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Course Details:
Course Details:
Grades: 9-12
Length: two semesters
Credit: 1 (0.5 per semester)
Prerequisite: Algebra 1 or teacher recommendation
Course Description:
Course Description:
The fundamental purpose of the course in Geometry is to formalize and extend students’ geometric experiences using more precise definitions and developing careful proofs. In Geometry, students take the basic distance and angle-preserving properties of rigid motions and similarity transformations as axiomatic, establish triangle congruence, and similarity criteria, then use them to prove a wide variety of theorems and solve problems involving, for example, triangles, other polygons, and circles.
Students study geometric measurement and solve problems involving length, area and volume, learning more sophisticated arguments for the circumference, area, and volume formulas that they learned in earlier grades. They use similarity of right triangles with given angle measures to define sine, cosine, and tangent in terms of side ratios. They prove theorems and solve problems about circles, segments, angles, and arcs.
Throughout the course, students use coordinates to connect geometry with algebra, and engage in mathematical modeling using geometric principles.
Geometry Honors:
Students will master all of the topics from Geometry listed above, with a variety of additional topics. These additional topics (content objectives) are documented within each unit in RED under the “can be covered” section.
Mathematical Topics
Mathematical Topics
(Recommended Order)
Semester 1
Semester 1
Constructions
Rigid Transformations
Congruence
Similarity
Semester 2
Semester 2
Right Triangle Trigonometry
Solid Geometry
Coordinate Geometry
Circles
Course/ Grade Competencies
Course/ Grade Competencies
Semester 1
Semester 1
M1.9-12.1: The learner will write, apply, and provide a rationale for a mathematical model representing a given situation.
M1.9-12.2: The learner will interpret and use symbols to express relationships and justify reasoning when solving problems.
M2.9-12.1: The learner will justify how to apply properties of real number systems to variable expressions in a variety of contexts.
M3.9-12.3: The learner will compare the effectiveness or logic of two plausible arguments or models.
M5.9-12.1: The learner will apply properties of arithmetic and algebra to simplify and manipulate symbolic expressions or models.
M5.9-12.2: The learner will write and apply algebraic modes to represent and answer questions about a given situation.
M5.9-12.3: The learner will interpret, analyze, and use relations and functions applied in a variety of contexts, including real-world phenomena.
M6.9-12.1: The learner will apply geometric theorems and postulates to solve problems, create arguments, and support their reasoning.
M6.9-12.2: The learner will use geometric theorems and postulates to construct and apply viable arguments.
M6.9-12.3: The learner will create and use a formal geometric construction, using appropriate tools, to illustrate geometric properties.
Semester 2
Semester 2
M1.9-12.1: The learner will write, apply, and provide a rationale for a mathematical model representing a given situation.
M1.9-12.2: The learner will interpret and use symbols to express relationships and justify reasoning when solving problems.
M2.9-12.1: The learner will justify how to apply properties of real number systems to variable expressions in a variety of contexts.
M3.9-12.1: The learner will use computational strategies and algorithms and provide rationale for their use.
M3.9-12.2: The learner will reason quantitatively when analyzing, representing, and solving problems.
M3.9-12.3: The learner will compare the effectiveness or logic of two plausible arguments or models.
M4.9-12.1: The learner will provide rationale for solving measurement problems that require making conversions among various units and measurement systems, or applying the effect of a scale factor.
M5.9-12.1: The learner will apply properties of arithmetic and algebra to simplify and manipulate symbolic expressions or models.
M5.9-12.2: The learner will write and apply algebraic modes to represent and answer questions about a given situation.
M5.9-12.3: The learner will interpret, analyze, and use relations and functions applied in a variety of contexts, including real-world phenomena.
M6.9-12.1: The learner will apply geometric theorems and postulates to solve problems, create arguments, and support their reasoning.
M6.9-12.2: The learner will use geometric theorems and postulates to construct and apply viable arguments.
M6.9-12.3: The learner will create and use a formal geometric construction, using appropriate tools, to illustrate geometric properties.
