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Pythagorean Theorem: Right On!

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Lesson Plan

Pythagorean Theorem: Right On!

Students will be able to recall the Pythagorean Theorem and apply it to solve for unknown sides in right triangles, as well as recognize its utility in real-world contexts.

Understanding the Pythagorean Theorem is fundamental for advanced geometry, trigonometry, and various STEM fields. It helps students develop critical thinking and problem-solving skills applicable to many real-world situations.

Audience

10th Grade Students

Time

30 minutes

Approach

Review, direct instruction, guided practice, and independent application.

Materials

Pythagorean Theorem Review Slide Deck, Warm-Up: Missing Side Mania!, Pythagorean Practice Worksheet, and Pythagorean Practice Answer Key

Prep

Teacher Preparation

15 minutes

Step 1

Warm-Up: Missing Side Mania!

5 minutes

  • Project or distribute the Warm-Up: Missing Side Mania!.
    - Instruct students to solve the quick problems to activate prior knowledge about right triangles and basic algebra.
    - Briefly review answers as a class.

Step 2

Revisiting the Theorem

10 minutes

  • Use the Pythagorean Theorem Review Slide Deck to review the Pythagorean Theorem (a² + b² = c²).
    - Emphasize the importance of identifying the hypotenuse (c) and the legs (a and b).
    - Go through example problems together, demonstrating how to set up and solve for unknown sides.

Step 3

Guided Practice & Application

10 minutes

  • Distribute the Pythagorean Practice Worksheet.
    - Guide students through the first few problems, offering support and clarifying misconceptions.
    - Encourage students to work individually or in pairs on the remaining problems.
    - Circulate around the room to provide individualized assistance.

Step 4

Cool-Down: Quick Check

5 minutes

  • Have students complete a quick cool-down activity (e.g., exit ticket question: 'Explain one real-world situation where the Pythagorean Theorem would be useful.').
    - Collect the cool-downs to gauge student understanding and inform future instruction.
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Slide Deck

Pythagorean Theorem: Right On!

Revisiting the Fundamentals of Right Triangles
Get ready to put your right-angle thinking caps on!

Greet students and introduce the day's review. Briefly mention the importance of the Pythagorean theorem. This slide serves as a title slide.

What's a Right Triangle Got to Do With It?

Key Features:
- One 90-degree angle
- Legs: The two shorter sides (a and b)
- Hypotenuse: The longest side, opposite the right angle (c)

Ask students what they remember about right triangles and the Pythagorean Theorem. Facilitate a brief discussion to activate prior knowledge.

The Big Formula!

a² + b² = c²

  • a and b are the lengths of the legs.
  • c is the length of the hypotenuse (always the longest side!).

This theorem only works for right triangles!

Introduce the formula. Emphasize that 'a' and 'b' are interchangeable for legs, but 'c' MUST be the hypotenuse.

Example 1: Finding the Hypotenuse

Problem: A right triangle has legs of 3 units and 4 units. What is the length of the hypotenuse?

Solution Steps:

  1. Identify a, b, and c.
    • a = 3, b = 4, c = ?
  2. Plug values into the formula: $3^2 + 4^2 = c^2$
  3. Calculate: $9 + 16 = c^2$
  4. Simplify: $25 = c^2$
  5. Solve for c: $c = /\text{25}$
  6. Answer: $c = 5$ units

Go through a simple example of finding the hypotenuse. Work through it step-by-step on the board or projector.

Example 2: Finding a Leg

Problem: A right triangle has a hypotenuse of 13 units and one leg of 5 units. What is the length of the other leg?

Solution Steps:

  1. Identify a, b, and c.
    • a = 5, b = ?, c = 13
  2. Plug values into the formula: $5^2 + b^2 = 13^2$
  3. Calculate: $25 + b^2 = 169$
  4. Isolate b²: $b^2 = 169 - 25$
  5. Simplify: $b^2 = 144$
  6. Solve for b: $b = /\text{144}$
  7. Answer: $b = 12$ units

Go through an example of finding a leg. Remind students to rearrange the formula if needed. Work through it step-by-step.

Where Do We See This?

The Pythagorean Theorem isn't just for textbooks! It's used in:

  • Construction: Measuring diagonals, ensuring square corners.
  • Navigation: Calculating distances.
  • Art & Design: Creating balanced compositions.
  • Sports: Analyzing trajectories and distances.

Think about it: where have you seen right triangles in the real world?

Discuss real-world applications. Ask students for their own examples. This helps make the theorem relevant.

Time to Practice!

You've got this! Now let's apply what we've reviewed with the Pythagorean Practice Worksheet.

Remember to:

  • Identify the legs and the hypotenuse.
  • Show your work.
  • Don't be afraid to ask questions!

Conclude the review and transition to the worksheet. Encourage students to ask questions.

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Warm Up

Warm-Up: Missing Side Mania!

Directions: Take a few minutes to solve for the missing side in each right triangle below. Show your work!

