In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.
Here’s the sequence of my proposed lesson:
1. Setting the Problem:
Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties: QR is 4 cm long. The perpendicular line from P to QR is 3 cm.
Pose this question: Can you draw Myra’s triangle?
Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.
2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.
3. Introducing the idea of congruence: Question: If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]
Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?
Possible answers:
- QR is 4 cm long. The perpendicular line PQ is 3 cm.
- QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
- PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.
4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?
- In triangle ABC, AB and BC are each 5 cm long.
- ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.
I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.