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Proof: Meaningful and as a Problem-Based Instructional Task

Proof: Meaningful and as a Problem-Based Instructional Task. First, inductive approach to find a pattern, make a conjecture, solve the problem Then prove deductively Then explain, discuss, compare, analyze, evaluate. Fido Problem and Proof Launch.

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Proof: Meaningful and as a Problem-Based Instructional Task

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  1. Proof:Meaningful and as a Problem-Based Instructional Task • First, inductive approach to find a pattern, make a conjecture, solve the problem • Then prove deductively • Then explain, discuss, compare, analyze, evaluate

  2. Fido Problem and ProofLaunch 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)

  3. 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?

  4. 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.)

  5. 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.)

  6. Fido Conjecture The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices of a triangle.

  7. 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.

  8. 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?

  9. 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?

  10. 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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