Basic Review for the Pythagorean Theorem Lesson

Triangles and Rectangles --

Observe the following figure of a rectangle. What can you tell about the two triangles formed by the diagonal?

Are they the same size? Are they right triangles? Can you explain it? The answer should be pretty obvious from what we've covered. And a final question, what is the area of any of the triangles with respect to the area of the rectangle? I hope that it is clear to you that it is exactly half, as we have divided the rectangle in exactly two halves. The area of the rectangle is calculated as

a x b

so the area of each triangle would be:

Squaring Patterns --

We will now work a little with the very interesting patterns from the sequence of the squares of whole numbers, i.e. the sequence of the squares of 1, 2, 3, 4, 5. . . . Two things that you should get out of this exercise are:

We start with a square with a side that measures 1 unit and has an area of 1 x 1 = 1 square units:

The square for the next size needs a side that measures 2 units. So we add to the original one, one pink square to the right, one pink one below, and a purple one on the lower right corner to get:

This one has an area of 2 x 2 = 4 square units. The next size up needs a side that measures 3 units. So we add to the previous one, two pink squares to the right, two pink ones below, and again one purple one on the lower right corner. Here is the next square that has an area of 9 square units:

By following the same pattern we can get the next square in the sequence that has an area of 16 square units:

Now let's put a square growth table together:

(A) Pink Squares to add	(B) Total Squares to Add	(C) Size of Side	(D) Size of Area	(E) Area Increase vs. Previous
--	--	1	1	--
2	3	2	4	3
4	5	3	9	5
6	7	4	16	7

Let's make sure that you understand the table. We start with a square with a side of size 1. To go to the next one, i.e., to the line with the square with a side of size 2, we add two pink squares plus the one in the corner for a total of 3 squares. The area of this square is 4. The area increased by (4 - 1) = 3 square units as compared to the first square.

I am sure that you have no trouble seeing the pattern in columns A, B and E. But can you see how to get the number in column B based on the number in the C column of the previous row? That is, how can you get the 5 from the 2, or the 7 from the 3? Try to figure that out. I suggest that you get a piece of paper (or you can do it in a computer program) and complete this table until you get to 20 in column (C).