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					           Proof:
Meaningful and as a Problem-
  Based Instructional Task
• First, inductive approach to find a
  pattern, make a conjecture
• Then prove deductively
• Then explain, discuss, compare,
  analyze, evaluate
       Fido Problem and Proof
               Launch
    Fido guards a yard. Put Fido on a leash and
    secure the leash in the yard so that all of the
    yard is guarded and the shortest possible
    leash used.
•   Can you find a solution if the yard is shaped
    like a circle? If so, what is the solution?
•   How about if the yard is square-shaped?
•   Other shapes?
•   Now think about a triangular-shaped yard …
(Adapted from PSSM, pp. 354-358)
            The Problem
A yard is shaped like a right triangle.
Fido will be on a leash and will guard the
  yard.
Put Fido on a leash and secure the leash
  somewhere in the yard.
Make sure Fido can reach every corner of
  the yard.
Use the shortest leash possible.
Where should you secure the leash?
     What we did previously
         (or do it now)
Work collaboratively on this problem.
Use geometry software to help you.
Propose a solution.
Make a conjecture for a mathematical
  theorem.
(Today we will prove, in several ways.)
     What we did previously
         (or do it now)
• What is the solution to the problem?
  (See GSP sketch.)
• What is the geometry theorem?
  (See next slide.)
        Fido Conjecture
The midpoint of the hypotenuse of a
right triangle is equidistant from the
three vertices of a triangle.
               Explore
• Examine the 4 diagrams (attached).
• Write a complete proof associated with
  each diagram.
• Put one proof on chart paper and be
  prepared to present and explain.
              Summarize
Present & explain proofs. Discuss the following:
• Describe the general strategy in each proof.
• What are some advantages and disadvantages
  of each general method of proof?
• How are the strategies and proofs similar and
  different?
• Are some proofs easier or more convincing to
  you than others? Why?
• What mathematical ideas are used in these
  proofs?
    Fido Proof: lesson format
• Launch – Recall and discuss Fido problem
  (or do it now), state the conjecture
• Explore – Do and explain 4 proofs, in groups
• Summarize – Groups present, discuss
  summary questions
• Modify/Extend – locus problem, PSSM p.
  358
• Check for Understanding – Teacher task:
  How and what would you assess?
           Proof:
Meaningful and as a Problem-
  Based Instructional Task
• First, inductive approach to find a
  pattern, make a conjecture
• Then prove deductively
• Then explain, discuss, compare,
  analyze, evaluate

				
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