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New Proof of the Pythagorean Theorem
15
Transcript
Reasoning and Sense Making with Pythagoras
New Proof
Previous Proofs of the Pythagorean Theorem
- Published by Burk (1996)
- Scaling triangles:
- Students get to choose their original right triangle and then the scale the triangle by the 3 side lengths of the original - legs a and b as well as the hypotenuse c.
- These shapes were used by he students to create geometric figures.
- Euclid's visual aid used in Elements
- In 1968, Elisha S. Loomis published more than 350 unique proofs of the theorem
- These proofs, though excellent and effective, are not as accessible for a student with lower mathematical skills because they are heavily focused on areas of squares rather than lengths of triangles.
A common misconception...
What might students come up with?
It is important that students note that two pieces "lining up" is not a good enough explanation or proof - we have to show they are truly the same or that they match based on the geometrical and algebraic rules we know.
not congruent just because they "match up"
What are the benefits to this proof?
While inspecting each of the following example student solutions, describe how we can be sure this solution works or doesn't work in our proof:
- Appeals to the less experienced mathematicians, gives them the chance to use geometry skills they understand to make connections.
- Can be worked out with manipulatives - great for kinesthetic learners!
- Shows how many mathematical ideas can be proved in multiple ways, both in the small scale and large scale
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