Download MTH 341

Mathematics Class Visit Record
Teacher: Firas Hindeleh
Observer: Karen Novotny
Date of lesson: 10/1/2014
Course: MTH 341 Euclidean
Geometry
Topic: Consequences of the
Alternate Interior Angle
Theorem.
Course experience (# of times
previously taught): Six times
An explanation of the use and purpose of this form is provided on the last page.
Lesson Preparation (teacher, 1 paragraph):
We have been studying Neutral Geometry for the past two weeks. Students explored and proved the Alternate
Interior Angle theorem (AIAT) in Neutral Geometry during a laboratory activity last week. In this class we will
define corresponding angles and interior angles on the same side of a transverse. Students will apply the AIAT
to prove theorems involving corresponding angles and interior angles on the same side. They will also write a
constructive proof on how to construct a parallel line through a point not on a given line in Neutral Geometry.
Lesson Summary (observer, 1 page):
I arrived about seven minutes before class started and found Dr. Hindeleh connecting his computer to the
overhead display. In one window, he opened Non-Euclid, a free Java program on the Internet for investigating
geometry in the Poincaré disk model for both neutral and hyperbolic geometry. Dr. Hindeleh then switched the
display at the front board to that of the attached handout.
Promptly at the start of class, Dr. Hindeleh drew figures on the side board and explained a common student error
that he had observed when reviewing their last homework assignments. In the proof of the Angle-Angle-Side
Congruence Theorem for Triangles, to be completed in neutral geometry, several students illegitimately
assumed that the sum of the measures of the interior angles of any triangle is 180°. Dr. Hindeleh reminded the
students that they had already learned, using Non-Euclid, that this statement is not true in the Poincaré disk
model for neutral geometry. He then briefly outlined a proof by contradiction that was valid in neutral geometry.
By this time, approximately five minutes of class had elapsed.
After asking if students had any questions, Dr. Hindeleh passed out copies of the handout already displayed on
the front board and reminded the class of their lab exploration of parallel lines and the AIAT in neutral geometry
using Non-Euclid. He then directed their attention to their handouts and some new definitions (of corresponding
angles and interior angles on the same side). Meanwhile, he toggled the display to a Non-Euclid sketch that was
the same as the figure on the handout but was much larger. Using this sketch, he solicited responses from
students to complete the first page of the handout and to illustrate the new terms.
At approximately 11:15, Dr. Hindeleh asked the class: “Remind me – what does the AIAT say?” After receiving
a correct response, Dr. Hindeleh asked for a statement of the Converse of the AIAT. After discussing the
responses, writing the converse on the board, and considering whether or not it was valid in neutral geometry,
Dr. Hindeleh, began a discussion of the proof of the first of four consequences of the AIAT (Theorem N21 on
the attached handout). He first asked what the students would assume and then what they’d show to prove the
theorem. After receiving appropriate responses from several students, he began writing the proof on the board.
At several steps, one or more students offered editorial advice, and Dr. Hindeleh asked the class for their
opinions on how to proceed. In this manner, with specific help from approximately eight of the fifteen students,
the proof was successfully completed. Dr. Hindeleh modeled the writing guidelines for the class when writing
the proof on the board by centering and numbering important statements and equations and later referring to
them by their numbers. He also reminded the students that they needed a closing sentence to end the proof.
Dr. Hindeleh then explained that they might be able to recycle some of the proof of Theorem N21 for the proof
of the next theorem. So, again with help from various students, including at least two who hadn’t contributed
previously, this proof was similarly completed. At one point, a student asked if the AIAT could be used to
conclude that certain interior angles were congruent. Dr. Hindeleh asked this student questions about the
hypothesis and conclusion of the AIAT, so that she eventually concluded that she was using the theorem
incorrectly. In a similar manner, the class worked together to prove the next consequence of the AIAT (Theorem
22) with Dr. Hindeleh asking leading questions and transcribing the student responses on the board in neat easyto-read writing with proper mathematical notation. At one point in this proof, a student asked a question and,
before Dr. Hindeleh could respond, another student answered the question. His answer was then supplemented
by a third student. Dr. Hindeleh asked if all students agreed and when all responses were “yes” including from
the student asking the question, the discussion of the proof resumed and was concluded satisfactorily.
