De-mystifying Algebra

Remember when you were an infant and your mom walked out of the room?  You thought she'd ceased to exist.  You hadn't yet grasped the concept of three-dimensionality.  When you got a little older, you faced similar perceptual challenges.  You got tricked by the shapes of objects into misjudging their volume.  Grocery stores still use this to their advantage.  Look at the weight listed on different boxes of cereal or your favorite canned food.  Curiously, the larger containers often contain less product. But wouldn't you know it, they cost more!  It pays to know something of dimension.

Which leads us to our next lesson.  This one isn't a puzzle, just a way to feel a little more comfortable with algebraic equations.

By now you're probably wondering about that animation at the top of the page.  Don't worry too much about it right now.  But rest easy in the knowledge that by the end of this lesson, you'll look back at it with a whole new understanding of the Pythagorean Theorem.

That's right.  The good old Pythagorean Theorem.   You've probably been asked to memorize this little ditty concerning right triangles:

a2  +  b2  =  c2                   or               H2  +  B2  =  D2                  or even               x2  +  y2  =  h2

You probably made a dutiful effort to memorize the equation but felt mystified as to why it works.  Well, stay mystified. Mystery is good.   We're just going to take a look at it from a different perspective which may increase your confidence when working with it (which will happen surprisingly often!)

To help us, we'll start with a simple reminder that a right triangle is really just half a rectangle.