Download Angles of Polygons

Angles of Polygons
Objective To develop an approach for finding the angle
measurement sum for any polygon.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Practicing Expanded Notation
• Investigate and compare the measurement
sums of interior angles of polygons. Math Journal 1, p. 90
Student Reference Book, p. 396
Students practice place-value
concepts by reading and writing large
numbers and decimals in standard
and expanded notation.
[Geometry Goal 1]
• Measure angles with a protractor. [Measurement and Reference Frames Goal 1]
• Find maximum, minimum, and median for
a data set. [Data and Chance Goal 2]
• Draw conclusions based on collected data. [Data and Chance Goal 2]
Math Boxes 3 9
Math Journal 1, p. 91
Students practice and maintain skills
through Math Box problems.
Key Activities
Study Link 3 9
Students measure to find angle sums for
triangles, quadrangles, pentagons, and
hexagons. They use the pattern in these
sums to devise a method for finding the
angle sum for any polygon.
Math Masters, p. 92
Students practice and maintain skills
through Study Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
ENRICHMENT
Tessellating Quadrangles
Math Masters, p. 93
paper (8_12 " by 11") scissors tape cardstock (optional)
Students investigate whether all quadrangles
will tessellate.
EXTRA PRACTICE
Finding Angle Measures in Polygons
Math Masters, p. 94
Students find the sums of the interior angles
of polygons.
ELL SUPPORT
Describing Tessellations
Students describe the tessellations in the
Tessellation Museum.
Ongoing Assessment:
Recognizing Student Achievement
Use an Exit Slip (Math Masters,
page 414). [Geometry Goal 1]
Materials
Math Journal 1, pp. 85–89
Study Link 38
Math Masters, p. 414
transparency of Math Masters, p. 420
(optional) Class Data Pad Geometry
Template (or protractor and straightedge)
Advance Preparation
This 2-day lesson begins with the Math Message on Day 1 and Dividing Polygons into Triangles on Day 2.
For the Math Message, draw a line plot on the board for students to record the sums of the angles they find.
Label it from about 175° to 185°. The Lesson 38 Study Link asks students to collect tessellations. These will
be displayed in the Tessellation Museum. Include the class definitions for angles and triangles in this display.
For the optional Enrichment activity in Part 3, do the tessellation activity yourself in advance, with convex
and nonconvex quadrangles, so you can help students see how the angles fit.
Teacher’s Reference Manual, Grades 4–6 p. 203
Lesson 3 9
199
Mathematical Practices
SMP2, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
5.NBT.2, 5.NBT.3a
Mental Math and Reflexes
Math Message
Use your slate procedures for problems
such as the following:
Use a straightedge to draw a big
triangle on a sheet of paper. Measure its angles
and find the sum. Record the sum on the class
line plot.
47 ∗ 10,000 470,000
4.7 ∗ 1,000 4,700
0.47 ∗ 100 47
0.047 ∗ 10 0.47
356 ∗ 1,000 356,000
42.6 ∗ 100 4,260
0.862 ∗ 100 86.2
0.009 ∗ 1,000 9
0.109 ∗ 1,000 109
7.08 ∗ 10,000 70,800
0.084 ∗ 100 8.4
79.04 ∗ 1,000 79,040
Study Link 3 8 Follow-Up
Ask volunteers to share their tessellation
examples. Encourage them to include
the names of polygons and to explain how they
identified the patterns as tessellations.
1 Teaching the Lesson
Day 1 of this lesson, students should
complete the Math Message, the Study
Link 3-8 Follow-Up, and explore finding the
sums of the angle measures.
● On
● On
Day 2 of this lesson, students begin
with Dividing Polygons into Triangles and
explore further how to find the sums of the
angle measures in any polygon. Then have
students complete Part 2 activities.
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Survey the class to complete the line plot and identify the
following landmarks.
●
What is the maximum sum of the angle measures?
●
What is the minimum sum of the angle measures?
