Progress Report 1: Area




Introduction and instructions: The overall goal of this project is to create a succinct list of undefined terms, definitions, and axioms that will allow you to justify the standard formulas for the area of a region and establish the basic properties of area. As a first step, in this progress report you will describe a procedure for finding the area of a polygonal region. As you describe your procedure, note any assumptions you are making about the nature of area. For instance, do you use a criterion for determining when two figures have an equal area? If you do, remember what the criterion is and when you use it in your procedure; this will help you formulate your set of area axioms.

Answer each of the questions below, using the space provided. The space allotted should be sufficient for your answers; be concise. When submitting the completed report, fold it in half lengthwise. On the outside, write your group number, have each group member sign the write up and indicate his/her role. These signatures will be taken to mean that each group member had a good understanding of each answer and could present it with some assistance from the group.


Questions:

  1. On the geoboard, construct a triangle with no right angles of area 1.5, assuming the etched squares have area 1. Draw a diagram of your triangle, describe it in words, and justify the assertion that the area is 1 .5.

  2. Describe a general procedure for finding the area of an arbitrary polygonal region assuming:

    1. That you know how to find the area of a square.

    2. That you know how to find the area of a triangle.

  3. Given the following polygonal region,
    1. apply the procedures that you described in problem 2.a to the polygonal region.

    2. apply the procedures that you described in problem 2.b to the polygonal region.

  4. Using one of the procedures you described in problem 2, determine the formula for the area of a square, triangle, rectangle, and trapezoid.

  5. What definitions are you using for trapezoid and rectangle?

  6. Consider the fundamental properties of the surface the figure is a subset of, in particular:

    1. List as many properties of the surface that contains the figures addressed in problem 4 as you can.

    2. Does the method you use to compute area change if any of the properties that you listed change? If so, how and why?

    Group processing skills:


  7. Briefly discuss your groups performance with respect to two of the `Norms of Cooperative Behavior'. (See the handout on cooperative group learning.)

 

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