The Pythagorean Theorem
Lesson 2
Proof
Summary
Four groups of students
will
each be given a different GSP file that is expected to lead to one of
four proofs of the Pythagorean Theorem. Each GSP file has a
diagram particular to one of the four expected proof strategies, as
well as a hint that can be hidden as students begin to work on the
problem but shown if they get stuck. Of course, students may come
up with their own proof of the theorem, rather than the one the diagram
is expected to lead them to. For this reason, groups who produce
more than one valid proof of the theorem will receive extra
credit. Groups will present their proofs to the class at the end
of the unit.
I chose these four proofs because a) each is understandable by a high
school student, b) each takes advantage of a different aspect of GSP,
and c) each approaches the theorem in a different way.
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Proof 1, one
of the
simplest proofs of the Pythagorean Theorem,
combines geometric reasoning (the whole is equal to the sum of its
parts) and algebraic reasoning ((a + b)2 = a2 +
2ab + b2). Using
GSP, students can see a and b as changeable lengths, rather than as the
fixed lengths they might imagine in a by-hand drawing. Students
can also develop a geometric understanding of why (a + b)2 =
a2 + 2ab +
b2, using the "Show Rectangles" button.
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With
the diagram for Proof
2, students can not only discover a proof of
the Pythagorean Theorem but can also use GSP’s dynamic nature to
manipulate the triangle to make new conjectures. For instance,
students can change b until the center square disappears, then they can
determine what must be true about the original triangle in order for
the center square to reduce to a single point. Because GSP’s ease
of use encourages this type of exploration—students need only to drag a
point to notice new things—this proof diagram is better than manual
drawings.
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Proof
3 uses both GSP's
measuring capabilities and its Transform
features to lead students toward a proof. The two small green
triangles are reproduced to the right of the main diagram to suggest to
students that they are similar to the original right triangle; proving
this fact is an important part of this proof strategy. In
addition, the measurements at the top of the screen show students, as
they move the vertices of the triangle, that the square of side length
c consists of one rectangle of area a2 and another of area b2.
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| Proof
4 begins with an
unusual construction, so GSP’s action buttons
are useful in showing students where the diagram came from and how it
relates to the original triangle. Simply giving students such a
strange diagram with a wordy explanation about its construction would
likely confuse them; however, with GSP, students can watch as the
diagram is constructed, thereby gaining a better understanding of how
the blue triangle was created from the yellow one. |
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