Objectives
To introduce how to calculate degree measures
of sectors; and to use a protractor to draw circle graphs.
1
materials
Teaching the Lesson
Key Activities
Students use fractions, decimals, and percents to calculate the degree measures of sectors
in a circle graph. They use a protractor to draw each sector.
ٗ Math Journal 1, pp. 169–171
Щ— Student Reference Book,
pp. 59, 60, and 147
Щ— Study Link 5 2
б­њ
Key Concepts and Skills
• Apply place-value concepts to round decimals to the nearest whole number.
[Number and Numeration Goal 1]
• Recognize and use equivalent names for fractions, decimals, and percents.
[Number and Numeration Goal 5]
• Use appropriate strategies to construct circle graphs.
[Data and Chance Goal 1]
• Draw and measure angles to the nearest degree.
[Measurement and Reference Frames Goal 1]
Щ— Transparencies (Math Masters,
pp. 152 and 153; optional)
Щ— compass
Щ— Geometry Template/protractor
Щ— calculator
Щ— scissors
Щ— tape
Щ— board compass and protractor
(for demonstration purposes)
Key Vocabulary
sector
Ongoing Assessment: Recognizing Student Achievement Use journal page 169.
[Geometry Goal 1]
2
Ongoing Learning & Practice
Students calculate percents of numbers to solve sale-price problems.
Students practice and maintain skills through Math Boxes and Study Link activities.
materials
Щ— Math Journal 1, pp. 172 and 173
Щ— Study Link Master (Math Masters,
p. 154)
Щ— Geometry Template/protractor
Щ— calculator
3
materials
Differentiation Options
READINESS
Students use a diagram of the full-circle
protractor to find fractional parts of a circle
and their degree equivalencies.
ENRICHMENT
Students use diagonals and apply properties
of triangles to find the sums of angle
measures in polygons.
Щ— Student Reference Book, p. 233
Щ— Teaching Masters (Math Masters,
pp. 155 and 156)
Щ— color pencils or markers
Technology
Assessment Management System
Math Message
See the iTLG.
346
Unit 5 Geometry: Congruence, Constructions, and Parallel Lines
Getting Started
Mental Math
and Reflexes
Math Message
Complete the Math Message on journal page 169.
Students use a calculator to solve percent problems.
Suggestions:
28% of 18 5.04
12.5% of 62 7.75
ଙ
3% of 42 1.26
7.25% of 52 3.77
Study Link 5 2 Follow-Up
б­њ
Ask students to explain how they
determined the angle measures.
1 Teaching the Lesson
б­¤ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 1, p. 169)
Use Problems 1 and 2 of the Math Message to check students’
understanding of the number of degrees in a circle. Students will
need to know that there are 360 degrees in a circle when they
calculate the degree measure of each sector in a circle graph.
Math Message
Ongoing Assessment:
Recognizing Student Achievement
ଙ
Use the Math Message to assess students’ ability to apply the definitions of
supplementary and vertical angles. Students are making adequate progress if
they use the given angle measure (150В°) to find the measures of vertical angles
and their supplements.
[Geometry Goal 1]
Student Page
Date
LESSON
б­¤ Calculating the Degree
5 3
б­њ
WHOLE-CLASS
ACTIVITY
Measure of a Sector
(Math Journal 1, pp. 169 and 170; Student Reference Book, pp. 59, 60,
and 147; Math Masters, pp. 152 and 153)
ଙ
Time
Degree Measures of Sectors
Math Message
163
1. Find the measures of the following angles in circle O without using a protractor.
P
a. mР„QOR П­ 150РЉ
b. mР„POS П­
c. mР„POQ П­
d. mР„SOR П­
150
30
30
Q
O
В°
150В°
S
В°
R
В°
2. Find the sum of the angle measures in Problem 1.
Use Problem 3 on journal page 169 to review strategies for
renaming fractions as percents, an essential skill for this lesson.
Circulate and assist as needed.
150РЉ П© mР„POS П© mР„POQ П© mР„SOR П­
360 В°
3. Connie, Josh, and Manuel were running for student council representative.
The table below shows the number of votes that each candidate received.
Complete the table.
Candidate
Number of
Votes Received
Connie
7
Josh
6
59 60
147
Fraction of
Percent of
Votes Received Votes Received
7
бЋЏбЋЏ
25
28%
24%
48%
100%
Manuel
12
6
бЋЏбЋЏ
25
12
бЋЏбЋЏ
25
Total
25
25
бЋЏбЋЏ
25
4. Use the percents from the table above to calculate the degree measure of the
sector representing each candidate.
