Progress Report 2: Area Axioms




Introduction and instructions: In this report, you will create a list of axioms that will be used to justify the procedure your group developed for computing the area of a polygonal figure. As best you can, model the example of the incidence axioms taking care to identify undefined terms, defined terms and the axioms themselves; remember that the axioms should specify relationships between the undefined terms or undefined and defined terms. Also identify the statements in your procedure which you believe to be theorems.

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. After reviewing the procedure for computing the area of a polygonal region, make a list of:

    1. The undefined terms you used.

    2. The defined terms you used; make an initial attempt at any definitions you wish to introduce.

    3. The axioms you identified:

    4. List any conjectures as to the theorems that you used; you do not need to offer proofs of your conjectures at this point.

  2. Does the area function used by GSP appear to behave in a fashion consistent with your axioms? _________ In order to answer this question you should conduct some experiments using GSP to test whether or not GSP is consistent with your axioms.

  3. Use your set of axioms and your procedure for finding the area of a polygon to write justifications for your formulas for calculating the area of a 1) triangle, 2) parallelogram, and 3) trapezoid. (A justification is a short version of a proof. It should contain enough information to allow a reader to reconstruct the main steps of the proof and should include reasons for key points. At this point you have only your defined terms and axioms available as reasons. You may find as you work through these justifications that you need to add definitions or axioms.)

  4. Where are the gaps in your proofs (i.e. are there any statements you cannot justify via your axioms or definitions)?


    Group processing skills:


  5. Briefly discuss your groups performance with respect to two of the norms of cooperative behavior; discuss two norms that you did not discuss in the first progress report. (See the handout Cooperative Group Learning.)

 

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