Unit Name: Unit 5: Similarity, Right Triangle Trigonometry, and Proof

Lesson Plan Number & Title: Lesson 3: Developing Formal Proof and Definitions

Grade Level: High School Math II

Lesson Overview:Students will compare and contrast a two-column traditional proof with a paragraph proof that utilizes a transformational method. They will also develop definitions necessary to further the rigor of the proof process. This lesson is designed for approximately 90 minutes, but may need more time as it is fundamental to ensuing lessons.

Focus/Driving Question:How can transformations provide a springboard for proving geometric theorems?

West Virginia College- and Career-Readiness Standards:M.2HS.42Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Implementation may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.2HS.C.3. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

Manage the Lesson:This lesson is pivotal in comparing methods of proof and in demonstrating a proof that is based on transformations. It will introduce definitions that are not typical of traditional geometry courses that most teachers have taught previously. Definitions that are necessary for later proofs that utilize transformations are introduced in this lesson. For example, the role of orientation of points to lines plays a more significant role in transformational geometry than in a more traditional geometry course. There is always a tension between maintaining rigor and yet allowing accessibility to content. With this in mind, the definitions and theorems in this unit are supported with a series of apps that will clarify statements that may at first reading appear wordy and even cumbersome, but are necessary to preserve the integrity of the proofs.

The apps, particularly for inclusion students, will provide a visual representation of a definition or theorem and thus augment the meaning of what may have been difficult to comprehend for the student. Teachers are encouraged to utilize this tool to deepen student understanding.

Academic Vocabulary Development:Vocabulary necessary for future theorems will be a major focus of this lesson. Students will consider the following definitions:

alternate exterior angles alternate interior angles corresponding angles exterior angle half plane interior angle same-side interior angles side formed by a line transversal vertical angles (addressed previously)

The understanding of these definitions will be fostered using the Geometry Apps developed for this unit.

Launch/Introduction:Distribute 3.01 Writing Justifications (3.02 Answers to Writing Justifications) to introduce students to proofs written in paragraph form. This activity does not rely on extensive vocabulary or previous geometric content other than area of a square. It allows students to experience how proofs are constructed utilizing previously proven conclusions.

Investigate/Explore:Have students, using a note card; draw on a piece of patty paper a pair of intersecting lines. Ask them to name the angles that are formed and identify which angles are adjacent and which are nonadjacent. Then ask them, “What do you notice about the adjacent angles and about the nonadjacent angles?” They may want to measure the angles with a protractor or fold the patty paper to make conjectures. (This investigation could also include drawing the vertical angles in Geogebra and determining their measures to make the conjecture.) They may identify linear pairs, supplementary angles, and vertical angles (use the app “Definition of Vertical Angles”) and make the conjecture that “Vertical angles are congruent.”

Next, justify their conjecture using a paragraph proof, a two-column proof, and finally a paragraph proof using transformations. Use example slides from 3.03 PowerPoint on the Vertical Angles Theorem to demonstrate each method of proof. Then go the APPS MENU to access the app for the Vertical Angles Theorem to animate the transformation.

Several definitions (half plane, side formed by a line, transversal, exterior angle, interior angle, corresponding angles, alternate interior angles) and one theorem can now be examined using the apps. These definitions will prove critical in proving theorems related to parallel lines. After students have a working understanding of these definitions, ask them to identify what we might call “same-side interior angles” and “alternate exterior angles.” They should then write their own definitions for these angles. A quick pair activity will assess if students have developed understanding of angles formed by a transversal. Distribute 3.04 Identifying Angles, and have one student in a pair fold a sheet of a paper in half to create two columns. One student will list angles in one column and the second student in the other column. The teacher will instruct the students to write as many pairs of corresponding angles as they can, alternating from one student to the other with each student writing in his/her column. Encourage students to check his/her partner’s angles and offer help as needed. Then the teacher will move on to alternate interior angles, same-side interior angles, and alternate exterior angles. The teacher should walk about the classroom checking for understanding. Two optional activities (particularly with the inclusion student or

the kinesthetic learner in mind) follow to deepen understanding. Make several very large sets of lines cut by a transversal either by using masking tape on the classroom floor or using sidewalk chalk outdoors.

Students will be using their hands and feet to identify the angle pairs much like they have played a game of Twister. Begin, for example, by instructing a student to place his/her right hand on one corresponding angle and his/her left foot on the other. Give instructions to a second student on the same set of lines for a pair of alternate interior angles, and so on. This can be happening on several sets of lines simultaneously. Additionally, 3.05 Angle Identification (3.06 Answers to Angle Identification), which utilizes a coffee stir stick to mark the transversal, can provide support for the inclusion student.

Summarize/Debrief:Ask students to compare and contrast the methods used in proving the Vertical Angles Theorem. Distribute the graphic organizer 3.07 Comparing and Contrasting Methods of Proof. When students have completed this table, discuss their findings. The following ideas will probably surface in the discussion:

Two-column proofs require the student to be extremely precise thus making it less likely that an error is made. It is advantageous for a beginner writing a proof, as it compels the student to carefully analyze the reasons used. Unfortunately, two-column proofs can become unwieldy, and the big picture of the proof can be lost in the details.

Paragraph form allows a proof to be condensed into fewer statements thus making the central ideas of the proof more transparent. It demands that the reader have a deeper level of mathematical maturity in that he/she must be able to make the necessary connections that are more clearly stated in a two-column proof.

Transformational proofs rely on the prerequisite understanding of functional analysis and set theory. This is an advantage in that additional tools are available to utilize in the deductive process, but perhaps a disadvantage in that those understandings must have previously been developed. Transformational proofs are potentially a more intuitive approach than those traditionally utilized.

Although two-column proof is a part of a teacher’s background, these students will not have any preconceived notions about this method, so they may be open to experimentation with transformational approaches. It is important to note that not all of the proofs studied in this unit will employ transformations; however, when a proof is more easily demonstrated using transformations, the student and teacher can both grow from that experience.

Materials:3.01 Writing Justifications3.02 Answers to Writing Justifications3.03 PowerPoint on the Vertical Angles Theorem3.04 Identifying Angles3.05 Angle Identification3.06 Answers to Angle Identification 3.07 Comparing and Contrasting Methods of Proof

Career Connection:Although logical reasoning skills are advantageous in most professions, transformations particularly lend themselves to the Arts and Communication area such as graphic design and to the Information Technology area such as computer programming and game development.

Lesson Reflection:After the closing discussion, ask students to answer the following question on an exit card: “Is it easier to visualize and understand the transformations applied in proving the Vertical Angles Theorem than the approach used in the two-column proof?”

In lesson 1, teachers were provided with a guide to aid them in reflecting upon the lesson as they seek to improve their practice. Certainly, it may not be feasible to formally complete such a reflection after every lesson, but hopefully the questions can generate some ideas for contemplation.

3.01 Writing Justifications

3.02 Answers to Writing Justifications

3.03 PowerPoint on the Vertical Angles Theorem

3.04 Identifying Angles

3.05 Angle Identification

3.06 Answers to Angle Identification

3.07 Comparing and Contrasting Methods of Proof