Proving the Pythagorean Theorem
This lesson unit is intended to help you assess how well students are able to produce and evaluate geometrical proofs. In particular, it is intended to help you identify and assist students who have difficulties in:
- Interpreting diagrams.
- Identifying mathematical knowledge relevant to an argument.
- Linking visual and algebraic representations.
- Producing and evaluating mathematical arguments.
This lesson unit is structured in the following way:
- Before the lesson, students attempt the assessment task individually. You then review the work and formulate questions that will help them to improve.
- The lesson begins with a whole-class discussion of the geometric diagram from the assessment task. Students then work collaboratively on the task, in pairs or threes, to produce a better collective solution than those produced individually, justifying their work within their groups. In the same small groups, students critique examples of other students’ work.
- In a whole-class discussion, students explain and evaluate the arguments they have seen and used.
- Finally, students work alone on a new task, similar to the original assessment task.
- Each student will need some Grid Paper and a copy of Proving the Pythagorean Theorem and Proving the Pythagorean Theorem (revisited).
- Each small group of students will need a large sheet of paper, copies of the Sample Methods to Discuss, and the Comparing Methods of Proof sheet.
- Provide copies of Extension: Proving the Pythagorean Theorem using Similar Triangles as necessary. There is a projector resource to help with whole-class discussions.
Approximately 20 minutes before the lesson, an 80-minute lesson (or two 45 minute lessons), and 20 minutes in a follow-up lesson. Exact timings will depend on the needs of the 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
- MP4: Model with mathematics
- 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:
- H.A-CED: Create equations that describe numbers or relationships
- H.A-SSE: Interpret the structure of expressions
- H.G-CO: Prove geometric theorems
- H.G-SRT: Prove theorems involving similarity