Dr. Hindeleh had previously promised that the culmination of this class period discussion would be a proof of
the following statement:
Theorem N24. In Neutral Geometry, for any line l and any point P not on l, there exists at least one line
through P that is parallel to l.
Before beginning the proof, Dr. Hindeleh asked the class what kind of proof would be appropriate for this
theorem, and many students responded that a constructive proof was in order. Then Dr. Hindeleh displayed a
blank sketch from Non-Euclid on the board, and he handed his wireless mouse to a student. He then asked this
student what she’d assume and show. As she explained that she would assume an arbitrary line l and an arbitrary
point P not on l, she used the mouse to construct these on the sketch. Continuing in this manner, she guided the
class through the construction. At various points, Dr. Hindeleh asked for justification and fellow students
assisted by providing rationale and suggestions for improvement. Meanwhile, Dr. Hindeleh wrote the proof on
the board, and, at one point, he again asked leading questions to allow the students to identify an error in
justification. To finish up the proof, one student suggested using Theorem N21, which they had proven earlier in
the class period, instead of the AIAT as it would save time. His suggestion was greeted with positive enthusiasm
by the rest of the class and Dr. Hindeleh, who finished writing the proof on the board just before class ended.
Lesson Response (observer, 1 paragraph):
I found this class period to be an exceptional example of active student learning masterfully led by Dr. Hindeleh.
During the course of this 50-minute class period, I counted eleven of the fifteen students who specifically
contributed to the class discussion. The four students who didn’t contribute verbally did shake their heads to
indicate “yes” or “no” for some of Dr. Hindeleh’s questions, and they appeared to be taking notes. Dr. Hindeleh
did not call on students by name, although he did pick two specific students to use his mouse to manipulate or to
create Non-Euclid sketches. I found this technique, particularly for the constructive proof, to be highly effective.
In addition, Dr. Hindeleh was careful to make sure that no one student monopolized the discussion by twice
suggesting that different students should take the lead or otherwise contribute to the discussion. At another time,
when several students offered suggestions on how to proceed, Dr. Hindeleh acknowledged that all the ideas had
merit, but that he’d proceed with one student’s suggestion as that student had answered first. Dr. Hindeleh
clearly knew all of his students’ names, and he was careful to thank them by name for their responses. In this
class, Dr. Hindeleh has developed a productive learning environment in which his students feel comfortable
asking questions, sharing ideas, and answering both Dr. Hindeleh’s questions and those of their peers.
Lesson Reflection (teacher, 1 paragraph):
I was pleased with the outcomes of this class. Students were engaged in the discussion, contributing to building
the four proofs, and taking more responsibility in the teaching and learning process. My role was to moderate
the discussion and to transcribe students’ input in a way that emphasizes the proof-writing guidelines they
learned in MTH 210. Students need to typeset those proofs in their journal project to reinforce the proof-writing
skills. We established a better understanding to what a Constructive Proof is. Students had a hands-on
experience with communicating their thoughts and writing the proof while constructing the figures. I have a
very strong group of students in this class who were excited to contribute towards the discussion. This might
have left a few slower students out. In the future I should pay more attention to those students and give them
even a simple task to complete, this will boost their confidence and give them a sense of ownership.
By signing below, I agree that my portions of this record accurately reflect my perception of the observation.
__________________________________
Teacher
__________________________________
Observer
Description of the Mathematics Class Visit Record
As teaching is a primary responsibility of faculty, documenting a faculty member’s teaching is an important
task. The Department of Mathematics views teaching as work that extends beyond the classroom and involves
assessment, planning, and evaluation, as well as instruction. This class visit form strives to create an image of
the teacher’s classroom as well as the work that surrounds it.
