Additional Resources

Materials Required

Estimated Time Needed

(Times are approximate and will depend on the needs of the students.)

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Applying Angle Theorems
During the
  • Collaborative Problem Solving: The Pentagon Problem (20 minutes)
      • Organize students into small groups of two or three.
      • Give each group a copy of The Pentagon Problem and a copy of the Geometrical Definitions and Properties sheet.
      • Display slide P-1 Instructions for The Pentagon Problem (1).
      • Introduce the task and explain what you are asking students to do.
        • Mrs. Morgan is a teacher in another school. She wrote this problem on the board for her students.
        • I'm giving you some work written by four of her students.
        • The students all used different methods to solve the problem.
        • I want you to use each student's method in turn to solve The Pentagon Problem.
      • Display slide P-2 Instructions for The Pentagon Problem (2).
        • To get started, choose one of the methods and work together to produce a solution. Make sure everyone in your group understands how that method works. Then move on to the next method.
        • Write all your reasoning in detail, and make sure you justify every step.
      • As students work you have two tasks: to note student difficulties and to support student problem solving.

      Note student difficulties.

      Look for difficulties students have with particular solution methods. Which solution method(s) do they find it most difficult to interpret and use? What is it that they find difficult? Notice also the ways they justify and explain to each other. Do they justify assumptions? Do they explain all their calculations with reference to theorems and definitions? You can use this information to focus whole-class discussion towards the end of the lesson.

      Support student problem solving.

      Try not to focus on numerical procedures for deriving answers. Instead, ask students to explain their interpretations and use of the different methods. Raise questions about their assumptions, and prompt for explanations based in angle theorems to encourage precision in students' reasoning. Refer them to the Geometrical Definitions and Properties sheet as needed.

  • Collaborative Analysis of Sample Responses to Discuss (15 minutes)
      • As students complete their solutions, give each group a copy of each of the four Sample Responses to Discuss.
      • You could also display the slide P-3 Instructions for Sample Responses to Discuss.
        • Four students in another class used Annabel, Carlos, Brian and Diane's methods to solve the problem like you just did.
        • Here are copies of the other students' work. None of this work is perfect!
        • For each student's solution:
          • Explain whether the reasoning is correct and complete.
          • Correct the method when necessary.
          • Use the method to calculate the measure of the missing angle x, giving detailed reasons for all your answers.
      • During small-group work, note student difficulties and support student problem solving as before. In particular, think about what students are finding most difficult and use this to focus the next activity; a whole-class discussion.
  • Whole-Class Discussion: Comparing Solution Methods (15 minutes)
      • Organize a whole-class discussion comparing the sample solutions methods.
      • To help with this discussion, show slide P-4 through P-7 displaying the Sample Responses to Discuss.
      • Using your understanding of your students' difficulties from the assessment task and their work during the lesson, choose one of the sample responses to discuss.
      • Ask one group to present their analysis of that response. Ask for comments and reactions from other students.
        • [Celia] What went wrong in Megan's solution?
        • Why did Brian draw that line?
        • Can you explain what assumption Katerina made? Was it a correct assumption?
        • [Trevor] Can you explain that in another way?
      • Then look at another solution method.
      • Finally, compare methods.
        • Which student's work provided the most complete reasoning?
        • Which student's work was most difficult to understand?

      The intention is that students will begin to realize the power of using different methods to solve the same problem, and to appreciate the need for, and nature of, adequate reasons for each assertion.

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