# Acquisition Lesson Planning Form Acquisition

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```					                             Acquisition Lesson Planning Form
Plan for the Concept, Topic, or Skill – Not for the Day
Key Standards addressed in this Lesson: MA1G3a
Time allotted for this Lesson: 2 Hours

Essential Question: LESSON 1 – INTERIOR AND EXTERIOR ANGLES OF POLYGONS

How do you find the sum of the interior and exterior angles of a polygon? How do you find
the measure of each interior and exterior angle of a regular polygon?

Activating Strategies: (Learners Mentally Active)

Card Match Activity: A set of cards (Pages 4-7) will be made for each of the following
polygons: quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon, and
dodecagon. Each set will include 4 cards: the name of the polygon, a drawing of the polygon
(some regular and some not), the number of sides for the polygon, and the number of interior
angles for the polygon. As the students enter the class, they are to pick one of the cards
without looking. The students should find the students who have the cards that correspond to
their polygon, and form a group that way.

Acceleration/Previewing: (Key Vocabulary)

   Convex Polygon, Concave Polygon, Interior Angle Sum Theorem

   Use the Frayer Models (Pages 8 & 9) for Polygon, Regular Polygon, Convex Polygon
and Concave Polygon

Teaching Strategies: (Collaborative Pairs; Distributed Guided Practice; Distributed
Summarizing; Graphic Organizers)

   Vocabulary Teaching Strategies: Word Wall: Have a place in your room for the word
wall. As you reach each term, have it written on a card and let the student place it on
the wall. This works well with a KWL chart. Then after discussing it, let the students
define it in their own words. It’s important to continually reference the vocabulary in the
unit. Brief daily reminders of the terms should be implemented. The first couple of
minutes of each class could be used to let the students review the terms on the wall. If
you are using a KWL chart, let the students decide where the term belongs and move it
as they see fit.

   Protractor Activity: Work in the small groups from the activating strategy. Using a
ruler, students will draw an example of their polygon. It should be convex. Encourage
students to make it large so it is easy to measure the angles. Each student should use
a protractor to measure each interior angle of the polygon. Find the sum. Then draw
one exterior angle at each vertex. Use the protractor to measure the exterior angles
and find the sum. First compare results with the group. Then share results with the
entire class, generalizing where possible.

   In collaborative pairs, students investigate interior angles using the Interior Angle
Investigation (Page 10). Encourage them to generalize as much as possible. Use
Think-Pair-Share. Summarize with the entire group.
Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 1
   Working in small groups, students should complete the Graphic Organizer #1
(Page 11 & 12) Interior Angle Sum. Use Think-Pair-Share within each group following
the questions after the organizer. Have each group discuss a polygon and its angles
with the class.

   Students do the practice problems (Page 13) individually as an assessment of their
knowledge at this point in time.

   Students follow this same process with the exterior angle sum investigation (Page 14)
and with the graphic organizer #2 (Page 15) on interior and exterior angles. Do the
guided practice problems (Page 16) in pairs.

   Task (Page 20): Robotic Gallery Guards Learning Task: Use the same groups
formed by the activating strategy. Work the task with the teacher guiding the learning.
Solution should be presented to the entire class on chart paper along with a detailed
explanation from the group.

Distributed Guided Practice/Summarizing Prompts: (Prompts Designed to Initiate
Periodic Practice or Summarizing)

   What patterns do you notice in the table from the graphic organizer?

   What does each part of the formula for the sum of the interior angles represent?

   What is the interior angle sum for a 60-gon?

   Four angles of a pentagon are 98º, 75º, 108º, and 56º. How would you find the
measure of the fifth angle?

   What is another name for a regular triangle……. a regular quadrilateral?

   Is it possible for a polygon to be equiangular but not equilateral? Why or why not?

   Is the sum of the interior angles of a regular polygon different from a polygon with the
same number of sides that is not regular? What about the sum of the exterior angles?

   If you know the sum of the interior angles of a regular polygon, how do you find the
measure of one interior angle?

Extending/Refining Strategies:

Pairs: Think-Pair-Share: Pose the following problems and let the students see if they can
devise a method to solve them.

1. If the measure of one interior angle of a regular polygon is 165º, how many sides does
the polygon have?

2. In a second regular polygon, the measure of one exterior angle is 60º. How many
sides does the polygon have?
Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 2
Summarizing Strategies: Learners Summarize & Answer Essential Question

Summarizers (Pages 17 & 18): Use at the end of the lesson.