●
What is the median sum of the angle measures?
x
x
175°
or
less
176°
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
177°
178°
179°
180°
181°
182°
183°
x
184°
185°
or
more
Sample data
Expect a range of sums, because measurements are never exact,
and because some of the sides of students’ drawings may not be
straight or meet exactly. Explain that if a triangle is accurately
drawn, and its angles are measured with precision, the sum of the
angle measures will always be 180°.
Explain that students can prove this statement using their
triangles as models. Ask students to tear the three angles off their
triangles as shown below.
Students’ angle measures might seem to total
slightly more or less than 180° because their
original triangles might not be accurate.
Next have students arrange their three angles next to each other
so they line up. (See margin.) Ask students what type of angle
200
Unit 3
Geometry Explorations and the American Tour
Student Page
they have formed, A straight angle and what the measure of a
straight angle is. 180° This shows that the sum of the three angles
of the triangle is 180°. Ask students to leave their angles arranged
in a straight angle on their desks, and then check each other’s
triangles. When students return to their desks, ask what they
observed about the angles in triangles. All the angles will form
straight angles; the sum of the angle measures in a triangle will
always total 180°. Record this property on the Class Data Pad.
Date
LESSON
39
Time
Angles in Quadrangles and Pentagons
1.
Circle the kind of polygon your group is working on:
quadrangle
pentagon
2.
Below, use a straightedge to carefully draw the kind of polygon your group is working
on. Your polygon should look different from the ones drawn by others in your group,
but it should have the same number of sides.
Drawings vary.
NOTE Precise language would call for writing and saying: the sum of the
measures of the angles instead of the sum of the angles. But it is common in
mathematics to use the shorter phrase.
▶ Finding the Sums of Angles
in Polygons
SMALL-GROUP
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VING
VIN
IN
ING
3.
Answers vary for individual angle measurements.
4.
Find the sum of the angles in your polygon.
The total sum of the angles should be
close to 360° for quadrangles and 540°
for pentagons.
(Math Journal 1, pp. 85 and 86)
Draw an example of a convex polygon and a concave polygon on
the board.
Measure the angles in your polygon. Write each measure for each angle.
Math Journal 1, p. 85
NOTE These figures will be used later in the lesson.
Ask students what true statements they can make about the
interior angles in the two figures. Expect responses to vary, but
structure your follow-up questions to guide students to recognize
that the concave polygon has one angle that is a reflex angle.
Remind students that the names for angles refer to the angles’
measures, not to whether the angle is or is not an interior angle.
Ask students to fold a blank sheet of paper into fourths. Open it and
label each box in the top row Polygons and Not Polygons. Label the
boxes in the bottom row Convex Polygons and Concave Polygons.
Assign small groups of three to five students to work together to
draw at least two examples of each figure. As groups finish, they
should examine other students’ examples. Circulate and assist.
Transition to the journal activity by first surveying the class
for questions or observations about their drawings. Then assign
groups to work on quadrangles or pentagons. Ask students to
circle the name of the figure they are going to work on, listed at
the top of journal page 85.
Ask students to complete Problems 1–7 on journal pages 85 and
86. Problem 7 provides data on hexagons for the next activity. As
you circulate, consider asking students, who are waiting for the
group to finish Problem 5, to go on to Problem 7 until the others
are done.
Student Page
Date
LESSON
39
5.
Time
Angles in Quadrangles and Pentagons
cont.
Record your group’s data below.
Group Member’s
Name
Sketch of
Polygon
Sum of
Angles
Drawings of
polygons vary. The
sum of the angles
varies but should be
close to 360° for
quadrangles and
540° for pentagons.
6.
Find the median of the angle sums for your group.
The total sum of the angles should be close to
360° for quadrangles and 540° for pentagons.
7.
If you have time, draw a hexagon. Measure its angles with a protractor.
Find the sum.
Drawings vary.
