Evaluating Conditions for Congruency
This lesson unit is intended to help you assess how well students are able to:
- Work with concepts of congruency and similarity, including identifying corresponding sides and corresponding angles within and between triangles.
- Identify and understand the significance of a counter-example.
- Prove and evaluate proofs in a geometric context.
In this lesson, students work on the concept of congruency whilst developing their understanding of proof in a geometric context.
- Before the lesson, students complete a task designed to help you assess their current levels of understanding. You analyze their responses and write questions to help them improve their work.
- The lesson begins with a whole-class discussion about establishing conditions for congruency from triangle properties. Students work alone to decide on the truth of a conjecture about congruency conditions for triangles. Then in pairs they share ideas and produce and justify a joint response. Working in the same pairs, they analyze sample responses produced by other students. In a whole-class discussion, students develop their understanding of proof in this context.
- In a follow-up lesson students use what they have learned to improve their responses to the initial assessment task before attempting a second, similar task.
- Each student will need a copy of Finding Congruent Triangles and Finding Congruent Triangles (revisited), a rule, compasses, a protractor, the cut-up Card Set: Must the Two Triangles be Congruent? and Instructions: Must the Two Triangles be Congruent?, a mini-whiteboard, pen, and eraser and some paper to work on.
- Each small group of students will need the Sample Student Proofs, some paper to work on and a glue stick.
- A board compass, protractor and rule would be useful.
20 minutes before the lesson, a 100-minute lesson (or two 55-minute lessons), and 20 minutes in a follow-up lesson. Timings are approximate and will depend on the needs of your students.
This lesson involves a range of mathematical practices from the standards, with emphasis on:
- MP1: Make sense of problems and persevere in solving them
- 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:
- H.G-CO: Understand congruence in terms of rigid motions
- H.G-CO: Make geometric constructions
- H.G-SRT: Prove theorems involving similarity