Candidate
Percent of
Votes Received
Degree Measure of Sector
(to nearest degree)
28%
0.28 ‫ ء‬360 ϭ 100.8Њ ഠ101Њ
Manuel
24%
48%
86РЉ
173РЉ
Total
100%
360РЉ
Connie
Josh
169
Math Journal 1, p. 169
Lesson 5 3
б­њ
347
Student Page
Date
Time
LESSON
Draw and shade in a sector of a circle on the board. Explain that
in previous lessons, students used the Percent Circle on their
Geometry Template to draw each wedge-shaped piece, or sector,
of a circle graph. In this lesson, they will learn how to use a
protractor to draw each sector.
Drawing Circle Graphs with a Protractor
5 3
б­њ
Mr. Li surveyed the students in his class to find out what kinds of pets they
owned and how many of each kind they had. The results are shown in the
first two columns of the table below.
Fraction
Decimal
of Total Equivalent
Number (to nearest
of Pets thousandth)
Percent
of Total
Number
of Pets
59 60
147
Degree
Measure
of Sector
Kind
of Pet
Number
of Pets
Dog
8
8
бЋЏбЋЏ
24
0.333
33бЋЏ3бЋЏ%
Cat
6
6
бЋЏбЋЏ
24
0.25
25%
Guinea pig
or hamster
3
3
бЋЏбЋЏ
24
0.125
12.5% 0.125
‫ ء‬360° ϭ
45РЉ
Bird
3
3
бЋЏбЋЏ
24
0.125
12.5% 0.125
‫ ء‬360° ϭ
45РЉ
4
4
бЋЏбЋЏ
24
Other
0.167
1. Complete the table above.
1
бЋЏбЋЏ
3
1
‫ ء‬360Њ ϭ
1
бЋЏбЋЏ
4
1
бЋЏбЋЏ
6
16.7%
Copy the table in Problem 4 on the board or use a transparency of
Math Masters, page 152. (See margin.) Discuss solution strategies
for finding the degree measure of each sector.
90РЉ
‫ ء‬360° ϭ
б­џ One approach is to express the data as fractions and find those
fractions of 360В°. For example, Connie received 7 out of 25
7
votes, or бЋЏ25бЋЏ of the votes.
60РЉ
‫ ء‬360° ϭ
Pets Owned by Mr. Li’s Students
Study the first row.
7
бЋЏбЋЏ
25
2. At the right, or on a separate
Bird
sheet of paper, use a compass and
a protractor to make a circle graph
of the data in the table. If you need to,
tape your completed circle graph on
this page. Write a title for the graph.
Dog
Cat
4
7
бЋЏбЋЏ
of 360В° П­ бЋЏ2бЋЏ
5 ‫ ء‬360° ϭ 100 5 °
6
Other
Similarly, Josh received бЋЏ25бЋЏ of the votes.
Guinea pig
or hamster
6
бЋЏбЋЏ
25
2
6
бЋЏбЋЏ
of 360В° П­ бЋЏ2бЋЏ
5 ‫ ء‬360° ϭ 86 5 °
12
Manuel received twice as many votes as Josh (бЋЏ2бЋЏ
5 ), so the
2
measure of the sector representing Manuel is 86ᎏ5ᎏ° ‫ ء‬2 ϭ
4
172бЋЏ5бЋЏВ°. Since the protractors that students use are marked in
whole-degree increments, round each degree measure to the
4
2
4
nearest whole degree: 100бЋЏ5бЋЏВ° в†’ 101В°; 86бЋЏ5бЋЏВ°в†’ 86В°; 172бЋЏ5бЋЏВ° в†’ 173В°.
170
Math Journal 1, p. 170
б­џ Another approach is to express the data as fractions, rename
the fractions as decimals, and then multiply the decimal by
360В°. For example:
7
ᎏᎏ ϭ 7 Ϭ 25 ϭ 0.28 and 0.28 ‫ ء‬360° ϭ 100.8°
25
NOTE The table on Math Masters, page 152
is identical to the table in Problem 4 on journal
page 169.
6
бЋЏбЋЏ
25
ϭ 6 Ϭ 25 ϭ 0.24 and 0.24 ‫ ء‬360° ϭ 86.4°
12
бЋЏбЋЏ
25
ϭ 12 Ϭ 25 ϭ 0.48 and 0.48 ‫ ء‬360° ϭ 172.8°
Similarly, each degree measure should be rounded to the nearest
whole degree.
Student Page
Date
Time
LESSON
Drawing Circle Graphs with a Protractor
5 3
б­њ
cont.
3. Sixth-grade students at Hawthorn School took a survey about after-school
activities. Students answering the survey named the activity on which they
spent the most time after school. The results are shown in the table below.