Sum of the angles in a hexagon =
720⬚
Math Journal 1, p. 86
Lesson 3 9
201
Student Page
Date
Time
LESSON
39
8.
Angles in Quadrangles and Pentagons
cont.
Record the class data below.
Sum of the Angles
in a Quadrangle
Group
Sum of the Angles
in a Pentagon
Group Median
Group
Group Median
The group median should be close to 360° for
quadrangles and 540° for pentagons.
Find the class median for each polygon. For the triangle, use the median
from the Math Message.
9.
The group median should
be close to 180° for a
triangle, 360° for a
quadrangle, 540° for a
pentagon, and 720° for a
hexagon.
Sums of Polygon Angles
Polygon
Class Median
triangle
quadrangle
pentagon
hexagon
What pattern do you see in the Sums of Polygon Angles table?
10.
Sample answer: As the number of sides increases by 1,
the sum of the angle measures increases by 180°.
Math Journal 1, p. 87
▶ Finding the Median for
WHOLE-CLASS
DISCUSSION
the Sums of Angles
(Math Journal 1, p. 87)
Bring the class together and use the board or a transparency of
Math Masters, page 420 to collect the group’s median angle sums;
first from the quadrangle group and then from the pentagon
group. Ask students to record this data in the tables for Problem 8
on journal page 87.
Next ask students to use the group medians to find the class
median for quadrangles and pentagons. Then record this data in
the table for Problem 9. For the triangle row, enter the class
median from the Math Message line plot.
Collect data from students who did Problem 7 on journal page 86,
listing the sums on the Class Data Pad or the overhead projector.
Ask students to find the median of these sums and record it in the
table for Problem 9.
The class medians should be close to 180° for a triangle, 360° for
a quadrangle, 540° for a pentagon, and 720° for a hexagon.
Ask students to complete Problem 10 on the journal page. As they
look for patterns in the Sums of Polygon Angles table, ask them to
think about how the contents of each column in the table are
related. Ask: What are the differences between a triangle and a
quadrangle? Do the numbers in the class median column increase
or decrease and by how much? The quadrangle has one more side
than the triangle; the median sum of their angles increases by
about 180°. Circulate and assist.
▶ Dividing Polygons
Student Page
Date
Time
LESSON
39
1.
into Triangles
(Math Journal 1, pp. 87 and 88)
Angles in Heptagons
A heptagon is a polygon with 7 sides.
Predict the sum of the angles in a heptagon.
2.
WHOLE-CLASS
DISCUSSION
900⬚
Draw a heptagon below. Measure its angles with a protractor. Write each measure
in the angle. Find the sum.
Sum of the angles in a heptagon ⫽
900⬚
Survey the class for the patterns that students found in the Sums
of Polygon Angles table. Ask: Why do you think the medians for the
sums of polygon angles increase by 180°? Use the following points
to guide the discussion.
The sum of the angles of a triangle equals 180°.
Quadrangles divide into 2 triangles. The sum of the angles of
the quadrangle equals 2 ∗ 180, or 360°.
3. a.
b.
Is your measurement close to your prediction?
Answers vary.
Why might your prediction and your measurement be different?
Sample answer: Because the angle
measurement might not be exact
for each angle in the heptagon
Use dotted lines to divide the polygons on the board from the
earlier discussion so the two triangles can be seen. (See above.)
Ask: How many triangles do you think could be drawn in a
pentagon? 3 What would be the sum of angles? 3 ∗ 180° is 540°.
Math Journal 1, p. 88
202
Unit 3
Geometry Explorations and the American Tour
Student Page
In a hexagon? 4; 4 ∗ 180° is 720°. Summarize by stating that as
the number of sides in a polygon increases by 1, the sum of the
angle measures increases by 180°.
Date
Time
LESSON
39
1.
Angles in Any Polygon
Draw a line segment from vertex A of this octagon to each of the other vertices
except B and H.
B
A
H
2.
