Lesson Plan: The Cartesian Detective

Solving Complex Problems by Drawing a Picture

Materials Needed:

• Pencil and Eraser
• Plain or Graph Paper (Graph paper is highly recommended)
• A ruler (for neatness and scale)
• Calculator
• Prealgebra Text (ISBN: 978-1-934124-21-5) for reference

Lesson Details

• Subject: Pre-Algebra
• Grade Level: High School (designed for a 15-year-old student)
• Topic: Problem-Solving Strategies & Pythagorean Theorem
• Time Allotment: 60 minutes

1. Learning Objectives (Your Mission)

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

• Translate multi-step word problems into accurate, scaled diagrams.
• Identify "hidden" right triangles within a complex scenario.
• Apply the Pythagorean theorem (a² + b² = c²) to find the direct distance between two points.
• Explain how the components of your diagram and equation (e.g., the 'legs' of the triangle) represent the quantities described in the problem.

2. Alignment with Standards (The Case Files)

This lesson focuses on developing critical thinking skills that align with the following Common Core State Standards:

• A-CED.1: You will be creating a visual model (a picture) that directly translates the context of a word problem into a geometric structure, which then leads to an equation.
• A-SSE.1: You will interpret the parts of your diagram—the horizontal and vertical paths—as individual components that combine to form the hypotenuse, giving structure and meaning to the Pythagorean theorem in a real-world context.

3. Instructional Strategies & Lesson Procedure (The Investigation)

Part 1: The Briefing (10 minutes)

Let's start with a warm-up to get our detective minds working. Imagine you're standing in the middle of a city grid.

• Warm-Up Question: "A secret package is hidden 8 blocks east and 6 blocks north of your current position. If you could travel in a straight line (say, with a drone), what is the shortest distance your drone would have to travel to get to the package?"
• Discussion:
  • Sketch this out. You'll see it forms a right triangle. The paths "east" and "north" are the legs.
  • Solve it together using the Pythagorean theorem: 8² + 6² = c². You'll find c = 10 blocks.
  • Key Idea: Today’s main strategy is simple but powerful: Draw a Picture. Complex problems often become simple when we can see them. Your pencil and paper are your most important detective tools.

Part 2: Guided Investigation (20 minutes)

Now let's tackle our first official case. We will work through this one together, step-by-step.

Case #1: W swims 60 miles north, then 30 miles east, then 30 miles north, then 150 miles west. How far is W from the starting point?

1. Start with a Dot: On your paper, make a dot and label it "Start."
2. Trace the Path: Let's draw each leg of the journey.
   • Draw a line straight up from "Start" (for 60 miles north).
   • From there, draw a line to the right (for 30 miles east).
   • Then, another line straight up (for 30 miles north).
   • Finally, a long line to the left (for 150 miles west). Mark the "End" point.
3. Analyze the Clues: The question asks for the distance from the starting point. This is the straight line connecting "Start" to "End." Let's draw that line. It should be the hypotenuse of a right triangle!
4. Find the Triangle: Where is the triangle? We need to figure out the total north-south change and the total east-west change.
   • Total North Movement: W went 60 miles north + 30 miles north = 90 miles north. This is the vertical leg of our triangle.
   • Total East/West Movement: W went 30 miles east and 150 miles west. Since they are opposite directions, we subtract: 150 - 30 = 120 miles west. This is the horizontal leg of our triangle.
5. Solve the Case: Now we have a simple right triangle with legs of 90 and 120.
   • 90² + 120² = c²
   • 8100 + 14400 = c²
   • 22500 = c²
   • c = √22500 = 150 miles.
   Conclusion: W is 150 miles from the starting point. See how drawing the picture helped us organize the information?

Part 3: Solo Mission (15 minutes)

Excellent work. Now you're ready for a more complex case on your own. Read it carefully, take your time, and draw everything. The picture is the key!

Case #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?

Detective's Tips:

• Establish a Baseline: All the east-west movements can be placed on a single horizontal line first. Start with one point (like B) and place the others relative to it.
• Add the Second Dimension: Once you have all the east-west points (B, A, C, D, E) figured out, use that information to place the north-south points (F, G).
• Find the Final Triangle: The question asks for the distance between B and G. This will be the hypotenuse of a right triangle whose legs are the total east-west distance and the total north-south distance between B and G.

Part 4: The Debriefing & Your Own Case (15 minutes)

Let's review your work on Case #2. Walk me through your diagram and your calculations. What was the most challenging part?

(Solution check: The final triangle should have a horizontal leg of 180m and a vertical leg of 30m. Distance BG ≈ 182.5 m)

Final Assessment (Creativity and Application):

Your final mission is to create your own "Cartesian Detective" case. Write a word problem that involves at least four steps of movement (like the ones we did). Your problem must require drawing a picture and using the Pythagorean theorem to solve. After writing the problem, you must create the solution key on a separate page, including a neat diagram and the step-by-step math.

4. Differentiation and Inclusivity (Special Ops Training)

• Extra Support: If drawing feels tricky, use graph paper. Let each square represent 10 miles or 10 meters. This keeps the lines straight and the proportions clear. We can also solve the problem one sentence at a time, drawing as we go.
• Advanced Challenge (Extension): Ready for a 3D case? Imagine a spider in one corner of a room that is 30 feet long, 12 feet wide, and 8 feet high. What is the shortest distance the spider can crawl (on the walls, floor, or ceiling) to get to the opposite corner of the room? (Hint: You have to "unfold" the room into a 2D picture to find the true shortest path!)