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Created byAnge Evans

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Evolving Designs: A Semester-Long Geometry Portfolio Project

Grade 10Math14 days

In this semester-long project, 10th-grade students act as designers to build and refine an original aesthetic composition using the fundamental "DNA" of geometry. Starting with basic notation and moving through rigid motions and complex compass-and-straightedge constructions, students document the iterative evolution of their work in a comprehensive portfolio. The experience culminates in a formal "Architect’s Manifesto," where students apply geometric theorems and logical reasoning to prove the mathematical integrity and precision of their final design.

Geometric ConstructionsRigid MotionsMathematical ProofIterative DesignSymmetryVisual Arts Integration

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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we use the structural 'DNA' and logical precision of geometry to build, transform, and justify the evolution of an original aesthetic design?

Essential Questions

Supporting questions that break down major concepts.

How can we use the formal language and logic of geometry to build, transform, and refine an evolving aesthetic design?
How do fundamental geometric elements—like points, lines, and angles—serve as the structural 'DNA' of a complex visual composition?
How do rigid motions and symmetry allow us to manipulate and replicate shapes while preserving their mathematical properties?
In what ways does the precision of compass-and-straightedge constructions bridge the gap between an artistic idea and a mathematically accurate design?
How can geometric proofs and reasoning be used to justify the relationships and patterns found within our original creations?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:

Students will construct a foundational geometric design using precise terminology, including points, lines, segments, and angles, to serve as the structural 'DNA' of their semester-long project.
Students will demonstrate mastery of rigid motions—translations, rotations, and reflections—by transforming their original designs while preserving congruence and identifying lines of symmetry.
Students will execute complex geometric constructions, such as bisecting angles, creating parallel/perpendicular lines, and inscribing/circumscribing circles, to add layers of mathematical complexity to their artwork.
Students will synthesize geometric reasoning and proof to justify the relationships within their designs, explaining how specific properties (e.g., vertical angles or parallel line theorems) are manifested in their work.
Students will maintain an iterative portfolio that documents the evolution of their geometric thinking, reflecting on how new mathematical concepts allow for more sophisticated design choices.

Common Core State Standards for Mathematics

HSG-CO.D.12

Primary

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Reason: This standard is the core of the project, as students must use constructions to build their portfolio design throughout the semester.

HSG-CO.A.2

Primary

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., rigid motion versus dilations).Reason: Students are required to use rigid motions and transformations to evolve their design in Unit 2.

HSG-MG.A.3

Primary

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).Reason: The entire project is a design-based task where geometry provides the constraints and the structural framework for the artistic output.

HSG-CO.A.1

Secondary

Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Reason: This provides the 'basic language' foundation mentioned in Unit 1, ensuring students use correct terminology in their portfolio.

HSG-CO.D.13

Secondary

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.Reason: Aligns with the Unit 4 requirement to inscribe regular polygons as part of the advancing design complexity.

HSG-C.A.3

Supporting

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Reason: Supports the Unit 4 focus on advanced constructions involving circles and triangles within the student's design.

HSG-CO.A.5

Supporting

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Reason: Supports the practical application of transformations in the design evolution during Unit 2.

Entry Events

Events that will be used to introduce the project to students

Project: Neo-Metropolis Foundation

Students receive a mysterious parcel of 'digital land' and a brief from a futuristic urban planning committee looking for a new city hub. They must draft a 'Foundation Blueprint' using basic geometric primitives, knowing that the city’s survival depends on their ability to later add symmetrical districts, complex intersections, and inscribed sanctuaries as their geometric knowledge expands.

The 'Legacy Brand' Reveal

A local streetwear designer or branding expert visits (or sends a video) to explain that the most expensive logos and sneaker silhouettes are built on invisible 'geometric skeletons.' Students are tasked with creating their own 'Legacy Mark'—a simple geometric base that they will 'evolve' into a complex visual brand as they unlock more advanced math throughout the semester.

The 'Infinite Seed' Mural Challenge

Students are shown a gallery of historical masterpieces, from Islamic tiling to stained glass, alongside a 'blank' geometric kernel. They are challenged to plant an 'Infinite Seed'—a small, precise geometric core—and promised that by the end of the term, they will use transformations and constructions to grow this single seed into a massive, mathematically perfect mural.

