Lesson Plan: Wobbledogs, Diagrams, and Distance - A Pythagorean Adventure

Materials Needed

• Computer with internet access
• Pre-Algebra Textbook (ISBN: 978-1-934124-21-5 by Rusczyk, Patrick, Boppana)
• Access to the Desmos Graphing Calculator (www.desmos.com/calculator)
• Access to the game Wobbledogs (or video clips of gameplay)
• Graph paper and pencil
• A calculator

1. Learning Objectives

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

• Translate a multi-step word problem describing movement into an accurate visual diagram.
• Identify a right triangle within a complex path on a diagram by consolidating directional movements (e.g., total north/south and total east/west).
• Apply the Pythagorean theorem (a² + b² = c²) to calculate the straight-line distance between a starting and ending point.
• Use the Desmos graphing calculator to model and verify the paths described in word problems.

2. Alignment with Standards and Curriculum

• Textbook Alignment: Chapter 15 — Problem-Solving Strategies (focus on "Draw a Picture").
• CCSS.MATH.CONTENT.A-SSE.A.1: Interpret expressions that represent a quantity in terms of its context. (Students will see a² + b² as the sum of squared horizontal and vertical distances to find the total straight-line distance).
• CCSS.MATH.CONTENT.A-CED.A.1: Create equations in one variable and use them to solve problems. (Students will create an equation using the Pythagorean theorem based on the context of the word problem).

Lesson Activities

Part 1: The Wobbledog Warm-Up (10 minutes) - Engagement

Let's start by opening up Wobbledogs! Spend a few minutes observing how your dogs move around their environment. Notice their quirky, unpredictable paths.

Discussion Prompt: "If a dog starts in one corner and waddles its way over to its food bowl in another corner, it takes a very roundabout path. But what is the shortest possible distance between where it started and the food bowl? It would be a straight line, right? Today, we're going to solve problems just like this, where we turn a complicated, multi-step path into a simple, straight-line distance."

Part 2: From Wobbles to Diagrams (15 minutes) - Instructional Strategy

1. Read the Textbook: Open your Pre-Algebra text to Chapter 15. Let's read the introduction to the "Draw a Picture" strategy. The main idea is that our brains are fantastic at understanding visual information. Turning confusing words into a simple sketch is one of the most powerful tools in all of math.
2. Introduce the Key Tool: When we draw diagrams for distance problems involving directions like north, south, east, and west, a very special shape almost always appears: the right triangle.
3. Review the Pythagorean Theorem: Remember our essential formula for right triangles? a² + b² = c².
   • 'a' and 'b' are the legs (the horizontal and vertical sides).
   • 'c' is the hypotenuse (the long, diagonal side).
   This theorem is the perfect tool for finding that straight-line "shortcut" distance we were just talking about!

Part 3: Guided Practice - The Swimming Wobbledog (20 minutes) - Application

Let's tackle the first problem together, step-by-step. We will use paper first, then check our work in Desmos.

Problem 1: A Wobbledog (let's call her W) swims 60 miles north, 30 miles east, 30 miles north, 150 miles west. How far is W from her starting point?

1. Draw the Picture (on graph paper):
   • Start with a dot labeled "Start."
   • Draw a line straight up 6 units (let 1 square = 10 miles). Label it "60".
   • From there, draw a line 3 units to the right. Label it "30".
   • From that point, draw a line 3 units up. Label it "30".
   • Finally, draw a line 15 units to the left. Label it "150". Mark the final spot "End."
2. Simplify the Diagram: Now, let's figure out the total movements.
   • Total North/South movement: W went 60 miles north + 30 miles north = 90 miles north.
   • Total East/West movement: W went 30 miles east and 150 miles west. Since they are opposite, we subtract: 150 - 30 = 120 miles west.
3. Find the Triangle: Our simplified drawing shows that the end point is 90 miles north and 120 miles west of the start point. If you draw lines for these two net movements and connect the start and end points, you have a perfect right triangle!
   • Leg 'a' = 90 miles
   • Leg 'b' = 120 miles
   • The distance we want to find is the hypotenuse, 'c'.
4. Apply the Theorem & Solve:

   a² + b² = c²

   90² + 120² = c²

   8100 + 14400 = c²

   22500 = c²

   c = √22500

   c = 150 miles

   So, the Wobbledog is exactly 150 miles from where she started!

5. Verify with Desmos: Go to the Desmos calculator. Let's plot the path using coordinates. Start at (0,0).
   • 60 miles north -> (0, 60)
   • 30 miles east -> (30, 60)
   • 30 miles north -> (30, 90)
   • 150 miles west -> (-120, 90)
   Look at the final point (-120, 90). The triangle connects (0,0), (0,90), and (-120,90), or (0,0), (-120,0), and (-120,90). The legs are clearly 120 and 90! It works!

Part 4: Independent Practice - The Position Puzzle (20 minutes) - Problem-Solving

Now it's your turn to be the lead problem-solver. Use the same strategy: Draw, Simplify, Find the Triangle, and Solve.

Problem 2: A is 50m east of B and 30m west of C. D is 60m east of C, and 40m east of E. F is 50m north of E and 80m north of G. To the nearest tenth of a meter, how far apart are B and G?

Hints to get you started:

• This problem is about relative positions. The best way to start is to place one point on your graph paper and work from there. Let's place B at the origin (0,0).
• Carefully calculate the (x, y) coordinates for every single point (A, C, D, E, F, G) based on B being at (0,0).
• Once you have the coordinates for B and G, you can find the horizontal distance (the change in x) and the vertical distance (the change in y). These are the legs of your right triangle!

(Work through the problem. We will check the answer together after you're done.)

Solution Check: B is at (0,0) and G is at (100, -30). The horizontal leg is 100m and the vertical leg is 30m. Using a² + b² = c², we get 100² + 30² = c², which is 10900 = c². So c = √10900 ≈ 104.4 meters.

Part 5: Creative Challenge - Design a Wobbledog Obstacle Course! (15 minutes) - Creativity & Innovation

Time to combine your math skills and your love for Wobbledogs!

1. Open a new graph in Desmos.
2. Design a simple "obstacle course" path for a Wobbledog by plotting and connecting at least 5 points.
3. Your path must include at least one perfect right-angle turn.
4. Calculate two things:
   • The total distance the dog walks if it follows your path exactly from start to finish.
   • The shortcut distance (the hypotenuse) across the right-angle turn you created.
5. Give your course a fun, Wobbledog-themed name and be ready to explain your calculations.

Assessment & Reflection

• Check for Understanding: Your correct, step-by-step solutions for the two word problems show that you've mastered the process.
• Creative Application: Your Desmos obstacle course demonstrates you can apply the Pythagorean theorem to a situation you created yourself.
• Reflection Question: How does drawing a picture make a word problem like this less intimidating?

Differentiation and Extensions

• Need Support? We can use the "grid" feature on Desmos and physical graph paper to make the coordinate plotting even clearer. We can also do more simple path problems before tackling the harder ones.
• Ready for a Challenge? Create your own complex word problem with at least 6 steps of movement. Or, research the 3D distance formula and design a path for a Wobbledog that also involves moving up and down between different levels!