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Geometry Problems: Circles and Triangles
  • Introduction: Mathematical Goals
    • This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties:

      1. Solving problems by determining the lengths of the sides in right triangles
      2. Finding the measurements of shapes by decomposing complex shapes into simpler ones.

      The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches.

  • Formative Assessment Task : Circles and Triangles
    • Students are presented a diagram of a complex shape composed of triangles and a circle and are asked to calculate the exact ratio of the areas of the two triangles. They then are asked to draw a second circle inscribed inside the small triangle and to calculate the exact ratio of the areas of the two circles.

  • Common Core State Standards
    • Common Core State Standards for Mathematical Practices Common Core State Standards for Content

      This lesson involves a range of mathematical practices from the standards, with emphasis on:

      1. Make sense of problems and persevere in solving them.
      2. Reason abstractly and quantitatively.
      3. Construct viable arguments and critique the reasoning of others.
      1. Attend to precision.
      2. Look for and make use of structure.

      This lesson involves mathematical content in the standards from across the grades, with emphasis on the following Domain.Cluster:

      • G-SRT: Understand similarity in terms of similarity transformations.
      • G-SRT: Define trigonometric ratios and solve problems involving right triangles.
      • G-C: Understand and apply theorems about circles.
  • Lesson Structure
    • The lesson is structured in the following way:

      • Before the lesson, students attempt the problem individually. You then review their work and create questions for students to answer in order to improve their solutions.
      • During the lesson,
        • Students work collaboratively in small groups to produce an improved solution to the same problem.
        • Working in the same small groups, students comment on and evaluate some solutions produced by students in another class.
        • In a whole-class discussion, students explain and compare the alternative solution strategies they have seen and used.
      • After the lesson, students review the work they did on their individual solutions and write about what they learned.
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