Constructions
Constructions
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Know and be able to use precise definitions of geometric terms.
Make formal geometric constructions.
Construct an equilateral triangle, a square, an angle bisector, and a perpendicular bisector.
Can be Covered:
The learner will:
Construct a regular polygon.
Standards:
Standards:
AKSS
G-CO.1, G-CO.12,
G-CO.13
Mathematical Practices
All mathematical practices are present in each unit.
Rigid Transformations
Rigid Transformations
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M3.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Develop and apply precise definitions for translations, reflections, and rotations.
Describe rigid motions that take one figure onto another.
Prove angles of a triangle add up to 180 degrees.
Construct and apply a sequence of rigid motions.
Standards:
Standards:
AKSS
G-CO.2, G-CO.3,
G-CO.4, G-CO.5
Mathematical Practices
All mathematical practices are present in each unit.
Congruence
Congruence
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M3.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Specify sequences of rigid motion that will carry a figure onto another.
Understand that there can be more than one sequence of rigid motion that carries a figure onto another figure.
Use the definition of congruence in terms of rigid motion to decide if two figures are congruent.
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Be able to explain how the criteria for triangle congruence (ASA, SAS, SSS) follow from the definition of congruence in terms of rigid motion.
Prove theorems about triangles, lines, angles, and parallelograms.
Can be Covered:
The learner will:
Prove congruence for quadrilaterals.
Develop an understanding of AAS triangle congruence.
Standards:
Standards:
AKSS
G-CO.6, G-CO.7,
G-CO.8, G-SRT.4
Mathematical Practices
All mathematical practices are present in each unit.
Similarity
Similarity
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M2 – Numbers and Number Systems: The learner will develop an applied knowledge of numbers and number systems to solve problems.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M5 – Algebraic Functions, Patterns and Relations: The learner will utilize patterns, relations, and functions to compare, interpret, and analyze situations.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M2.9-12.1
M3.9-12.3
M5.9-12.1
M5.9-12.2
M5.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Understand dilations, using center and scale factor to describe them.
Use the definition of similarity to decide if two figures are similar.
Use the properties of similarity to understand the Angle Similarity Theorem.
Prove why triangles are similar.
Prove why all circles are similar.
Find unknown side lengths of similar triangles.
Apply similarity to quadrilaterals.
Can be Covered:
The learner will:
Understand Side Side Side and Side Angle Side Triangle Similarity Theorem.
Prove the Law of Sines and Cosines and use them to solve problems.
Standards:
Standards:
AKSS
G-SRT.1, G-SRT.2,
G-SRT.3, G-SRT.4,
G-SRT.5, G-C.1
Mathematical Practices
All mathematical practices are present in each unit.
Right Triangle Trigonometry
Right Triangle Trigonometry
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M2 – Numbers and Number Systems: The learner will develop an applied knowledge of numbers and number systems to solve problems.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M4 - Measurement: The learner will explain reasoning when applying and modeling geometric principles.
M5 – Algebraic Functions, Patterns and Relations: The learner will utilize patterns, relations, and functions to compare, interpret, and analyze situations.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M2.9-12.1
M3.9-12.1
M3.9-12.2
M3.9-12.3
M4.9-12.1
M5.9-12.1
M5.9-12.2
M5.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Using similarity, show that side ratios in right triangles are properties of angles.
Define the trigonometric ratios (sin, cos, tan) for acute angles.
Explain and use the relationship between sine and cosine of complementary angles.
Use trigonometric ratios to solve a variety of modeling problems.
Can be Covered:
The learner will:
Understand relationships in special right triangles (30-60-90 & 45-45-90 triangles).
Standards:
Standards:
AKSS
G-SRT.6, G-SRT.7,
G-SRT.8
Mathematical Practices
All mathematical practices are present in each unit.
Solid Geometry
Solid Geometry
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M2 – Numbers and Number Systems: The learner will develop an applied knowledge of numbers and number systems to solve problems.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M4 - Measurement: The learner will explain reasoning when applying and modeling geometric principles.
