# Evaluating Conditions for Congruency

## Mathematical goals

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.

## Introduction

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.

## Materials required

- 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 - A board compass, protractor and rule would be useful.

## Time needed

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.

## Lesson Type

## Mathematical Practices

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