The Great Wobbledog Rescue: A Pythagorean LEGO Challenge

Materials Needed:

• Computer or tablet with internet access
• Wobbledogs game (www.wobbledogs.com)
• Access to Desmos online graphing calculator (www.desmos.com)
• LEGO Education Spike Prime set
• Art of Problem Solving (AoPS) Pre-Algebra textbook
• Art of Problem Solving (AoPS) Introduction to Algebra textbook
• A flat surface for building (floor or large table)
• A small box or stack of books to act as a "platform"
• Paper and pencil for sketching ideas

1. Learning Objectives

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

• Apply the Pythagorean theorem (a² + b² = c²) to solve a real-world, multi-step problem.
• Design and construct a physical model (a ramp) using the LEGO Spike Prime set based on mathematical calculations.
• Utilize Desmos to visualize the relationship between the sides of a right triangle and dynamically model potential solutions.
• Translate an abstract mathematical concept into a creative, hands-on engineering project.

2. Alignment with Standards

• Common Core State Standards for Mathematics (CCSS.MATH.CONTENT.8.G.B.7): Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Lesson Activities

Part 1: The Mission Briefing & Conceptual Deep Dive (15 Minutes)

1. The Hook: Open Wobbledogs and spend a few minutes playing. Choose a favorite dog and imagine a scenario: "Oh no! Your wobbliest dog has climbed onto this high platform (use the box or stack of books) and can't get down! It's too wobbly to jump. Your mission is to engineer a rescue ramp."
2. Conceptual Review: Open the AoPS Pre-Algebra textbook to the chapter on the Pythagorean Theorem (Chapter 15). Instead of just reading the formula, focus on the visual proof showing that the area of the squares on the two shorter sides (legs) equals the area of the square on the longest side (hypotenuse). Discuss why it works. This builds a deeper, more intuitive understanding than just memorizing a² + b² = c².

Part 2: The Digital Blueprint - Modeling in Desmos (20 Minutes)

Before we build, we need a plan. We will use Desmos to create an interactive blueprint for our rescue ramp.

1. Set up the Scene: The platform creates a right triangle with the floor. The height of the platform is leg 'a'. The distance from the platform to the base of the ramp is leg 'b'. The ramp itself will be the hypotenuse, 'c'.
2. Create in Desmos:
   • Go to Desmos.com/calculator.
   • Create a slider for the height, 'a'. Set its range from 10 to 30 (representing LEGO units).
   • Create a slider for the base length, 'b'. Set its range from 10 to 50.
   • In a new expression line, define the hypotenuse: c = sqrt(a^2 + b^2). Desmos will now automatically calculate the required ramp length!
   • Challenge: Can you graph the points (0,0), (b,0), and (0,a) and connect them to visualize the triangle changing as you move the sliders? This makes the connection between the numbers and the shape explicit.
3. Experiment: Play with the 'a' and 'b' sliders. Watch how 'c' changes. Ask questions like, "What makes the ramp steeper?" or "If we want the ramp to be exactly 25 units long, what are some possible heights and base lengths we could use?"

Part 3: The Build - Prototyping with LEGO Spike Prime (45 Minutes)

Now, let's bring the digital blueprint to life. This is where math meets engineering.

1. Choose Your Dimensions: Set the platform (box/books) on the building surface. Measure its height in LEGO beam units (or just estimate). This is your value for 'a'. Choose a starting point for your ramp on the floor. Measure the distance to the base of the platform. This is 'b'. Plug these values into your Desmos calculator to find the target length 'c' for your ramp.
2. The Engineering Problem: The calculated length 'c' will almost certainly be a decimal (e.g., 28.28 units). LEGO beams come in integer lengths. This is the core challenge! How can you build a sturdy, smooth ramp that is *very close* to the required length?
   • Encourage creative thinking: Can you slightly overlap beams? Can you use angled connectors to get a more precise length? How do you ensure the structure is strong enough to support a Wobbledog?
3. Build the Ramp: Using the LEGO Spike Prime parts, construct the ramp. Focus on structural integrity and meeting the length requirement as closely as possible. This is an open-ended design task.
4. (Optional Extension) Add a Program: Use the Spike Prime app to write a simple program. For example, you could program the hub's light matrix to display the calculated length 'c', or use a motor and a sensor to create a "gate" at the bottom of the ramp that opens when the Wobbledog approaches.

Part 4: The Test and The Remix (15 Minutes)

1. The Rescue: Place your finished ramp against the platform. Gently slide a LEGO figure (or a printed picture of your Wobbledog) down the ramp. Was the rescue a success? Is the ramp stable? Is it the right length?
2. The Remix: What if the platform was taller? Go back to your Desmos graph, increase the 'a' slider, and see the new required length for 'c'. Discuss how you would have to change your LEGO design to accommodate this. This reinforces the dynamic nature of the theorem and the link between the model and the real object.

4. Assessment and Extension

• Formative Assessment: The primary assessment is the successful completion of the project. I will observe the problem-solving process, especially how the student tackles the "decimal length" LEGO problem. Success is defined by:
  • A completed Desmos model that accurately reflects the problem.
  • A physical LEGO ramp that is structurally sound and based on the Desmos calculations.
  • The ability to articulate *why* their design choices were made.
• Summative Extension (The 3D Challenge): Open the AoPS Introduction to Algebra textbook. Find a problem on the "Distance Formula in Three Dimensions" (this is just the Pythagorean theorem with a z-axis: d² = x² + y² + z²).
  • The Challenge Problem: "Imagine your Wobbledog is in one corner of the Spike Prime box. You need to build a direct ramp from the opposite bottom corner to the opposite top corner, going through the middle of the box. How long would that ramp have to be?"
  • This pushes the student to visualize and apply the theorem in 3D space, moving from a concrete application to a more abstract, spatial reasoning problem. Guide them through sketching the problem and applying the formula.