Ask partners to work together to solve Problems 1–3 on journal
page 88. They can use the pattern in the table or try dividing a
heptagon into triangles to make their prediction. Circulate
and assist.
Bring the class together to discuss results. Ask: Do your
predictions match your measurements? Why might they be
different? The angle measurement(s) may not be exact for each
angle in the heptagon.
▶ Finding Angle Sums for
PARTNER
ACTIVITY
Any Polygon
How many triangles did you divide the octagon into?
6
1,080⬚
3.
What is the sum of the angles in this octagon?
4.
Ignacio said the sum of his octagon’s angles is 1,440°. Below is the picture he drew
to show how he found his answer. Explain Ignacio’s mistake.
Ignacio should have drawn lines from one vertex to each
of the other vertices in his octagon, instead of drawing a
line between each vertex and a point in the interior of his
octagon.
5.
A 50-gon is a polygon with 50 sides. How could you find the sum of the angles
in a 50-gon? A
50-gon can be divided into 48 triangles. The
sum of the angles would be 48 180°.
Sum of the angles in a 50-gon =
8,640⬚
Math Journal 1, p. 89
(Math Journal 1, p. 89; Math Masters, p. 414)
Ask students to state the relationship between the number of
sides of a polygon and the number of triangles that the polygon
can be divided into. The number of triangles is 2 less than the
number of sides. Explain that some polygons are impractical to
draw because they have so many sides that it’s hard to draw them
accurately. In these instances, the number of triangles can be
determined by subtracting 2 from that polygon’s number of sides.
Ask partners to work together to complete journal page 89.
Ongoing Assessment:
Recognizing Student Achievement
Exit Slip
Student Page
Date
Time
LESSON
39
Use an Exit Slip (Math Masters, page 414) to assess students’ understanding
of angle measures and relationships in polygons. Have students write a
response to the following: Explain how to find the sum of the measures of the
angles in polygons without using a protractor. Students are making adequate
progress if they indicate that they are able to use the sum of the measures of the
angles in a triangle to calculate the angle sums for at least one other polygon.
Some students may generalize finding the sum of angles for all polygons.
Practicing Expanded Notation
396
Use the place-value chart on page 396 of the Student Reference Book to help you
write the following numbers in expanded notation.
1.
6,456 ⫽
2.
64.56 ⫽
3.
98,204 ⫽
4.
982.04 ⫽
5. a.
[Geometry Goal 1]
6,000 ⫹ 400 ⫹ 50 ⫹ 6
60 ⫹ 4 ⫹ 0.5 ⫹ 0.06
90,000 ⫹ 8,000 ⫹ 200 ⫹ 4
900 ⫹ 80 ⫹ 2 ⫹ 0.04
Build a 4 digit numeral. Write
3 in the hundredths place,
4 in the tens place,
6 in the ones place, and
9 in the tenths place.
4 6.9 3
b.
6.
Write this number in expanded notation.
40 ⫹ 6 ⫹ 0.9 ⫹ 0.03
Write the following expanded notation in standard form.
600 ⫹ 50 ⫹ 4 ⫹ 0.2 ⫹ 0.07 ⫹ 0.009
7. a.
654.279
Build an 8-digit number. Use these clues.
The digit in the place with the greatest value is equal to 4 ⫹ 0.
The digit in the place with the least value is equal to 32.
The number in the hundreds place is the first counting number.
The number in the tenths place multiplied by 54 is zero.
The number in the tens place is the square root of 9.
The number in the ones place is the square root of 4.
The number in the hundredths place is the product of the number in the tens place
and the number in the ones place.
The number in the thousands place is equal to 9 ⫺ 22.
45,132.069
b.
Write this number in expanded notation.
40,000 ⫹ 5,000 ⫹ 100 ⫹ 30 ⫹ 2 ⫹ 0.06 ⫹ 0.009
Math Journal 1, p. 90
Lesson 3 9
203
Student Page
Date
Time
LESSON
39
Write five names for 1,000,000.