Complete the table.
Activity
Number
of Students
Fraction
of Students
Music
12
12
бЋЏбЋЏ
60
Math Club
28
28
бЋЏбЋЏ
60
Art
5
Decimal
Equivalent
Percent of
Students
(to nearest
percent)
59 60
147
Size of
Sector
0.2
20%
72РЉ
47%
168РЉ
5
бЋЏбЋЏ
60
0.4–
6
0.08–
3
8%
30РЉ
13%
48РЉ
5%
18РЉ
7%
24РЉ
Sports
8
8
бЋЏбЋЏ
60
0.1–
3
Computers
3
3
бЋЏбЋЏ
60
None
4
4
бЋЏбЋЏ
60
0.05
–
0.06
When students have completed Problem 4 on journal page 169,
demonstrate how to make the circle graph of the data. Draw
a circle on the board or use the circle on the transparency
(Math Masters, p. 152) and mark off sectors with a protractor.
In order to make the graph as accurate as possible, graph
the smallest category (Josh’s votes) first; graph the largest
category (Manuel’s votes) last.
Allow students to make their circle graphs on a separate sheet of
paper as you demonstrate the appropriate procedures. When the
graphs have been completed and titled, have students check them
with the Percent Circle.
4. In the space below, or on a separate sheet of paper, use a compass and
Votes Received in the
Student Council Election
a protractor to make a circle graph of the data in the table. If you need to,
tape your completed circle graph on this page. Write a title for the graph.
After-School Activities of Sixth Graders
Art
Sports
Computers
e
Non
Music
Manuel
Math
Club
Connie
171
Math Journal 1, p. 171
348
Unit 5 Geometry: Congruence, Constructions, and Parallel Lines
Josh
Student Page
Next, work through the table on journal page 170 with the class.
Have students complete the table as you do the same on the board
or transparency (Math Masters, p. 153). The first row of the table
has already been filled in. Because some data are easier to work
with as fractions and other data as decimals, ask students to
share and explain the approach that they find most efficient for
finding the degree measures of each sector.
When students have completed the table, ask them to use a
compass and a protractor to make a circle graph of the survey
data. Students may construct the graph on the bottom of journal
page 170 or on a separate sheet of paper.
Date
Time
LESSON
Calculating Sale Price
5 3
б­њ
Study each method for calculating sale price.
Two-Step Method
Regular Price: $35.50
Discount: 20%
Find the sale price.
One-Step Method
Regular Price: $35.50
Discount: 20%
Find the sale price.
Step 1: Find the discount in
dollars: 20% of $35.50.
0.2 ‫ ء‬$35.50 ϭ $7.10
The discount is 20%, so the
amount of the regular price that
remains is 100% – 20%, or 80%
of the regular price.
Step 2: Subtract the discount
amount from the
regular price.
Find 80% of $35.50.
80% of $35.50 ϭ 0.8 ‫ ء‬$35.50
П­ $28.40
Regular Price ПЄ Discount П­ Sale Price
The sale price is $28.40.
$35.50 ПЄ $7.10 П­ $28.40
The sale price is $28.40.
Use either method to find the sale price.
1. Regular Price: $99.00
Discount: 30%
Sale Price:
2. Regular Price: $45.00
$69.30
3. Regular Price: $435.00
б­¤ Drawing a Circle Graph
PARTNER
ACTIVITY
Students find fraction, decimal, and percent equivalencies,
calculate degree measures of sectors, and use a compass and a
protractor to create a circle graph of the after-school activities
of sixth graders.
Remind students that to make the graph as accurate as possible,
they should graph the smallest category (Computers) first; they
should graph the largest category (Math Club) last.
2 Ongoing Learning & Practice
$38.25
4. Regular Price: $348.50
$413.25
5. Regular Price: $4,380
Discount: 18%
Sale Price:
(Math Journal 1, p. 171)
б­¤ Calculating Sale Price
Discount: 5%
Sale Price:
Discount: 15%
Sale Price:
Discount: 20%
Sale Price:
$278.80
6. Regular Price: $25,125
$3,591.60
Discount: 12%
Sale Price:
$22,110
172
Math Journal 1, p. 172
NOTE A small error in a big percentage is
relatively less important than a small error
in a small percentage. For example, if a
category is only 2% of the total and the
sector is off by 1%, that is a fairly large
error (50%). However, if a category is
34% of the total and the sector is off by
1%, the error is much less significant (only
about 3%).
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 172)
The problems on journal page 172 provide practice using a
one- or two-step method for finding a sale price.
