Wobbledog Wanderings: A Pythagorean Adventure
Subject: Pre-Algebra
Topic: Problem-Solving with the Pythagorean Theorem
Key Concepts: Drawing diagrams, Pythagorean theorem, coordinate planes, displacement vs. distance.
Common Core State Standards:
- A-SSE.1: Interpret expressions that represent a quantity in terms of its context. (Seeing the structure of the path in the geometric model.)
- A-CED.1: Create equations and inequalities in one variable and use them to solve problems. (Translating the word problem into the Pythagorean theorem equation.)
Materials Needed
- Pre-Algebra Textbook (ISBN: 978-1-934124-21-5), open to Chapter 15
- Computer or tablet with internet access
- Access to Desmos Graphing Calculator (www.desmos.com/calculator)
- Access to Wobbledogs game (www.wobbledogs.com)
- Pencil and paper (graph paper is a bonus!)
- Calculator
Lesson Plan (Approx. 60-75 minutes)
Part 1: The Hook - Wobbly Paths (5 minutes)
Let's start by thinking about your Wobbledogs. When they move around, they don't exactly walk in a straight line, do they? They wiggle, bounce, and wander all over the place. Imagine you want to know how far your favorite dog is from its food bowl after one of its strange journeys.
It might have traveled a very long, winding path (total distance), but what we really want to know is the direct line from its start point to its end point. This direct-line distance is called displacement. Today, we're going to use a powerful math strategy to figure that out, and it starts with the main idea from Chapter 15: Draw a Picture!
Part 2: The Strategy - Visualizing the Problem (15 minutes)
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Textbook Connection: Let's quickly review the core idea in Chapter 15 of your textbook. The main strategy for complex problems is often the simplest: draw a picture. It helps us turn confusing words into a clear visual map. For paths and distances, this is the most important first step.
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Introducing the Tool: The Pythagorean Theorem. When a path involves movements that are at right angles to each other (like North/South and East/West), we can find the direct-line distance using the Pythagorean Theorem: a² + b² = c².
- 'a' and 'b' are the lengths of the two shorter sides of a right-angled triangle (the "legs").
- 'c' is the length of the longest side, the one opposite the right angle (the "hypotenuse"). This is our direct-line distance!
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Desmos Visualization: Let's see this in action. Open Desmos.
- Plot the point (0,0). This is our starting point.
- Now, imagine something moves 4 units East (right) and 3 units North (up). What point would it be at? (Answer: (4,3)). Plot that point.
- You can see the path creates a right triangle! The horizontal "leg" is 4 units long, and the vertical "leg" is 3 units long. The direct distance is the hypotenuse.
- Using the theorem: 4² + 3² = c². This becomes 16 + 9 = c², so 25 = c². The square root of 25 is 5. The direct distance is 5 units! Desmos helps us see the picture that the formula describes.
Part 3: Guided Practice - The Swimmer's Journey (15 minutes)
Let's tackle this problem together, step-by-step. We will draw it on paper and then check our drawing in Desmos.
Problem 1: W swims 60 miles north, 30 miles east, 30 miles north, 150 miles west. How far is W from the starting point?
- Draw a Picture: On your paper, mark a starting point. Draw an arrow up labeled "60". From there, draw an arrow right labeled "30". From that point, draw another arrow up labeled "30". Finally, draw a long arrow left labeled "150".
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Simplify the Movements (See Structure): This looks messy. Let's combine the movements to make it simpler.
- How far North did W go in total? (60 miles + 30 miles = 90 miles North)
- What was the total East/West movement? (30 miles East is positive, 150 miles West is negative. 30 - 150 = -120, or 120 miles West).
- Draw the Final Triangle: Now, our simplified picture is just two movements from the start: 90 miles North and 120 miles West. This forms a perfect right triangle!
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Apply the Theorem: The legs of our triangle are 90 and 120.
- a² + b² = c²
- 90² + 120² = c²
- 8100 + 14400 = c²
- 22500 = c²
- c = √22500
- c = 150 miles
So, even though W swam 270 miles in total, they are only 150 miles from where they started!
- (Optional) Check in Desmos: Plot the path: (0,0) to (0,60) to (30,60) to (30,90) to (-120,90). The final point is (-120, 90). The distance from (0,0) to (-120, 90) is the hypotenuse of a triangle with legs 120 and 90. Perfect!
Part 4: Independent Practice - Mapping the Locations (15 minutes)
Your turn! Use the same "Draw a Picture" strategy for this one. It has more steps, so drawing it carefully is the key to success. I recommend using B as your starting point (0,0).
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?
(Allow time for independent work. Provide hints only if needed, such as "Where would you place C on your map relative to A?")
Click here for the step-by-step solution to check your work
- Set up coordinates: Let B = (0, 0).
- Place points:
- A is 50m east of B -> A = (50, 0)
- A is 30m west of C -> C is 30m east of A -> C = (80, 0)
- D is 60m east of C -> D = (140, 0)
- D is 40m east of E -> E is 40m west of D -> E = (100, 0)
- F is 50m north of E -> F = (100, 50)
- F is 80m north of G -> G is 80m south of F -> G = (100, 50 - 80) = (100, -30)
- Identify the triangle: We need the distance from B (0, 0) to G (100, -30). The horizontal leg (a) is 100m. The vertical leg (b) is 30m.
- Apply the Theorem:
- 100² + 30² = c²
- 10000 + 900 = c²
- 10900 = c²
- c = √10900 ≈ 104.403...
- Answer: B and G are approximately 104.4 meters apart.
Part 5: Creative Application - The Wobbledog Challenge! (20 minutes)
This is where it all comes together. Time to apply your skills to the Wobbledogs!
- Observe: Open Wobbledogs and pick one dog to watch for about 30-60 seconds.
- Record its Path: On your paper, describe its path using simple directions and estimated "steps" (you can decide how long one "step" is). For example: "Walked 5 steps forward, turned right and went 3 steps, walked backward 2 steps, turned left and went 6 steps..."
- Draw the Path: Just like we did in the first problem, draw out this wobbly path on your paper or in Desmos.
- Simplify: Combine all the forward/backward movements and all the left/right movements to find the final position.
- Calculate Displacement: You should now have a final right triangle! Use the Pythagorean theorem to calculate your Wobbledog's displacement—the direct-line distance from where it started to where it ended up.
- Share Your Findings: Explain the path your Wobbledog took and how you calculated its final displacement. Was it farther from its starting point than you expected?
Part 6: Wrap-Up & Reflection (5 minutes)
Let's think about what we did today.
- How did drawing a picture turn a confusing word problem into a solvable one?
- The Pythagorean theorem is used in construction, navigation, and video game design. Can you think of another example in a game or in real life where it would be useful?
- What was more helpful for you in understanding the problem: drawing by hand or plotting the points on Desmos? Why?
Great job today! You took a core Pre-Algebra strategy and applied it to complex problems and even a quirky physics game. That's what real problem-solving is all about.