The 'Level Up' Map Blueprint

Entering the role of Lead Level Designers for a new tactical strategy game, students must draft the 'Origin Map' for a multiplayer arena. They start with basic boundaries and lines of sight, but are told that to create 'Power-Up Zones' and 'Fortifications,' they will eventually need to master the secrets of inscribed circles and parallel constructions.

The 'Resilient Prototype' Portfolio

Students investigate 'Design Disasters' where simple geometric errors led to catastrophic failures in engineering and fashion. They are then challenged to create a 'Resilient Prototype' of a product of their choice, beginning with a simple geometric chassis that they will stress-test and refine using rigid motions and geometric proofs as the semester progresses.

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Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.

Activity 1

Activity 1: The Origin Blueprint (Building the DNA)

In this initial phase, students will establish the "Genetic Code" of their project. Using the 'Legacy Brand' or 'Neo-Metropolis' theme, students create a foundational geometric skeleton. This activity focuses on the precise placement and labeling of the most fundamental building blocks of geometry. Every element placed must be intentional and documented using correct mathematical notation.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Select your project theme (e.g., City Planning, Branding, or Mural Art).

2. On a clean plane, plot at least five distinct points that will serve as the vertices or anchors of your design.

3. Connect these points using a variety of lines, segments, and rays. Your design must include at least one pair of intersecting lines and one set of supplementary or complementary angles.

4. Incorporate exactly one circle and one arc, labeling the center and radius clearly.

5. Create a 'Geometric Key' document that lists every element you’ve drawn using proper notation (e.g., ∠ABC) and explains its geometric type.

Final Product

What students will submit as the final product of the activityA "Base Blueprint" on vellum or digital software, accompanied by a 'Geometric Key' (a glossary) that lists and defines at least 10 specific geometric elements used in the design (e.g., Ray AB, Obtuse Angle CDE, etc.).

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG-CO.A.1: Students must demonstrate the precise use of points, lines, segments, rays, and angles. It also touches on HSG-MG.A.3 by using these elements to solve an initial design problem (creating the "Origin Mark").

Activity 2

Activity 2: The Dynamic Echo (Transformational Growth)

Now that the foundation is set, students will "evolve" their design by applying rigid motions. This activity teaches students how to create complexity through repetition and movement without changing the size or shape of their original elements. This represents the 'growth' of the city or the 'expansion' of the brand.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Identify a specific 'sub-shape' or 'cluster' within your Activity 1 blueprint.

2. Perform a Translation: Move your sub-shape to a new location in the design to create a sense of rhythm. Document the vector of movement.

3. Perform a Reflection: Create a line of symmetry within your design and reflect a portion of your work across it to create balance.

4. Perform a Rotation: Rotate a geometric element 90 or 180 degrees around a fixed point to add dynamic energy.

5. Write a 'Congruence Justification'—a paragraph explaining why the transformed shapes are still congruent to the originals based on the properties of rigid motion.

Final Product

What students will submit as the final product of the activityAn "Evolution Map" showing the original 'Origin Mark' and its transformed versions. Each transformation must be accompanied by a 'Transformation Rule' written in function notation (e.g., T(x,y) -> (x+5, y-2)).

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG-CO.A.2 and HSG-CO.A.5: Students represent transformations in the plane and describe them as functions. They must show that rigid motions preserve distance and angle (congruence).

Activity 3

Activity 3: The Infrastructure Grid (Precise Constructions)

In this activity, students transition from 'sketching' to 'constructing.' Using a compass and straightedge (or digital tools), students will add structural 'infrastructure' to their design. This represents the roads of the metropolis or the grid system of a brand mark, requiring absolute precision that cannot be achieved by freehand drawing.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Select a major 'arterial' line segment from your current design.

2. Construct a Perpendicular Bisector of this segment to find the exact midpoint. Use this midpoint as the center for a new design element.

3. Choose a point not on your main line and construct a Parallel Line passing through that point. This creates 'lanes' or 'borders' in your design.

4. Construct an Angle Bisector for one of the angles created in Activity 1 to create a perfectly symmetrical 'sub-district' or 'logo detail.'