M5 – Algebraic Functions, Patterns and Relations: The learner will utilize patterns, relations, and functions to compare, interpret, and analyze situations.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M2.9-12.1
M3.9-12.1
M3.9-12.2
M3.9-12.3
M4.9-12.1
M5.9-12.1
M5.9-12.2
M5.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Identify the shapes of two-dimensional cross-sections of three-dimensional objects and 3D objects from rotation of 2D shapes.
Understand the effects of dilation on area and volume.
Derive volume formulas using dissections and Cavalieri’s Principle.
Apply volume formulas to solve problems involving surface area to volume ratios and density.
Can be Covered:
The learner will:
Informally prove the formula for the volume of a sphere using Cavalieri’s principle.
Standards:
Standards:
AKSS
G-GMD.1, G-GMD.3,
G-GMD.4
Mathematical Practices
All mathematical practices are present in each unit.
Coordinate Geometry
Coordinate Geometry
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M2 – Numbers and Number Systems: The learner will develop an applied knowledge of numbers and number systems to solve problems.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M4 - Measurement: The learner will explain reasoning when applying and modeling geometric principles.
M5 – Algebraic Functions, Patterns and Relations: The learner will utilize patterns, relations, and functions to compare, interpret, and analyze situations.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M2.9-12.1
M3.9-12.1
M3.9-12.2
M3.9-12.3
M4.9-12.1
M5.9-12.1
M5.9-12.2
M5.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Describe functions as transformations using coordinate transformation notation.
Describe transformations in the coordinate plane.
Derive equations for circles using definition of a circle.
Derive equations for parabolas using focus and directrix.
Understand the relationship between an equation and the graph, especially for linear and quadratic equations.
Understand slopes of parallel and perpendicular lines in a coordinate plane.
Use coordinates to make conjectures and prove geometric theorems algebraically.
Can be Covered:
The learner will:
Derive the formula for an ellipse and hyperbolas.
Standards:
Standards:
AKSS
G-GPE.1, G-GPE.2,
G-GPE.4, G-GPE.5
Mathematical Practices
All mathematical practices are present in each unit.
Circles
Circles
Graduate-Level Competency:
Graduate-Level Competency:
M1 – Symbolic Expression: The learner will be able to reason abstractly and utilize symbolic expressions and mathematical models.
M2 – Numbers and Number Systems: The learner will develop an applied knowledge of numbers and number systems to solve problems.
M3 – Reasoning and Strategic Thinking: The learner will use evidence to support authentic application of concepts and support mathematical arguments.
M4 - Measurement: The learner will explain reasoning when applying and modeling geometric principles.
M5 – Algebraic Functions, Patterns and Relations: The learner will utilize patterns, relations, and functions to compare, interpret, and analyze situations.
M6 - Geometry: The learner will solve problems involving spatial reasoning and model geometric concepts in applied contexts.
Course/ Grade Competencies:
Course/ Grade Competencies:
M1.9-12.1
M1.9-12.2
M2.9-12.1
M3.9-12.3
M4.9-12.1
M5.9-12.1
M5.9-12.2
M5.9-12.3
M6.9-12.1
M6.9-12.2
M6.9-12.3
Content Objectives:
Content Objectives:
Must be Covered:
The learner will:
Use the Pythagorean Theorem to derive an equation for a circle given center and radius.
Use similarity to derive the length of the arc of the circle.
Derive a formula for the area of sector.
Describe the relationship between central and inscribed angles and their arcs.
Describe relationships and ratios of lengths of intersecting chords.
Use relationships about inscribed angles to solve problems about inscribed polygons.
Solve problems involving properties of circles.
Prove properties of angles of inscribed polygons.
Proves that a radius and a tangent to a circle at the same point are perpendicular.
Can be Covered:
The learner will:
Prove that an inscribed angle that subtends a diameter is a right angle.
Standards:
Standards:
AKSS
G-C.2, G-C.3,
G-C.5, G-GPE.1
Mathematical Practices
All mathematical practices are present in each unit.
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