FYS 100 Lesson Plan: The Pythagorean Theorem
(Metacognitive/Reiterative Activity)
Author: Judy Silver
Domains: Mathematical and Abstract Thinking; Multicultural and International Thinking; Social,
Ethical, and Historical Thinking
Topics covered: The Pythagorean Theorem, mathematical proofs, metacognitive/reiterative
thinking
Desired Outcomes:
Introduction: Today we are going to discuss metacognitive thinking. [Ask students if they
know the definition of metacognitive.] Metacognitive thinking is basically “thinking about
thinking”. It can take many forms; and "it includes knowledge about when and where to use particular
strategies for learning or for problem solving.” [1] In particular, we will think about how mathematical
thinking is done.
When I was in graduate school, I was fascinated by the many techniques that other grad
students used to solve problems and create proofs. Not only did the students come up with a
variety of proofs for the same theorem, but many different techniques were used to find those
proofs.
For example, some students would sit and just think for a long time. Then they would slowly
write down the steps to get from the premise to the conclusion in the theorem. But others
would use a forward/backward approach. They would use scratch paper to quickly write down
everything they could think of that would be implied by the theorem’s premises. Then they
would write down everything they could think of that would imply the theorem’s conclusion.
Looking over both lists, they would then look for a connection. If there was no common link,
they would look for implications of the things they had already written down. They would
continue this process until they could find a chain of implications that would connect the
premise and the conclusion.
It’s a matter of style! People naturally think in many different ways. This is similar to what you
have heard about the five basic learning styles (aural, verbal, visual, logical, and kinesthetic or
physical).
Consider the Pythagorean Theorem, for example. In 1927, a man named Elisha Loomis
published a book entitled The Pythagorean Proposition.[2] In that book he claimed to show 367
proofs of the Pythagorean Theorem! Let’s see if you can find a few of those proofs, and find
out what types of proofs appeal to you personally.
Activities:
1. Write down a statement of the Pythagorean Theorem.
2. Try to find ten different proofs of the Pythagorean Theorem and turn in copies of those
proofs. Be sure to include at least one example of a proof that was created by someone
who did not live in the United States.
3. Be prepared to tell the class which proof you prefer and why.
4. Do you understand the theorem better after looking at various types of proofs?
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1. Santrock, John W. Information Processing. A Topical Approach to Life-Span Development. (pp. 272275). McGraw-Hill, New York. 2008.
2. Loomis, Elisha Scott. The Pythagorean Proposition 2nd Edition. Its Demonstrations Analyzed and
Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs". National Council of
Teachers. 1968. (Currently out of print.)