Teacher: | What do the interior angles of a triangle add up to? |
Student: | 180°. |
Teacher: | Are you sure? |
Student: | Yes, absolutely. |
Teacher: | How do you know this is so? |
Student: | Because … you tell us and it says so in the book. |
Teacher: | OK, I want you to now think carefully for a moment about how you would prove mathematically that the sum of the angles of a triangle equals 180°. This is a mathematical proof, so your reasoning has to be rigorous and convincing. Imagine you are a mathematician trying to convince the prime minister of India – how would you go about it? Think about what you know already about triangles and mathematical proof. Discuss your ideas first with your partner. I will give you five minutes to do so. |
| [Five minutes later.] |
| So, what will you say to the prime minister to convince him that the sum of the interior angles of a triangle always equals 180°? |
Student 1: | We would say that when you measure the interior angles of a triangle and add these up, then you always end up with 180°. You can do this with any number, and any kind of triangle. |
Student 2: | We would say: draw a triangle on a piece of paper. Cut it out. Tear off the corner and place these corners next to each other. They always fit on a straight line. As we know that angles on a straight line add up to 180°, we can deduce that the sum of the interior angles of a triangle also equals 180°. This will always be so for any triangle. |
Teacher: | What do you think, class – are you convinced these ideas are mathematical proofs? Will the prime minister be convinced? Should he be? |