Geometry

Title: Geometry

Target Audience: High school students (grades 9–10) who have successfully completed Algebra I or an equivalent introductory algebra course.

Prerequisites:

• Algebra I
• Comfort with basic equations, graphing, and logical reasoning
• Curiosity about shapes, patterns, and spatial relationships

Course Summary

Geometry is the art and science of shapes, space, and structure — from the perfect symmetry of snowflakes to the soaring arcs of bridges. In this course, you’ll explore the elegant rules that govern our visual world and learn how to reason like a mathematician.

Through a blend of interactive simulations (MicroSims), hands-on geometric constructions, and real-world design challenges, you’ll discover how geometric ideas explain both natural forms and human-made creations — from honeycombs to roller coasters, from kaleidoscopes to architecture.

You’ll use dynamic geometry tools and visual MicroSims to:

• Move shapes and watch how transformations preserve (or change) them.
• Explore triangles, circles, and solids in 3D.
• Model real-world objects like ramps, wheels, or prisms.
• Use coordinate geometry to bridge algebra and geometry.

By the end of the course, you’ll not only know why geometric relationships are true — you’ll be able to prove them and use them creatively to solve meaningful design and modeling problems.

Topics Covered

Unit 1: Foundations of Geometry

Focus: Points, lines, planes, reasoning, and proof.

• Geometric terms and notation
• Building arguments with inductive and deductive reasoning
• Measuring and classifying angles
• Constructing figures using compass and straightedge
• Exploring logic with interactive Proof Builder MicroSim

Example MicroSim: A “Logic Flow Game” where students drag and connect reasoning statements to build valid proofs.

Unit 2: Transformations and Congruence

Focus: Motions that preserve shape and size.

• Translations, rotations, reflections
• Compositions of transformations
• Defining congruence in terms of rigid motion
• Triangle congruence (SSS, SAS, ASA, AAS, HL)

Example MicroSim: An animation where students move a triangle with sliders to test which transformations keep it congruent.

Unit 3: Parallel and Perpendicular Lines

Focus: Relationships between lines and angles.

• Angle pairs formed by transversals
• Proving lines are parallel or perpendicular
• Using coordinates to verify geometric properties

Example MicroSim: A “Line Explorer” where students adjust slopes and see parallel and perpendicular relationships update in real time.

Unit 4: Triangles and Polygons

Focus: Properties, relationships, and inequalities.

• Properties of isosceles, equilateral, and right triangles
• Triangle inequality theorem
• Interior and exterior angles of polygons
• Regular polygons and tessellations

Example MicroSim: A “Polygon Tiling Lab” that lets students create patterns using regular polygons and test which ones tile perfectly.

Unit 5: Similarity and Trigonometry

Focus: Scaling, proportional reasoning, and right triangle relationships.

• Similarity transformations and proofs
• Indirect measurement
• Right triangles, Pythagorean theorem
• Trigonometric ratios: sine, cosine, tangent
• Using trig to solve elevation, shadow, and distance problems

Example MicroSim: A “Climbing the Mountain” simulator — adjust angle and distance sliders to find height using trigonometry.

Unit 6: Circles

Focus: Tangents, arcs, angles, and equations.

• Parts of a circle: chords, tangents, secants
• Inscribed and central angles
• Arc length and area of sectors
• Equations of circles in the coordinate plane

Example MicroSim: A “Rolling Wheel” animation showing how arc length connects rotation to distance traveled.

Unit 7: Area and Volume

Focus: Measurement and visualization in two and three dimensions.

• Area of polygons and circles
• Surface area and volume of prisms, pyramids, cones, and spheres
• Cavalieri’s Principle and cross-sections

Example MicroSim: A “Shape Slicer” visualization where students drag a plane through a 3D solid to see cross-sections form dynamically.

Unit 8: Coordinate Geometry

Focus: Connecting algebra and geometry on the Cartesian plane.

• Distance, midpoint, and slope formulas
• Partitioning segments
• Equation of a circle
• Parallel and perpendicular line conditions

Example MicroSim: A “Graph Motion” lab where students drag points to dynamically update distances and line equations.

Unit 9: Modeling with Geometry

Focus: Applying geometric ideas to the real world.

• Designing with scale drawings
• Optimizing packaging, structures, and materials
• Modeling physical shapes and curves
• Using geometry in art, engineering, and architecture

Example MicroSim: A "Bridge Design Challenge" where students balance material use and strength by adjusting beam geometry.

Topics NOT Covered

To maintain focus on high school geometry fundamentals, this course does not include:

• Advanced trigonometry (law of sines, law of cosines beyond introduction)
• Analytic geometry beyond basic coordinate plane work
• Non-Euclidean geometry (hyperbolic, spherical)
• Formal axiomatic systems (Hilbert's axioms, Euclid's postulates in depth)
• Vector geometry and linear transformations
• Calculus-based geometry (derivatives of curves, optimization with calculus)
• Advanced solid geometry (polyhedra theory, Euler's formula beyond basics)
• Projective geometry and perspective drawing theory
• Fractal geometry and chaos theory
• Differential geometry

Learning Objectives (Grouped by Bloom's Taxonomy)

**Remembering

• Recall geometric definitions and notation for points, lines, planes, and angles.
• Identify properties of triangles, circles, and polygons.
• List the formulas for perimeter, area, surface area, and volume.

Understanding

• Explain how rigid motions define congruence.
• Describe relationships between parallel and perpendicular lines.
• Interpret how transformations affect shapes and their coordinates.
• Classify triangles and polygons based on their properties.

Applying

• Use transformations to test congruence and similarity.
• Apply the Pythagorean theorem to solve distance and height problems.
• Use trigonometric ratios to find unknown sides and angles.
• Compute the area and volume of complex composite shapes.

Analyzing

• Compare and contrast congruent and similar figures.
• Deconstruct a geometric proof into its logical steps.
• Examine relationships among angles, arcs, and chords in circles.
• Analyze cross-sections and 3D models to understand structure.

Evaluating

• Critique geometric arguments for logical soundness.
• Determine when a geometric model is appropriate for a real-world situation.
• Assess design efficiency in geometric modeling problems.
• Evaluate measurement error and precision in geometric constructions.

Creating

• Design and build geometric models using simulations or physical materials.
• Create original proofs for geometric relationships.
• Develop real-world solutions using geometric modeling (e.g., optimal ramp angle, packaging design).
• Construct animations or interactive MicroSims demonstrating geometric concepts.

Course Experience

Students will engage in a mix of collaborative projects, visual proofs, and interactive MicroSims, including:

• A Virtual Compass Lab for digital constructions.
• A Proof Playground for drag-and-drop reasoning.
• A Trigonometry Simulator for exploring height and distance.
• A 3D Shape Explorer with rotating solids and volume calculations.

The emphasis is on exploration, reasoning, and creativity — making geometry both a rigorous mathematical discipline and an inspiring artistic journey.

Standards

This course was designed around the Common Core State Standards for Mathematics (CCSSM), High School — Geometry Domain Published by the National Governors Association (NGA) and Council of Chief State School Officers (CCSSO) in 2010.

Common Core State Standards for Mathematics: High School — Geometry. National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010. https://www.corestandards.org/Math/Content/HSG