Lesson Plan: Tiling with Pythagoras - A Proof Without Words

Subject: Geometry

Grade Level: High School (Approx. Age 15)

Time Allotment: 70-80 minutes

Materials Needed

• Colored paper or cardstock (at least 2 different colors)
• Scissors
• A ruler
• Pencil and eraser
• A computer with internet access
• Access to the Desmos Graphing Calculator

1. Learning Objectives

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

• Physically construct Annairizi of Arabia's visual proof of the Pythagorean Theorem using paper shapes.
• Verbally explain how the arrangement of shapes visually demonstrates that a² + b² = c².
• Build a dynamic and interactive version of the proof in Desmos, using sliders to explore the relationship for various right triangles.
• Apply the Pythagorean Theorem to solve a creative, real-world design problem based on the proof's structure.

2. Alignment with Standards

• CCSS.MATH.CONTENT.HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. This lesson focuses on a deep, conceptual understanding of the theorem before applying it.

Lesson Activities

Part 1: The Hook - Can You Prove It With No Words? (10 minutes)

Let's start with a question: Do you think you can prove a fundamental rule of math without writing a single equation? Mathematicians have been doing this for centuries with something called "proofs without words." They are elegant, visual puzzles that show a mathematical truth.

Today, we're going to explore a beautiful one for the Pythagorean Theorem, credited to Annairizi of Arabia around 900 AD. It’s all about how shapes fit together.

1. Look at the classic diagram: Draw a large square with a smaller, tilted square inside it, where the corners of the inner square touch the sides of the outer square.
2. Initial thoughts: What shapes do you see? You should see one large square, one smaller square, and four right triangles. This is the basis of our proof! Let's build it ourselves.

Part 2: The Hands-On Proof - Tiling Time! (25 minutes)

This is where we bring the proof to life. We'll show that the area of the two smaller squares (a² and b²) on the legs of a right triangle equals the area of the square on the hypotenuse (c²).

1. Create Your Triangles:
   • On a piece of colored paper, draw and cut out four identical right triangles.
   • Make them a good size, for example, with the two shorter sides (the legs) being a = 3 inches and b = 4 inches.
   • Don't measure the longest side (the hypotenuse, 'c'). We'll let the proof reveal its relationship.
2. Arrange the Tiles:
   • Now, arrange your four triangles on a different colored sheet of paper to form a large square, with the hypotenuse of each triangle forming the outer edge.
   • You will notice a square-shaped hole in the very center. Tape the triangles down.
3. Analyze the Picture:
   • The Big Square: The side length of this large square is 'c' (the hypotenuse of your triangle). So, what is its area? It must be c².
   • The Pieces Inside: The area of the big square is also the sum of all the pieces inside it. What are those pieces?
     • The four triangles you put down.
     • The one square hole in the middle.
   • Let's find the area of the hole: Measure the side of the inner square. You'll find it's the difference between the longer leg and the shorter leg of your triangle (b - a). For our 3-4-c triangle, the side of the hole is 4 - 3 = 1 inch. So, the area of the hole is (b - a)².
4. The "Aha!" Moment:
   • Let's write down the area of the big square in two different ways:
     1. Area = (Side) × (Side) = c²
     2. Area = (Area of 4 triangles) + (Area of middle square) = 4 * (½ * a * b) + (b - a)²
   • Since these are both the area of the same big square, they must be equal! Let's simplify the second expression with algebra:

     c² = 4(½ab) + (b - a)(b - a)

     c² = 2ab + (b² - 2ab + a²)

     c² = 2ab + b² - 2ab + a²

     c² = a² + b²

You just visually and algebraically proved the Pythagorean Theorem! The way the pieces tile together forces the relationship to be true.

Part 3: The Digital Proof - Let's Go Dynamic with Desmos (20 minutes)

Paper is great, but it's static. Let's build a version of this proof that you can change and play with in real-time. This will prove to you that it works for *any* right triangle, not just the 3-4-5 one we built.