Student Page
Date
Time
LESSON
Math Boxes
5 3
б­њ
1. Estimate the degree measure of Р„QRS.
2. Use your full-circle protractor to draw an
Sample estimate:
б­¤ Math Boxes 5 3
б­њ
INDEPENDENT
ACTIVITY
angle measuring 330РЉ. Label it Р„NOP.
В°
Estimate 200
Then use your full-circle protractor to
measure Р„QRS to the nearest degree.
N
O
Q
R
(Math Journal 1, p. 173)
P
S
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 5-1. The skills in Problems 5 and 6
preview Unit 6 content.
В°
Р„QRS measures about 240 .
230–232
232
3. Solve mentally.
2
a. бЋЏбЋЏ of 45 П­
9
4. Rewrite each fraction pair using a common
denominator.
10
6
of 100 П­ 80
75% of 32 П­ 24
1
b. 33 бЋЏбЋЏ% of 18 П­
3
б­¤ Study Link 5 3
б­њ
4
c. бЋЏбЋЏ
5
INDEPENDENT
ACTIVITY
(Math Masters, p. 154)
Home Connection Students make a circle graph.
They then interpret the graph and draw conclusions.
Remind students to take home a protractor or their
Geometry Template.
d.
49 50
87
3
2бЋЏбЋЏ
4
5. Marta has
rolls of ribbon. If there are
1
3 бЋЏбЋЏ yards of ribbon on each roll, about how
3
Sample answers:
3
8
a. бЋЏбЋЏ and бЋЏбЋЏ
10
25
15
бЋЏбЋЏ
50
and
16
бЋЏбЋЏ
50
7
19
b. бЋЏбЋЏ and бЋЏбЋЏ
15
45
21
бЋЏбЋЏ
45
and
19
бЋЏбЋЏ
45
4
3
c. бЋЏбЋЏ and бЋЏбЋЏ
5
9
36
бЋЏбЋЏ
45
and
15
бЋЏбЋЏ
45
6
d. бЋЏбЋЏ
27
12
бЋЏбЋЏ
54
and
7
бЋЏбЋЏ
54
and
7
бЋЏбЋЏ
54
6. Insert parentheses to make each
sentence true.
many yards of ribbon does Marta have?
Circle the best estimate.
a. 20 Ϫ 16 / 2 ‫ ء‬15 ϭ 30
1
A. бЋЏбЋЏ yard
2
b. 12 ‫ ء‬6 Ϫ 22 / 5 ϭ 10
B. 6 yards
c. 72 / 8 П© 4 / 6 П­ 1
((
) )
((
)
( (
1
C. 6 бЋЏбЋЏ yards
2
D. 9 yards
79
)
))
( (
))
d. 95 ПЄ 10 / 3 П© 2 П­ 93
90
247
173
Math Journal 1, p. 173
Lesson 5 3
б­њ
349
Study Link Master
Name
Date
STUDY LINK
Time
б­њ
The table below shows a breakdown, by age group, of adults who listen to
classical music.
1.
3 Differentiation Options
Circle Graphs
53
a.
Calculate the degree measure of each sector to the nearest degree.
b.
Use a protractor to make a circle graph. Do not use
the Percent Circle. Write a title for the graph.
49 147
Age
Percent of
Listeners
Degree
Measure
Age of Adult
Classical Music Listeners
18–24
11%
25–34
18%
35–44
24%
45–54
20%
40РЉ
65РЉ
86РЉ
72РЉ
40РЉ
58РЉ
18–
65+ 24
55 yrs yrs
25–34
yrs–64
yrs
45–54
yrs 35–44
yrs
55–64
11%
65П©
16%
б­¤ Finding Fractions of 360В°
5–15 Min
(Math Masters, p. 155)
Source: USA Today, Snapshot
On average, about 8 million adults listen to classical music on
the radio each day.
2.
INDEPENDENT
ACTIVITY
READINESS
a.
Estimate how many adults between the
ages of 35 and 44 listen to classical
music on the radio each day.
About
1,920,000 adults
b.
Estimate how many adults at least
45 years old listen to classical
music on the radio each day.
About
3,760,000 adults
(unit)
To provide experience finding fractions of 360В°, have students use
Math Masters, page 155 and coloring pencils or markers. Ask
students to shade fractional parts of a full-circle protractor
diagram. Then have them use the shaded regions to determine
the number of degrees in each fractional part.
(unit)
INDEPENDENT
ACTIVITY
ENRICHMENT
Practice
Order each set of numbers from least to greatest.
б­¤ Finding Sums of Angle
ПЄ7, 0, 0.07, 0.7, 7
3.
