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Geometry Problems: Circles and Triangles
Before the
  • Assessment Task: Circles and Triangles (15 minutes)
    • Have students do this task in class or for homework a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and to find out the kinds of difficulties students have with it. Then you will be able to target your help more effectively in the follow-up lesson.

      • Give out the assessment task Circles and Triangles, a pencil, and a ruler. Issue calculators if students ask for them.
      • Introduce the task briefly and help the class to understand the problem and its context.
        • Read through the task and try to answer it as carefully as you can.
        • Show all your work so that I can understand your reasoning.
        • Don't worry too much if you don't understand everything because there will be a lesson [tomorrow] using this task.

      It is important that students are allowed to answer the questions without assistance, as far as possible. If students are struggling to get started then ask questions that help them understand what is required, but make sure you do not do the task for them.

      Students who sit together often produce similar answers and then when they come to compare their work, they have little to discuss. For this reason we suggest that, when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual seats. Experience has shown that this produces more profitable discussions.

      When all students have made a reasonable attempt at the task, tell them that they will have time to revisit and revise their solutions later.

  • Assessing Students' Responses
    • Collect students' responses to the assessment task. We suggest that you do not write scores on students' work. The research shows that this is counterproductive, as it encourages students to compare scores, and distracts their attention from what they might do to improve their mathematical work.

      Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given in the table below. These have been drawn from common difficulties observed in trials of this lesson unit.

      We suggest that you write your own lists of questions, based on your own students' work, using these ideas. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will help the majority of students. These can then be written on the board at the beginning of the lesson.

      Common Issues: Suggested Questions and Prompts:

      Student has difficulty getting started.

      • What do you know about the angles or lines in the diagram? How can you use what you know? What do you need to find out?
      • You may find it helps to give a name to some of the lengths. Try r for the radius of the circle, x for the side of the big triangle, and so on.
      • Can you add any helpful construction lines to your diagram? What do you know about these lines?
      • Can you find relationships between the lengths from what you know about geometry?

      Student works out the ratio by measuring the dimensions of the triangles.

      • What are the advantages/disadvantages of your method?
      • Are your measurements accurate enough? How do you know?

      Student does not explain the method clearly.

      • For example: The student does not explain why triangles are similar. Or: The student does not explain why triangles are congruent.
      • Would someone unfamiliar with your type of solution easily understand your work?
      • How do you know these triangles are similar/congruent?
      • It may help to label points and lengths in the diagram.

      Student has problems recalling standard ratios.

      • The student recalls incorrectly or makes an error using the special ratios for a 30°, 60°, 90° triangle (1, √3 , 2).
      • What do you know about cos 30°? What do you know about sin 30°? How can you use this information?
      • Use the Pythagorean Theorem to check/calculate the ratio of the sides of the triangle.

      Student uses perception alone to calculate the ratio.

      • For example: The student rotates the small triangle about the center of the circle and assumes that the diagram alone is enough to show the ratio of areas is 4:1.
      • What math can you use to justify your answer?

      Student makes a technical error.

      • For example: The student makes an error manipulating an equation.
      • Check to see if you have made any algebraic errors.

      Student uses ratios of lengths rather than ratios of areas.

      • For example: When finding the ratio of the areas of the two circles, the student obtains an incorrect answer because they find the ratio of the radii, rather than the ratio of the squares of the radii.
      • What is the formula for the area of the circle? How can you use it to find the ratio of the areas of the circles?

      Student produces correct solutions.

      • Can you solve the problem using a different method? Which method do you prefer? Why?
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