Discovering the Pythagorean Theorem
This lesson unit is intended to help you assess how well students are able to:
- Use the area of right triangles to deduce the areas of other shapes.
- Use dissection methods for finding areas.
- Organize an investigation systematically and collect data.
- Deduce a generalizable method for finding lengths and areas (The Pythagorean Theorem.)
- Before the lesson, students attempt a task individually. You review their work and create questions for students to answer in order to improve their solutions.
- A whole-class introduction poses the problem of finding the areas of ‘tilted squares’ drawn on a square grid. Students share different approaches for calculating areas and are offered three generalizable methods that they might use. Students are asked to find possible areas of ‘tilted squares’ at a specific tilt. This requires a systematic approach.
- In a whole-class discussion results are shared and organized.
- In a follow-up lesson, students receive your comments on the assessment task and use these to attempt the similar task, approaching it with insights gained from the lesson.
- Each student will need a copy of the task sheets Square Areas, Tilted Squares, Proving the Pythagorean Theorem, Square Areas (revisited), and multiple copies of the Dotted Grid Paper (on demand).
- Each small group of students will need copies of the sheet Some Different Approaches.
- There are projector resources to help introduce activities and support whole-class discussions.
20 minutes before the lesson, an 80-minute lesson (or split into two shorter lessons), and 20 minutes in a follow-up lesson (or for homework.) Timings given are only approximate. Exact timings will depend on the needs of your class.
This lesson involves a range of mathematical practices from the standards, with emphasis on:
- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning
Mathematical Content Standards
This lesson asks students to select and apply mathematical content from across the grades, including the content standards: