Constructing The Pythagorean Tree in Sketchpad

Hey you all!

Long time, no see, huh? It's been really busy semester in all ways. So today I will talk about The Pythagorean Tree and how we use it in our classrooms to impress our students. We already know what a The Pythagorean Tree and what it is meant to be. It is always easy to be a teacher just give the formulas and expect students to understand how it works and to apply the formula to the problems. However, if we want our students to internalize the math concepts we need to go beyond the formulas. As a prospective teacher, I believe that students must be taught about proof of formulas. Proofs make it easier to understand any concept. Anyway, let's turn back today's topic.

1.     The objective of this topic is;

·        M.8.3.1.5. Student forms the Pythagorean relationship, solves the related problems.

2.     Pedagogical Explanation:

As I mentioned earlier, it is important to engage our students to the topics. Proofs of concepts accelerate the learning process of students. So to achieve this internalizing, we need our students to understand where the formula comes from. This also helps them to relate the topics with real world. So, let’s look at what Pythagorean Theorem says. The Pythagorean Theorem states that: "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."

So what we mean with that,is; 

We all know this already, right? We want our students to know why does it work like this, right? It comes from the area of a square.  

Figure 1

So what is figure 1 saying us?

Area of Whole Square

It is a big square, with each side having a length of a+b, so the total area is:

A = (a+b)(a+b)

Area of The Pieces

Now let's add up the areas of all the smaller pieces:

First, the smaller (tilted) square has an area of		A = c2
And there are four triangles, each one has an area of		A =½ab
So all four of them combined is		A = 4(½ab) = 2ab
So, adding up the tilted square and the 4 triangles gives:		A = c2+2ab

Both Areas Must Be Equal

The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:

(a+b)(a+b) = c²+2ab

NOW, let us rearrange this to see if we can get the Pythagorean Theorem:

Start with:		(a+b)(a+b)	=	c2 + 2ab
Expand (a+b)(a+b):		a2 + 2ab + b2	=	c2 + 2ab
Subtract "2ab" from both sides:		a2 + b2	=	c2

That's it! We're done. This proof came from China over 2000 years ago!

You can show this proof with an activity in class. 

Now, we can talk about the Pythagorean Tree. We iterate the shape we have in Pythagorean Theorem as many times as we want to construct the Pythagorean Tree. It is basically a fractal which comes from the theorem. The relationship for the area between iterations gives the tree its name: it is the same equation as the Pythagorean Theorem.

Figure 2

If you keep iterating the shape in the theorem you will see the shape in figure 2. The angles of the triangle affect the shape.

Figure 3

In figure 3 you can see how does the angle affect the Pythagorean Tree. Also, you will see and construct this change in sketchpad tutorial.

3.     User Manual:

In the tutorial video, I show the construction steps in the sketchpad. With the help of the video, you can construct the tree in class. At this stage, you can ask your students about the connection between side lengths or areas of squares. It would help your students to engage with concepts. As I said above, after the construction, you can change the position of point E to change the angles of the triangle to be able to see what happens. 

4.     Construction Steps:

I create this video with the help of my colleague Bahar Sönmez. You can see her blog from here .  

Please don't forget to leave a comment below. It helps me to improve myself 😊