What would it feel like to be Pythagoras? Teachers can empower their students to channel Pythagoras and offer the students an opportunity to discover the Pythagorean Theorem individually! This gives a great feeling of power and accomplishment, as well as giving the student a deep understanding of the relationship between the legs (two shorter sides) of a right triangle and its hypotenuse. Using a set of tangrams, the ancient Chinese puzzle that has so many uses in mathematics, students can manipulate the puzzle pieces in recreating Pythagoras’ experience. Each student needs to have four sets of self-made tangrams to complete the activity.
The students will determine that the small triangle in the tangram set is one square unit for the purposes of this activity. The area of the smallest triangle is 1/16 the area of the whole tangram square. Students will build squares adjacent to each side of the three different size triangles in the tangram set. (see photos- in each photo the white triangle is the central focus, with the adjacent squares in color.) Students will then determine the area in units of each square. (see table) Students will discover that in each case, the legs of the small square, when squared and added to the square of the square adjacent to the other leg, will equal the square adjacent to the hypotenuse. This is expressed by a2 + b2 = c2.
The number of shapes needed to create the squares that lie adjacent to the sides of the triangle are as follows:
|Number of small triangles needed to make a square that lies adjacent to the legs of the triangle.||Number of small triangles needed to make a square that lies adjacent to the hypotenuse of the triangle|
This is an excellent introduction to square roots and rational numbers because it is hands on, and can be completed with both real and e-manipulatives, making it accessible to all students. The numbers are small and even, removing much of the mathematics anxiety that can go along with understanding higher level math. Students will discover the relationship between the three squares by manipulating and counting the units from each side, then compute to discover that the area of the two “leg squares” add up to the area of the “hypotenuse square”. They will realize that this means that there is also a relationship between the sides of the right triangle. A discussion of rational and irrational numbers would be appropriate at this point in the lesson, as the sides of the triangles may be irrational, such as √2.
I would present this activity to my students by having them follow the above listed steps. Prior to this lesson the students would create their own set of tangrams. This will engage them and help them internalize the relationship between the seven pieces of the puzzle. Students would require a working knowledge of many other geometric terms. This is particularly necessary as students write to explain each step.