The pythagorean theorem expresses a nonintuitive relationship – that’s where a lot of its power comes from.
Unfortunately, the way it is usually taught involves a formula (or two or even three depending on how it’s taught) and learning which numbers to plug in where.
And it almost never sticks.
This is the lesson I use to introduce the whole thing:
The kids cut out the squares, and spend the period trying to make right triangles out of the sides of the squares.
On its own, it’s a horrible lesson. It’s almost impossible to tell if the triangle is really a right triangle or not, resulting in a lot of close but not quite guesses, which can be frustrating for the students. It requires me to filter out the correct solutions (there are 7 of them) for the students.
The pattern in the triplets is obvious if you get it. This year, out of 80 kids, I had one who could find it on their own without help. There isn’t enough of an aha moment to make it stick.
I still teach it though.
By the labeling, and subsequent use of the squares and counting of the sides, the students learn to associate a square and it’s root. It gets them to think about right triangles and the squares associated with the sides. And it gets them ready for this next sheet:
The first two triangles show off the pythagorean association. The next three allow them to find the third side on their own. (i have them add the two smaller squares while I cover the larger square with my hand. When they get to the third problem, they’ve forgotten that there isn’t anything there.) The last three involve finding one of the smaller squares – during which my most frequently asked question of the students is: “Does it make sense to have a larger number for a smaller square?”.
The final problem has a triangle without the squares – the kids get to draw them in on their own. The next couple of days involve variations on the theme, building to word problems, none of which ever see even a hint of a^2^+b^2^=c^2^. Everything is done by first drawing squares on the triangle, which eventually disappear to be replaced by just the length of the sides and its square.
I’ve done the second sheet without the first, and it was a struggle, as were all of the lessons after it. They didn’t understand why the squares were stuck to the triangles, or why or how to calculate the areas of the squares. That crappy first lesson with all the struggles and frustrations turns the everything after it into a feast of exploration and aha moments.