Honors Geometry

Course Description

This is a college preparatory course, designed for the high achieving student. Students will apply both inductive and deductive methods of reasoning to a specific mathematical structure. Connections to postulates and theorems are made through practical applications to everyday, real-life problems. Students will study in-depth geometric vocabulary, logic and proofs, parallel and perpendicular lines, congruent and similar triangles, right triangles and trigonometry, polygons and quadrilaterals, circles, volume and surface area, and transformations in the coordinate plane.

Course Big Ideas

Logic, inductive, and deductive reasoning can be used to prove postulates and theorems.
Use definitions, postulates and theorems to prove relationships about parallel lines and triangles.
Understand triangle properties, relationships, and congruence to analyze more complex shapes and explore real-world problems.
Explore the creation and application of formulas and shapes to determine why we use them.
Look in depth on how shapes are related to each other.
Use transformations and symmetry to explore various 2-D shapes.

Course Essential Questions

How are definitions, postulates and theorems used in logic statements and proofs for parallel lines and triangles?
Why are parallel and perpendicular lines important to understand properties of shapes?
How are triangles related to formulas and other shapes?
How to investigate 2-D and 3-D shapes?
How do we write, convert, and use formulas related to various geometric shapes?

Course Competencies

Understand angles created by parallel or perpendicular lines and how we recognize these properties in shapes.
Understand side and angle relationships in triangles and how they are related to more complex shapes.
Understand characteristics of shapes and how to apply knowledge of 2-D shapes to 3-D shapes.
Understand where formulas come from and explore how to write, convert, and use these formulas related to various geometric shapes.
Understand transformations and symmetry to relate congruent and similar figures.

Course Assessments

Skill Based Summative Unit Quizzes and Tests
Check for understanding Mini-Quizzes
Formative Assessments: homework, IXL, All Things Algebra Activities

How can we represent and name points, lines, and planes accurately?
How do points, lines, and planes relate to one another in space?
How does the midpoint formula help in dividing a line segment into equal parts?
How does the distance between two points help us understand their relationship on a graph?
How is the distance formula derived from the Pythagorean Theorem?
What properties exist between angles, and how can we use them to find unknown measurements?
How do we accurately find the measure of whole line segments given algebraic expressions for each part of the segment?

Identify the hypothesis and conclusion of an if-then statement.
Convert statements into if-then statements.
Changing conditional statements to determine the truth value of related statements.
Use properties of equality to complete multi-step algebraic proofs (two column format).
Use definitions, postulates, and properties of congruence to write segment and angle proofs (two column format).

Unit Essential Questions

Lesson Essential Questions

How can counterexamples be used to test the validity of geometric conjectures formed by inductive reasoning?
What is the difference between a conjunction, disjunction, and negation in geometric statements?
What is a conditional statement, and how is it structured?
How do the converse, inverse, and contrapositive of a conditional statement differ from the original?
How are conditional statements used in geometric proofs and theorems?
How can we use properties of equality to justify each step in solving an equation in a two column proof?
How do properties of angle pairs (like complementary, supplementary, vertical, and adjacent angles) help us construct valid proofs?
How can we use definitions, postulates, and theorems to prove relationships between line segments?

Materials/Resources

Vocabulary

Conjecture
Counterexample
Negation
Compound Statements
Conjunction
Disjunction
Conditional Statements
Inverse
Converse
Contrapositive
Biconditional
Hypothesis
Conclusion
Distributive Property
Substitution Property
Reflexive Property
Symmetric Property
Transitive Property
Two-Column Proof
Congruence Properties
Definition of Congruence
Definition of Midpoint
Segment Addition Postulate
Angle Addition Postulate

Determine whether lines are parallel, perpendicular, or neither given slope.
Calculate slope given ordered pairs of lines and determine if they are parallel, perpendicular, or neither.
Convert equations of lines from standard form to slope-intercept and determine if they are parallel, perpendicular, or neither.
Write the equation of a line in slope-intercept form using the point-slope equation.
Determine the type of angle relationships given parallel lines cut by a transversal.
Determine if angle relationships are congruent or supplementary based on their position on parallel lines cut by a transversal.
Use the converse of a conditional statement to determine if lines are parallel.

