**Course: Geometry (& GT)**In this course, students will develop an understanding of transformational, Euclidean, and coordinate geometry with extensive real-world application. Students will study logic, inductive and deductive reasoning, geometric definitions, postulates, and the proofs of theorems. Other topics include an introduction to trigonometry and vectors. Course requirements are rigorous with an emphasis on mathematical reasoning and communication

**.****Unit 1: Transformations, Similarity, and Congruence**

**Part 1: Transformations and the Coordinate Plane**Students are expected to use coordinates to prove simple geometric theorems algebraically. Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. For example, students will prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

Students will also experiment with transformations in the plane. In Grade 8, students had experience with transformations: translations, reflections, rotations, and dilations. Comparisons of transformations will provide the foundation for understanding similarity and congruence. Additionally, students will prove theorems, using a variety of formats, and solve problems about triangles, quadrilaterals, and other polygons. Students will be exposed to multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. They also apply reasoning to complete geometric constructions and explain why they work. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning.

*Part 2: Similarity*Students apply their earliest experience with dilations and proportional reasoning to build a formal understanding of similarity. The only transformations that do not absolutely preserve congruence are dilations. Focus will be on teh power of dilations to effect size change in geometric figures. Dilations preserve angle measure while enlarging or reducing lengths proportionally. Students will be able to determine if two figures are similar, essentially a dilation of each other, by comparing their angles for congruence and their sides for proportionality. Students will discover that the other transformations (reflections, rotations, shifts, translations) produce a similarity in a 1:1 ratio and thus are a special subset of similarity transformations, namely transformations that produce congruent figures. (A metaphor for this would be that of a square being a specifically defined rectangle in that way that a congruence is a specifically defined similarity).

*Part 3: Congruence*Students will build on their prior experience with rigid motions, including translations, reflections and rotations, in order to develop notions about what it means for two objects to be congruent. Rigid motions are at the foundation of the definition of congruence, and students should recognize that they preserve distance and angle. Students will continue to develop their understanding that congruence is a special case of similarity with a 1:1 ratio. Students should experiment with rigid motions in the coordinate plane in order to carry given polygons onto themselves, in order to verify major properties of parallelograms, and in order to determine the congruence of given polygons. Students should be encouraged to use a variety of tools, such as protractors, compasses, MIRA boards, patty paper, graph paper, and geometry software in order to transform given figures to prove congruence. In addition, students should use a variety of methods for proof, both formal and informal. Students will apply their understanding of congruence to copy segments and angles using a variety of tools and construction methods.

**Unit 2: Triangles, Proof, and Trigonometry**

**Part 1: Triangles**Students will use their knowledge of similarity and congruence to build an understanding of similar and congruent triangles. They will use similarity transformations to establish the AA and other Triangle Similarity Theorems and rigid motions to establish the SSS, SAS, and ASA Triangle Congruence Theorems. Students will understand that congruence is a special case of similarity, where the ratio of side lengths is 1:1. They will use multiple formats to prove various theorems about triangles throughout the unit, including those pertaining to congruence, mid-segments, and medians.

*Part 2: Right Triangle Trigonometry*Students will use their knowledge of similarity of right triangles (and other triangles, in Geometry GT) to establish an understanding of the trigonometric ratios of angles in these triangles. They will understand the interrelationships between the trigonometric functions. They will use these ratios, and the Pythagorean theorem to solve right triangles, given various initial information.

In Geometry GT, students will further their exploration or triangles using the Law of Sines and the Law of Cosines. They will both solve triangles that are not right triangles and they will develop and use a general formula for the area of triangles which uses trigonometry.

**Unit 3: Circles, Proof, and Constructions**

In this unit, students will build on their understanding of similarity to investigate relationships between circles. In addition, students will explore and prove relationships between parts of circles, to include radii, tangents, secants, and chords. Students should understand how these parts relate to segment lengths and angle measures, and how this relates back to similarity. Through multiple constructions, students will explore properties of other figures and how this relates to circles. Students will justify the formulas for circumference and area, and use them to explore arc length, define radians, and derive the formula for the area of a sector. Using their understanding of the Cartesian coordinate system, students will use distance formula to write equations of circles given a radius and center. Students should be able to justify whether or not a given point lies on a given circle using their understanding of coordinate geometry.

**Unit 4: Extending to Three Dimensions**

Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. 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.

**Unit 5: Applications of Probability**

Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions