## Geometry

Link to the Geometry Course in Canvas

What is geometry? Geometry is generally more concrete than other math subjects. We will study lots of things that you can actually see, draw, and touch like triangles, circles, cubes, pyramids, and other shapes. However, studying the properties of these shapes still requires algebraic principles. Two general topics in geometry that will be very new to most students are proofs and trigonometry. Like any subject, we will start at the basics and build to more complex ideas as we go. There are over 30 standards that we will learn about in geometry, but the "BIG 5" areas are listed below.

This is from the Arizona Math Standards adopted in 2016.

See https://www.azed.gov/standards-practices/k-12standards/mathematics-standards/ if you would like more information.

*For the high school Geometry course, instructional time should focus on
five critical areas: *

* *

**1. Establishing
criteria for congruence of geometric figures based on rigid motions.**

**2. Establishing
criteria for similarity of geometric figures based on dilations and
proportional reasoning.**

**3. Develop
understanding of informal explanations of circumference, area, and volume
formulas. **

**4. Proving
geometric theorems.**

**5. Solve problems
involving right triangles.**

** **

(1) Students have prior experience with drawing triangles based on given measurements and performing rigid motions including translations, reflections, and rotations. They have used these to develop notions about what it means for two objects to be congruent. Students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. They apply reasoning to complete geometric constructions throughout the course and explain why these constructions work.

(2) Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of geometric figures, use similarity to solve problems (including utilizing real-world contexts), and apply similarity in right triangles to understand right triangle trigonometry. When studying properties of circles, students develop relationships among segments on chords, secants, and tangents as an application of similarity.

(3) Students’ experience with three-dimensional objects is extended to developing informal explanations of circumference, area, and volume formulas. Radians are introduced for the first time as a unit of measure – which prepares students for work done with the Unit Circle in the Algebra II course. Students have opportunities to apply their understanding of volume formulas to real-world modeling contexts. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line.

(4) Students prove theorems—using a variety of formats including deductive and inductive reasoning and proof by contradiction—and solve problems about triangles, quadrilaterals, circles, and other polygons. Relating back to work in previous courses, students apply the Pythagorean Theorem in the Cartesian coordinate system to prove geometric relationships and slopes of parallel and perpendicular lines. Continuing in the Cartesian coordinate system, students graph circles by manipulating their algebraic equations and apply techniques for solving quadratic equations – all of which relates back to work done in the Algebra I course.

(5) Students define the trigonometric ratios of sine, cosine, and tangent for acute angles using the foundation of right triangle similarity. Students use these trigonometric ratios with the Pythagorean Theorem to find missing measurements in right triangles and solve problems in real-world contexts – which prepares students for work done with trigonometric functions in the Algebra II course.