Jason Guzman Math Problem: Pythagorean Theorem

THE PROBLEM

The Pythagorean Theorem is a relation in geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Prove that this relationship is true by creating a right triangle out of paper using one of the configurations below. Cut out a triangle and cut out the squares built off the legs of the triangle and the square built off the hypotenuse. If the Pythagorean theory holds true you should be able to place the squares built off both legs over the square for the hypotenuse perfectly.

(6,8,10)     (5,12,13)      (7,24,25)    (12, 18, 20)    (10, 24, 26)      (3,4,5)        (14,24,25)  )    (8,15,17)   

PLANNING

Ways to Solve the Problem

a)      To solve this problem students used graph paper to cut draw a triangle with sides measuring 3, 4,& 5 units.

b)      Next I cut out 3 squares having sides that were the lengths of each of the sides of the triangle respectively.

c)       Students tried to place the squares respective to the legs of the triangle over the square respective to the hypotenuse but only one would fit.

d)      They then cut the second square into single units and were able to cover the square respective to the hypotenuse in this way.

e)      An alternate approach students used to solve the problem of the second square respective to the leg not fitting over the square respective to the hypotenuse was  to cut out a shape of the of the area remaining uncovered . Students notices that this missing piece while not a square had the same number of units as the second square respective to the leg.

Problems Students May Encounter

The most difficult part of this problem is for students to understand the definition of the Pythagorean Theorem, the language associated with a right triangle and what the question is asking them to do. It helps to have students read the question and get started thinking about the problem and then stop them to clarify any questions they have about terminology.

Some Prompts/Questions to help Students Unpack the Question

a)      Describe the Pythagorean Theorem in your own words.
b)      Base on the Pythagorean Theorem what do you know about a right triangle?
c)       What is this problem asking you to do?
d)      Draw and label everything you know based on the Pythagorean Theorem, what you know about right triangle and based on this specific problem.

Math Practices This Problem Can Develop

a)     Understanding a problem (rereading, marking-up)

b)    Drawing a picture/ labeling a diagram

REFLECTION

When teaching the Pythagorean Theorem in the past I asked students to memorize the formula. Only some students succeeded in this approach. In this lesson students get a chance to see how and why the Pythagorean Theorem works by proving it with manipulatives. This gives students a more concrete understanding and reference point to refer back when solving similar problems in the future.