Download Triangles: Finding Interior Angle Measures

Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 38498
Triangles: Finding Interior Angle Measures
In this lesson plan, students will start with a hands-on activity and then experiment with a GeoGebra-based computer model to investigate and
discover the Triangle Angle Sum Theorem. Then they will use the Triangle Angle Sum Theorem to write and solve equations and find missing angle
measures in a variety of examples.
Subject(s): Mathematics
Grade Level(s): 8, 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Computers for Students,
Internet Connection, Basic Calculators, LCD Projector,
Overhead Projector, Adobe Acrobat Reader, Microsoft
Office, Java Plugin, GeoGebra Free Software (Download
the Free GeoGebra Software)
Instructional Time: 1 Hour(s)
Freely Available: Yes
Keywords: triangle, interior angles, sum of interior angles, Triangle Angle-Sum Theorem, GeoGebra, exterior angle
of a triangle, adjacent angles, straight angle, supplementary angles, alternate interior angles of parallel lines.
Instructional Design Framework(s): Demonstration, Confirmation Inquiry (Level 1)
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Finding Interior Angles of Triangles GP Solutions.docx
Investigating the Triangle Angle Sum Theorem.ggb
Finding Interior Angles of Triangles GP.docx
Finding Interior Angles of Triangles Solutions.docx
Finding Interior Angles of Triangles.docx
Guiding Question 3.docx
Investigating the Triangle Angle Sum Theorem.docx
Investigating the Triangle Angle Sum Theorem.pdf
Investigating_the_Triangle_Angle_Sum_Theorem.html
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will discover and apply the Triangle-Angle Sum Theorem to find missing angle measures.
Students will use computer-generated models (using the dynamic geometry software, GeoGebra) along with paper models to investigate the Triangle-Angle Sum
Theorem.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students must be proficient with the following geometric vocabulary and concepts:
page 1 of 3 Triangle, angle, straight angle, straight line, interior angle, exterior angle, vertical angles (Vertical Angle Theorem), supplementary angles, alternate interior angles of
parallel lines.
Students should have some prior experience with math journal activities i.e. summarizing their learning experience in writing.
Guiding Questions: What are the guiding questions for this lesson?
1. How is the Triangle Angle Sum Theorem based on the measure of a straight angle? (The sum of the interior angles of any triangle equals 180. Similarly, the
measure of a straight angle (line) equals 180. When the 3 interior angles of a triangle are added together, they form a straight angle (or line)
2. How can we find missing angle measures in triangles and how can this knowledge be used to solve real world problems? (Using the Triangle Angle Sum Theorem,
write an equation showing the sum of the three angles equals 180, with the unknown angle being represented by the variable. The teacher can confirm that without
using algebra, the two known interior angles are added together, then that sum is subtracted from 180 degrees. Real world problems resemble geometric models.)
3. Where do we see examples of the Triangle Angle Sum Theorem outside of the classroom? (Real world representations may include questions involving geography,
architecture, and construction. See "Guiding Question #3" word document for examples and applicable websites.)
Teaching Phase: How will the teacher present the concept or skill to students?
First, students will discover the Triangle Interior Angle Theorem using paper models. Using paper, pencil, straight edges, and scissors, students will draw and cut out
triangles of various sizes and shapes. Following the teacher's lead on overhead or document camera and LCD projector, have students label the triangle's vertices, A, B,
and C. Fold vertex B so that it intersects with segment AC (creating a parallel line to AC). Next, have students fold in Vertex A and Vertex C to coincide at Vertex B. Ask
students for their observations
Key observation: The triangle's 3 interior angles add up to be 180 degrees (or form a straight angle or straight line). Common misconception: A rectangle and/or square
has been created.
Then, using an Internet browser (Safari is preferred) and computers, students will continue to investigate and discover angle measures by exploring the Investigating
the Triangle Angle Sum Theorem dynamic worksheet via computer and internet connection.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
While teacher circulates among the students and monitors student work, students will explore the Investigating the Triangle Angle Sum Theorem dynamic worksheet via
computer and internet connection. During the computer-based activity, students will record their findings on the PDF worksheet of the same name.
Upon completion of the computer activity, students, working independently or in small groups, will complete the "Finding Interior Angles of Triangles Guided Practice"
worksheet. (Solutions are provided). Students will share answers in teacher directed, whole class discussion to ensure mastery.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
For homework or if time permits in class, students will complete practice problems involving numeric and algebraic values. See worksheet: Finding Interior Angles of
Triangles.docx.
Solutions are also provided: Finding Interior Angles of Triangles Solutions.docx
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students will record their findings of the paper model and GeoGebra investigation in their math journals. To reinforce writing skills, students will write a paragraph
consisting of 5-7 sentences that includes the following information:
An introductory sentence
A sentence stating the Triangle Angle Sum Theorem (The sum of the measures of the interior angles of any triangle is 180.)
