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Lesson 3�9 199

Advance PreparationThis 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 3�8 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

Key Concepts and Skills• Investigate and compare the measurement

sums of interior angles of polygons.

[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]

Key ActivitiesStudents 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.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). [Geometry Goal 1]

MaterialsMath Journal 1, pp. 85–89

Study Link 3�8

Math Masters, p. 414

transparency of Math Masters, p. 420

(optional) � Class Data Pad � Geometry

Template (or protractor and straightedge)

Practicing Expanded Notation 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.

Math Boxes 3�9Math Journal 1, p. 91

Students practice and maintain skills

through Math Box problems.

Study Link 3�9Math Masters, p. 92

Students practice and maintain skills

through Study Link activities.

ENRICHMENTTessellating QuadranglesMath Masters, p. 93

paper (8 1

_ 2 " by 11") � scissors � tape �

cardstock (optional)

Students investigate whether all quadrangles

will tessellate.

EXTRA PRACTICE

Finding Angle Measures in PolygonsMath Masters, p. 94

Students find the sums of the interior angles

of polygons.

ELL SUPPORT

Describing TessellationsStudents describe the tessellations in the

Tessellation Museum.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Angles of PolygonsObjective To develop an approach for finding the angle

measurement sum for any polygon.

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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200 Unit 3 Geometry Explorations and the American Tour

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

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?

175°orless

176° 177° 178° 179° 180° 181° 182° 183° 184° 185°or

more

x x xxxx

x

xx

xx

xx

xx

xx

xxx x

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.

Next have students arrange their three angles next to each other so they line up. (See margin.) Ask students what type of angle

Math MessageUse 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.

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.

Mental Math and Reflexes Use your slate procedures for problems such as the following:

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

● On 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 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.

Students’ angle measures might seem to total

slightly more or less than 180° because their

original triangles might not be accurate.

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Angles in Quadrangles and PentagonsLESSON

3�9

Date Time

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.

3. Measure the angles in your polygon. Write each measure for each angle.

4. Find the sum of the angles in your polygon.

Drawings vary.

Answers vary for individual angle measurements.

The total sum of the angles should be

close to 360° for quadrangles and 540°

for pentagons.

Math Journal 1, p. 85

Student Page

Angles in Quadrangles and Pentagons cont.LESSON

3�9

Date Time

5. Record your group’s data below.

6. Find the median of the angle sums for your group.

7. If you have time, draw a hexagon. Measure its angles with a protractor.

Find the sum.

Sum of the angles in a hexagon = 720�

Group Member’s Sketch of Sum ofName Polygon Angles

The total sum of the angles should be close to360° for quadrangles and 540° for pentagons.

Drawings vary.

Drawings of polygons vary. Thesum of the anglesvaries but should beclose to 360° forquadrangles and540° for pentagons.


Student Page

Lesson 3�9 201

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.

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

SMALL-GROUP ACTIVITY

in Polygons(Math Journal 1, pp. 85 and 86)

Draw an example of a convex polygon and a concave polygon on the board.

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.

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Angles in Quadrangles and Pentagons cont.LESSON

3�9

Date Time

8. Record the class data below.

9. Find the class median for each polygon. For the triangle, use the median

from the Math Message.

10. What pattern do you see in the Sums of Polygon Angles table?

Sum of the Anglesin a Quadrangle

Group Group Median

Sum of the Anglesin a Pentagon

Group Group Median

The group median should

be close to 180° for a

triangle, 360° for a

quadrangle, 540° for a

pentagon, and 720° for a

hexagon.

The group median should be close to 360° for quadrangles and 540° for pentagons.

Sample answer: As the number of sides increases by 1,

the sum of the angle measures increases by 180°.

Sums of Polygon AnglesPolygon Class Median

triangle

quadrangle

pentagon

hexagon


Student Page

Angles in HeptagonsLESSON

3�9

Date Time

1. A heptagon is a polygon with 7 sides.

Predict the sum of the angles in a heptagon.

2. 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 �

3. a. Is your measurement close to your prediction?

b. Why might your prediction and your measurement be different?

900�

900�

Answers vary.

Sample answer: Because the angle

for each angle in the heptagonmeasurement might not be exact


Student Page


▶ Finding the Median for WHOLE-CLASSDISCUSSION

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 WHOLE-CLASSDISCUSSION

into Triangles(Math Journal 1, pp. 87 and 88)

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°.

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°.


Angles in Any PolygonLESSON

3�9

Date Time

1. Draw a line segment from vertex A of this octagon to each of the other vertices

except B and H.

2. How many triangles did you divide the octagon into?

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.

5. A 50-gon is a polygon with 50 sides. How could you find the sum of the angles

in a 50-gon?

