Developing Formulas (3)

A few years ago, I was in a room of 100 or so educators who were asked to draw a picture to represent the Pythagorean Rule. Maybe 5 or 6 people were actually successful. I would say half didn't even know the Pythagorean Rule. Others drew a triangle (sometime right-angled, sometimes not) a bit like this:

But this doesn't illustrate what the Pythagorean Rule is, namely that the sum of the areas of the squares on the two shorter sides are identical to the area of the square on the longer side:

At the time I made the point that if we don't understand this, then we don't truly understand the Pythagorean Rule. I reckon that this is the result of teaching the Pythagorean Rule purely from an algebraic point of view. I was once shocked to see a textbook from the UK that told students to memorise these three formulas for the Pythagorean Rule:

a²+b²=c²

c²-a²=b²

c²-b²=a²

But this is not how the rule was discovered. It was discovered geometrically long before algebra was invented. Here in snowy, frigid Ontario, the curriculum has got it right: in Grade 8 the focus is on a geometric understanding which is followed up in Grade 9 by connecting this to an algebraic understanding. If this is the way it was discovered, then it should be the way we develop it with our students.
So let's not just give students the abstract formulae.
Instead, have them try some decomposition activities like the following (notice how one of the triangles is not right-angled: I always like to offer a counter-example!)

This can be followed by this video which has been doing the rounds on social media (I don't know who first created it so apologies for not giving this person credit):

Now this isn't a proof as such but merely one example. Using a dynamic geometry software such as Geometer's Sketchpad or Geogebra

helps us generalise that it doesn't matter what the original right-angled triangle looks like, the sum of the two smaller squares adds up to the area of the larger square. This is the point where we should bring in algebra to PROVE that it works for all right-angled triangles

But what happens if we draw other shapes on the sides of the triangle such as semi-circles?

It appears to hold true for semi-circles too. Can we prove this algebraically? Spoiler alert:  Yes we can, and it is a lovely wee proof too.
If we get our students to develop a geometric understanding of the Pythagorean Rule, they will be in a better position to use the algebraic representation to tackle problems like the one above or the one below (which is in my Top 10 favourite math problems of all time).