# High School – Geometry

## Monmouth County Private High School – Geometry

This year-long class is designed to enable high school students to learn the fundamentals of Geometry through a combination of hands-on activities, group discussions, and lectures. Students will build their understanding by proving a set of precepts that can then be relied upon to prove more advanced concepts. Students will spend much of the class time working in small groups to brainstorm methods to solve problems of increasing complexity.

The students will understand:

• common Geometry terms and concepts.
• the nature of points, lines, and angles and how they interact.
• the difference between equals and congruence.
• the properties of parallel lines.
• similarity.
• how to create Geometric proofs.
• how to calculate properties of polygons.
• the Pythagorean Theorem.
• Trigonometry of Right Triangles.

### Process

Fall Session
The students began the course with an introduction to the basic vocabulary and notation used to describe angles and line segments. Words such as reflex and midpoint were defined and examples displayed and understood. Once the students understood the names of items a brief history of geometry ensued. As various theorems and postulates are discovered through experiments, students maintain a series of notecards to refer back to later when working on proofs. Geometry Discussions

The students on the geometry class spent many of the first weeks understanding triangles. By studying lengths of sides and interior angles the students were able to discover the nature and rules that govern this important shape. Students did hands-on experiments using cut paper, spaghetti pieces, compasses and protractors to allow each student to fully understand the intricacies of triangles. These understandings were then applied to other polygons, which could always be broken down into a series of triangles.

Much of the work in the first session was done in small groups of four students. Each group was responsible for solving problems and challenges posed by the teacher. Within each group, individuals took on very specific roles but all students were responsible for fully understanding and performing the work. Some of the challenges faced by the groups were drawing an equilateral triangle, squares and hexagons using only a compass and calculating the height of the flagpole based on its shadow.

Near the end of the first session, each student was tasked with designing the location of a small (500-1000 square feet) house on a triangular plot of land. Students needed to locate the house foundation corners taking setbacks into account. The final task for the project was to calculate the length of one roof rafter based on the width of the house.

Winter Session
The second session began with each student finishing and turning in the plot of land assignment. The first major topic of discussion was the nature of parallelism. Students studied and discussed basic terminology of parallel lines such as transversal, same side interior angles, alternate interior angles, and corresponding angles.

The next course of study covered the concept of similar and congruent shapes. Students learned the definition of what makes a shape similar to another. This led to common methods to proving similarity between triangles such as Side Side Side, Side Angle Side, and Angle Angle principles. Students worked individually and in small groups to determine if given triangles were similar and, if so, to calculate the scale factor. Side Side Side, Side Angle Side, and Angle Angle principles

To prepare the students to start writing proofs, pupils learned to solve logic problems. These problems required each student to use both deductive and inductive reasoning to come to a solution. Students worked both individually and in small groups to better understand the basis of proof writing. As the class progressed, students used their understanding of triangles and parallel lines to solve proofs given by the teacher.

Once each student had several weeks of practice under their belts, they were given a Geometry Review to complete. This review consisted of many questions summarizing all of the subject matter covered to that point during the school year. Students were allowed time in class in addition to being able to finish the problems at home.

Near the end of the Winter Session, the students began learning the basics of trigonometry. Starting with a unit circle, students learned the difference and relationship between sine, cosine, and tangent in right triangles. Student-built triangles in class and calculated side lengths of triangles created outside and in the classroom by the teacher. Once complete, students began the study of non-right triangles, using the Laws of Sines and Law of Cosines.

Spring Session
The Spring session started with calculating the measurements of sides and angles of triangles without right angles. These were studied by fostering a deep understanding of the Law of Sines and the Law of Cosines.

### Outcome

To date the students have learned about and demonstrated proficiency in:

• common Geometry terms and concepts.
• the nature of points, lines, and angles and how they interact.
• the difference between equals and congruence.
• the Pythagorean Theorem and its uses.
• calculating the sides of similar shapes.
• the properties of parallel lines.
• writing geometric proofs.
• using trigonometry to calculate the sides and angles of right triangles.

Reference
Geometry: An Interactive Journey to Mastery (2014) published by The Great Courses

TEACHER: Richard Knab