5. Leave all construction 'ghost marks' (the light pencil arcs) visible to prove the mathematical process.

Final Product

What students will submit as the final product of the activityA "Precision Overlay" added to the portfolio. This layer must show visible construction marks (arcs and intersecting lines) that prove the student used a compass and straightedge rather than a ruler to find midpoints and parallels.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG-CO.D.12: Making formal geometric constructions with a compass and straightedge. It specifically targets the construction of parallel and perpendicular lines.

Activity 4

Activity 4: The Geometric Sanctuary (Advanced Hubs)

Students will now add the 'Sanctuaries' or 'Focal Points' to their design. This activity focuses on the relationship between circles and polygons. Students will learn to nest shapes within one another, creating the intricate, high-level detail found in professional architecture and sacred geometry.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Create a 'Central Plaza' or 'Focal Hub' by constructing a large circle.

2. Inscribe a regular hexagon or a square inside this circle using only a compass and straightedge. This ensures perfect regularity and symmetry.

3. Identify a triangle within your design. Construct the 'Circumscribed Circle' that passes through all three vertices of that triangle.

4. Inscribe a circle within a different triangle (the Incenter), creating a 'nested' effect.

5. Color-code the different components (e.g., all inscribed shapes in blue, all circumscribed in red) to highlight the mathematical relationships.

Final Product

What students will submit as the final product of the activityThe "Master Geometric Hub"—a highly detailed section of the portfolio featuring at least one inscribed regular polygon and one circumscribed triangle, serves as the centerpiece of the design.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG-CO.D.13 and HSG-C.A.3: Constructing equilateral triangles, squares, and hexagons inscribed in circles, as well as circumscribed circles of triangles.

Activity 5

Activity 5: The Architect's Manifesto (Proof & Synthesis)

The final step of the portfolio is not a drawing, but a defense. Students must prove that their design is mathematically sound. They will choose one specific relationship in their design (like two lines being parallel or two triangles being congruent) and write a formal or informal proof to justify it. This connects the visual design back to logical geometric reasoning.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Select one pair of lines in your design that you constructed to be parallel. Write a brief 'Logical Chain' explaining how the construction steps guaranteed they are parallel.

2. Identify two shapes in your design that are congruent. State the Transformation Sequence that maps one to the other.

3. Choose one 'Geometric Property' discovered in your design (e.g., 'The sum of these two angles is 180 degrees') and explain why this must be true using a known theorem.

4. Reflect on the 'Evolution': How did moving from basic points/lines to complex constructions change the way you viewed your original design?

5. Assemble all 5 activities into a final 'Geometry Portfolio' binder or digital presentation.

Final Product

What students will submit as the final product of the activityThe "Architect’s Manifesto"—a formal written reflection and proof document. It includes a 'Proof of Integrity' where students use theorems (like Alternate Interior Angles or SSS Congruence) to justify a specific part of their design.

Alignment

How this activity aligns with the learning objectives & standardsAligns with HSG-MG.A.3: Applying geometric methods to solve design problems and justify results. It also integrates Unit 2's focus on proof and reasoning.

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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Geometry Portfolio: Design & Logic Rubric

Category 1

Geometry Portfolio Domains

These categories evaluate the student's ability to apply geometric principles to an evolving design project, focusing on technical precision, mathematical notation, and logical justification.

Criterion 1

Geometric Notation & Terminology (Activity 1)

Accuracy and application of geometric terminology, labeling, and notation in the 'Geometric Key' and throughout the 'Base Blueprint'.

Exemplary

4 Points

All elements (points, lines, rays, angles, etc.) are labeled with flawless mathematical notation (e.g., line segments vs. lines). The Geometric Key is exhaustive, providing precise definitions and intentional placement for more than 10 elements.

Proficient

3 Points

Most elements are labeled correctly using proper notation. The Geometric Key includes at least 10 elements with accurate definitions and identifies their roles in the design.

Developing

2 Points

Geometric elements are present, but notation is inconsistent (e.g., confusing an angle with a triangle). The Geometric Key is incomplete or contains several inaccuracies in definitions.

Beginning

1 Points

Minimal use of proper geometric notation. Labels are missing or incorrect, and the Geometric Key lacks required elements or definitions.

Criterion 2

Transformational Logic & Congruence (Activity 2)

Execution of rigid motions (translations, reflections, rotations) and the ability to represent them using function notation and congruence justifications.

Exemplary

4 Points

Transformations are executed with high precision and innovative aesthetic integration. Function notation is flawlessly applied to all movements, and the 'Congruence Justification' provides a sophisticated explanation of isometry.