Problem 1

Right triangle with legs 3 and 4
(Assume a right angle is formed by the sides 3 and 4)

a = 3
b = 4
c = ?

Work:





Problem 2

Right triangle with hypotenuse 13 and one leg 5
(Assume a right angle is formed by the unknown side and side 5)

a = 5
b = ?
c = 13

Work:





Problem 3

Think about a real-world scenario where you might need to find the length of a missing side in a right triangle. Describe it briefly.

Description:





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Worksheet

Pythagorean Practice Worksheet

Directions: Use the Pythagorean Theorem ($a^2 + b^2 = c^2$) to solve for the unknown side in each right triangle. Round your answers to the nearest tenth if necessary. Show all your work!

Part 1: Finding the Hypotenuse

  1. A right triangle has legs measuring 6 cm and 8 cm. What is the length of the hypotenuse?

Work:





  1. The sides of a right triangle are 7 inches and 24 inches. Find the length of the hypotenuse.

Work:





Part 2: Finding a Leg

  1. A right triangle has a hypotenuse of 15 meters and one leg of 9 meters. What is the length of the other leg?

Work:





  1. One leg of a right triangle is 12 feet long, and the hypotenuse is 20 feet long. Find the length of the other leg.

Work:





Part 3: Real-World Applications

  1. A 10-foot ladder is leaning against a wall. If the base of the ladder is 6 feet from the wall, how high up the wall does the ladder reach?

Work:





  1. You're building a ramp for your skateboard. The ramp needs to be 13 feet long, and the horizontal distance it covers on the ground is 12 feet. How tall will the ramp be at its highest point?

Work:





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Answer Key

Pythagorean Practice Answer Key

Here are the solutions for the Pythagorean Practice Worksheet.

Part 1: Finding the Hypotenuse

  1. A right triangle has legs measuring 6 cm and 8 cm. What is the length of the hypotenuse?
  • Thought Process: We are given the lengths of the two legs (a = 6, b = 8) and need to find the hypotenuse (c). We will use the Pythagorean Theorem: $a^2 + b^2 = c^2$.
  • Solution:
    $6^2 + 8^2 = c^2$
    $36 + 64 = c^2$
    $100 = c^2$
    $c = /\text{100}$
    c = 10 cm
  1. The sides of a right triangle are 7 inches and 24 inches. Find the length of the hypotenuse.
  • Thought Process: Similar to problem 1, we have two legs (a = 7, b = 24) and need to find the hypotenuse (c). Use $a^2 + b^2 = c^2$.
  • Solution:
    $7^2 + 24^2 = c^2$
    $49 + 576 = c^2$
    $625 = c^2$
    $c = /\text{625}$
    c = 25 inches

Part 2: Finding a Leg

  1. A right triangle has a hypotenuse of 15 meters and one leg of 9 meters. What is the length of the other leg?
  • Thought Process: We are given one leg (a = 9) and the hypotenuse (c = 15). We need to find the other leg (b). Rearrange the Pythagorean Theorem to solve for a leg: $b^2 = c^2 - a^2$.
  • Solution:
    $9^2 + b^2 = 15^2$
    $81 + b^2 = 225$
    $b^2 = 225 - 81$
    $b^2 = 144$
    $b = /\text{144}$
    b = 12 meters
  1. One leg of a right triangle is 12 feet long, and the hypotenuse is 20 feet long. Find the length of the other leg.
  • Thought Process: We have one leg (a = 12) and the hypotenuse (c = 20). We need to find the other leg (b). Use $b^2 = c^2 - a^2$.
  • Solution:
    $12^2 + b^2 = 20^2$
    $144 + b^2 = 400$
    $b^2 = 400 - 144$
    $b^2 = 256$
    $b = /\text{256}$
    b = 16 feet

Part 3: Real-World Applications

  1. A 10-foot ladder is leaning against a wall. If the base of the ladder is 6 feet from the wall, how high up the wall does the ladder reach?
  • Thought Process: The ladder forms the hypotenuse (c = 10), the distance from the wall is one leg (a = 6), and the height up the wall is the unknown leg (b). Use $b^2 = c^2 - a^2$.
  • Solution:
    $6^2 + b^2 = 10^2$
    $36 + b^2 = 100$
    $b^2 = 100 - 36$
    $b^2 = 64$
    $b = /\text{64}$
    b = 8 feet
  1. You're building a ramp for your skateboard. The ramp needs to be 13 feet long, and the horizontal distance it covers on the ground is 12 feet. How tall will the ramp be at its highest point?
  • Thought Process: The ramp length is the hypotenuse (c = 13), the horizontal distance is one leg (a = 12), and the height of the ramp is the unknown leg (b). Use $b^2 = c^2 - a^2$.
  • Solution:
    $12^2 + b^2 = 13^2$
    $144 + b^2 = 169$
    $b^2 = 169 - 144$
    $b^2 = 25$
    $b = /\text{25}$
    b = 5 feet
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