1.
2.
Sample answers:
300,000 ⫹ 700,000
10 10 10 10 10 10
500,000 ⫹ 500,000
222222 5 55555
3,000,000 ⫺ 2,000,000
Use a straightedge to draw an angle
that is less than 90⬚.
Answers vary.
Write ⬍ or ⬎.
⬍
⬍
⬎
⬍
⬎
3.67
0.02
4.06
3.1
7.6
139
3.7
I have four sides. All opposite sides are
parallel. I have no right angles.
Draw me in the space below.
0.21
Sample answer:
4.
4.02
3.15
I am called a
7.56
parallelogram .
9 32
33
What is the measure of angle R?
5.
▶ Practicing Expanded Notation
143
6.
27°
20°
measure angle R ⫽
S
133 ⴗ
Students practice place-value concepts by reading and writing
large numbers and decimals in standard notation and in expanded
notation. Students can refer to the place-value chart in the Student
Reference Book, page 396. Remind students that decimals may also
be written as fractions. For example, in Problem 2, the expanded
5 +_
6 .
notation for 64.56 may be written as 60 + 4 + _
10
100
Solve.
2,400,000 ⫽ 3,000 800
900 60 ⫽ 54,000
36,000 ⫽ 40 900
100,000
20 5,000 ⫽
80
72,000 ⫽
900
Q
R
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 90; Student Reference Book, p. 396)
219
3.
2 Ongoing Learning & Practice
Math Boxes
207
▶ Math Boxes 3 9
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 91)
18
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 3-6. The skill in Problem 6
previews Unit 4 content.
Math Journal 1, p. 91
Writing/Reasoning Have students write a response to
the following: John called his drawing in Problem 4 a
parallelogram, and Jack called his drawing a rhombus.
Who was correct? Sample answer: A rhombus is a parallelogram
with 4 equal sides. Both are correct.
NOTE Student use of vocabulary (quadrangle, quadrilateral, parallel sides,
congruent sides, and/or opposite sides) and explanation of the concepts in their
solution will provide information about how students are integrating these
geometry concepts.
▶ Study Link 3 9
Study Link Master
Name
Date
STUDY LINK
39
1.
(Math Masters, p. 92)
Time
Sums of Angle Measures
Describe one way to find the sum of the angles
in a quadrangle without using a protractor. You
might want to use the quadrangle at the right to
illustrate your explanation.
207
Sample answer: Draw a line between two of
the vertices to create two triangles. Since
the sum of the angles in each triangle is
180ⴗ, the sum of the angles in a quadrangle
is 360ⴗ.
2. The sum of the angles in a quadrangle is 360⬚ .
tear
C
3.
Follow these steps to check your answer to Problem 2.
a.
With a straightedge, draw a large quadrangle
on a separate sheet of paper.
b.
Draw an arc in each angle.
c.
Cut out the quadrangle and tear off part of
each angle.
B
D
tear
tear
A
tear
d.
Tape or glue the angles onto the back of this
page so that the angles touch but do not overlap.
C
D
B
A
Practice
5,238
22,152
3,020
8 R3
4.
3,007 ⫹ 1,251 ⫹ 980 ⫽
5.
4,310 ⫺ 1,290 ⫽
6.
3,692 º 6 ⫽
7.
67 ⫼ 8 →
Math Masters, p. 92
204
Unit 3
INDEPENDENT
ACTIVITY
Geometry Explorations and the American Tour
Home Connection Students describe one way to find
the sum of the angles of a quadrangle without using a
protractor. They investigate finding the sum by tearing
off the angles and putting them together around a point.
Teaching Master
Name
3 Differentiation Options
ENRICHMENT
▶ Tessellating Quadrangles
Date
LESSON
Time
A Quadrangle Investigation
39
The sum of the angles in a quadrangle is equal to 360⬚. Since there are 360⬚
in a circle, you might predict that every quadrangle will tessellate. Follow the
procedure below to investigate this prediction.