In the first section, the teacher writes a paragraph that describes his or her preparation for the lesson to be
observed. This paragraph should include a listing of the objective(s) for the lesson along with a description of
the teacher’s experience with the course and lesson. It may also include a description of how class activities
were chosen or developed, or a short description of the reasoning behind the objective(s).
In the second section, the observer writes a brief summary (no more than one page) of the observed class. The
focus of this summary is not to provide a transcript that conveys every detail of the class, but rather to capture
the essence of the lesson, with selected details that help the reader understand the teacher’s classroom. In the
summary, the observer may wish to comment on the mode of instruction, student-teacher communication,
student-student interaction, student engagement, the teacher’s assessment techniques, and any other salient
features of the class.
In the third section, the observer writes a one-paragraph response to the observation. In this paragraph, the
observer reflects on what he or she thought was important or noteworthy about the class. This reflection may
include questions, issues, praise, or commentary.
In the fourth and final section, the teacher writes a paragraph reflecting upon the observation. This paragraph
may include consideration of the lesson objectives, reflection about what happened during instruction,
explanation to assist the reader in understanding the lesson, or a response to what the observer has written.
The observation record is a collaboration between the observer and teacher for the purpose of communicating
the teacher’s teaching, and the final form will be signed by both. Communication between the teacher and
observer is encouraged throughout the process, in particular before and after the observation and in the final
editing of the record. Observations for the personnel process are forwarded with the candidate’s materials and
should strive to be both clear and concise.
Hindeleh1
Mathematics 341
Theorems in Neutral Geometry, Part IV
We continue to work in Neutral Geometry, using only the SMSG postulates for neutral geometry
and theorems we’ve already proved.
While still working in Neutral Geometry, we have recently begun to study the role of parallel lines.
Parallel lines, by definition, are those that have no point in common. We can work in neutral
geometry with such lines, but in so doing we cannot assume the parallel postulate. Henceforth,
when we say “the Euclidean parallel postulate”, we mean the following:
For every line l and any point P not on l, there exists a unique line m through P parallel
to l.
Today we will ultimately prove that in Neutral Geometry, for every line l and any point P not on
l, there exists at least one line through P parallel to l (!).
Before we begin, we need to make several important definitions:
Defintions: Let l, m be two distinct lines that are cut by a transverse t (see figure). Then
• The pair of angles on the same side of the transverse t and both above (or below) the lines
l, m respectively are called Corresponding Angles.
6
EGD and 6 GHB,
Example:
• The pair of angles on opposite sides of the transverse t and are between the lines l, m respectively are called Alternate Interior Angles.
6
Example:
DGH and 6 GHA,
• The pair of angles on the same side of the transverse t and are between the lines l, m respectively are called Interior Angles on the same side.
Example:
1
6
DGH and 6 GHB,
The worksheet is adapted from Dr. Boelkins
Theorem N19. In neutral geometry, there exist no quadrilaterals with four right angles.
Theorem N20. (The Alternate Interior Angle Theorem) If two lines l and m are intersected by a transversal
t such that a pair of alternate interior angles formed is congruent, then the lines l and m are
parallel. (Proved in Lab 4)
Theorem N21. Two lines perpendicular to the same line are parallel.
Proof.
Theorem N22. If two lines are intersected by a transversal such that a pair of corresponding angles is congruent, then the lines are parallel.
Proof.
Theorem N23. If two lines are intersected by a transversal such that a pair of interior angles on the same
side of the transversal is supplementary, then the lines are parallel.
Proof.
Theorem N24. In Neutral Geometry, for any line l and any point P not on l, there exists at least one line
through P that is parallel to l.
Proof.
Exercise. How would you phrase the converse of the Alternate Interior Angle Theorem? Is the converse
true in Neutral Geometry?