Goal-directed Instructional Design Plan – Geometry – Angle Relationships Within Triangles Author – Ethan Fahy 1.

A problem or a need – The continual problem that I face in teaching geometry is that the course depends too much upon lecture as the only means of educating the students. Although lecture is usually the format that teachers find most comfort and control in using, its effectiveness is limited in the classroom where students learn best from a variety of different methods. I believe that as a classroom teacher it is my job to find different ways to present the material to help the students develop a deeper understanding and begin to make connections to previous concepts. I also believe that geometry, more so than other math courses, lends itself to a variety of different platforms, tools, and activities that could allow students to experience the material in different formats in order to improve their comprehension. The lecture format also tends to create a disconnect between the concepts and applications outside of the classroom because the students begin to believe that the teacher holds all the knowledge and they are only able to make connections that teacher presents during lecture.

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A real-world performance – At the beginning of class, I will give the overview of what the lesson objectives are and how they directly apply to both the current and future chapters. The theorems build throughout the chapter as the students continue to learn more about properties of specific triangles. The theorems also connect to upcoming chapters because most properties of regular polygons are derived from the properties of triangles. I will also explain what professions use concepts of triangles and angle measures. After the students have completed the lesson they will have the skills to solve real applications of problems involving missing angles in triangles. They will also be able to apply these theorems to show triangle congruency. These problems are assigned in their homework set for the evening and will be reviewed the next day. By the conclusion of the lesson, the students will have developed and demonstrated their knowledge through Geometer’s Sketchpad. They will have applied their skills through practice problems and real world problems from the homework set. Finally they will be able to assess their success based on their homework on whether or not they are able to apply these theorems accurately through iteration problems and real world applications.

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An instructional objective a. Discover three specific properties of triangles (Triangle Sum Theorem, Third Angles Theorem, and the Exterior Angles Theorem) through a series of prompts and exploration using Geometer’s Sketchpad. b. Apply the theorems to actual diagrams and solve for missing information. c. Build on prior knowledge of concepts dealing with triangles such as acute, right, and obtuse by using their principles along with the new theorems to solve equations. d. Use their knowledge of the three theorems and apply it to real life applications solving for missing angles and completing proofs. Curriculum Standard – Michigan Merit Curriculum

e. G1.2.1 Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles. f. G1.2.2 Construct and justify arguments and solve multistep problems involving angle measure, side length, perimeter, and area of all types of triangles. NETS – S Standards g. Creativity and Innovation - apply existing knowledge to generate new ideas, products, or processes. h. Communicate and Collaborate - communicate information and ideas effectively to multiple audiences using a variety of media and formats. i.

Technology Operations and Concepts - understand and use technology systems

j. Technology Operations and Concepts - transfer current knowledge to learning of new technologies. 4.

A set of essential content –The students will each be assigned to a computer with Geometer’s Sketchpad on it and each student will be given a guided note sheet to complete sequentially. They will be paired with a classmate on a separate computer next to them; however, they will work independently to discover three specific theorems and properties that exist within triangles. After both students have explored a specific theorem, they will discuss their findings with their partner to make sure they get the same results. The students will first explore the triangle sum theorem by constructing a triangle and measuring its angles. Then they will manipulate the triangle by moving the vertices while looking for a trend in the sum of all three angles as it moves. At the conclusion they will make a conjecture as to what they believe the theorem is stating. They will do a similar activity with the exterior angles theorem. Due to time constraints they will be given the third angles theorem. After they have discovered the theorems and discussed them with their partner then they are to attempt the examples on the note sheet that correspond to each theorem to make sure they can apply their knowledge.

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An evaluation consisting of a test or observation – The first form of assessment will take place as they are completing the worksheet. I will be able to observe, ask questions, and answer student questions as I walk around the room and watch student progress. By placing the emphasis of the learning on the students rather than the teacher, it allows me to more easily observe the class. My first more formal way to assess student progress will come from their homework assignment when it is collected the following day. The evening will give students an opportunity to further investigate the topics if they are confused. My last forms of assessment will come from the quiz after the next section and the test at the end of the chapter. Both of these assessments will provide substantial evidence as to whether the Geometer’s Sketchpad activity was meaningful for student learning. (Activity Handout)

6. A method to help participants learn – To begin my lesson I would start by providing the students with the lesson objectives. They are responsible for learning three specific properties that pertain to triangles and to be able to solve for missing parts of triangles using these principles. I would then explain the connection that this has to life outside of the classroom. These theorems are often applied to surveyors, engineers, and

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architects. By knowing these properties, it allows the person who is taking measurements to find all the angles, both interior and exterior, by only measuring angles that they have easy access to. A simple example would be if you are looking for angles of elevation or depression for structures that are large such as mountains, buildings, and pyramids. After I have explained the objectives and applications of these lessons, I will allow them to begin their exploration using Geometer’s Sketchpad with their partner and guided note sheet. By completing their guided note sheet they will be learning the theorems and completing practice problems along the way to test their understanding. In the last five minutes of class I will review the note sheet and provide them with answers so they can assess their comprehension. They will then complete their homework and obtain more thorough feed back the following day when we review it. 7. Motivation: The importance of the lesson will be stressed at the beginning of the lesson when the instructor explains it applications. The applications will also be given so that students will know what occupations actually use the theorems often and they can see practical applications. This understanding for the students will help them develop a positive mentality in their approach because they will be able to learn and work with problems that apply to problems around them. By understanding these principles thoroughly students will be able to identify the many properties that carry over from previous chapters and serve as building blocks for upcoming sections. The larger regular polygons such as squares, pentagons, and hexagons, are all based on the principles of the triangle. Students will be able to break down large polygons to a group of triangles and then solve things such as angle sums for both exterior and interior angles. An abstract motivator is on old song by a band called They Might Be Giants. The song should lighten the mood and hopefully get people to laugh a little about triangles. The song talks about fights between Triangle Man, Particle Man, and Person Man. Well according to the song Triangle man wins both fights. They never say why but my theory is because triangles are considered the strongest geometric shape. http://www.youtube.com/watch?v=sNT8SMlqLJA 8. Socialization - In order to develop a successful lesson, it is always important to consider the socialization that will occur between each individual student and the material. The design of this lesson is intended to allow the student to interact with the material, their peers, and the instructor. The interactions between the student and instructor will occur at the first and last five minutes of the lesson and also throughout the lesson when either the student or instructor has a specific question. The student to student interactions will occur throughout the guided lesson as they compare their findings and help explain how to apply the theorems on the note sheet. The student to content interactions will occur through Geometer’s Sketchpad as they explore and define the theorems. 9. Audience -My Lesson is intended for Freshmen – Juniors in high school. Students need to be currently enrolled in geometry and should have completed Algebra 1 or at least have the ability to solve linear and quadratic equations of one variable. Students need to have experience working with Geometer’s Sketchpad and be familiar

with its basic commands. Students also need to understand basic principles of triangles, such as they have 3 sides and they come in a variety of different sizes and shapes. 10. Technology Needs – The technology needs for this lesson are a computer lab and the Geometer Sketchpad software on each computer.