Establishing the Pythagorean Theorem
area on the geoboard, more geoboard areas, areas of more complex shapes
Students use their knowledge of geoboards and area to explore a variety of possible squares on a 5 x 5 geoboard. As they explore the areas and side lengths of some squares, it becomes clear that some of the sides are not whole numbers, and that they are not easily measurable. Here are a few examples:
The dimensions of the 1st and 3rd squares are not easy to measure directly, even smaller units or fractions still need estimation and rounding.
In previous explorations and discussions
students have determined that we can get the length of a square's
side by taking the sq. root of the area (2 = 
4
in the second example above). Here, however, we see that some of the
squares' sides are not simple to determine, as they are in the second
example. We have a symbol - -
which represents the side length of a square to help us out. The
sides of the first square are 10
("square root of ten") in length, and those of the third square,
5.
Square roots are exact numbers, although we cannnot always determine what their decimal equivalents are, except by rounding and estimating.
We can make use of this relationship in another way. Here is a line on a dot grid. How can we use the square root relationship to determine its length?
Try thinking of the line as the side of a square..
If the length of the side of the square is
29,
so is the length of the original line!
We can perform an interesting dissection of these 'tilted' squares and rearrange the pieces to form two smaller squares whose areas are together equal to the area of the dissected square...
We can cut many pairs of congruent triangles to dissect the square - so one square can be arranged into a variety of differently-sized pairs of smaller squares.
We can also do the same in reverse - dissect two adjacent squares to form a larger area equal to the sum of the smaller squares' areas.
You may have noticed that the squares' dimensions have a relationship to the dimensions of the legs of the right triangles we've been cutting and moving...What do you see in the above animations?
You're right if you noticed that the legs of the right triangles form the side dimensions of the smaller squares, and that the length of the hypotenuse forms the side dimension of the largest square.
The fact that the largest square, c2, can be formed into two smaller squares, a2 and b2, visually 'proves' the Pythagorean Theorem:
The sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse.
Usually, people define the Pythagorean theorem as a2 + b2 = c2, but this misses the point that it is a geometric relationship, not just an algebraqic statement. Still, it is a useful concept, and allows us to determine missing dimensions on right triangles, and was one of the earliest mathematical constructs to achieve widespread use, as far back as the ancient Babylonian, Sumerian and Egyptian cultures some 5,000 years ago.
Here are a few examples from one of the student worksheets. Try to determine the missing lengths using the concepts represented on the models above.
