Spiral Geometry: Building a Theodorus Wheel
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Spiral Geometry: Building a Theodorus Wheel

Grade 8Math3 days
★★★★★
5.0 (1 rating)
In the 'Spiral Geometry: Building a Theodorus Wheel' project, 8th-grade students explore geometric transformations such as dilations, translations, rotations, and reflections by constructing a Theodorus wheel. Through hands-on activities and the use of coordinate geometry tools, students develop a deep understanding of geometric properties, symmetry, and patterns. The project includes interactive entry events, like using virtual reality and engaging in a mathematical race, to spark curiosity and real-world connections. Ultimately, students create and display their Theodorus wheels in a gallery, reflecting on their learning journey and the transformations they applied.
Geometric TransformationsTheodorus WheelCoordinate GeometrySymmetryDilationsTranslationsRotations
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we use the principles of dilations, translations, rotations, and reflections to construct a Theodorus wheel, and what does this reveal about the patterns and properties within our geometric designs?

Essential Questions

Supporting questions that break down major concepts.
  • What are the properties and characteristics of a Theodorus wheel?
  • How do dilations, translations, rotations, and reflections affect the design and structure of geometric figures?
  • In what ways can coordinating geometry be used to create and analyze patterns?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Students will understand and apply the concepts of dilations, translations, rotations, and reflections in constructing geometric figures.
  • Students will be able to identify and describe the properties of a Theodorus wheel and its components.
  • Students will develop skills in using coordinate geometry to create and analyze patterns within geometric figures.
  • Students will apply their knowledge of geometric transformations to explore symmetry and similarity in the context of a Theodorus wheel.

Common Core Standards

8.G.A.3
Primary
Describe the effect of dilations, translations, rotations, and reflections on two dimensional figures using coordinatesReason: The project involves creating and analyzing a Theodorus wheel, which directly relates to understanding and applying transformations such as dilations, translations, rotations, and reflections.
8.G.A.1
Secondary
Verify experimentally the properties of rotations, reflections, and translationsReason: Students can experiment with these transformations while constructing their Theodorus wheel to understand their geometric properties.
8.G.B.5
Supporting
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.Reason: Understanding angles and similarity is essential for the construction and analysis of the Theodorus wheel, as it involves connecting right triangles.

Entry Events

Events that will be used to introduce the project to students

The Digital Landscape

Students will interact with virtual reality headsets to visualize a digital landscape where shapes and figures can be transformed. This immersive experience challenges students’ understanding of dilations, translations, and reflections, making them curious about how these transformations can be applied to real-world objects, encouraging inquiry into technology and geometry.

The Mathematical Race

Kick off the project with a race using bicycle wheels of different sizes, challenging students to calculate speed and distance covered by each wheel. This physical demonstration of geometry in action immediately engages students and introduces concepts of rotations and dimensions, providing them numerous inquiries to explore how geometry relates to movement and measurements.
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Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Reflective Transformations Gallery

Students create a gallery exhibition of the Theodorus wheel, incorporating elements of reflection and rotation to analyze the final pattern they've created.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Guide students to finalize their constructed Theodorus wheel by adding reflective symmetry and exploring rotational patterns.
2. Set up a gallery space where students can display their wheels, highlighting unique features and geometric transformations.
3. Facilitate a gallery walk, encouraging peer feedback and discussion on how reflections and rotations are showcased in the displayed works.
4. Conclude with reflective essays where students write about their learning journey through these transformations, focusing on personal discoveries in symmetry and geometric design.

Final Product

What students will submit as the final product of the activityA public gallery of Theodorus wheels, accompanied by reflective essays that encapsulate students' understanding of geometric transformations and their impact on design.

Alignment

How this activity aligns with the learning objectives & standardsThis final activity addresses 8.G.A.3 by examining the holistic application of transformations—rotations and reflections particularly—in producing the Theodorus wheel and reflects upon 8.G.A.1 through the gallery discussion and essays.
Activity 2

Theodorus Wheel Blueprinting Workshop

As part of building the Theodorus wheel, students create precise blueprints using coordinate geometry principles. They will apply transformations such as dilations and translations to strategize their design.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce the concept of the Theodorus wheel and its historical significance in mathematics.
2. Provide students with graph paper and coordinate geometry tools to start designing their Theodorus wheel using basic right triangles.
3. Guide students to apply dilations and translations to expand their design blueprint, ensuring each triangle in the wheel follows this transformation pattern.
4. Encourage peer review sessions where students present their blueprints for feedback and make necessary adjustments based on feedback.