1. Open Desmos: Go to desmos.com/calculator.
2. Create Sliders:
   • In the expression list, type `a=3`. Click the blue circle to make it a slider. Set its range from 1 to 10.
   • Do the same for `b=4`. Set its range from 1 to 10.
3. Draw the Triangles with Polygons: We will define the corners (vertices) of the four triangles. This is the trickiest part, but it's like giving the computer coordinates for a connect-the-dots puzzle. Type these four polygon commands into four separate expression lines:
   • `polygon((0,a), (b,a), (0,0))` (This is the bottom-left triangle)
   • `polygon((b,a), (b,0), (a+b,0))` (Bottom-right)
   • `polygon((b,a), (a,a+b), (a+b,a+b))` (Top-right)
   • `polygon((a,a+b), (0,a), (0,0))` (Top-left - wait, that's not right)

   Let's correct the coordinates for a better arrangement:

   • Triangle 1 (Bottom): `polygon((0,0), (a,0), (0,b))` Oops, that's not a right triangle with sides a and b. Let's use a standard setup.

   Let's try a clearer coordinate system. Our big square will have corners at (0,b), (a,0), (a+b, a), and (b, a+b).

   1. Triangle 1: `polygon((0,b), (a,0), (0,0))`
   2. Triangle 2: `polygon((a,0), (a+b,a), (a+b,0))`
   3. Triangle 3: `polygon((a+b,a), (b,a+b), (a+b,a+b))`
   4. Triangle 4: `polygon((b,a+b), (0,b), (0,a+b))`

   (Teacher Note: This part may require some fiddling and is a great problem-solving exercise. A simpler way is to place one triangle at the origin and build from there).

   A simpler, more robust way for the student:

   1. Triangle 1: `polygon((0,0),(a,0),(0,b))` ... *Still not a right triangle with legs a,b.* Okay, let's fix this for good. The key is defining the vertices properly based on the arrangement we made on paper.

      The vertices of the outer `c by c` square will be at (0,a), (b,0), (a+b,b), (a, a+b). Let's draw the 4 triangles inside this boundary.

      • Triangle 1 (bottom left): `polygon( (0,a), (b,0), (0,0) )`
      • Triangle 2 (bottom right): `polygon( (b,0), (a+b,b), (a+b,0) )`
      • Triangle 3 (top right): `polygon( (a+b,b), (a,a+b), (a+b,a+b) )`
      • Triangle 4 (top left): `polygon( (a,a+b), (0,a), (0,a+b) )`
   2. Play with the sliders for 'a' and 'b'. Watch how the four triangles and the central square dynamically resize, but always fit together perfectly to form the larger square on the hypotenuse. You've just created a living, breathing proof!

Part 4: Application & Creativity - The Garden Designer (15 minutes)

Now, let's use this. You're not just a math student; you're a landscape architect.

The Challenge: You are designing a square patio. The design is made of four identical triangular flower beds surrounding a square fountain in the center, exactly like the proof we just built.

• The client specifies that the longest side of each triangular flower bed (the hypotenuse, 'c') must be 13 feet.
• They also want one of the shorter sides of the flower bed (let's call it 'b') to be 12 feet.

Your Task:

1. Find the length of the other short side, 'a'.
2. What is the total area of the entire square patio? (Hint: It's c²)
3. What is the area of the central fountain? (Hint: It's (b-a)²)

Solve this on paper. You can even check your work by plugging your final 'a' and 'b' values into the Desmos model you built to see if it looks right!

Differentiation and Extension

• Need a little help? If the Desmos coordinates are tricky, start with a pre-made graph and focus on analyzing it instead of building it from scratch. For the final problem, try drawing a picture first to label all the sides you know.
• Ready for a challenge? Research "President Garfield's Proof" of the Pythagorean Theorem. It involves a trapezoid. Can you build a dynamic Desmos model for that proof as well? Which proof do you find more intuitive or beautiful, Annairizi's or Garfield's? Write a short paragraph comparing them.

Assessment

• Formative (During the lesson): Your verbal explanation of the hands-on proof and the successful functioning of your Desmos model show me you're on the right track.
• Summative (End of lesson): Your correct solution to the "Garden Designer" problem will demonstrate that you can apply the Pythagorean Theorem in a creative context derived from the proof itself.