Lesson 2
Proof
Lesson Plan
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Math
1
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Time: |
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Two
50-minute class periods (one now, one later), or part of two 90-minute
blocks (one now, one later)
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Goals: |
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Students
will develop a proof of the Pythagorean Theorem. |
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Students
will be able to explain their own group's proof strategy and understand
other groups' strategies. |
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Students
will recognize the importance of communication and rigorous proof in
mathematics.
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NCTM Content &
Process Standards
Addressed: |
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Geometry
(explore relationships among classes of two-dimensional objects, make
and test conjectures about them, solve problems involving them;
establish the validity of geometric conjectures using deduction, prove
theorems, and critique arguments made by others)
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Measurement
(understand and use formulas for the area of geometric figures)
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Reasoning
and Proof (recognize reasoning and proof as fundamental aspects of
mathematics; develop and evaluate mathematical arguments and proofs)
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Communication
(communicate mathematical thinking coherently to peers, teachers, and
others; analyze and evaluate the mathematical thinking and strategies
of others)
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GPS Content &
Process Standards
Addressed: |
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MM1G2a
(make conjectures) |
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MM1P1a,d
(build new knowledge through problem solving; reflect on problem
solving)
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MM1P2a,c,d
(recognize the importance of mathematical proof; develop and evaluate
proofs; use various types of proofs)
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MM1P3a,b,c,d
(communicate mathematically)
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Supplies and Resources: |
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Four
GSP files (see files)
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Computer
projector for proof presentations
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Assessment: |
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Presentation
- after Lesson 4 |
Overview: |
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Introduction
(5-10 minutes)
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We
messed around on GSP yesterday, and none of us could find a right
triangle for which a2 + b2 ≠ c2.
Why doesn't this constitute a
proof?
What proof methods do you think would work to prove the Pythagorean
Theorem? Would a geometric proof make more or less sense than an
algebraic one? What about a combination of the two?
Students get in three to four groups, either of their own choosing or
of the teacher's. Students are assigned a proof method, either by
random selection or by the teacher's choosing (The author feels that
Proof 1 is easier than Proof 2, which is easier than Proof 3, which is
easier than Proof 4.).
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Exploration
(35-40 minutes)
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Students
explore the diagram for their assigned proof (see files) to try to develop a proof of the
Pythagorean Theorem. Certainly, students may come up with a proof
other than the one suggested by the diagram. Groups that write
more than one valid proof (without copying other groups) should receive
extra credit.
Each group will finish, write up, and present their proof to the class
after Lesson 4.
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Wrap-Up (5
minutes)
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There
are hundreds of proofs of the Pythagorean Theorem (see, for example, this
book). Discussion:
Why would we want more than one proof of a theorem?
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