Final Product

What students will submit as the final product of the activityA detailed and precise blueprint of a Theodorus wheel, showcasing understanding of dilations, translations, and applying coordinate geometry.

Alignment

How this activity aligns with the learning objectives & standardsThis activity is aligned with 8.G.A.3 as it involves using coordinates to describe and execute dilations and translations on the designs, as well as 8.G.B.5 when students explore angles and similarity in triangle patterns.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Geometric Transformations and Design Rubric

Category 1

Understanding of Geometric Transformations

Assess students' comprehension and application of dilations, translations, rotations, and reflections in their Theodorus wheel designs.
Criterion 1

Application of Transformations

Evaluates how effectively students apply geometric transformations, including dilations, translations, rotations, and reflections, to their designs.

Exemplary
4 Points

Demonstrates sophisticated application of geometric transformations with innovative approaches in creating a Theodorus wheel.

Proficient
3 Points

Demonstrates thorough application of geometric transformations with accurate execution in creating a Theodorus wheel.

Developing
2 Points

Demonstrates emerging understanding of geometric transformations with some inaccuracies in creating a Theodorus wheel.

Beginning
1 Points

Demonstrates minimal understanding of geometric transformations with significant inaccuracies in creating a Theodorus wheel.

Criterion 2

Identification of Properties

Measures students' ability to identify and describe the properties of the Theodorus wheel, including symmetry, angles, and patterns.

Exemplary
4 Points

Provides comprehensive and insightful descriptions of the properties and patterns in the Theodorus wheel, highlighting symmetry and angles.

Proficient
3 Points

Provides clear descriptions of the properties and patterns in the Theodorus wheel, with accurate identification of symmetry and angles.

Developing
2 Points

Provides basic descriptions of the properties and patterns in the Theodorus wheel, with partial identification of symmetry and angles.

Beginning
1 Points

Provides minimal descriptions of the properties and patterns in the Theodorus wheel, struggling to identify symmetry and angles.

Category 2

Design and Craftsmanship

Evaluate the precision and creativity in the designs and execution of Theodorus wheel blueprints and final products.
Criterion 1

Precision and Accuracy

Assesses the precision and accuracy in students' blueprints and final Theodorus wheel products.

Exemplary
4 Points

Demonstrates outstanding precision and accuracy in design, with detailed and flawless execution in blueprints and final products.

Proficient
3 Points

Demonstrates consistent precision and accuracy in design, with only minor errors in blueprints and final products.

Developing
2 Points

Shows some precision and accuracy in design, with notable errors in blueprints and final products.

Beginning
1 Points

Shows limited precision and accuracy in design, with frequent errors in blueprints and final products.

Criterion 2

Creative Integration

Evaluates the creative integration of geometric transformations in students' designs and gallery displays.

Exemplary
4 Points

Exhibits innovative and aesthetically pleasing integration of transformations in designs and gallery displays.

Proficient
3 Points

Shows solid and aesthetically pleasing integration of transformations in designs and gallery displays.

Developing
2 Points

Integrates transformations in designs and gallery displays with some creative effort, but lacks aesthetic consistency.

Beginning
1 Points

Struggles to integrate transformations creatively or aesthetically in designs and gallery displays.

Category 3

Reflective Insight

Assesses the depth of insight and reflection in students' essays on their learning journey with geometric transformations and design.
Criterion 1

Reflective Depth

Measures the depth and quality of reflection demonstrated in students' essays.

Exemplary
4 Points

Provides profound and insightful reflections with detailed personal discoveries and learning experiences.

Proficient
3 Points

Offers clear and thoughtful reflections with detailed personal discoveries and learning experiences.

Developing
2 Points

Presents basic reflections with some personal discoveries, but lacks depth and detail.

Beginning
1 Points

Provides minimal reflections with limited personal discoveries or exploration.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

How did creating a Theodorus wheel enhance your understanding of geometric transformations such as dilations, translations, rotations, and reflections?

Text
Required
Question 2

Which transformation did you find most challenging to apply in your design, and how did you overcome it?

Text
Required
Question 3

On a scale of 1 to 5, how confident are you in applying geometric transformations to different figures after this project?

Scale
Required
Question 4

In what ways did peer feedback influence your project outcome?

Text
Optional
Question 5

Which essential question from the project do you feel was the most important in guiding your learning, and why?

Multiple choice
Required
Options
What are the properties and characteristics of a Theodorus wheel?
How do dilations, translations, rotations, and reflections affect the design and structure of geometric figures?
In what ways can coordinating geometry be used to create and analyze patterns?