Lesson Title: Wobbledog Biomechanics & Robotic Prototyping

Materials Needed:

• Access to a computer with the game Wobbledogs (www.wobbledogs.com)
• Lego Education Spike Prime set
• Notebook or digital document for calculations and notes
• Pencil and ruler (or digital equivalent for measuring on-screen)
• AoPS Pre-Algebra textbook (for reference on Pythagorean Theorem)
• AoPS Introduction to Algebra textbook (for reference on variables and expressions)

Subject Area Integration:

Geometry, Algebra, Robotics, Engineering Design, and Digital Literacy

Guiding Question:

How can we use mathematical principles like the Pythagorean theorem and algebra to analyze a digital creature and build a physical, robotic model that mimics its structure?

Learning Objectives:

By the end of this lesson, you will be able to:

1. Apply the Pythagorean theorem to calculate unknown distances in a real-world (or digital-world!) context.
2. Translate mathematical calculations into a physical design using the Lego Spike Prime system.
3. Use algebraic variables to create a general model that can adapt to changes.
4. Document and explain the process of moving from analysis to a functional prototype.

Lesson Activities

Part 1: The Wobbledog Biomechanics Lab (Approx. 45 minutes)

Goal: To find and apply a right triangle in your Wobbledog's anatomy and use the Pythagorean theorem to analyze its structure.

1. Observation & Data Collection:
   • Open Wobbledogs and find a dog with a relatively stable, clear stance.
   • Take a screenshot of your dog from a side-on view. Paste this image into a document or just have it open for reference.
   • Identify the Triangle: Look at your dog's legs and body. Imagine a right triangle formed by its stance. A great example is:
     • Leg 'a': The vertical distance from the ground to where the leg attaches to the body (the "height").
     • Leg 'b': The horizontal distance from where the paw touches the ground to a point directly under the leg's attachment to the body.
     • Hypotenuse 'c': The actual length of the dog's leg segment.
2. Measurement & Calculation:
   • Using an on-screen ruler or by estimating, assign unit values to the lengths of 'a' and 'b'. You can use any unit you like (pixels, centimeters, or even "Lego bumps"), as long as you are consistent.
   • In your notebook, write down the formula: a² + b² = c².
   • Plug in your measured values for 'a' and 'b'.
   • Solve for 'c'. This value represents the calculated length of your Wobbledog's leg. This is your core design parameter!
     (Reference AoPS Pre-Algebra, Chapter 15, for a refresher on the Pythagorean Theorem if needed.)

Part 2: Robotic Prototyping (Approx. 60-90 minutes)

Goal: To build a static model of your Wobbledog's leg and body structure using the Lego Spike Prime set, based on your calculations.

1. Design & Translate:
   • Look at your calculated values for 'a', 'b', and 'c'. Your challenge is to build a right-triangle structure using Lego beams that is proportional to these values.
   • For example, if you calculated a=3, b=4, and c=5, you might use a 3-unit long beam, a 4-unit long beam, and a 5-unit long beam to form your triangle. You will likely need to scale up your numbers to build something larger (e.g., a=6, b=8, c=10).
2. Build the Skeleton:
   • Using beams, pins, and connectors from your Spike Prime set, construct the triangle you designed. This represents one leg assembly.
   • Now, build a "body" for your Wobbledog and attach at least two of these leg assemblies to it. The goal is not to make it walk yet, but to create a stable structure that can stand on its own, just like your digital dog.
   • Does it stand? Is it stable? If not, what adjustments do you need to make? This is the engineering design process in action!

Part 3: The Algebraic Wobbledog - Advanced Challenge (Approx. 45 minutes)

Goal: To create a flexible, algebraic model for your dog's legs that could adapt to mutations, just like in the game.

1. From Numbers to Variables:
   • In Wobbledogs, dogs grow and their body proportions change. Let's model this with algebra.
   • Let's say the horizontal distance ('b' from Part 1) is our primary variable, which we will now call x.
   • Let's assume for stability, the height of the leg attachment ('a') must always be 75% of x (or 0.75x).
   • In your notebook, write the Pythagorean theorem using these new algebraic expressions for 'a' and 'b': (0.75x)² + (x)² = c²
2. Create a General Formula:
   • Simplify the expression. Solve for 'c' in terms of 'x'. You will get a formula like c = kx, where 'k' is a constant number you calculate.
   • Your Challenge: What is the value of 'k'? Show your work.
     (Reference AoPS Introduction to Algebra for help with simplifying expressions with variables and square roots.)
   • This new formula is powerful! It means that no matter how your Wobbledog's body length 'x' changes, you can instantly calculate the required leg length 'c' to maintain its stable structure.
3. Spike Prime Extension (Optional):
   • Can you program the Spike Prime hub to do this calculation for you?
   • Write a simple program where you can input a value for 'x', and the hub's light matrix displays the calculated value for 'c'.

Conclusion & Reflection (15 minutes)

Discuss or write down your answers to the following questions:

• How did using the Pythagorean theorem help you accurately build your Lego model?
• What was the biggest challenge in translating your on-screen measurements to a physical Lego structure?
• Explain in your own words why the algebraic formula you created in Part 3 is more powerful than the single calculation you did in Part 1.
• How could you use motors from the Spike Prime set to make your model move while maintaining the geometric principles you explored today? (A thought experiment for a future lesson!)