Sum of the angles in a 50-gon = 8,640�

1,080�

6

A

B

H

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.

A 50-gon can be divided into 48 triangles. The

sum of the angles would be 48 � 180°.


Student Page

Practicing Expanded NotationLESSON

3�9

Date Time

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. 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.

.

b. Write this number in expanded notation.

6. Write the following expanded notation in standard form.

600 � 50 � 4 � 0.2 � 0.07 � 0.009

7. a. 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.

, .

b. Write this number in expanded notation.

40,000 � 5,000 � 100 � 30 � 2 � 0.06 � 0.009

96023154

654.279

40 � 6 � 0.9 � 0.03

3964

900 � 80 � 2 � 0.04

90,000 � 8,000 � 200 � 4

60 � 4 � 0.5 � 0.06

6,000 � 400 � 50 � 6

396


Student Page

Lesson 3�9 203

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°.

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(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: Exit Slip �

Recognizing Student Achievement

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.

[Geometry Goal 1]


STUDY LINK

3�9 Sums of Angle Measures

207

Name Date Time

tear

tear

tear

tear

D

C

B

A

DB

C

A

1. 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.

Sample answer: Draw a line between two of the vertices to create two triangles. Since the sum of the angles in each triangle is 180�, the sum of the angles in a quadrangle is 360�.

2. The sum of the angles in a quadrangle is .

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.

d. Tape or glue the angles onto the back of this

page so that the angles touch but do not overlap.

4. 3,007 � 1,251 � 980 � 5. 4,310 � 1,290 �

6. 3,692 º 6 � 7. 67 � 8 → 8 R322,152

3,0205,238

360�

Practice

Math Masters, p. 92

Study Link Master

Math Boxes LESSON

3�9

Date Time

4. I have four sides. All opposite sides are

parallel. I have no right angles.

Draw me in the space below.

I am called a .parallelogram

Sample answer:

6. Solve.

� 3,000 � 800

� 60 � 54,000

� 40 � 900

20 � 5,000 �

72,000 � � 90080

100,000

36,000

900

2,400,000

1. Write five names for 1,000,000. 2. Use a straightedge to draw an angle

that is less than 90�.

Answers vary.

3. Write � or �.

3.67 3.7

0.02 0.21

4.06 4.02

3.1 3.15

7.6 7.56��

�

�

�

5. What is the measure of angle R?

measure angle R ��133

139219

9 3233 143

Q

RS

27°

20°

207 18

Sample answers:

300,000 � 700,000

10 � 10 � 10 � 10 � 10 � 10

500,000 � 500,000

5 � 5 � 5 � 5 � 5

2 � 2 � 2 � 2 � 2 � 2 � 5 �

3,000,000 � 2,000,000


Student Page


2 Ongoing Learning & Practice

▶ Practicing Expanded Notation

INDEPENDENT ACTIVITY

(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 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 notation for 64.56 may be written as 60 + 4 + 5 _ 10 + 6 _ 100 .

▶ Math Boxes 3�9


(Math Journal 1, p. 91)

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.

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


(Math Masters, p. 92)

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.

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LESSON

3�9

Name Date Time

A Quadrangle Investigation

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. Fold a piece of paper (8�1

2�'' by 11'') into six parts by first folding it into thirds

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.

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.

A

B CD

A

B CD A

B CD

A

B CD A

B CD

A

B CD

A

B CD

Math Masters, p. 93

Teaching Master

LESSON

3�9

Name Date Time

Angle Measures in Polygons

1. Fill in the chart below using this pattern.

Polygons

Number of Number ofSum of Angles

Sides Triangles

4 2 2 º 180� � 360�

5 3 3 º 180� �

6 4 4 º 180� �

7 5 º 180� �

13 º 180� �

26 º �

51 º �

63 º �

85 º � 14,940�180�8383

10,980�180�6161

8,820�180�4949

4,320�180�2424

1,980�1111

900�5

720�

540�

2. Use expressions to complete the statement.

If n equals the number of sides in a polygon, equals the number

of triangles within the polygon, and equals the

sum of the angles in the polygon.

(n � 2) � 180�

n � 2

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.

Math Masters, p. 94

Teaching Master

Lesson 3�9 205

3 Differentiation Options

ENRICHMENT


▶ Tessellating Quadrangles 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.

Tessellations with concave (nonconvex) quadrangles

EXTRA PRACTICE


▶ Finding Angle Measures 5–15 Min

in Polygons(Math Masters, p. 94)

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.

ELL SUPPORT PARTNER ACTIVITY

▶ Describing Tessellations 5–15 Min

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.

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angles of polygons - everyday math - login of polygons objective to develop an approach for finding...

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