„CAT: 62В°
b. Р„DOG: 135В°
C
A
PARTNER
ACTIVITY
Drawing Angles
D
O
T
30В° angle
Students will practice measuring angles throughout the year
in Math Boxes problems and in Ongoing Learning & Practice
activities.
L
G
(Math Journal 1, p. 69; Math Masters, pp. 419, 78, and 79)
69
Math Journal 1, p. 69
NOTE Students used the full-circle and
the half-circle protractors in Fourth Grade
Everyday Mathematics. With half-circle
protractors expect students to be able to
measure to within about 2В° of the measured
approximation. With full-circle protractors,
expect that students might be less precise,
measuring to within about 5В° of the
approximation.
Remind students that all measurements are approximations. They
should always measure carefully but recognize that a measure
obtained with a measuring tool does not give an exact size for the
object being measured. The goal is to be as precise as possible, but
the degree of precision is dependent on the accuracy of the
measuring tool and how effectively it is used.
Assign journal page 69. Remind students to use the half-circle
protractor to complete Problems 4 and 6, and the full-circle
protractor to complete Problem 5. Ask them to estimate each angle
before measuring. It is important that students practice with both
types of protractors. You might want to display or use overhead
transparencies of the Geometry Template and journal pages
(Math Masters, pp. 419, 78, and 79). Circulate and assist.
Student Page
Date
Time
LESSON
Watching Television
3 4
б­њ
Ongoing Assessment: Informing Instruction
Adeline surveyed the students in her class to find out how much television they watch
in a week. She made the following graph of the data.
Watch for students who have difficulty measuring Angle T because the figure is
so small. Suggest that students extend the angleвЂ™s sides with a straightedge
before measuring the angle with a protractor. Demonstrate this approach and
illustrate the pitfalls of extending the sides without using a straightedge.
Hours of Television Watched per Week
Number of Students
8
7
6
5
4
3
2
1
0
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Hours per Week
Find each data landmark.
1. a. minimum:
d. median:
13
22
29
20.96
b. maximum:
c. range:
e. mean:
f. mode:
16
23
all of the values in order from smallest to
largest, and then found the middle value.
2. Explain how you found the median.
3. a. Which data landmark best represents the number of hours a typical student
watches televisionвЂ”the mean, median, or mode?
b. Why?
70
Math Journal 1, p. 70
174
Unit 3 Geometry Explorations and the American Tour
Incorrect
Correct
Extending the sides of an angle
Student Page
Ongoing Assessment:
Recognizing Student Achievement
Exit Slip
а¬™
Date
Time
LESSON
Math Boxes
34
б­њ
1. Estimate and solve.
2. Find the landmarks for this set of numbers:
99, 87, 85, 32, 57, 82, 85, 99, 85, 65, 78,
87, 85, 57, 85, 99
463
2,500
2,541
Estimate:
Ask students to respond to the following question on an Exit Slip (Math Masters,
page 414) or half-sheet of paper. Which is easier to useвЂ”the full-circle protractor
or the half-circle protractor? Why? Students are making adequate progress if
their answers demonstrate an understanding of how to use both protractor types.
[Measurement and Reference Frames Goal 1]
Solution:
Minimum:
Range:
5,046
ПЄ 2,491
Median:
2,500
2,555
Estimate:
Solution:
13вЂ“17
119
3. Solve.
2 Ongoing Learning & Practice
4. Estimate and solve.
xП­
36 П­ p вЂ« ШЎвЂ¬4
pП­
200 П­ 50 вЂ« ШЎвЂ¬m
mП­
tП­
28 ПЄ b П­ 13
bП­
37
9
4
15
15
473.894
a.
530
533.129
Estimate:
Solution:
b.
78.896
ПЄ 29.321
50
49.575
Estimate:
Solution:
219
5. Write the name of an object in the room
б­¤ Interpreting a Bar Graph
99
32
67
85
Maximum:
6. Solve.
that is about 10 inches long.
INDEPENDENT
ACTIVITY
5.8 вЂ« ШЎвЂ¬76 П­
Write the name of an object in the room
that is about 10 centimeters long.
159 вЂ« ШЎвЂ¬7 П­
0.4 вЂ« ШЎвЂ¬231 П­
185
Students find landmarks for data represented by the bar graph on
journal page 70. They describe how to find the median, and tell
whether the mean, median, or mode best represents the data for a
typical student.
2,108
440.8
1,113
92.4
634.12
34 вЂ« ШЎвЂ¬62 П­
(Math Journal 1, p. 70; Student Reference Book, p. 121)
34вЂ“36
76.4 вЂ« ШЎвЂ¬8.3 П­
19 20
38вЂ“40
71
Math Journal 1, p. 71
Ongoing Assessment: Informing Instruction
Watch for students who have difficulty finding the mean. It might be helpful to
review the steps on Student Reference Book, page 121. Ask students to
summarize the process for finding the mean in their own words and have them
list the steps on an index card. For example: 1. Count the numbers in the data
set; 2. Add the numbers in the data set; and 3. Divide the sum by the count.
Students should check their steps by using them to solve the Check Your
Understanding problem at the bottom of the Student Reference Book page.
Date
34
б­њ
measure of Р„CAT П­
2.
mР„BAR П­
3.
4.
5.
6.