Ticket Out the Door # 1 and # 2 (Page 19) : May be used during the lesson to determine
progress.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 3
Cards for Activator

4 Interior Angles

Pentagon                                         5 Sides

5 Interior Angles

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 4
Hexagon                                          6 Sides

6 Interior Angles

Heptagon                                         7 Sides

7 Interior Angles

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 5
Octagon                                          8 Sides

8 Interior Angles

Nonagon                                          9 Sides

9 Interior Angles

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 6
Decagon                                         10 Sides

10 Interior Angles

Dodecagon                                        12 Sides

12 Interior Angles

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 7
Frayer Models

Definition                                                     Characteristics

Polygon

Examples                                                        NonExamples

Definition                                                     Characteristics

Regular
Polygon

Examples                                                         NonExamples

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 8
Frayer Models

Definition                                                     Characteristics

Convex
Polygon

Examples                                                        NonExamples

Definition                                                     Characteristics

Concave
Polygon

Examples                                                        NonExamples

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 9
Interior Angle Investigation
Consider the quadrilateral to the right.                                                     F
Draw diagonal EG (the dotted line).                                                      5
A diagonal is a segment connecting a vertex                E
1
The quadrilateral is now divided into two triangles.
What are they?

Angles 1, 2, and 3 represent the interior angles       D 2
of triangle DEG and angles 4, 5, and 6 represent
the interior angles of triangle FEG.

m<1 + m<2 + m<3 = _______.
3
4
m<4 + m<5 + m<6 = _______.
G
m<1 + m<2 + m<3 + m<4 + m<5 + m<6 = _______.

What is the relationship between the sum of the angles in the quadrilateral and the sum of
the angles in the two triangles?

This procedure can be used to determine the sum of the interior angles in any polygon.
1. Draw the polygon.
2. Select one vertex.
3. Draw all possible diagonals from that vertex.
4. Determine the number of triangles formed.
5. Multiply the number of triangles by the sum of the interior angles of any triangle.

Try it with the polygons below.

Sum of the interior angles of a                    Sum of the interior angles of a

hexagon is _________.                               pentagon is ________.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 10
Graphic Organizer #1 Interior Angle Sum
Number
Number of           Number of              Interior
of
Polygon              Sketch            Sides of           Degrees in              Angle
Triangles
Polygon            a Triangle              Sum
Formed

Triangle

Pentagon

Hexagon

Heptagon

Octagon

Decagon

Dodecagon

n-gon

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 11
Consider the table on the previous page and answer the following questions:
1. What patterns do you notice in the table? When you know the number of triangles formed,
how do you find the sum of the interior angles of the triangle?

2. In the last row of the table you should have developed a formula for finding the sum of the
interior angles of a polygon. Use this formula to find the sum of the interior angles of a 20-gon.

3. Write a sentence explaining how to find the sum of the interior angles of a polygon.

4. The measures of the angles in a convex quadrilateral are 2x, 2(2+1), x-5, and 3(x-2).
a. Sketch and label the figure.

b. What is the sum of the interior angles of a convex quadrilateral?

c. Find x

d. Find each angle measure.

5. The measures of the angles in a convex pentagon are 120º, 2y-5, 3y-25, 2y, and 2y.
a. Sketch and label the figure.

b. What is the sum of the interior angles of a convex quadrilateral?

c. Find y

d. Find each angle measure.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 12
Problems for Practice:

1. Classify the following as either a polygon or not a polygon. If it is a polygon, further
classify is as convex or concave.

2. Find the sum of the interior angles of the following convex polygons:

24-gon:

13-gon:

3. The four interior angles of a quadrilateral measure x-5, 3(x+8), 3x+6, and 5x-1. Find the
measures of the four angles.

4. Find each angle measure in the figure below.

12x

13x +10

5x+40                               10x+5
0
6x         8x +80

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 13
Exterior Angle Exploration

An exterior angle of a polygon is formed by extending a
side of the polygon outside the figure.

4

6    5                                        3

7
1    2
8

1. How many exterior angles does the quadrilateral
have? Name them.

2. Using your protractor, find the measure of <1.

3. What kind of angles are <1 and <2? What is their
sum?

4. Find the measure of <2.

5. Find the measure of each of the remaining exterior
angles in the same way.

6. What is the sum of the exterior angles?

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 14
Angles of Regular Polygons

Measure                           Measure
Interior                 Exterior
Regular                           of Each                           of Each
Angle                    Angle
Polygon                           Interior                          Exterior
Sum                      Sum
Angle                             Angle
Triangle
Pentagon
Hexagon
Heptagon
Octagon
Decagon
Dodecagon
n-gon

Find the measure of one interior angle of a regular
18-gon.