Unit Essential Questions

Lesson Essential Questions

What do different types of slopes (positive, negative, zero, and undefined) tell us when comparing two or more lines?
How can we determine if two lines are parallel or perpendicular by looking at their slopes?
How do lines and points define planes, and what relationships exist between lines when they intersect or lie within the same plane?
How can we identify corresponding, alternate interior, alternate exterior, and same-side interior angles?
How do we use algebra to find missing angle measures in parallel line diagrams?
How can angle relationships help us prove that two lines are parallel?

CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence.
CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures.
CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

Classify triangles by their side lengths and angle measures.
Calculate unknown angle measures using the triangle sum and exterior angle sum theorems.
Determine side and angle measurements using the properties of isosceles and equilateral triangles.
Write two column proofs proving triangles are congruent by SSS, SAS, ASA, AAS, or HL postulates.
Write two column proofs proving parts (sides and angles) are congruent using the theorem Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

Unit Essential Questions

Lesson Essential Questions

How can we classify triangles based on their side lengths and angle measures?
How can we determine the measure of an unknown angle in a triangle if we know the other two angles?
Given two equal sides in a triangle and the included angle, how can you determine if the triangle is isosceles?
Why can we say all sides and angles are equal in an equilateral triangle?
How can we use definitions, postulates, and properties of congruence to prove that two triangles are congruent (SSS, SAS, ASA, AAS, HL)?
How does CPCTC help us prove that parts (angles and sides) of congruent triangles are congruent?

What does it mean for two triangles to be similar?
How can you use angle and side relationships to determine if triangles are similar?
What are the different methods to prove triangles are similar (AA, SSS, SAS)?
Why does knowing two angles are congruent guarantee triangle similarity?
How can you use proportions to find missing side lengths in similar triangles?
How can similar triangles be used to find distances or heights indirectly?
How does the scale factor between similar triangles affect side lengths and perimeters?

Use Pythagorean Theorem to find missing side lengths of right triangles.
Determine if three side lengths can form a triangle, then classify the type of triangle, using the triangle inequality theorem.
Calculate side lengths of special right triangles using their specific properties.
Use trigonometric ratios to find side lengths and angle measurements of right triangles.
Determine side or angle measurements using angles of elevation or depression.
Use Law of Sines to find sides and angles for triangles that are not right triangles.

Unit Essential Questions

How can we use the Pythagorean Theorem to solve real-world and mathematical problems involving right triangles?
How do trigonometric ratios (sine, cosine, and tangent) help us find unknown sides or angles in right triangles?
How do inverse trigonometric functions help us solve right triangles?
Why are special right triangles (30°-60°-90° and 45°-45°-90°) useful in solving problems without a calculator?

Lesson Essential Questions

How do you determine if a triangle is a right, acute, or obtuse triangle using side lengths?
What are some real-life scenarios where the Pythagorean Theorem is useful?
Why are special right triangles helpful shortcuts in problem-solving?
How do you determine which trigonometric ratio to use (sine, cosine, or tangent)?
How can you use trigonometric ratios to find a missing side or angle?
How can you use a diagram to set up and solve a right triangle problem?
When and why do we use the Law of Sines instead of other trigonometric methods?

CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence.
CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.
CC.2.3.HS.A.13 Analyze relationships between two-dimensional and three-dimensional objects.

Apply the interior angle sum formula to any polygon to find angle and side measurements.
Use the property of exterior angles of polygons to a sum of 360º.
Apply the properties of parallelograms to find unknown side and angle measurements.
Apply the properties of parallelograms to the remainder of the quadrilateral family consisting of rectangles, rhombi, and squares.
Classify the type of quadrilateral on the coordinate plane.
Explore the properties of trapezoids and kites.

