Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 Exploring Reasoning and Proof
 Introduction | Triangular and Square Numbers | Conjecturing with Algebraic Symbols | Proving the Conjecture | Reflection Questions | Summary | Your Journal

Think about the Triangle and Square Interactive Activity and reflect on the following questions. When you've formulated an answer to each question, select "Show Answer" to see our response.

 Question: What is the relationship between the diagram demonstrating the conjecture and the algebraic argument? Show Answer
 Sample Answer: The diagram prompts inductive thinking by helping you to look for a pattern and to develop strategies that can be used to prove the conjecture algebraically. Mathematics often builds along these lines. Inductive work, such as creating a diagram or examining cases, prompts insights that can then be tested deductively. The diagrams or individual cases do not constitute proof in a formal sense, but they can lay the groundwork for developing a proof for a conjecture, aid in visualizing, or show that a particular approach is not valid or mathematically productive.
 Question: What role does communication play in working on a proof activity like this? Show Answer
 Sample Answer: A proof is a structured deductive argument demonstrating that a conjecture is valid. It typically moves step-by-step, each one of which is explained. Clear communication about what is to be proved, how that is represented, how the steps of the proof are described, and how the final conclusion is reached is vital. The structure and methods must be convincing and persuasive to an outside party who is reviewing the proof. In this sense, formal proof is the most rigorous and advanced form of mathematical communication.
 Question: If you were asked to go back and find a complete different way to prove this conjecture visually and algebraically, could you? Show Answer
 Sample Answer: Answers will vary. It is often difficult to find alternative ways to reason or to find proof once we have succeeded with an approach. But successful teaching requires openness to such alternatives. In fact, many conjectures in the high school curriculum and in mathematics generally can be proven many different ways, calling on different aspects of mathematics. Whether we are capable of generating completely new approaches to the same task on our own or not, we are likely to encounter students who derive both valid and erroneous methods which differ from ours or their classmates'. Remembering that there is seldom "one correct" method to finding proof is helpful in the classroom.
 Question: How could you extend this problem? Show Answer
 Sample Answer: One possible discussion point is that this proof not only proves that the sum of two consecutive triangular numbers is a square number, it provides a means to determine which square number will result in a specific case. A further possible extension would be to look for other characteristics relating triangular numbers and square numbers. Explorations of other figurative numbers such as hexagonal numbers and their relationship to these could be developed. Both the algebraic proof and the graphical demonstration can be the basis for further exploration.
 Question: How does finding proof relate to problem solving? Show Answer
 Sample Answer: Organizing information, trying different approaches, and applying established knowledge and procedures to new mathematical situations are all key parts of both problem solving and proof.

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