Wilmington University
Pathwise Lesson Plan Format
Teacher/Student
Teacher
Grade:
Subject:
Date:
Tara M. Ryan
9
Science
4/2/10
1. Briefly describe the students in this class.
http://www.sciencenetlinks.com/lessons.php?Grade=9-12&BenchmarkID=2&DocID=0
2. What are your goals for the lesson? What do you want the students to learn?
To explore the nature of logic, evidence and proofs in the context of mathematics. The nature of
logic and evidence are topics that should come up frequently in science, history, social studies and
mathematics.
3. Why are these goals suitable for this group of students?
These students are already familiar with the Pythagorean Theorem.
4. How do these goals support the district’s curriculum, state frameworks, and/or content
standards?
Standard 6 – Reasoning and Proof: Students will develop their Reasoning and Proof ability by solving
problems in which there is a need to investigate
significant mathematical ideas in all content areas; to justify their thinking; to reinforce and extend their logical
reasoning abilities; to reflect on and clarify their
own thinking; to ask questions to extend their thinking; and to construct their own learning. All students in
grades K–12 will be able to:
• Understand that reasoning and proof are fundamental aspects of mathematics
• Make and investigate mathematical conjectures
• Develop and evaluate mathematical arguments and proofs
• Select and use various types of reasoning (such as: compare; analyze; make an inference-inductively and
deductively; evaluate) and methods of proof
5. How do these goals relate to broader curriculum goals in the discipline as a whole or in other
disciplines?
6. How do you plan to engage students in the content? What will you do? What will the
students do? (include time estimates).
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Mindset or “Warm-Up” Activity (5-10 minutes)
o Play the Pythagorean Puzzle on The Proof page, part of the NOVA Online
website. http://www.pbs.org/wgbh/nova/proof/puzzle/theorem.html
State objective or purpose of the lesson
o The nature of logic and evidence are topics that will be seen in science,
history, social studies and mathematics.
Provide Instructional Input (30 minutes)
o Have the students read the interview of Solving Fermat: Andrew Wiles on the
Proof website.
o Have them read Math’s Hidden Woman, the true story of Sophie Germain, an
18th-century woman who assumed a man’s identity in order to attempt to
prove Fermat’s Last Theorem.
o Ask the students the following
o What is the difficulty in proving Fermat’s Theorem? (because there are an
infinite number of equations, and an infinite number of possible values for x,y,
and z, the proof has to prove that no solutions exist within this infinity of
infinities.)
o Why did Wiles and Germain devote so much of their time to trying to solve
this mathematical puzzle? Was there a practical application to solving the
theorem? (The discussion should emphasize the non-practical aspects of this
particular endeavor. Both of these stories emphasize that both Wiles and
Germain were motivated by the intellectual challenge of a purely abstract
problem.)
o Does Wiles believe that his proof is the same as Fermat’s would have been?
Does it matter? Does Wiles believe that there is only one way to prove the
theorem? (Wiles does not believe that he found the same solution that
Fermat had purported to find because he used techniques that were not
available in Fermat’s time. In fact, he believes that Fermat’s probable proof
was likely flawed.)
Model
o How do non-mathematical proofs, such as in law or mystery novels, compare
to mathematical proofs?
o Show a finished product of what is expected from the student.
Guided Practice(30 minutes)
o Follow the discussion by having students use the webpage Proofs in
Mathematics from the Interactive Mathematics Miscellany and Puzzles for
guided, firsthand experience in solving mathematical proofs. Working in small
groups, students should select and explore one of the problems on this page
and share the “proof” with the class.
o Activities students will complete individually or in groups to practice use of
knowledge or skills learned in the lesson.
7. What difficulties do students typically experience in this area, and how do you
plan to anticipate these difficulties?
Describe any special concerns you may have about this material, the methodology, or the
equipment that you will be incorporating into this lesson as related to the students in this class.
Explain how you plan to address these concerns.
8. What instructional materials or other resources, if any, will you use?
List all equipment and materials needed. Include:
 Computers
 Seatwork (handouts, manipulatives, etc.)
9. How did you plan to assess student achievement of the goals? What procedures
will you use? (Attached any tests or performance tasks, with accompanying scoring
guides or rubrics.)
Use the activities on the Using the Pythagorean Puzzle page to assess student understanding of the
applications of Pythagorean Theorem.
10. How do you plan to use the results of the assessment?
Having the students share the “proof” they selected with the other students in the class.