70РЉ
50РЉ
mР„RAT П­ 110РЉ
mР„CAB П­ 130РЉ
mР„BAT П­ 60РЉ
mР„CAR П­ 180РЉ
T
B
80 90 100
70 100 90 80 110 1
70 20
60 0 110
60 1
2
50 0 1
50 30
13
C
0 10
180 170 1620 3
01 0
50 4
14 0
0
1.
180
0 170 0
0 16 10
15 20
0 30
14 0
4
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 3-2. The skill in Problem 6
previews Unit 4 content.
A
R
Find the approximate measure of each angle at the right.
mР„MEN П­
8.
mР„DEN П­
9.
30
E
12
0
24
11.
0
90
0
0
21
Home Connection Students practice measuring angles
with half-circle and full-circle protractors.
0
33
60
10.
M
T
30
0
(Math Masters, p. 80)
120РЉ
90РЉ
mР„MET П­ 50РЉ
mР„MED П­ 150РЉ
mР„TEN П­ 170РЉ
7.
270
INDEPENDENT
ACTIVITY
180
б­њ
204вЂ“205
Find the approximate measure of each angle at the right.
(Math Journal 1, p. 71)
Time
Angle Measures
0
б­њ
INDEPENDENT
ACTIVITY
15
б­¤ Math Boxes 3 4
Name
N
D
Practice
12.
5,844
13.
8,243
15.
60 П¬ 5 П­
238
ПЄ 129
109
12
16.
50 П¬ 6 в†’
14.
234
вЂ« ШЎвЂ¬22
5,148
8 R2
Math Masters, p. 80
Lesson 3 4
б­њ
175
Teaching Master
Name
Date
LESSON
Time
Points, Lines, and Angles
34
б­њ
3 Differentiation Options
141
Identify the terms and objects in the riddles below. Use the words and phrases
from the Word Bank to complete the table.
Word Bank
point
line segment
ray
line
angle
parallel lines
parallel line segments
intersecting lines
vertices perpendicular lines perpendicular line segments
б­¤ Identifying Points, Lines,
vertex
Clues
What Am I?
1
I am a location in space. It takes only one letter to name me.
2
My length cannot be measured, but I am named by
two of my points.
(Student Reference Book, p. 141; Math Masters, p. 81)
ray
angle
3
I do not curve. I have only one end point.
I am measured in degrees. I have a vertex. My sides
are two rays.
5
Perpendicular
line segments
There are always at least two of us. We have endpoints. Parallel line
We always stay the same distance apart.
segments
I am the point where two rays meet to form an angle.
vertex
Two of us meet.
Intersecting lines, perpendicular
lines, or perpendicular line segments
Our lengths cannot be measured. When two of us meet, Perpendicular
we form right angles.
lines
I am the endpoint where two sides of a polygon meet.
vertex
My length can be measured. I have two endpoints.
Line segment
Our lengths cannot be measured. There are always at
Parallel lines
To review vocabulary and concepts related to angles, have
students solve riddles about points, lines, line segments,
and angles.
We have endpoints. When two of us meet, we form
one or more right angles.
7
8
9
10
11
12
5вЂ“15 Min
and Angles
point
Line or ray
4
6
SMALL-GROUP
ACTIVITY
ENRICHMENT
б­¤ Measuring Baseball Angles
least two of us. We always stay the same distance apart.
INDEPENDENT
ACTIVITY
15вЂ“30 Min
(Math Masters, p. 82)
Math Masters, p. 81
To apply studentsвЂ™ understanding of angle properties and angle
measurements, have students solve a baseball problem that
involves addition and subtraction of angle measures.
As students complete the assignment, discuss answers and
strategies. The field has a 90В° angle within which a batted ball
is put in play. Each of the four infielders covers 13В°, for a total of
4 вЂ« ШЎвЂ¬13В°, or 52В° and the pitcher covers 6В°. That leaves 90В° ПЄ 52В°
ПЄ 6В°, or 32В°, uncovered, which suggests that on average a little
more than one-third of hard-hit ground balls should get past the
infield. Ask the baseball players and fans in the class whether
that conclusion is consistent with their experiences.
ELL SUPPORT
Teaching Master
Name
Date
LESSON
б­¤ Building a Math Word Bank
Time
Baseball Angles
б­њ
p
rtsto
sho over
c
can
lin
ul
e
lin
fo
ul
fo
3r
d
ca bas
n
e
co ma
ve n
r
1s
t
ca base
nc m
ov an
er
can cover
pitcher
13В°
6В°
To provide language support for angles, have students use the
Word Bank Template found in the Differentiation Handbook. Ask
students to write the terms acute angle, right angle, obtuse angle,
straight angle, and reflex angle, draw pictures relating to each
term, and write other related words. See the Differentiation
13В°
13В°
13В°
batter
The playing field for baseball lies between the foul lines, which form a 90РЉ angle.
Suppose that each of the four infielders can cover an angle of about 13РЉ on a hard-hit
ground ball, and that the pitcher can cover about 6РЉ. (See the diagram above.)
Source: Applying Arithmetic, Usiskin, Z. and Bell, M. В© 1983 University of Chicago
1.
How many degrees are left for the batter to hit through?
32РЉ
Math Masters, p. 82
176
30+ Min
(Differentiation Handbook)
2n
db
can asem
cov an
er
e
34
SMALL-GROUP
ACTIVITY
Unit 3 Geometry Explorations and the American Tour
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