And, because this is where most math teachers derive their only sense of joy in the world, let’s mess with a student’s world again: how far apart are C and A?

Such a question is clearly not as simple as the distance from A to B or B to C, but we all have students that will swear A and C are 5 blocks apart because they drew a line between them and saw 5 unit squares were crossed. It’s a valid point, therefore the introduction of a measurement tool, maybe one made of blocks can be used to disprove this theory.

Now, here is the perfect time to work on a proof for the Pythagorean Theorem, which begins with finding the area of the green square below (try it before reading on):

To begin, I would think about the side lengths of the large square and each of the right triangles:

If we find the area of the large square, then subtract the areas of the four triangles, the area that remains should be just that of the green square:

But, let’s take this one step further and say, what are the side lengths of the green square? Well, remember the length and width of a square are the same, so that means each side of the square must be 5 units because 5 x 5 = 25. But, now let’s think about the picture again with some labels:

Notice how the following is now true:

Therefore, we have shown:

But this does not satisfy the mathematicians definition of proof. Just because it works for a 3, 4, 5 triangle doesn’t mean it works in all cases. Before we generalize, feel free to try our strategy on another square:

After proving our strategy works for a 5, 12, 13 triangle, maybe we can feel a bit more confident, but this is still only two cases out of an infinite number of possibilities. It’s time to get more general:

Now, the areas are defined as:

And we want to somehow show that

turns into

Well, obeying the laws of algebra, let’s see if we can make that happen:

Sure enough, simplification has shown that no matter what our side lengths are, the Pythagorean Theorem holds, thus we have completed our proof!

It’s important for us to show students that the material they are learning was not simply understood, but creative individuals and communities had to think about patterns they understood and try to develop new insights. In this case, understanding the laws of area allowed us to derive a property relating the side lengths of right triangles. And even more magical is the idea that mathematicians are currently pursuing truths that we do not yet know exist, making mathematics an ever-growing field for students to participate in themselves if they so choose.

Check out the following links to research tasks inspired by the Pythagorean Theorem:

- Midpoints: With the PT, there is an opportunity to develop the strategies to find the midpoints of line segments and verify results.
- From Pythagoras to Fermat: Fermat’s Last Theorem was only recently solved and is a logical extension from the PT.
- Traveling Salesman: Another open-ended problem, this can inspire some unique strategies with a wide range of applications.