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Geometry Problems: Circles and Triangles
During the
  • Improve Individual Solutions to Circles and Triangles (10 minutes)
      • Return students' papers, and give each student a mini-whiteboard, pen, and eraser.
        • Recall what we were looking at in a previous lesson. What was the task?
        • I have read your solutions, and I have some questions about your work.
      • If you have not added questions to individual pieces of work, write your list of questions on the board and ask students to select questions appropriate to their own work.
      • Ask students to spend a few minutes answering your questions.
      • It is helpful if they do this using mini-whiteboards, so that you can see what they are writing.

        • I would like you to work on your own to answer my questions for about ten minutes.
  • Collaborative Small-Group Work on Circles and Triangles (15 minutes)
      • When students have made a reasonable attempt at the task on their own, organize them into groups of two or three.
      • Give each group a large, fresh piece of paper and a felt-tipped pen.
      • Ask students to have another go at the task, but this time, ask them to combine their ideas and make a poster to show their solutions.
        • Put your own work aside until later in the lesson. I want you to work in groups now.
        • Your task is to work together to produce a solution that is better than your individual solutions.
      • While students work in small groups, you have two tasks, to note their different approaches to the task and to support their reasoning.

      Note different student approaches to the task.

      • What mathematics do students choose to use?
      • Have they moved on from the mathematical choices made in the assessment task?
      • Do they measure the lengths of the sides of the triangles?
      • Do they draw construction lines?
      • Do they use similar triangles?
      • Do they use algebra?
      • Do they use proportion?
      • Do students attempt to use the special ratios for 30°, 60°, 90° triangles (1 : √3 : 2)? If so, how do they do this?
      • When finding the ratio of the areas of the two triangles, do they find the ratio of the squares of the bases, or do they use an alternative method?
      • When finding the ratio of the areas of the two circles, do students find the ratio of the squares of the radii, or do they use an alternative method?
      • Do students fully explain their solutions?
      • Note any errors, and think about your understanding of students' strengths and weaknesses from the assessment task. You can use this information to focus whole-class discussion towards the end of the lesson.

      Support student problem solving.

      Try not to make suggestions that move students towards a particular approach to the task.

      Instead, ask questions that help students to clarify their thinking.

      Focus on supporting students' strategies rather than finding the numerical solution. You may find the questions on the previous page helpful.

      • If the whole class is struggling on the same issue, write relevant questions on the board.

      You may find that some students think the empirical approach (measuring the diagram) is best.

        • Will your answer change if you measure in inches rather than millimeters?

      This question may focus students' attention on the lack of units of measure in the solution and the problem of accuracy.

        • What are the strengths/weaknesses of this approach?
        • Are your measurements exact?
        • Do you think that, if we asked another group that used this same method, they would come up with exactly the same answer as you?
  • Collaborative Small-Group Analysis of Sample Responses to Discuss (20 minutes)
      • When students have had sufficient time to attempt the problem in their group, give each group copies of the Sample Responses to Discuss.

      This task gives students an opportunity to evaluate a variety of possible approaches to the task without providing a complete solution strategy.

      • You may decide there is not enough time for each group to work through all four pieces of work.

      In that case, be selective about what you hand out.

      For example, groups that have successfully completed the task using one method will benefit from looking at different approaches.

      Other groups that have struggled with a particular approach may benefit from seeing a student version of the same strategy.

        • Here are some different solutions to the problem.
        • Compare these solutions with your own.
        • Imagine you are the teacher. Describe how the student approached the problem.
        • Write your explanation on each solution.
        • What do you like/dislike about the work?
        • What isn't clear about the work?
        • What questions would you like to ask this student?
      • To encourage students to do more than check to see if the answer is correct, you may wish to use the projector resource Analyzing Sample Responses to Discuss.
      • During the group work, check to see which of the explanations students find more difficult to understand.
  • Plenary Whole-Class Discussion: Comparing Different Solution Methods (15 minutes)
      • Organize a whole-class discussion comparing the four given solutions.
      • Collect comments and ask for explanations.
        • We are going to look at and compare the four solutions.
        • Can you explain Bill's method?
        • Why does Carla draw another triangle in the inner circle?
      • Encourage students to challenge explanations while keeping your own interventions to a minimum.
        • Do you agree with Tyler's explanation? [If yes] Explain again in your own words. [If no] Explain what you think then.
      • Finally, ask students to evaluate and compare methods.
        • Which one did you like best? Why?
        • Which approach did you find most difficult to understand? Why?
        • Did anyone come up with a method different from these?
      • Some issues that might be discussed, with suggested questions and prompts, are given below:

      Anya uses measurement.

      Strengths: It is easy to do. It gives you a feeling for the answer.

      Anya's calculations are correct. She has rounded to two decimal places.

      Weaknesses: You only know it is true for the particular case you measure. It's not exact. It doesn't tell you why it's true. Anya does not calculate the areas of the circles, or their ratio (About: 252 : 112 = 5 : 1).

      • Do you think Anya's answer is accurate?
      • Would an answer rounded to four decimal places be better?
      • What do you think the answer should be?

      Bill uses algebra and ratios.

      Strengths: Bill's method does not depend on the size of the diagram.

      You can use this method for all sorts of problems.

      Weaknesses: Bill's work is difficult to follow. There are gaps in his explanation, and it is quite difficult work. Bill does not answer the question, as he does not calculate the ratio of the areas of the triangles.

      He does not explain why the side lengths in the triangle are in the ratios he writes down, which is based on these trigonometric ratios:

      sin30° = c = 1
      r 2
      cos30° = b = 3
      r 2
      tan60° = a = 3

      You could ask students to explain where the ratios in Bill's solution come from, and then to use the lengths to complete the solution.

      Why does c = 1 ?
      r 2
      Why does b = 3 ?
      r 2

      Why does a = r√3?

      Why does Bill multiply by 6?

      What is the ratio of the areas of the two triangles?

      Carla uses transformations - rotation and enlargement.

      Strengths: It is simple. It is clear, even elegant. It is easy to do.

      Weaknesses: You have to see it! There are some gaps in the explanation that need to be completed.

        • How do you know that, if you rotate the small triangle, it hits the midpoints of the large triangle?
        • How do you know the four small triangles are congruent?
        • How do you know the four small triangles are equilateral?
        • How do you know the circle has been enlarged in the same ratio as the triangle?

      Darren uses algebra and similar triangles.

      Strengths: Darren's method does not depend on the size of the diagram. Darren has labeled the diagram: this makes his work easier to understand.

      Weaknesses: Darren's work is difficult to follow at times. He has failed to explain part of his work.

        • Are triangles OBC and OEF similar? How do you know?
        • What does Darren mean by "double × double"?
        • Can you use math to show Darren's answer is correct?
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