Find the measure of one exterior angle of a regular
14-gon.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 15
Guided Practice Problems

1. An exterior angle of a regular polygon measures 72o. What is the measure
of its corresponding interior angle?

2. What is the measure of one interior angle of a regular 40-gon?

3. What is the measure of one exterior angle of a regular 40-gon?

4. If the exterior angles of a quadrilateral measure 93 o, 78 o, 104 o, and x o,
find the value of x.

5. If an exterior angle of a regular polygon measures 45 o, how many sides
does the polygon have?

6. If an interior angle of a regular polygon measures 160º, how many sides
does the polygon have?

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 16
Summarizer
Sum of the Interior Angles of             Measure of Each Interior Angle
any polygon:                              of any regular polygon:

Regular
Polygon

Sum of the Exterior Angles of             Measure of Each Exterior Angle
any polygon:                              of any regular polygon

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 17
Polygons Graphic Organizer/Summarizer

What is a polygon?

Examples                                Non-examples

What is a regular polygon?

What is a concave polygon?              What is a convex polygon?

What is the sum of the interior angles of a polygon?
Interior Angle Sum Theorem

What is the measure of an interior angle of a regular polygon?

What is the sum of the exterior angles of a convex polygon?
Exterior Angle Sum Theorem

What is the measure of an exterior angle of a regular polygon?

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 18
Ticket Out the Door                              Name: ________________

The measures of the angles in a convex quadrilateral are 2x,
2(x + 10), x – 5, and 3(x – 2). Sketch and label the figure. What is
the sum of the interior angles of a quadrilateral? Find x. How long
is each side?

Ticket Out the Door                              Name: ________________

The measures of the angles in a convex pentagon are 120º, 2y – 5,
3y – 25, 2y, and 2y. Sketch and label the figure. What is the sum
of the interior angles of a pentagon? Find y. How long is each
side?

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 19

The Asimov Museum has contracted with a company that provides Robotic Security Squads to
guard the exhibits during the hours the museum is closed. The robots are designed to patrol
the hallways around the exhibits. The robots are equipped with cameras and sensors that
detect motion.

Each robot is assigned to patrol the area around a specific exhibit. They are designed to
maintain a consistent distance from the wall of the exhibits. Since the shape of the exhibits
change over time, the museum staff must program the robots to turn the corners of the exhibit.

Below, you will find a map of the museum’s current exhibits. One robot is assigned to patrol
each exhibit. There is one robot, Captain Robot, CR, who will patrol the entire area.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 20
1. When a robot reaches a corner, it will stop, turn through a programmed angle, and then
continue its patrol. Your job is to determine the sum of the angles that R1, R2, R3, and R4
will need to turn as they patrol their area. Keep in mind the direction in which the robot is
traveling and make sure it always faces forward as it moves around the exhibits.

Exhibit A

R1

a) What angles will R1 need to turn? What is the total of these turns?

Exhibit A

R1

b) What angles will R2 need to turn? What is the total of these turns?

c) What angles will R3 need to turn? What is the total of these turns?

d) What angles will R4 need to turn? What is the total of these turns?

1. What angles will CR need to turn? What is the total of these turns?

2. What do you notice about the sum of the angles? Do you think this will always be true?

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 21
3. Determine the sum of the measures of the interior angles of the paths the robots travel.
Use this information to help you write a function that gives the sum of the interior angles of

4. Looking at your results from #2 and #3, can you find a way to prove your conjecture about
the sum of the exterior angles?

5. The museum intends to create regular polygons for its next exhibition; how can the
directions for the robots be determined for a regular pentagon? Hexagon? Nonagon? N-
gon?

6. A sixth exhibit was added to the museum. The robot patrolling this exhibit only makes 15º
turns. What shape is the exhibit? What makes it possible for the robot to make the same
turn each time?

7. Robot 7 makes a total of 360 º during his circuit. What type of polygon does this exhibit
create?

8. The sum of the interior angles of Robot 7’s exhibit is 3,420 º. What type of polygon does
the exhibit create?

9. What is different about the path CR travels versus the paths of R1, R2, R3 and R4? Do
your earlier generalizations hold for CR’s path?

10. A counter-clockwise turn is considered positive while a clockwise turn is negative.
Investigate CR’s path using this information. How does this affect the application of your
earlier generalizations on the polygon traced by CR?

11. The museum is now using Exhibit Z. Complete a set of instructions for RZ that will allow
the robot to make the best circuit of this circular exhibit. Defend why you think your
instructions are the best.

Accelerated Math 1 Unit 2 Lesson 1 Interior & Exterior Angles Page 22

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