This document outlines the course plan for a Modern Geometry course. The course aims to discuss non-Euclidean geometries like finite, non-Euclidean, and projective geometry. Over 18 weeks, topics will include spherical, hyperbolic, and projective geometry as well as transformations and inversions. Assessment will include quizzes, group activities, and exams worth 100% of the grade. The course intends for students to demonstrate understanding of key concepts in non-Euclidean geometries.

1. 1st
Revision
GOV. ALFONSO D. TAN COLLEGE
Bachelor of Secondary Education Major in Mathematics (BSEd)
Outcomes – Based Teaching and Learning Plan in ME 107
Alfonsos as Lux Mundi: Serving Humanity with Empowered Mind, Passionate Heart and Virtuous Soul
Course Title Modern Geometry Course Code Math 107
Credit Units 3 units Course Pre-/Co-requisites Plane & Solid Geometry, Logic & Set Theory
Course Description
(CMO 75 s. 2017)
The course is an enrichment of the course on Euclidean Geometry. It discusses the properties and applications of other types of geometries
such as finite geometry, non-Euclidean geometry and projective geometry.
Institute Intended
Learning Outcomes
(IILO)
Graduates of BSEd programs are teachers who:
a. Articulate the rootedness of education in philosophical, socio-cultural, historical, psychological, and political contexts
b. Demonstrate mastery of subject matter/discipline
c. Facilitate learning using a wide range of teaching methodologies and delivery modes appropriate to specific learners and their
environment
d. Develop innovative curricula, instructional plans, teaching approaches, and resources for diverse learners
e. Apply skills in the development and utilization of ICT to promote quality, relevant, and sustainable educational practices
f. Demonstrate a variety of thinking skills in planning, monitoring, assessing, and reporting learning processes and outcomes
g. Practice professional and ethical teaching standards sensitive to the local, national, and global realities
h. Pursue lifelong learning for personal and professional growth through varied experiential and field-based opportunities
Program Intended
Learning Outcomes
(PILO)
At the end of this program, graduates will have the ability to:
a. Exhibit competence in mathematical concepts and procedures.
b. Exhibit proficiency in relating mathematics to other curricular areas.
c. Manifest meaningful and comprehensive pedagogical content knowledge (PCK) of mathematics.
d. Demonstrate competence in designing, constructing and utilizing different forms of assessment in mathematics.
e. Demonstrate proficiency in problem-solving by solving and creating routine and non-routine problems with different levels of
complexity.
f. Use effectively appropriate approaches, methods, and techniques in teaching mathematics including technological tools.
g. Appreciate mathematics as an opportunity for creative work, moments of enlightenment, discovery and gaining insights of the world.

2. 1st
Revision
Course Intended
Learning Outcomes
(CILO)
At the end of this course, the students should be able to:
a. Demonstrate competence in unravelling various information about Non-Euclidean geometry
b. Define and describe the different concepts of Spherical Geometry.
c. Analyze and use different theorems in Hyperbolic geometry to solve problems
d. Demonstrate and illustrate different geometric figures under projective geometry
MIDTERM Essential Learning
Intended Learning Outcomes
(ILO)
Suggested
Teaching/Learning
Activities (TLAs)
Assessment
Tasks (ATs)Week Content Standards
Declarative
Knowledge
Functional Knowledge
1-3 Demonstrate familiarity with
the Introduction to Non-
Euclidean Geometry
Introduction and Orientation
(Vision and Mission, Classroom
Rules and Grading System)
Introduction to Non-Euclidean
Geometry
 Planet Earth and the
Longitude Problems
Spherical Geometry
 Facts from Spherical
Geometry
 Great Circles
 Lunes
 Spherical biangles Spherical
triangles
 Angle Sums and surface area
in spherical geometry
-Discussing the process in solving
planet and longitude problems
Discussing the facts from Spherical
Geometry
Illustrating and explaining great circles,
lunes and spherical triangles
Solving from the sums of angles and
surface area in spherical geometry
Solve planet earth and longitude problems
Describe spherical geometry based on its
facts
Illustrate the great circles, lunes and spherical
triangles
Calculate the angle sums and surface area in
spherical geometry
Group Activity with output
Group Power Point
Presentation
Brainstorming
Group Demonstration
Group Interactive Discussion
Quiz
Rubrics with defined
criteria for grouped
performance
4 -6
Demonstrate Understanding
in Hyperbolic Geometry
Neutral Geometry
 Rectangles in Neutral
Geometry
 The All-or-Nothing Theorem
for Angle Sums
 Similar Triangles
 Alternate-Interior Angles
 Weak Exterior Angle
Theorem
 Poincare Circle
 Inversion in a Circle
Discussing Neutral geometry and
theorem for angle sums
Differentiating Alternate-Interior Angles
from Weak Exterior Angle theorem
Illustrating Poincare and inversion in a
circle
Illustrate how rectangles work in Neutral
Geometry
Explain All-or-Noting Theorem for Angle
Sums
Illustrate the Alternate-Interior Angles
Analyze and prove the Weak Exterior Angle
Theorem
Explain and illustrate Poincare Circle and
Inversion in a Circle
Group Activity with output
Group Power Point
Presentation
Brainstorming
Group Demonstration
Group Interactive Discussion
Quiz
Rubrics with defined
criteria for grouped
performance
6 - 9
 The Saccheu – Legendre
Theorem
 Saccheu Quadrilaterals
Explaining the Saccheu Legendre
Theorem and Quadrilaterals
Discussing Hyperbolic Distance,
Solve problems using Saccheu – Legendre
Theorem and Quadrilaterals
Solve problems involving Hyperbolic
Group Activity with output
Group Power Point
Presentation
Brainstorming
Quiz
Rubrics with defined
criteria for grouped

