Beyond the Formula - Pythagorean Theorem

When tackling this theorem in the class room, student often get caught up in the formula. They memorize it in a few minutes and soon figure out how to plug numbers in and get the "answer." Before you know it, this ends up as a trend through quite a bit of their schooling. Seriously, go into a calculus class and nearly any student can tell you how to find a derivative, but when asking what a derivative is, you'll get a far less enthusiastic response.

So, how do we address this? Through lots of visual examples.

Let's start with the actual formula and a picture.

a² + b² = c²

With these two things you get a student to see that each letter (or variable) is actually just a line that makes the side of a triangle. 

Here is where many student check out and just start plugging in numbers to get their "answer." But it's here where we need them to really thing about what's happening. We'll do that by zooming in on side a and using some guiding questions. 

Side a is just a line. 

 a       

So we ask the students. What are we doing with a? What does it mean to square something? And when you say square. REALLY PUSH THAT YOU MAKE A SQUARE. Then add in the visual literally making a square.

That way, when we do it for each side and put it all together, the students can see what the formula is actually having them do. Taking a side. And squaring it.

 

Hopefully, by this point a student would understand what the formula is actually doing to each side of the right triangle.  

So we can finish with the simple meaning of the Pythagorean Theorem. If you take sides a & b and make them into squares, those two squares put together are equal to the square you make from side c.

To make the concept more fluid watch the following 40 second clip. 

 

^{At the end, the goal is to simply have the student really understand what the formula is actually doing in a physical sense as oppose to, "Numbers go here. Answer comes out here."}

^{For further work on vocabulary and concrete examples visit:}

 ^{http://www.mathsisfun.com/pythagoras.html} 

^Sources:

^{All examples taken from:}

^{http://www.mathsisfun.com/pythagoras.html}

^{http://www.math-aids.com/Pythagorean_Theorem/Definition.html}