One or two sentences that summarize student's individual findings of the "Investigating the Triangle Angle Sum Theorem" worksheet
One or two sentences that explain how the paper-folding model exhibits the Triangle Angle Sum Theorem
and a conclusion (A sentence that should restate the main idea or introductory sentence.).
Summative Assessment
As a culminating activity, teacher will pose the following questions after the lesson to ensure that students have mastered the learning objectives:
1. If a triangle has angle measures equal to 30 and 50 degrees, find the measure of the third angle. (100 degrees)
2. If an isosceles triangle has base angles that each equal 50 degrees, find the measure of the third angle, the vertex angle. (80 degrees)
3. If a triangle has angle measures equal to x, 3x, and 5x, find the value of x and the measures of the three angles. (x=20; Angle measures 20, 60, 100 degrees
respectively).
Teacher will use an overhead projector or document camera/LCD projector to display the 3 questions above. Teacher will remind students to set up each example by
writing an equation based upon the Angle Sum Theorem. They should solve each example algebraically, without using a calculator. Students will answer questions
individually and then verbally share how they solved each example when called on. Teacher will remediate if necessary, or if all students have mastered these tasks, will
conclude the lesson. Individual practice during class or for homework will be assigned. Students who do not master the objectives should receive remedial instruction
and further practice.
Formative Assessment
Teacher will pose the following questions prior to the lesson to ensure that students have mastered prior concepts:
1. How is a straight angle formed and what is its measure? (A straight angle is formed by two opposite rays that share a common endpoint; a straight angle measures
180 degrees.
2. Compare the interior angles and exterior angles of triangles. Draw a picture illustrating interior vs. exterior angles. (Check student work. Interior angles are inside
the triangle; exterior angles are formed by the extension of the sides of a triangle and are outside the triangle's boundaries. Prompt students to recall that adjacent
interior and exterior angles of each vertex are supplementary. )
3. Draw and label several triangles including acute, obtuse, right, isosceles, and equilateral triangles. Be sure to label sides and vertices. (Check student work.)
Feedback to Students
Students will receive feedback while sharing their independent work with the entire class through a teacher-led, whole-class discussion. Teacher can remediate or move
forward depending on mastery of skills by students.
The .pdf worksheet, "Investigating the Triangle Angle Sum Theorem" will be completed in class using laptops and internet browser; This worksheet provides
opportunities for the teacher to observe students' written work and assess their understanding.
page 2 of 3 In addition, students will complete a series of two worksheets, "Finding Interior Angles of Triangles". The first provides guided practice and the latter will provide
independent practice in class or for homework. Solutions are provided for both worksheets.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations: If special needs students do not meet the lesson objectives, the teacher will work directly with them in a small group or individually. Care should
be taken to use angle measures that are compatible (or easy to work with; ie. whole numbers, multiples of 10, no fractional values).
Use of a calculator would be recommended in the initial stage of the lesson so that the focus would shift from the calculations to mastery of the concept.
Students who have difficulty with manual dexterity and struggle with manipulating and folding paper could, instead of folding, tear off the vertices of the triangle and
then position the 3 angles adjacent to one another to form a line.
Extensions: Advanced students can investigate triangle interior angles further by calculating the interior angles of well-known structures (i.e. the Transamerica
Pyramid Building in San Francisco, CA, or the pyramids in Egypt) via a web-quest. See Guiding Question #3 word document for suggestions.
Suggested Technology: Document Camera, Computer for Presenter, Computers for Students, Internet Connection, Basic Calculators, LCD Projector, Overhead
Projector, Adobe Acrobat Reader, Microsoft Office, Java Plugin, GeoGebra Free Software
Special Materials Needed:
Internet connection and student computers
Paper, pencils, straight edge, and scissors
Protractors (if students will measure angles)
Copies of worksheet, "Investigating the Triangle Angle Sum Theorem"
Copies of worksheet, "Find Interior Angles of Triangles Guided Practice"
Copies of worksheet, "Find Interior Angles of Triangles"
This lesson uses the GeoGebraTube resource "Investigating the Triangle Angle Sum Theorem," accessible at http://www.geogebratube.org/student/m28392
Additional Information/Instructions
By Author/Submitter
This resource supports the following the Mathematical Practice Standards:
MAFS.K12.MP.3.1 Construct viable arguments and critique the reasoning of others
MAFS.K12.MP.4.1 Model with mathematics
MAFS.K12.MP.5.1 Use appropriate tools strategically
SOURCE AND ACCESS INFORMATION
Contributed by: Joan OBrien
Name of Author/Source: Joan OBrien
District/Organization of Contributor(s): Broward
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.G.1.5:
Description
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles
created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example,
arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
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