7, 0.07, ПЄ7, 0.7, 0
4.
0.25, 0.75, 0.2, бЋЏбЋЏ, бЋЏбЋЏ, 0.06, 0.18, бЋЏбЋЏ
5 4
10
1
бЋЏбЋЏ
10
4 4
1
0.06, , 0.18, 0.2, 0.25, 0.75, бЋЏ45бЋЏ, бЋЏ44бЋЏ
5–15 Min
Measures in Polygons
Math Masters, p. 154
(Math Masters, p. 156; Student Reference Book, p. 233)
To further extend their knowledge of angle relationships, students
find the sums of angle measures in various polygons. They use a
pattern to calculate the sums of angle measures in a heptagon,
nonagon, and dodecagon.
Teaching Master
Teaching Master
Name
Date
Time
Fractions of 360Ш‡
53
б­њ
53
б­њ
Shade each fractional part. Then record the number of degrees in each shaded region.
1
2
Shade бЋЏбЋЏ of the circle.
280 29
260
0
250
24
0
280 29
270 30
0
260
250
0
23
0
23
270 30 31
00 0
280 29
250
280 29
0
260
250
24
0
180 20
1
23
0
13
0
170
31
0
6 Вє180РЉП­1,080РЉ
10
8
8 Вє180РЉП­1,440РЉ
Study your completed table. Use any patterns you notice to write a formula
to find the sum of the angle measures in any polygon (n-gon).
0
13
0
14
0
0
22
15
0
15
0
160
190 20
of 360РЉ П­
0
0
0
160
12
100 11
0
2. a.
Formula
160
280 29
6
40
90
0
180 20
1
170
0
22
190 20
0
15
270 30
0
260
30
0
260
14
0
22
190 20
270 30 31
00 0
10 20
14
0
0
180 20
1
170
0
22
190 20
250
24
0
0
180 20
1
170
24
8
160
0
0
160
0
23
15
15
0
23
РЉ
0
0
0
0
4 Вє180РЉП­ 720
70 80
280 29
14
24
14
0
22
190 20
31
0
180 20
1
170
350
0
0 360В°
33
60
270 30
0
4
50
12
0
13
0
100 11
0
24
6
0
90
0
РЉ
0
70 80
60
260
250
0
0
12
50
0
180 20
1
170
23
3 Вє180РЉП­ 540
100 11
13
0
340
0
270
В°
0
22
190 20
Math Masters, p. 155
350
3
Decagon
3
4
3
бЋЏбЋЏ
4
5
Octagon
В°
Shade бЋЏбЋЏ of the circle.
32
В°
2 Вє 180РЉ П­ 360РЉ
Hexagon
120
of 360РЉ П­
40
30
40
90
0
30
30
70 80
0
of 360РЉ П­
0
0 360В°
60
12
100 11
6.
2
Pentagon
50
90
1
бЋЏбЋЏ
3
4
10 20
33
10 20
34
0
0
33 360В°
13
0
70 80
60
1
бЋЏбЋЏ
12
0
50
В°
1
12
0 350
0
13
0
0
32
Shade бЋЏбЋЏ of the circle.
0
diagonal
В°
1
3
40
Sum of Angle Measures
Example: Quadrangle
Shade бЋЏбЋЏ of the circle.
350
340
60
of 360РЉ П­
32
12
0
30
90
of 360РЉ П­
10 20
0
0
33 360В°
Number of Number of
Sides (n)
Triangles
Polygon
100 11
0
1
бЋЏбЋЏ
4
4.
Draw diagonals from the given vertex to separate each polygon into triangles.
Then complete the table.
40
90
12
100 11
0
30
70 80
90
В°
1
6
32
1.
10 20
350
0
0
33 360В°
60
70 80
60
180
of 360РЉ П­
350
340
5.
270 30 31
00 0
0
32
160
31
0
340
40
15
30
50
0
0
33 360В°
Shade бЋЏбЋЏ of the circle.
1
бЋЏбЋЏ
6
Shade бЋЏбЋЏ of the circle.
50
1
бЋЏбЋЏ
2
1
4
Sums of Angle Measures in Polygons
0
0
32
2.
10 20
Time
A diagonal is a line segment that connects two vertices of a polygon and is not a side.
You can draw diagonals from one vertex to separate polygons into triangles.
14
350
340
3.
Date
LESSON
LESSON
1.
Name
Unit 5 Geometry: Congruence, Constructions, and Parallel Lines
b.
(n ПЄ 2) Вє 180
Use the formula to find the sums of the angle measures in a
В°
В°
heptagon.
nonagon.
dodecagon.
900
Math Masters, p. 156
1,260
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