Geometry (Credit Recovery)

Syllabus:

Semester 1:

Section 1 - Geometry Foundations

Objectives:

Section 1 presents the critical definitions of many geometric terms and figures needed in further study of geometric topics. How to construct a line segment, segment bisector, a copy of an angle, and an angle bisector using a straightedge and compass is presented. Points, lines, planes and angles are studied.

After this section the student will be able to:

• Define basic geometric terms.
• Recognize and name geometric figures using proper notation.
• Find the length of line segments and determine congruency.
• Construct geometric figures using a straightedge and compass.
• Classify and name angles.
• Find measures of unknown angles.
• Identify complementary and supplementary angles.
• Identify vertical angles and linear pairs.

Lessons:

• History of Geometry
• Points, Lines and Planes
• Line Segments
• Midpoint and Distance
• Introduction to Angles
• Angle Relationships

Section 2 - Geometric Reasoning

Objectives:

Section 2 introduces the student to proofs. Students will practice logical reasoning by writing 2-column, flow chart, paragraph, and algebraic proofs.

• After this section the student will be able to:
• Discern between inductive and deductive reasoning.
• Make conjectures and draw conclusions.
• Write conditional statements.
• Understand the components to writing a proof.
• Identify and use the properties of equality.
• Write algebraic proofs.
• Prove conjectures about line segments and angles.

Lessons:

• Reasoning and Proofs
• Algebraic Proofs
• Geometric Proofs
• More Proofs

Section 3 - Parallel and Perpendicular Lines

Objectives:

Section 3 studies the relationships between parallel lines and the angles formed by a transversal. Properties of perpendicular lines are explored. Students will construct a parallel line to another line. Construct a perpendicular line to another line through a point not on the line and from a point on the line.

After this section the student will be able to:

• Define parallel and perpendicular lines.
• Identify angles formed by transversals.
• Construct a set of parallel and perpendicular lines.
• Prove theorems involving parallel lines.
• Use theorems to find unknown angle measures.
• Investigate properties and theorems of perpendicular lines
• Construct a perpendicular line to a line from a point on the line.
• Determine if two lines are parallel, perpendicular or neither.
• Write equations of parallel and perpendicular lines.

Lessons:

• Lines and Angles
• Parallel Lines and Transversals
• Perpendicular Lines
• Parallel and Perpendicular Lines and Coordinate Plane

Section 4 - Geometric Transformations

Objectives:

Section 4 introduces students to geometric transformations. Translation, reflection, and rotations of geometric figures are defined and explored. Students perform transformation on a pre-image based on a given rule.

After this section the student will be able to:

• Describe the types of transformations.
• Identify rigid and non-rigid transformations.
• Describe the transformation of a geometric figure using coordinate notation.
• Describe the transformation of a geometric figure using vector notation.
• Write the rule for transforming a pre-image to a given image.
• Transform geometric figures based on a given rule.
• Identify sequences of transformations to a pre-image.
• Apply a sequence of transformation to a preimage.

Lessons:

• Introduction to Geometric Transformations
• Translations
• Reflections
• Rotations
• Sequences of Transformations

Section 5 - Congruent Triangles

Objectives:

Section 5 focuses on triangles and proving congruence of triangles. How to construct an equilateral triangle is shown. A wide variety of triangle theorems are presented and used to complete proofs. Coordinate proofs are formalized.

After this section the student will be able to:

• Classify triangles by their angles and sides.
• Construct an equilateral triangle.
• Know and use theorems about triangles.
• Use transformations to prove figures are congruent.
• Use properties of congruence to solve problems.
• Use theorems, postulates, definitions and properties to prove triangles are congruent.
• Use theorems to find unknown side length and angle measures.
• Find distance between two points on a coordinate plane.
• Find the slope between two points on a coordinate plane.
• Prove conjectures by finding distance and angle measures on a coordinate plane.

Lessons:

• Introduction to Triangles
• Congruence and Transformations
• Proving Triangles Congruent
• Using Congruent Triangle Theorems
• Introduction to Coordinate Proofs

Section 6 - Properties of Triangles

Objectives:

Section 6 studies the properties of triangles. Points of concurrency, midsegments of triangles and triangle inequalities are presented.

After this section the student will be able to:

• Define and recognize point of concurrency in a triangle
• Use theorems about concurrency to solve problems
• Construct a circumcircle of a triangle and a circle inscribed in a triangle
• Define and recognize midsegments of a triangle
• Recognize relationships between the medial triangle and its triangle
• Use midsegment theorems to solve problems
• Use theorems to compare angle measure and side lengths in a triangle.
• Determine if given segment lengths can form a triangle
• Use the Hinge theorem to compare two triangles

Lessons:

• Points of Concurrency
• Midsegment of Triangles
• Triangle Inequalities

Semester 2:

Section 1 - Similarity

Objectives:

In this section, students:

• Explore the relationship between dilation and similarity; 
• Apply the concepts of similarity to understand the trigonometric ratios; 
• Use Trigonometric concepts to solve problems; 
• Prove theorems about similar figures and proportions in triangles.

Lessons:

• Dilations & Similarity
• Triangle Similarity
• Proportionality Theorems

Section 2 – Right Triangle Trigonometry

Objectives:

In this section, students:

• Understand the relationship between similar triangles and trigonometry; 
• Use the geometric mean to solve problems; 
• Solve right triangles using trig ratios, 
• Use the Law of sines and cosines to solve all triangles; 
• Solve real world problems.

Lessons:

• Similarity in right triangles
• Trigonometric Ratio
• Solve right triangles
• Law of sines and cosines

Section 3 - Quadrilaterals

Objectives:

In this section, students:

• State the properties of a parallelogram; 
• Prove parallelogram theorems; 
• Determine if geometric figures are parallelograms; 
• Classify figures in the coordinate plane; 
• Prove the slope criteria for parallel lines.

Lessons:

• Parallelograms
• Classify quadrilaterals
• Geometry in the coordinate plane

Section 4 – Exploring Circles

Objectives:

In this section, students:

• Derive the equation of a circle by completing the square; 
• prove theorems about circles; 
• Calculate the area of a sector and arc length; 
• Find the measures of inscribed angles; 
• Find unknown segment lengths.

Lessons:

• Introduction to circles
• Arcs of a circle
• Angle relationships in circles
• Segment relationships in circles

Section 5 - Geometric Measurements

Objectives:

In this section, students

• Calculate perimeter and area of polygons on the coordinate plane; 
• Identify 2D cross sections of 3D objects; 
• Identify 3D objects generated by rotating 2D figures; 
• Derive volume formulas using Cavalieri’s Principle, dissection and reasoning; 
• Calculate the volume of spheres, pyramids, cylinders and cones; 
• Calculate the volume of similar figures;
• Use geometric modeling to solve real world problems.

Lessons:

• Measurements on the coordinate plane
• Exploring 3D figures
• Volume of geometric solids
• Geometric modeling

Section 6 - Probability

Objectives:

In this section, students:

• Calculate experimental and theoretical probability. 
• Make and use frequency tables. 
• Use permutation and combinations to solve problems. 
• Find compound and conditional probabilities. 
• Create and use simulations to model real world problems.

Lessons:

• Introduction to probability
• Permutations & combinations
• Compound probability
• Conditional probability
• Modeling with simulations