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The Pythagorean Theorem: Square Areas

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.)


The lesson is structured in the following way:

  • 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. The method is generalized into the Pythagorean theorem.
  • Finally, students work individually on a new assessment task.

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 two 40-minute 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.

Mathematical Practices

This lesson involves a range of mathematical practices from the standards, with emphasis on:

Mathematical Content

This lesson asks students to select and apply mathematical content from across the grades, including the content standards:


Lesson (complete)

Projector Resources

A draft Brief Guide for teachers and administrators (PDF) is now available, and is recommended for anybody using the MAP Classroom Challenges for the first time.

We have assigned lessons to grades based on the Common Core State Standards for Mathematical Content. During this transition period, you should use your judgement as to where they fit in your current course.

The Beta versions of the MAP Lesson Units may be distributed, unmodified, under the Creative Commons Attribution, Non-commercial, No Derivatives License 3.0. All other rights reserved. Please send any enquiries about commercial use or derived works to

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