Unit Essential Questions

Lesson Essential Questions

How does the interior angle sum formula for a polygon relate to triangle sum?
What role do diagonals play in understanding the structure of polygons?
What methods can be used to find the interior or exterior angle of a regular polygon?
How do interior and exterior angles relate to each other in polygons?
What are the properties of parallelograms?
How are parallelograms related to rectangles, rhombi, and squares?
What type of symmetry do the quadrilaterals have?
How do the properties of a trapezoid differ from an isosceles trapezoid?
How do the sides and diagonals in a kite differ from the other quadrilaterals?

Graph and label figures and its image under a given translation on the coordinate plane.
Graph and label figures and its image under a reflection in the given line on the coordinate plane.
Identify lines of reflection used to map a preimage to its image.
Graph and label figures and its image under the given rotation about the origin or other fixed point as the center.
Graph and label figures and its image under a dilation with a given scale factor and the origin as the center or a fixed point as the center.
Identify the center and scale factor given dilation.
Determine if figures have a line, point, and/or rotational symmetry.

Unit Essential Questions

How do translations affect the properties and position of geometric figures, and how can we describe and predict these changes using mathematical rules?
Where do we see reflections in the real world, and how does understanding them help us interpret or design those situations?
How do rotations transform geometric figures, and how can we describe and predict their effects using mathematical reasoning?
How do dilations affect the size, shape, and position of geometric figures, and what relationships remain constant under dilation?

Lesson Essential Questions

How does translating a figure affect its position and properties on the coordinate plane?
How does reflecting a figure across a line affect its position and orientation?
What information is needed to accurately perform a rotation? (center, angle, direction)
How do you describe a 90°, 180°, or 270° rotation on a coordinate plane?
What happens to the size and shape of a figure when it is dilated with a scale factor greater than 1 or less than 1?
Why is the center of dilation important in determining the location of the image?
How does a line of symmetry show congruence within a shape?

CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.
CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.
CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.
CC.2.3.HS.A.13 Analyze relationships between two-dimensional and three-dimensional objects.
CC.2.3.HS.A.14 Apply geometric concepts to model and solve real world problems.

Know

Understanding/Key Learning

Understanding the relationship between a circle's radius and its key measurements—diameter, circumference, and area—is fundamental to real-world problems.
The position of the vertex of an angle and the arc(s) it intercepts on a circle will determine how we calculate measurements of unknown angles and segment lengths.
Understanding the standard form equation for a circle and the geometric features each part represents.

Do

Identify and name parts of circles with symbolic notation.
Calculate the area and circumference of a circle.
Use the properties of central angles to find unknown angles and arcs of circles.
Calculate the arc length which is a portion of the circle's circumference.
Using congruent chord and arc properties of a circle, find segment and angle measures using trigonometry functions.
Use the properties of inscribed angles to find the measurement of the angle and its corresponding arc.
Calculate the tangent line of a circle using the Pythagorean Theorem.
Determine the measure of angles and arcs formed by intersecting lines in circles.
Calculate the segment length in circles formed by intersecting lines.
Write the equation of a circle in standard form given the center and radius, two endpoints, or a graph.
Graph circles given the standard form of an equation for a circle.

Unit Essential Questions

How can we use formulas to find the area and circumference of circles, and why are these measurements important in real-world contexts?
How do the relationships between angles and arcs in a circle help us understand and solve geometric problems?
How can we use the properties of segments in circles to solve problems involving lengths and relationships within and around circles?
How can we use the standard equation of a circle to identify and describe its center and radius, and solve problems involving points on or inside the circle?