3. 1st
Revision
 Hyperbolic Distance
 Hyperbolic Laws of Cosines
 Hyperbolic Axioms
Hyperbolic Laws of Consines and
hyperbolic axioms
Distance, hyperbolic laws of cosines and
Hyperbolic axioms
Group Demonstration
Group Interactive Discussion
performance
FINAL
11 – 14
Demonstrate
Understanding in Projective
Geometry
Projective Geometry
 Planar Geometry and the 2D
projective plane
 Homogeneous representation
of Lines
 Homogeneous of points
 Intersection of Parallel lines
 Ideal points and the line at
infinity
 A visual way to think of P2
Discussing Planar Geometry and 2D
projective plane
Explaining homogenous representation
of lines and points
Illustrating the intersection of parallel
lines, ideal points and the line at infinity
Visualizing the P2
Simplify and solve expressions in Planar
Geometry and 2D projective plane
Show the Homogeneous representation of
lines and points
Draw the intersection of parallel lines
Discuss ideal points and the line at infinity
Present the visual to think of P2
Group Activity with output
Group Power Point
Presentation
Brainstorming
Group Demonstration
Group Interactive Discussion
Quiz
Rubrics with defined
criteria for grouped
performance
15-18
Demonstrate Projective
Transformations
 Transformation of Lines
 The hierarchy of
transformations
 Isometric Transformation
 Similarity Transformation
 Affine Transformation
 Projective Transformation
 Cross ratios and vanishing
points
Discussing the transformation of lines
Discussing and differentiating the
Isometric, Similarity, Affine and
Projective Transformations
Elaborating Cross ratios and vanishing
points
Simplify and solve equations involving
transformation of line
Explain and differentiate Hierarchy of
Transformations: Isometric, Similarity, Affine,
and projective
Solve problems cross ratios and vanishing
points
Group Activity with output
Group Power Point
Presentation
Brainstorming
Group Demonstration
Group Interactive Discussion
Quiz
Rubrics with defined
criteria for grouped
performance
Demonstrate Understanding
in Inversion
Inversion
 Dynamic Investigation
 Properties of Inversion
 Applications of Inversion
 Tilings of the Hyperbolic
lane
Discussing Dynamic investigation
Discussing the ways in construction of
inversion
Discussing the properties of inversion
and its various theorems
Discussing the ways to apply inversion
Demonstrating how to create a n-gon
to tile the hyperbolic plane
Perform various exercises involving inversion
Prove the following theorems involving
inversion
Apply the concept on inversion in constructing
different geometrical figures
Group Activity with output
Group Power Point
Presentation
Brainstorming
Group Demonstration
Group Interactive Discussion
Quiz
Rubrics with defined
criteria for grouped
performance
Basic Readings
Not available as of this moment of time
Extended Readings Stan Birtchfield. Introduction to Projective Geometry
Chapter 3. Projective Geometry
W. Ewald (1954 –), Bulletin (New Series) of the American Mathematical Society, Vol. 40 (2002), pp. 125 – 126.
Module 6. Non-Euclidean Geometry

4. 1st
Revision
Simeon Ball and Zsuzsa Weiner. (2011). An Introduction to Finite Geometry
Chapter 4. Introduction to Hyperbolic Geometry
Course Assessment As identifiedin the Assessment Task
Course Policies LanguageofInstructions
English
Attendance
As identifiedin the student handbook
Homework,Quizzes,Written Reports,ReactionPapersand Portfolio
Special Requirement
GradingSystem
SummativeQuiz 30%
SummativePerformance 40%
PeriodicalExam 30%
100%
Classroom RulesandRegulations
Respect
Committee Members CommitteeLeader : Alemar C. Mayordo
Members : Elton John B. Embodo
RogielouP. Andam
Clint Joy Quije
ZarleneM.Tigol
Consultation Schedule FacultyMember : EltonJohn B. Embodo
ContactNumber : 09107619989
E-mailaddress : eltonjohn439@yahoo.com
ConsultationHours:
TimeandVenue :
Course
Title
A.Y. Term of
Effectivity
Prepared by Checked by Noted by Approved by Pages
Modern
Geometry
2019– 2020 ELTONJOHNB.EMBODO,MAEd
Instructor
ELTON JOHN B. EMBODO, MAEd
Program Coordinator, Math
ALEMAR C. MAYORDO, MAED
ITE, OIC-Dean
LOVE H. FALLORAN, Ph.D
VP for Academics
4