6. Explain a proof of the Pythagorean Theorem and its converse.
We're about to start working with the Pythagorean Theorem, which most students are somewhat familiar with, probably. (Remember it? It says, basically, that in any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.)
And, while it's called a theorem, that doesn't mean we're unsure about it. It's a fact. The Pythagorean Theorem works every single time. In fact, so does its converse, which means that we can switch the rule: if the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse, then it's a right triangle. (Most of the time, the converse of a theorem isn't true—but it's true this time!)
And you believe us, right? Well, you shouldn't! Hasn't geometry taught you anything? Don't believe anything until you prove it.
Students should understand the Pythagorean Theorem and explain simple proofs of it. A good one to start out with might be drawing a 3-4-5 triangle along with squares on the side lengths, like this.
Then, using a = 3, b = 4, and c = 5, does the Pythagorean Theorem hold up? In other words, is it true that a2 + b2 = c2? Rather than squaring the numbers, students can count up the number of boxes in each side's square and see that 9 + 16 = 25. Having a protractor to verify that the triangle does have a right angle might also be helpful.
Asking students to explain the importance of certain figures and diagrams is very helpful (since that's sort of what geometry is, anyway), and drawing on prior knowledge like the distance formula, perpendicular lines, and theorems about angles is always encouraged. Given the right information, they should be able to use the Pythagorean theorem to conclude whether a triangle is right or not.
So Pythagoras was right. His theorem works. (He wouldn't be so famous if it didn't work, now would he?)