Proficient

3 Points

At least one translation, reflection, and rotation are clearly executed. Each is accompanied by a correct transformation rule in function notation, and the congruence paragraph accurately explains why shapes remain identical.

Developing

2 Points

Transformations are present but may lack one type (e.g., missing a rotation). Function notation is attempted but contains errors in coordinates or symbols. Justification of congruence is vague.

Beginning

1 Points

Transformations are poorly defined or do not preserve congruence. Function notation is missing, and no justification for the relationship between shapes is provided.

Criterion 3

Construction Accuracy & Process (Activity 3)

The technical skill in using a compass and straightedge to create bisectors, parallel lines, and perpendicular lines, evidenced by 'ghost marks'.

Exemplary

4 Points

Constructions are perfectly precise with all 'ghost marks' (arcs and intersections) preserved as evidence of process. The infrastructure grid shows advanced layering of parallel and perpendicular elements that exceed basic requirements.

Proficient

3 Points

Compass and straightedge are used correctly to construct parallel/perpendicular lines and bisectors. Construction marks are visible and prove the mathematical method rather than 'eyeballing' with a ruler.

Developing

2 Points

Required constructions are attempted but show lack of precision (e.g., lines are not perfectly parallel). Some construction marks are missing, making the process difficult to verify.

Beginning

1 Points

Design elements appear free-handed or drawn with a ruler only. No evidence of compass-and-straightedge process (no arcs) is visible.

Criterion 4

Advanced Hubs & Circular Relationships (Activity 4)

Mastery of complex circular constructions, including inscribed regular polygons and circumscribed/inscribed circles for triangles.

Exemplary

4 Points

Successfully inscribes multiple complex regular polygons and constructs both circumscribed and inscribed circles with perfect accuracy. Color-coding is used effectively to illuminate complex mathematical relationships.

Proficient

3 Points

Accurately constructs one inscribed regular polygon and one circumscribed circle. The relationship between the circle and the vertices/sides is clearly demonstrated through precise points of tangency or intersection.

Developing

2 Points

Attempted inscribed or circumscribed shapes, but they do not meet the mathematical requirements (e.g., the circle does not touch all vertices). Technical execution with the compass is shaky.

Beginning

1 Points

Inscribed or circumscribed elements are missing or do not follow geometric principles. Polygons are not regular, or circles are placed arbitrarily.

Criterion 5

Logical Reasoning & Proof (Activity 5)

The ability to synthesize geometric theorems to justify the structural integrity and properties of the final design.

Exemplary

4 Points

Provides a sophisticated 'Architect’s Manifesto' with a formal logical chain for multiple design features. Correctly applies advanced theorems (e.g., Parallel Line Theorems, Congruence Postulates) to prove design integrity.

Proficient

3 Points

Successfully writes a 'Proof of Integrity' using at least one theorem to justify a relationship in the design (e.g., proving lines are parallel). Reflection shows clear connection between math and design choices.

Developing

2 Points

The 'Architect’s Manifesto' is present but the logical reasoning is circular or lacks specific theorem references. Reflection on the evolution of the design is surface-level.

Beginning

1 Points

Manifesto is missing or provides no mathematical justification for the design. No attempt is made to link geometric theorems to the visual work.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning

Question 1

How did the transition from 'freehand sketching' to 'mathematical construction' change the way you perceived your design's evolution? Did the constraints of geometry limit your creativity or provide a new kind of structure for it?

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Required

Question 2

To what extent do you now see geometry as a foundational 'DNA' for design rather than just a collection of abstract formulas?

Scale

Required

Question 3

Which specific unit of geometry do you feel was the most transformative for the aesthetic quality of your project?

Multiple choice

Required

Options

Activity 1: Basic Elements (Points, Lines, Angles)

Activity 2: Transformations (Rigid Motions & Symmetry)

Activity 3: Infrastructure (Parallel & Perpendicular Constructions)

Activity 4: Advanced Hubs (Inscribed & Circumscribed Circles)

Activity 5: The Manifesto (Logical Proofs & Synthesis)

Question 4

How does being able to mathematically 'prove' the relationships in your design (like congruence or parallelism) change the way you view the final product compared to a project that is purely artistic?

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Optional