Discovering the Pythagorean Theorem

Mathematical goals

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.

Materials required

  • 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.

Time needed

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.