1
1.
Fold a piece of paper (8ᎏᎏ'' by 11'') into six parts by first folding it into thirds
2
and then into halves.
2.
Using a straightedge, draw a quadrangle on the top layer of the folded paper.
Label each of the four vertices with a letter inside the figure—for example, A,
B, C, and D.
3.
Cut through all six layers so that you have six identical quadrangles. Label the
vertices of each quadrangle in the same manner as the quadrangle on top.
INDEPENDENT
ACTIVITY
15–30 Min
(Math Masters, p. 93)
To apply their understanding of interior angle measures
of polygons, have students investigate whether all
quadrangles will tessellate. Ask students whether they
think the following statement is true. Because the sum of the
angles in a quadrangle is equal to the number of degrees in a
circle, all quadrangles should tessellate. Answers vary. Explain
that in this activity, students will investigate whether the
statement is true.
Ask students to tape their results onto a separate piece of paper.
Challenge students to tessellate concave (nonconvex) quadrangles.
Concave quadrangles will tessellate, but it is more difficult to
place the pieces so all four angles meet. Make sure all angles are
correctly labeled to facilitate the process.
A
A
B
B
D
D
C
A
C
B
A
B
D
C
D
C
A
B
A
B
D
C
D
C
A
B
D
C
4.
Arrange the quadrangles so that they tessellate.
5.
When you have a tessellating pattern, tape the final pattern onto a separate
piece of paper. Color it if you want to.
6.
Talk with other students who did this investigation. Were their quadrangles
a different shape than yours? Do you think that any quadrangle will tessellate?
Option To make a pattern that has more than six quadrangles, draw your original
quadrangle on a piece of cardstock, cut it out, and use it as a stencil. By tracing
around your quadrangle, you can easily cover a half-sheet of paper with your
pattern. Label the angles on your stencil so you can be sure you are placing all
four angles around points in the tessellation. Color your finished pattern.
Math Masters, p. 93
Tessellations with concave (nonconvex) quadrangles
EXTRA PRACTICE
▶ Finding Angle Measures
INDEPENDENT
ACTIVITY
5–15 Min
in Polygons
Teaching Master
(Math Masters, p. 94)
Name
Date
LESSON
Algebraic Thinking Students complete a table that shows the
relationship between the number of the sides of a polygon and the
number of interior triangles. Then they summarize the pattern by
writing numerical expressions.
39
Angle Measures in Polygons
The measure of the interior angles of a triangle is 180⬚. The number of triangles
within a polygon is 2 less than the number of sides of the polygon.
1.
Fill in the chart below using this pattern.
Polygons
Number of
Sides
ELL SUPPORT
▶ Describing Tessellations
Time
Number of
Triangles
Sum of Angles
PARTNER
ACTIVITY
4
2
2
º
180⬚ ⫽
360⬚
5
3
3
º
180⬚ ⫽
540ⴗ
5–15 Min
6
4
4
º
180⬚ ⫽
720ⴗ
7
5
5
º
180⬚ ⫽
900ⴗ
13
11
11
180⬚ ⫽
1,980ⴗ
26
24
24
º
180ⴗ ⫽ 4,320ⴗ
51
49
49
º
180ⴗ ⫽ 8,820ⴗ
63
61
61
º
180ⴗ ⫽ 10,980ⴗ
85
83
83
º
180ⴗ ⫽ 14,940ⴗ
To provide language support for polygon angles, have students look
at the Tessellations Museum and describe some of the tessellations
in the museum to a partner. Encourage them to use mathematical
terminology (for example, polygons) and to describe components
such as colors, shapes, and patterns.
2.
º
Use expressions to complete the statement.
n ⫺ 2 equals the number
(n ⫺ 2) 180ⴗ equals the
If n equals the number of sides in a polygon,
of triangles within the polygon, and
sum of the angles in the polygon.
Math Masters, p. 94
Lesson 3 9
205