Lesson Essential Questions

What is the significance of π (pi) in understanding circles?
How are the radius, diameter, and circumference of a circle related?
How do we find the area and circumference of a circle, and what do these measurements represent?
How do central angles, arcs, and chords help describe parts of a circle?
How does arc length relate to the circumference of a circle?
How is the measure of an inscribed angle related to the arc it intercepts?
What happens when multiple inscribed angles intercept the same arc?
How do inscribed angles help us understand the properties of cyclic (inscribed) quadrilaterals?
What is the relationship between a tangent and a radius at the point of contact?
How are angles formed by chords, tangents, and secants related to arcs?
How can you identify the center and radius of a circle by looking at its equation?
How do you graph a circle given its equation?
In what ways can the equation of a circle be used to solve problems involving distance and location?

Materials/Resources

Vocabulary

Circle
Radius
Chord
Diameter
Secant
Tangent
Point of Tangency
Area of a Circle
Circumference
Central Angles
Arc
Arc Measures
Arc Length
Semicircle
Minor Arc
Major Arc
Inscribed Angles
Intercepted Arc
Inscribed Quadrilaterals
Pythagorean Theorem
Circumscribed
Multiplying Binomials
Factoring
Quadratic Equations
Standard Form of Circle
General Form of Circle
Completing the Square

CC.2.3.HS.A.1 Use geometric figures and their properties to represent transformations in the plane.
CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence.
CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.
CC.2.3.HS A.6 Verify and apply theorems involving similarity as they relate to plane figures.
CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.
CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.
CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.
CC.2.3.HS.A.12 Explain volume formulas and use them to solve problems.
CC.2.3.HS.A.13 Analyze relationships between two-dimensional and three-dimensional objects.
CC.2.3.HS.A.14 Apply geometric concepts to model and solve real world problems.

Know

Understanding/Key Learning

Understand that area represents the amount of space within a plane figure, and that complex shapes (composite figures and regular polygons) can be broken down into simpler geometric components (like rectangles, triangles, or circles).
All 3-dimensional objects have a surface area; however, lateral surface area must be removed for overlapping faces.
All 3-dimensional objects have a volume which is used in real-world applications such as filling a pool with water or saving space when shipping a package.
Understanding that similar figures and solids have the same shape but different sizes allows us to compare corresponding sides (or dimensions, areas, and volumes) with proportions.

Do

Calculate the area of plane figures given the formula.
Determine the area of a sector (portion of the whole circle area).
Given a diagram, calculate the area of composite figures and area of shaded regions of composite figures.
Calculate the area of regular polygons using trigonometric functions.
Calculate the surface area of prisms, cylinders, pyramids, spheres, hemispheres, and cones.
Calculate the volume of prisms, cylinders, pyramids, spheres, hemispheres, and cones.
Determine the density of a 3-dimensional object.
Compare area, volume, and surface area of similar figures and similar solids.
Determine the effects of changing a dimension of a 3-dimensional object.

Unit Essential Questions

How can we use formulas and reasoning to find the area of plane and composite figures in real-world and mathematical contexts?
How does understanding surface area help us design and build objects in the real world, like packaging, buildings, or art?
What strategies can we use to find the volume of prisms and pyramids, and why is this important?
How can we determine if two figures are similar, and what does similarity tell us about their side lengths, angles, and proportions?
What relationships exist between the dimensions, surface areas, and volumes of similar solids?

Lesson Essential Questions

How does the formula for area relate to the properties of a shape (like base, height, radius)?
How can we calculate and apply the area of sectors?
What is a composite figure, and how can we break it down into simpler shapes?
How can we use area formulas of basic shapes to find the area of a composite figure?
What strategies can we use to subtract areas when there are missing or hollow parts?
How can we find the area of a regular polygon in more than one way?
Why do we use different formulas for different 3D figures?
How can calculating surface area be useful in real-world contexts (e.g., packaging, painting, wrapping)?
How do we calculate the volume of prisms, cylinders, pyramids, cones, and spheres?
What is density, and how does it relate mass and volume in 3D objects?
How does changing one dimension of a shape affect its area?
How can we compare the volume of different shapes with similar dimensions?